diff --git "a/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_FGModuleCat_Basic.jsonl" "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_FGModuleCat_Basic.jsonl"
new file mode 100644--- /dev/null
+++ "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_FGModuleCat_Basic.jsonl"
@@ -0,0 +1,39 @@
+{"state":"R : Type u\ninst✝ : Ring R\nM : FGModuleCat R\n⊢ AddCommGroup ↑M","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  ","nextTactic":"change AddCommGroup M.obj","declUpToTactic":"instance (M : FGModuleCat R) : AddCommGroup M := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.68_0.zxIgMxHBoMNqCBA","decl":"instance (M : FGModuleCat R) : AddCommGroup M "}
+{"state":"R : Type u\ninst✝ : Ring R\nM : FGModuleCat R\n⊢ AddCommGroup ↑M.obj","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  ","nextTactic":"infer_instance","declUpToTactic":"instance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.68_0.zxIgMxHBoMNqCBA","decl":"instance (M : FGModuleCat R) : AddCommGroup M "}
+{"state":"R : Type u\ninst✝ : Ring R\nM : FGModuleCat R\n⊢ Module R ↑M","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  ","nextTactic":"change Module R M.obj","declUpToTactic":"instance (M : FGModuleCat R) : Module R M := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.72_0.zxIgMxHBoMNqCBA","decl":"instance (M : FGModuleCat R) : Module R M "}
+{"state":"R : Type u\ninst✝ : Ring R\nM : FGModuleCat R\n⊢ Module R ↑M.obj","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  ","nextTactic":"infer_instance","declUpToTactic":"instance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.72_0.zxIgMxHBoMNqCBA","decl":"instance (M : FGModuleCat R) : Module R M "}
+{"state":"R : Type u\ninst✝ : Ring R\n⊢ LargeCategory (FGModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : LargeCategory (FGModuleCat R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.76_0.zxIgMxHBoMNqCBA","decl":"instance : LargeCategory (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : Ring R\n⊢ LargeCategory (FullSubcategory fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.76_0.zxIgMxHBoMNqCBA","decl":"instance : LargeCategory (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : Ring R\n⊢ ConcreteCategory (FGModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : ConcreteCategory (FGModuleCat R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.83_0.zxIgMxHBoMNqCBA","decl":"instance : ConcreteCategory (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : Ring R\n⊢ ConcreteCategory (FullSubcategory fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.83_0.zxIgMxHBoMNqCBA","decl":"instance : ConcreteCategory (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : Ring R\n⊢ Preadditive (FGModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : Preadditive (FGModuleCat R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.87_0.zxIgMxHBoMNqCBA","decl":"instance : Preadditive (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : Ring R\n⊢ Preadditive (FullSubcategory fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.87_0.zxIgMxHBoMNqCBA","decl":"instance : Preadditive (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝³ : Ring R\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : Module.Finite R V\n⊢ Module.Finite R ↑(ModuleCat.of R V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by ","nextTactic":"change Module.Finite R V","declUpToTactic":"/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.106_0.zxIgMxHBoMNqCBA","decl":"/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R "}
