Delete Extracted

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_Opposite.jsonl

@@ -1,2 +0,0 @@
-{"state":"R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra S A\ninst✝¹ : SMulCommClass R S A\ninst✝ : IsScalarTower R S A\nr : R\nx : A\n⊢ (RingHom.toOpposite (algebraMap R A)\n          (_ : ∀ (x y : R), (algebraMap R A) x * (algebraMap R A) y = (algebraMap R A) y * (algebraMap R A) x))\n        r *\n      op x =\n    op x *\n      (RingHom.toOpposite (algebraMap R A)\n          (_ : ∀ (x y : R), (algebraMap R A) x * (algebraMap R A) y = (algebraMap R A) y * (algebraMap R A) x))\n        r","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Equiv\nimport Mathlib.Algebra.Module.Opposites\nimport Mathlib.Algebra.Ring.Opposite\n\n/-!\n# Algebra structures on the multiplicative opposite\n\n## Main definitions\n\n* `MulOpposite.instAlgebra`: the algebra on `Aᵐᵒᵖ`\n* `AlgHom.op`/`AlgHom.unop`: simultaneously convert the domain and codomain of a morphism to the\n  opposite algebra.\n* `AlgHom.opComm`: swap which side of a morphism lies in the opposite algebra.\n* `AlgEquiv.op`/`AlgEquiv.unop`: simultaneously convert the source and target of an isomorphism to\n  the opposite algebra.\n* `AlgEquiv.opOp`: any algebra is isomorphic to the opposite of its opposite.\n* `AlgEquiv.toOpposite`: in a commutative algebra, the opposite algebra is isomorphic to the\n  original algebra.\n* `AlgEquiv.opComm`: swap which side of an isomorphism lies in the opposite algebra.\n-/\n\n\nvariable {R S A B : Type*}\n\nopen MulOpposite\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\nvariable [Algebra R S] [Algebra R A] [Algebra R B] [Algebra S A] [SMulCommClass R S A]\nvariable [IsScalarTower R S A]\n\nnamespace MulOpposite\n\ninstance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n  toRingHom := (algebraMap R A).toOpposite fun x y => Algebra.commutes _ _\n  smul_def' c x := unop_injective <| by\n    simp only [unop_smul, RingHom.toOpposite_apply, Function.comp_apply, unop_mul, op_mul,\n      Algebra.smul_def, Algebra.commutes, op_unop, unop_op]\n  commutes' r := MulOpposite.rec' fun x => by\n    ","nextTactic":"simp only [RingHom.toOpposite_apply, Function.comp_apply, ← op_mul, Algebra.commutes]","declUpToTactic":"instance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n  toRingHom := (algebraMap R A).toOpposite fun x y => Algebra.commutes _ _\n  smul_def' c x := unop_injective <| by\n    simp only [unop_smul, RingHom.toOpposite_apply, Function.comp_apply, unop_mul, op_mul,\n      Algebra.smul_def, Algebra.commutes, op_unop, unop_op]\n  commutes' r := MulOpposite.rec' fun x => by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Opposite.40_0.Ep4LYRAHgoq5qI9","decl":"instance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n  toRingHom "}
-{"state":"R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra S A\ninst✝¹ : SMulCommClass R S A\ninst✝ : IsScalarTower R S A\nc : R\nx : Aᵐᵒᵖ\n⊢ unop (c • x) =\n    unop\n      ((RingHom.toOpposite (algebraMap R A)\n            (_ : ∀ (x y : R), (algebraMap R A) x * (algebraMap R A) y = (algebraMap R A) y * (algebraMap R A) x))\n          c *\n        x)","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Equiv\nimport Mathlib.Algebra.Module.Opposites\nimport Mathlib.Algebra.Ring.Opposite\n\n/-!\n# Algebra structures on the multiplicative opposite\n\n## Main definitions\n\n* `MulOpposite.instAlgebra`: the algebra on `Aᵐᵒᵖ`\n* `AlgHom.op`/`AlgHom.unop`: simultaneously convert the domain and codomain of a morphism to the\n  opposite algebra.\n* `AlgHom.opComm`: swap which side of a morphism lies in the opposite algebra.\n* `AlgEquiv.op`/`AlgEquiv.unop`: simultaneously convert the source and target of an isomorphism to\n  the opposite algebra.\n* `AlgEquiv.opOp`: any algebra is isomorphic to the opposite of its opposite.\n* `AlgEquiv.toOpposite`: in a commutative algebra, the opposite algebra is isomorphic to the\n  original algebra.\n* `AlgEquiv.opComm`: swap which side of an isomorphism lies in the opposite algebra.