+{"state":"R : Type u\ninst✝³ : Ring R\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : Module.Finite R V\n⊢ Module.Finite R V","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; ","nextTactic":"infer_instance","declUpToTactic":"/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.106_0.zxIgMxHBoMNqCBA","decl":"/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R "}
+{"state":"R : Type u\ninst✝ : Ring R\n⊢ HasForget₂ (FGModuleCat R) (ModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.114_0.zxIgMxHBoMNqCBA","decl":"instance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) "}
+{"state":"R : Type u\ninst✝ : Ring R\n⊢ HasForget₂ (FullSubcategory fun V => Module.Finite R ↑V) (ModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.114_0.zxIgMxHBoMNqCBA","decl":"instance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) "}
+{"state":"R : Type u\ninst✝⁶ : Ring R\nV W : Type u\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : Module.Finite R V\ninst✝² : AddCommGroup W\ninst✝¹ : Module R W\ninst✝ : Module.Finite R W\ne : V ≃ₗ[R] W\n⊢ ↑e ≫ ↑(LinearEquiv.symm e) = 𝟙 (of R V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ","nextTactic":"ext x","declUpToTactic":"/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.129_0.zxIgMxHBoMNqCBA","decl":"/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom "}
+{"state":"case w\nR : Type u\ninst✝⁶ : Ring R\nV W : Type u\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : Module.Finite R V\ninst✝² : AddCommGroup W\ninst✝¹ : Module R W\ninst✝ : Module.Finite R W\ne : V ≃ₗ[R] W\nx : (forget (FGModuleCat R)).obj (of R V)\n⊢ (↑e ≫ ↑(LinearEquiv.symm e)) x = (𝟙 (of R V)) x","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; ","nextTactic":"exact e.left_inv x","declUpToTactic":"/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.129_0.zxIgMxHBoMNqCBA","decl":"/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom "}
+{"state":"R : Type u\ninst✝⁶ : Ring R\nV W : Type u\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : Module.Finite R V\ninst✝² : AddCommGroup W\ninst✝¹ : Module R W\ninst✝ : Module.Finite R W\ne : V ≃ₗ[R] W\n⊢ ↑(LinearEquiv.symm e) ≫ ↑e = 𝟙 (of R W)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ","nextTactic":"ext x","declUpToTactic":"/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.129_0.zxIgMxHBoMNqCBA","decl":"/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom "}
+{"state":"case w\nR : Type u\ninst✝⁶ : Ring R\nV W : Type u\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : Module.Finite R V\ninst✝² : AddCommGroup W\ninst✝¹ : Module R W\ninst✝ : Module.Finite R W\ne : V ≃ₗ[R] W\nx : (forget (FGModuleCat R)).obj (of R W)\n⊢ (↑(LinearEquiv.symm e) ≫ ↑e) x = (𝟙 (of R W)) x","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; ","nextTactic":"exact e.right_inv x","declUpToTactic":"/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.129_0.zxIgMxHBoMNqCBA","decl":"/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ Linear R (FGModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : Linear R (FGModuleCat R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.147_0.zxIgMxHBoMNqCBA","decl":"instance : Linear R (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ Linear R (FullSubcategory fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.147_0.zxIgMxHBoMNqCBA","decl":"instance : Linear R (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ MonoidalCategory (FGModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : MonoidalCategory (FGModuleCat R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.157_0.zxIgMxHBoMNqCBA","decl":"instance : MonoidalCategory (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ MonoidalCategory (FullSubcategory fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.157_0.zxIgMxHBoMNqCBA","decl":"instance : MonoidalCategory (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ SymmetricCategory (FGModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : SymmetricCategory (FGModuleCat R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.166_0.zxIgMxHBoMNqCBA","decl":"instance : SymmetricCategory (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ SymmetricCategory (FullSubcategory fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.166_0.zxIgMxHBoMNqCBA","decl":"instance : SymmetricCategory (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ MonoidalPreadditive (FGModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : MonoidalPreadditive (FGModuleCat R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.170_0.zxIgMxHBoMNqCBA","decl":"instance : MonoidalPreadditive (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ MonoidalPreadditive (FullSubcategory fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.170_0.zxIgMxHBoMNqCBA","decl":"instance : MonoidalPreadditive (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ MonoidalLinear R (FGModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : MonoidalLinear R (FGModuleCat R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.174_0.zxIgMxHBoMNqCBA","decl":"instance : MonoidalLinear R (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ MonoidalLinear R (FullSubcategory fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.174_0.zxIgMxHBoMNqCBA","decl":"instance : MonoidalLinear R (FGModuleCat R) "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ Faithful (forget₂Monoidal R).toLaxMonoidalFunctor.toFunctor","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  ","nextTactic":"dsimp [forget₂Monoidal]","declUpToTactic":"instance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.183_0.zxIgMxHBoMNqCBA","decl":"instance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ Faithful (fullSubcategoryInclusion fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  ","nextTactic":"exact FullSubcategory.faithful _","declUpToTactic":"instance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.183_0.zxIgMxHBoMNqCBA","decl":"instance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ Functor.Additive (forget₂Monoidal R).toLaxMonoidalFunctor.toFunctor","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  ","nextTactic":"dsimp [forget₂Monoidal]","declUpToTactic":"instance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.189_0.zxIgMxHBoMNqCBA","decl":"instance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ Functor.Additive (fullSubcategoryInclusion fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  ","nextTactic":"exact Functor.fullSubcategoryInclusion_additive _","declUpToTactic":"instance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.189_0.zxIgMxHBoMNqCBA","decl":"instance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ Functor.Linear R (forget₂Monoidal R).toLaxMonoidalFunctor.toFunctor","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusion_additive _\n#align fgModule.forget₂_monoidal_additive FGModuleCat.forget₂Monoidal_additive\n\ninstance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  ","nextTactic":"dsimp [forget₂Monoidal]","declUpToTactic":"instance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.195_0.zxIgMxHBoMNqCBA","decl":"instance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R "}