\n-/\n\n\nvariable {R S A B : Type*}\n\nopen MulOpposite\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\nvariable [Algebra R S] [Algebra R A] [Algebra R B] [Algebra S A] [SMulCommClass R S A]\nvariable [IsScalarTower R S A]\n\nnamespace MulOpposite\n\ninstance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n  toRingHom := (algebraMap R A).toOpposite fun x y => Algebra.commutes _ _\n  smul_def' c x := unop_injective <| by\n    ","nextTactic":"simp only [unop_smul, RingHom.toOpposite_apply, Function.comp_apply, unop_mul, op_mul,\n      Algebra.smul_def, Algebra.commutes, op_unop, unop_op]","declUpToTactic":"instance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n  toRingHom := (algebraMap R A).toOpposite fun x y => Algebra.commutes _ _\n  smul_def' c x := unop_injective <| by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Opposite.40_0.Ep4LYRAHgoq5qI9","decl":"instance MulOpposite.instAlgebra : Algebra R Aᵐᵒᵖ where\n  toRingHom "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_Pi.jsonl

@@ -1,8 +0,0 @@
-{"state":"I : Type u\nR : Type u_1\nf✝ : I → Type v\nx y : (i : I) → f✝ i\ni : I\nr : CommSemiring R\ns : (i : I) → Semiring (f✝ i)\ninst✝ : (i : I) → Algebra R (f✝ i)\nsrc✝ : R →+* (i : I) → f✝ i := Pi.ringHom fun i => algebraMap R (f✝ i)\na : R\nf : (i : I) → f✝ i\n⊢ { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n          map_add' :=\n            (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n        a *\n      f =\n    f *\n      { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n          map_add' :=\n            (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n        a","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Equiv\n\n#align_import algebra.algebra.pi from \"leanprover-community/mathlib\"@\"b16045e4bf61c6fd619a1a68854ab3d605dcd017\"\n\n/-!\n# The R-algebra structure on families of R-algebras\n\nThe R-algebra structure on `∀ i : I, A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nuniverse u v w\n\nnamespace Pi\n\n-- The indexing type\nvariable {I : Type u}\n\n-- The scalar type\nvariable {R : Type*}\n\n-- The family of types already equipped with instances\nvariable {f : I → Type v}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nvariable (I f)\n\ninstance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ","nextTactic":"ext","declUpToTactic":"instance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Pi.40_0.iEzBlhDeTy24dhL","decl":"instance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) "}
-{"state":"case h\nI : Type u\nR : Type u_1\nf✝ : I → Type v\nx y : (i : I) → f✝ i\ni : I\nr : CommSemiring R\ns : (i : I) → Semiring (f✝ i)\ninst✝ : (i : I) → Algebra R (f✝ i)\nsrc✝ : R →+* (i : I) → f✝ i := Pi.ringHom fun i => algebraMap R (f✝ i)\na : R\nf : (i : I) → f✝ i\nx✝ : I\n⊢ ({ toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n          a *\n        f)\n      x✝ =\n    (f *\n        { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n          a)\n      x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Equiv\n\n#align_import algebra.algebra.pi from \"leanprover-community/mathlib\"@\"b16045e4bf61c6fd619a1a68854ab3d605dcd017\"\n\n/-!\n# The R-algebra structure on families of R-algebras\n\nThe R-algebra structure on `∀ i : I, A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nuniverse u v w\n\nnamespace Pi\n\n-- The indexing type\nvariable {I : Type u}\n\n-- The scalar type\nvariable {R : Type*}\n\n-- The family of types already equipped with instances\nvariable {f : I → Type v}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nvariable (I f)\n\ninstance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; ","nextTactic":"simp [Algebra.commutes]","declUpToTactic":"instance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Pi.40_0.iEzBlhDeTy24dhL","decl":"instance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) "}