+{"state":"R : Type u\ninst✝ : CommRing R\n⊢ Functor.Linear R (fullSubcategoryInclusion fun V => Module.Finite R ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusion_additive _\n#align fgModule.forget₂_monoidal_additive FGModuleCat.forget₂Monoidal_additive\n\ninstance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  ","nextTactic":"exact Functor.fullSubcategoryInclusionLinear _ _","declUpToTactic":"instance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.195_0.zxIgMxHBoMNqCBA","decl":"instance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R "}
+{"state":"K : Type u\ninst✝ : Field K\nV W : FGModuleCat K\n⊢ Module.Finite K (↑V →ₗ[K] ↑W)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusion_additive _\n#align fgModule.forget₂_monoidal_additive FGModuleCat.forget₂Monoidal_additive\n\ninstance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusionLinear _ _\n#align fgModule.forget₂_monoidal_linear FGModuleCat.forget₂Monoidal_linear\n\ntheorem Iso.conj_eq_conj {V W : FGModuleCat R} (i : V ≅ W) (f : End V) :\n    Iso.conj i f = LinearEquiv.conj (isoToLinearEquiv i) f :=\n  rfl\n#align fgModule.iso.conj_eq_conj FGModuleCat.Iso.conj_eq_conj\n\nend CommRing\n\nsection Field\n\nvariable (K : Type u) [Field K]\n\ninstance (V W : FGModuleCat K) : Module.Finite K (V ⟶ W) :=\n  (by ","nextTactic":"infer_instance","declUpToTactic":"instance (V W : FGModuleCat K) : Module.Finite K (V ⟶ W) :=\n  (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.212_0.zxIgMxHBoMNqCBA","decl":"instance (V W : FGModuleCat K) : Module.Finite K (V ⟶ W) "}
+{"state":"K : Type u\ninst✝ : Field K\n⊢ MonoidalClosed (FGModuleCat K)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusion_additive _\n#align fgModule.forget₂_monoidal_additive FGModuleCat.forget₂Monoidal_additive\n\ninstance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusionLinear _ _\n#align fgModule.forget₂_monoidal_linear FGModuleCat.forget₂Monoidal_linear\n\ntheorem Iso.conj_eq_conj {V W : FGModuleCat R} (i : V ≅ W) (f : End V) :\n    Iso.conj i f = LinearEquiv.conj (isoToLinearEquiv i) f :=\n  rfl\n#align fgModule.iso.conj_eq_conj FGModuleCat.Iso.conj_eq_conj\n\nend CommRing\n\nsection Field\n\nvariable (K : Type u) [Field K]\n\ninstance (V W : FGModuleCat K) : Module.Finite K (V ⟶ W) :=\n  (by infer_instance : Module.Finite K (V →ₗ[K] W))\n\ninstance closedPredicateModuleFinite :\n    MonoidalCategory.ClosedPredicate fun V : ModuleCat.{u} K => Module.Finite K V where\n  prop_ihom := @fun X Y hX hY => @Module.Finite.linearMap K X Y _ _ _ _ _ _ _ hX hY\n#align fgModule.closed_predicate_module_finite FGModuleCat.closedPredicateModuleFinite\n\ninstance : MonoidalClosed (FGModuleCat K) := by\n  ","nextTactic":"dsimp [FGModuleCat]","declUpToTactic":"instance : MonoidalClosed (FGModuleCat K) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.220_0.zxIgMxHBoMNqCBA","decl":"instance : MonoidalClosed (FGModuleCat K) "}
+{"state":"K : Type u\ninst✝ : Field K\n⊢ MonoidalClosed (FullSubcategory fun V => Module.Finite K ↑V)","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusion_additive _\n#align fgModule.forget₂_monoidal_additive FGModuleCat.forget₂Monoidal_additive\n\ninstance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusionLinear _ _\n#align fgModule.forget₂_monoidal_linear FGModuleCat.forget₂Monoidal_linear\n\ntheorem Iso.conj_eq_conj {V W : FGModuleCat R} (i : V ≅ W) (f : End V) :\n    Iso.conj i f = LinearEquiv.conj (isoToLinearEquiv i) f :=\n  rfl\n#align fgModule.iso.conj_eq_conj FGModuleCat.Iso.conj_eq_conj\n\nend CommRing\n\nsection Field\n\nvariable (K : Type u) [Field K]\n\ninstance (V W : FGModuleCat K) : Module.Finite K (V ⟶ W) :=\n  (by infer_instance : Module.Finite K (V →ₗ[K] W))\n\ninstance closedPredicateModuleFinite :\n    MonoidalCategory.ClosedPredicate fun V : ModuleCat.{u} K => Module.Finite K V where\n  prop_ihom := @fun X Y hX hY => @Module.Finite.linearMap K X Y _ _ _ _ _ _ _ hX hY\n#align fgModule.closed_predicate_module_finite FGModuleCat.closedPredicateModuleFinite\n\ninstance : MonoidalClosed (FGModuleCat K) := by\n  dsimp [FGModuleCat]\n  -- Porting note: was `infer_instance`\n  ","nextTactic":"exact MonoidalCategory.fullMonoidalClosedSubcategory\n    (fun V : ModuleCat.{u} K => Module.Finite K V)","declUpToTactic":"instance : MonoidalClosed (FGModuleCat K) := by\n  dsimp [FGModuleCat]\n  -- Porting note: was `infer_instance`\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.220_0.zxIgMxHBoMNqCBA","decl":"instance : MonoidalClosed (FGModuleCat K) "}
+{"state":"K : Type u\ninst✝ : Field K\nV W : FGModuleCat K\n⊢ (𝟙 (FGModuleCatDual K V) ⊗ FGModuleCatCoevaluation K V) ≫\n      (α_ (FGModuleCatDual K V) V (FGModuleCatDual K V)).inv ≫ (FGModuleCatEvaluation K V ⊗ 𝟙 (FGModuleCatDual K V)) =\n    (ρ_ (FGModuleCatDual K V)).hom ≫ (λ_ (FGModuleCatDual K V)).inv","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusion_additive _\n#align fgModule.forget₂_monoidal_additive FGModuleCat.forget₂Monoidal_additive\n\ninstance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusionLinear _ _\n#align fgModule.forget₂_monoidal_linear FGModuleCat.forget₂Monoidal_linear\n\ntheorem Iso.conj_eq_conj {V W : FGModuleCat R} (i : V ≅ W) (f : End V) :\n    Iso.conj i f = LinearEquiv.conj (isoToLinearEquiv i) f :=\n  rfl\n#align fgModule.iso.conj_eq_conj FGModuleCat.Iso.conj_eq_conj\n\nend CommRing\n\nsection Field\n\nvariable (K : Type u) [Field K]\n\ninstance (V W : FGModuleCat K) : Module.Finite K (V ⟶ W) :=\n  (by infer_instance : Module.Finite K (V →ₗ[K] W))\n\ninstance closedPredicateModuleFinite :\n    MonoidalCategory.ClosedPredicate fun V : ModuleCat.{u} K => Module.Finite K V where\n  prop_ihom := @fun X Y hX hY => @Module.Finite.linearMap K X Y _ _ _ _ _ _ _ hX hY\n#align fgModule.closed_predicate_module_finite FGModuleCat.closedPredicateModuleFinite\n\ninstance : MonoidalClosed (FGModuleCat K) := by\n  dsimp [FGModuleCat]\n  -- Porting note: was `infer_instance`\n  exact MonoidalCategory.fullMonoidalClosedSubcategory\n    (fun V : ModuleCat.{u} K => Module.Finite K V)\n\nvariable (V W : FGModuleCat K)\n\n@[simp]\ntheorem ihom_obj : (ihom V).obj W = FGModuleCat.of K (V →ₗ[K] W) :=\n  rfl\n#align fgModule.ihom_obj FGModuleCat.ihom_obj\n\n/-- The dual module is the dual in the rigid monoidal category `FGModuleCat K`. -/\ndef FGModuleCatDual : FGModuleCat K :=\n  ⟨ModuleCat.of K (Module.Dual K V), Subspace.instModuleDualFiniteDimensional⟩\n#align fgModule.fgModule_dual FGModuleCat.FGModuleCatDual\n\n@[simp] lemma FGModuleCatDual_obj : (FGModuleCatDual K V).obj = ModuleCat.of K (Module.Dual K V) :=\n  rfl\n@[simp] lemma FGModuleCatDual_coe : (FGModuleCatDual K V : Type u) = Module.Dual K V := rfl\n\nopen CategoryTheory.MonoidalCategory\n\n/-- The coevaluation map is defined in `LinearAlgebra.coevaluation`. -/\ndef FGModuleCatCoevaluation : 𝟙_ (FGModuleCat K) ⟶ V ⊗ FGModuleCatDual K V :=\n  coevaluation K V\n#align fgModule.fgModule_coevaluation FGModuleCat.FGModuleCatCoevaluation\n\ntheorem FGModuleCatCoevaluation_apply_one :\n    FGModuleCatCoevaluation K V (1 : K) =\n      ∑ i : Basis.ofVectorSpaceIndex K V,\n        (Basis.ofVectorSpace K V) i ⊗ₜ[K] (Basis.ofVectorSpace K V).coord i :=\n  coevaluation_apply_one K V\n#align fgModule.fgModule_coevaluation_apply_one FGModuleCat.FGModuleCatCoevaluation_apply_one\n\n/-- The evaluation morphism is given by the contraction map. -/\ndef FGModuleCatEvaluation : FGModuleCatDual K V ⊗ V ⟶ 𝟙_ (FGModuleCat K) :=\n  contractLeft K V\n#align fgModule.fgModule_evaluation FGModuleCat.FGModuleCatEvaluation\n\n@[simp]\ntheorem FGModuleCatEvaluation_apply (f : FGModuleCatDual K V) (x : V) :\n    (FGModuleCatEvaluation K V) (f ⊗ₜ x) = f.toFun x :=\n  contractLeft_apply f x\n#align fgModule.fgModule_evaluation_apply FGModuleCat.FGModuleCatEvaluation_apply\n\nprivate theorem coevaluation_evaluation :\n    letI V' : FGModuleCat K := FGModuleCatDual K V\n    (𝟙 V' ⊗ FGModuleCatCoevaluation K V) ≫ (α_ V' V V').inv ≫ (FGModuleCatEvaluation K V ⊗ 𝟙 V') =\n      (ρ_ V').hom ≫ (λ_ V').inv := by\n  ","nextTactic":"apply contractLeft_assoc_coevaluation K V","declUpToTactic":"private theorem coevaluation_evaluation :\n    letI V' : FGModuleCat K := FGModuleCatDual K V\n    (𝟙 V' ⊗ FGModuleCatCoevaluation K V) ≫ (α_ V' V V').inv ≫ (FGModuleCatEvaluation K V ⊗ 𝟙 V') =\n      (ρ_ V').hom ≫ (λ_ V').inv := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.267_0.zxIgMxHBoMNqCBA","decl":"private theorem coevaluation_evaluation :\n    letI V' : FGModuleCat K "}
+{"state":"K : Type u\ninst✝ : Field K\nV W : FGModuleCat K\n⊢ (FGModuleCatCoevaluation K V ⊗ 𝟙 V) ≫ (α_ V (FGModuleCatDual K V) V).hom ≫ (𝟙 V ⊗ FGModuleCatEvaluation K V) =\n    (λ_ V).hom ≫ (ρ_ V).inv","srcUpToTactic":"/-\nCopyright (c) 2021 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer\n-/\nimport Mathlib.CategoryTheory.Monoidal.Rigid.Basic\nimport Mathlib.CategoryTheory.Monoidal.Subcategory\nimport Mathlib.LinearAlgebra.Coevaluation\nimport Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\n#align_import algebra.category.fgModule.basic from \"leanprover-community/mathlib\"@\"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2\"\n\n/-!\n# The category of finitely generated modules over a ring\n\nThis introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`.\nIt is implemented as a full subcategory on a subtype of `ModuleCat R`.\n\nWhen `K` is a field,\n`FGModuleCatCat K` is the category of finite dimensional vector spaces over `K`.\n\nWe first create the instance as a preadditive category.\nWhen `R` is commutative we then give the structure as an `R`-linear monoidal category.\nWhen `R` is a field we give it the structure of a closed monoidal category\nand then as a right-rigid monoidal category.\n\n## Future work\n\n* Show that `FGModuleCat R` is abelian when `R` is (left)-noetherian.\n\n-/\n\nset_option linter.uppercaseLean3 false\n\nnoncomputable section\n\nopen CategoryTheory ModuleCat.monoidalCategory\n\nopen scoped Classical BigOperators\n\nuniverse u\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\n/-- Define `FGModuleCat` as the subtype of `ModuleCat.{u} R` of finitely generated modules. -/\ndef FGModuleCat :=\n  FullSubcategory fun V : ModuleCat.{u} R => Module.Finite R V\n-- Porting note: still no derive handler via `dsimp`.\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\n-- deriving LargeCategory, ConcreteCategory,Preadditive\n#align fgModule FGModuleCat\n\nvariable {R}\n\n/-- A synonym for `M.obj.carrier`, which we can mark with `@[coe]`. -/\ndef FGModuleCat.carrier (M : FGModuleCat R) : Type u := M.obj.carrier\n\ninstance : CoeSort (FGModuleCat R) (Type u) :=\n  ⟨FGModuleCat.carrier⟩\n\nattribute [coe] FGModuleCat.carrier\n\n@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier := rfl\n\ninstance (M : FGModuleCat R) : AddCommGroup M := by\n  change AddCommGroup M.obj\n  infer_instance\n\ninstance (M : FGModuleCat R) : Module R M := by\n  change Module R M.obj\n  infer_instance\n\ninstance : LargeCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance {M N : FGModuleCat R} : LinearMapClass (M ⟶ N) R M N :=\n  LinearMap.semilinearMapClass\n\ninstance : ConcreteCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Preadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nend