-{"state":"I : Type u\nR : Type u_1\nf✝ : I → Type v\nx y : (i : I) → f✝ i\ni : I\nr : CommSemiring R\ns : (i : I) → Semiring (f✝ i)\ninst✝ : (i : I) → Algebra R (f✝ i)\nsrc✝ : R →+* (i : I) → f✝ i := Pi.ringHom fun i => algebraMap R (f✝ i)\na : R\nf : (i : I) → f✝ i\n⊢ a • f =\n    { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n          map_add' :=\n            (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n        a *\n      f","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Equiv\n\n#align_import algebra.algebra.pi from \"leanprover-community/mathlib\"@\"b16045e4bf61c6fd619a1a68854ab3d605dcd017\"\n\n/-!\n# The R-algebra structure on families of R-algebras\n\nThe R-algebra structure on `∀ i : I, A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nuniverse u v w\n\nnamespace Pi\n\n-- The indexing type\nvariable {I : Type u}\n\n-- The scalar type\nvariable {R : Type*}\n\n-- The family of types already equipped with instances\nvariable {f : I → Type v}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nvariable (I f)\n\ninstance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; simp [Algebra.commutes]\n    smul_def' := fun a f => by ","nextTactic":"ext","declUpToTactic":"instance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; simp [Algebra.commutes]\n    smul_def' := fun a f => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Pi.40_0.iEzBlhDeTy24dhL","decl":"instance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) "}
-{"state":"case h\nI : Type u\nR : Type u_1\nf✝ : I → Type v\nx y : (i : I) → f✝ i\ni : I\nr : CommSemiring R\ns : (i : I) → Semiring (f✝ i)\ninst✝ : (i : I) → Algebra R (f✝ i)\nsrc✝ : R →+* (i : I) → f✝ i := Pi.ringHom fun i => algebraMap R (f✝ i)\na : R\nf : (i : I) → f✝ i\nx✝ : I\n⊢ (a • f) x✝ =\n    ({ toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n          a *\n        f)\n      x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Equiv\n\n#align_import algebra.algebra.pi from \"leanprover-community/mathlib\"@\"b16045e4bf61c6fd619a1a68854ab3d605dcd017\"\n\n/-!\n# The R-algebra structure on families of R-algebras\n\nThe R-algebra structure on `∀ i : I, A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nuniverse u v w\n\nnamespace Pi\n\n-- The indexing type\nvariable {I : Type u}\n\n-- The scalar type\nvariable {R : Type*}\n\n-- The family of types already equipped with instances\nvariable {f : I → Type v}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nvariable (I f)\n\ninstance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; simp [Algebra.commutes]\n    smul_def' := fun a f => by ext; ","nextTactic":"simp [Algebra.smul_def]","declUpToTactic":"instance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; simp [Algebra.commutes]\n    smul_def' := fun a f => by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Pi.40_0.iEzBlhDeTy24dhL","decl":"instance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) "}
-{"state":"R : Type u\nA : Type v\nB : Type w\nI✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nI : Type u_2\nsrc✝ : (I → A) →+* I → B := RingHom.compLeft (↑f) I\nc : R\n⊢ OneHom.toFun\n      (↑↑{\n            toMonoidHom :=\n              { toOneHom := { toFun := fun h => ⇑f ∘ h, map_one' := (_ : OneHom.toFun (↑↑src✝) 1 = 1) },\n                map_mul' :=\n                  (_ :\n                    ∀ (x y : I → A),\n                      OneHom.toFun (↑↑src✝) (x * y) = OneHom.toFun (↑↑src✝) x * OneHom.toFun (↑↑src✝) y) },\n            map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ :\n                ∀ (x y : I → A), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) })\n      ((algebraMap R (I → A)) c) =\n    (algebraMap R (I → B)) c","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Equiv\n\n#align_import algebra.algebra.pi from \"leanprover-community/mathlib\"@\"b16045e4bf61c6fd619a1a68854ab3d605dcd017\"\n\n/-!