Ring\n\nnamespace FGModuleCat\n\nsection Ring\n\nvariable (R : Type u) [Ring R]\n\ninstance finite (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n#align fgModule.finite FGModuleCat.finite\n\ninstance : Inhabited (FGModuleCat R) :=\n  ⟨⟨ModuleCat.of R R, Module.Finite.self R⟩⟩\n\n/-- Lift an unbundled finitely generated module to `FGModuleCat R`. -/\ndef of (V : Type u) [AddCommGroup V] [Module R V] [Module.Finite R V] : FGModuleCat R :=\n  ⟨ModuleCat.of R V, by change Module.Finite R V; infer_instance⟩\n#align fgModule.of FGModuleCat.of\n\ninstance (V : FGModuleCat R) : Module.Finite R V :=\n  V.property\n\ninstance : HasForget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : Full (forget₂ (FGModuleCat R) (ModuleCat.{u} R)) where\n  preimage f := f\n\nvariable {R}\n\n/-- Converts and isomorphism in the category `FGModuleCat R` to\na `LinearEquiv` between the underlying modules. -/\ndef isoToLinearEquiv {V W : FGModuleCat R} (i : V ≅ W) : V ≃ₗ[R] W :=\n  ((forget₂ (FGModuleCat.{u} R) (ModuleCat.{u} R)).mapIso i).toLinearEquiv\n#align fgModule.iso_to_linear_equiv FGModuleCat.isoToLinearEquiv\n\n/-- Converts a `LinearEquiv` to an isomorphism in the category `FGModuleCat R`. -/\n@[simps]\ndef _root_.LinearEquiv.toFGModuleCatIso\n    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]\n    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :\n    FGModuleCat.of R V ≅ FGModuleCat.of R W where\n  hom := e.toLinearMap\n  inv := e.symm.toLinearMap\n  hom_inv_id := by ext x; exact e.left_inv x\n  inv_hom_id := by ext x; exact e.right_inv x\n#align linear_equiv.to_fgModule_iso LinearEquiv.toFGModuleCatIso\n\nend Ring\n\nsection CommRing\n\nvariable (R : Type u) [CommRing R]\n\ninstance : Linear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance monoidalPredicate_module_finite :\n    MonoidalCategory.MonoidalPredicate fun V : ModuleCat.{u} R => Module.Finite R V where\n  prop_id := Module.Finite.self R\n  prop_tensor := @fun X Y _ _ => Module.Finite.tensorProduct R X Y\n#align fgModule.monoidal_predicate_module_finite FGModuleCat.monoidalPredicate_module_finite\n\ninstance : MonoidalCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\nopen MonoidalCategory\n\n@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) := rfl\n@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) := rfl\n\ninstance : SymmetricCategory (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalPreadditive (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\ninstance : MonoidalLinear R (FGModuleCat R) := by\n  dsimp [FGModuleCat]\n  infer_instance\n\n/-- The forgetful functor `FGModuleCat R ⥤ Module R` as a monoidal functor. -/\ndef forget₂Monoidal : MonoidalFunctor (FGModuleCat R) (ModuleCat.{u} R) :=\n  MonoidalCategory.fullMonoidalSubcategoryInclusion _\n#align fgModule.forget₂_monoidal FGModuleCat.forget₂Monoidal\n\ninstance forget₂Monoidal_faithful : Faithful (forget₂Monoidal R).toFunctor := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact FullSubcategory.faithful _\n#align fgModule.forget₂_monoidal_faithful FGModuleCat.forget₂Monoidal_faithful\n\ninstance forget₂Monoidal_additive : (forget₂Monoidal R).toFunctor.Additive := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusion_additive _\n#align fgModule.forget₂_monoidal_additive FGModuleCat.forget₂Monoidal_additive\n\ninstance forget₂Monoidal_linear : (forget₂Monoidal R).toFunctor.Linear R := by\n  dsimp [forget₂Monoidal]\n  -- Porting note: was `infer_instance`\n  exact Functor.fullSubcategoryInclusionLinear _ _\n#align fgModule.forget₂_monoidal_linear FGModuleCat.forget₂Monoidal_linear\n\ntheorem Iso.conj_eq_conj {V W : FGModuleCat R} (i : V ≅ W) (f : End V) :\n    Iso.conj i f = LinearEquiv.conj (isoToLinearEquiv i) f :=\n  rfl\n#align fgModule.iso.conj_eq_conj FGModuleCat.Iso.conj_eq_conj\n\nend CommRing\n\nsection Field\n\nvariable (K : Type u) [Field K]\n\ninstance (V W : FGModuleCat K) : Module.Finite K (V ⟶ W) :=\n  (by infer_instance : Module.Finite K (V →ₗ[K] W))\n\ninstance closedPredicateModuleFinite :\n    MonoidalCategory.ClosedPredicate fun V : ModuleCat.{u} K => Module.Finite K V where\n  prop_ihom := @fun X Y hX hY => @Module.Finite.linearMap K X Y _ _ _ _ _ _ _ hX hY\n#align fgModule.closed_predicate_module_finite FGModuleCat.closedPredicateModuleFinite\n\ninstance : MonoidalClosed (FGModuleCat K) := by\n  dsimp [FGModuleCat]\n  -- Porting note: was `infer_instance`\n  exact MonoidalCategory.fullMonoidalClosedSubcategory\n    (fun V : ModuleCat.{u} K => Module.Finite K V)\n\nvariable (V W : FGModuleCat K)\n\n@[simp]\ntheorem ihom_obj : (ihom V).obj W = FGModuleCat.of K (V →ₗ[K] W) :=\n  rfl\n#align fgModule.ihom_obj FGModuleCat.ihom_obj\n\n/-- The dual module is the dual in the rigid monoidal category `FGModuleCat K`. -/\ndef FGModuleCatDual : FGModuleCat K :=\n  ⟨ModuleCat.of K (Module.Dual K V), Subspace.instModuleDualFiniteDimensional⟩\n#align fgModule.fgModule_dual FGModuleCat.FGModuleCatDual\n\n@[simp] lemma FGModuleCatDual_obj : (FGModuleCatDual K V).obj = ModuleCat.of K (Module.Dual K V) :=\n  rfl\n@[simp] lemma FGModuleCatDual_coe : (FGModuleCatDual K V : Type u) = Module.Dual K V := rfl\n\nopen CategoryTheory.MonoidalCategory\n\n/-- The coevaluation map is defined in `LinearAlgebra.coevaluation`. -/\ndef FGModuleCatCoevaluation : 𝟙_ (FGModuleCat K) ⟶ V ⊗ FGModuleCatDual K V :=\n  coevaluation K V\n#align fgModule.fgModule_coevaluation FGModuleCat.FGModuleCatCoevaluation\n\ntheorem FGModuleCatCoevaluation_apply_one :\n    FGModuleCatCoevaluation K V (1 : K) =\n      ∑ i : Basis.ofVectorSpaceIndex K V,\n        (Basis.ofVectorSpace K V) i ⊗ₜ[K] (Basis.ofVectorSpace K V).coord i :=\n  coevaluation_apply_one K V\n#align fgModule.fgModule_coevaluation_apply_one FGModuleCat.FGModuleCatCoevaluation_apply_one\n\n/-- The evaluation morphism is given by the contraction map. -/\ndef FGModuleCatEvaluation : FGModuleCatDual K V ⊗ V ⟶ 𝟙_ (FGModuleCat K) :=\n  contractLeft K V\n#align fgModule.fgModule_evaluation FGModuleCat.FGModuleCatEvaluation\n\n@[simp]\ntheorem FGModuleCatEvaluation_apply (f : FGModuleCatDual K V) (x : V) :\n    (FGModuleCatEvaluation K V) (f ⊗ₜ x) = f.toFun x :=\n  contractLeft_apply f x\n#align fgModule.fgModule_evaluation_apply FGModuleCat.FGModuleCatEvaluation_apply\n\nprivate theorem coevaluation_evaluation :\n    letI V' : FGModuleCat K := FGModuleCatDual K V\n    (𝟙 V' ⊗ FGModuleCatCoevaluation K V) ≫ (α_ V' V V').inv ≫ (FGModuleCatEvaluation K V ⊗ 𝟙 V') =\n      (ρ_ V').hom ≫ (λ_ V').inv := by\n  apply contractLeft_assoc_coevaluation K V\n\nprivate theorem evaluation_coevaluation :\n    (FGModuleCatCoevaluation K V ⊗ 𝟙 V) ≫\n        (α_ V (FGModuleCatDual K V) V).hom ≫ (𝟙 V ⊗ FGModuleCatEvaluation K V) =\n      (λ_ V).hom ≫ (ρ_ V).inv := by\n  ","nextTactic":"apply contractLeft_assoc_coevaluation' K V","declUpToTactic":"private theorem evaluation_coevaluation :\n    (FGModuleCatCoevaluation K V ⊗ 𝟙 V) ≫\n        (α_ V (FGModuleCatDual K V) V).hom ≫ (𝟙 V ⊗ FGModuleCatEvaluation K V) =\n      (λ_ V).hom ≫ (ρ_ V).inv := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Basic.273_0.zxIgMxHBoMNqCBA","decl":"private theorem evaluation_coevaluation :\n    (FGModuleCatCoevaluation K V ⊗ 𝟙 V) ≫\n        (α_ V (FGModuleCatDual K V) V).hom ≫ (𝟙 V ⊗ FGModuleCatEvaluation K V) =\n      (λ_ V).hom ≫ (ρ_ V).inv "}