\n# The R-algebra structure on families of R-algebras\n\nThe R-algebra structure on `∀ i : I, A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nuniverse u v w\n\nnamespace Pi\n\n-- The indexing type\nvariable {I : Type u}\n\n-- The scalar type\nvariable {R : Type*}\n\n-- The family of types already equipped with instances\nvariable {f : I → Type v}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nvariable (I f)\n\ninstance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; simp [Algebra.commutes]\n    smul_def' := fun a f => by ext; simp [Algebra.smul_def] }\n#align pi.algebra Pi.algebra\n\ntheorem algebraMap_def {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]\n    (a : R) : algebraMap R (∀ i, f i) a = fun i => algebraMap R (f i) a :=\n  rfl\n#align pi.algebra_map_def Pi.algebraMap_def\n\n@[simp]\ntheorem algebraMap_apply {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]\n    (a : R) (i : I) : algebraMap R (∀ i, f i) a i = algebraMap R (f i) a :=\n  rfl\n#align pi.algebra_map_apply Pi.algebraMap_apply\n\n-- One could also build a `∀ i, R i`-algebra structure on `∀ i, A i`,\n-- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.\nvariable {I} (R)\n\n/-- `Function.eval` as an `AlgHom`. The name matches `Pi.evalRingHom`, `Pi.evalMonoidHom`,\netc. -/\n@[simps]\ndef evalAlgHom {_ : CommSemiring R} [∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] (i : I) :\n    (∀ i, f i) →ₐ[R] f i :=\n  { Pi.evalRingHom f i with\n    toFun := fun f => f i\n    commutes' := fun _ => rfl }\n#align pi.eval_alg_hom Pi.evalAlgHom\n\nvariable (A B : Type*) [CommSemiring R] [Semiring B] [Algebra R B]\n\n/-- `Function.const` as an `AlgHom`. The name matches `Pi.constRingHom`, `Pi.constMonoidHom`,\netc. -/\n@[simps]\ndef constAlgHom : B →ₐ[R] A → B :=\n  { Pi.constRingHom A B with\n    toFun := Function.const _\n    commutes' := fun _ => rfl }\n#align pi.const_alg_hom Pi.constAlgHom\n\n/-- When `R` is commutative and permits an `algebraMap`, `Pi.constRingHom` is equal to that\nmap. -/\n@[simp]\ntheorem constRingHom_eq_algebraMap : constRingHom A R = algebraMap R (A → R) :=\n  rfl\n#align pi.const_ring_hom_eq_algebra_map Pi.constRingHom_eq_algebraMap\n\n@[simp]\ntheorem constAlgHom_eq_algebra_ofId : constAlgHom R A R = Algebra.ofId R (A → R) :=\n  rfl\n#align pi.const_alg_hom_eq_algebra_of_id Pi.constAlgHom_eq_algebra_ofId\n\nend Pi\n\n/-- A special case of `Pi.algebra` for non-dependent types. Lean struggles to elaborate\ndefinitions elsewhere in the library without this, -/\ninstance Function.algebra {R : Type*} (I : Type*) (A : Type*) [CommSemiring R] [Semiring A]\n    [Algebra R A] : Algebra R (I → A) :=\n  Pi.algebra _ _\n#align function.algebra Function.algebra\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {I : Type*}\n\nvariable [CommSemiring R] [Semiring A] [Semiring B]\n\nvariable [Algebra R A] [Algebra R B]\n\n/-- `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps]\nprotected def compLeft (f : A →ₐ[R] B) (I : Type*) : (I → A) →ₐ[R] I → B :=\n  { f.toRingHom.compLeft I with\n    toFun := fun h => f ∘ h\n    commutes' := fun c => by\n      ","nextTactic":"ext","declUpToTactic":"/-- `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps]\nprotected def compLeft (f : A →ₐ[R] B) (I : Type*) : (I → A) →ₐ[R] I → B :=\n  { f.toRingHom.compLeft I with\n    toFun := fun h => f ∘ h\n    commutes' := fun c => by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Pi.112_0.iEzBlhDeTy24dhL","decl":"/-- `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps]\nprotected def compLeft (f : A →ₐ[R] B) (I : Type*) : (I → A) →ₐ[R] I → B "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\nI✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nI : Type u_2\nsrc✝ : (I → A) →+* I → B := RingHom.compLeft (↑f) I\nc : R\nx✝ : I\n⊢ OneHom.toFun\n      (↑↑{\n            toMonoidHom :=\n              { toOneHom := { toFun := fun h => ⇑f ∘ h, map_one' := (_ : OneHom.toFun (↑↑src✝) 1 = 1) },\n                map_mul' :=\n                  (_ :\n                    ∀ (x y : I → A),\n                      OneHom.toFun (↑↑src✝) (x * y) = OneHom.toFun (↑↑src✝) x * OneHom.toFun (↑↑src✝) y) },\n            map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ :\n                ∀ (x y : I → A), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) })\n      ((algebraMap R (I → A)) c) x✝ =\n    (algebraMap R (I → B)) c x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Equiv\n\n#align_import algebra.algebra.pi from \"leanprover-community/mathlib\"@\"b16045e4bf61c6fd619a1a68854ab3d605dcd017\"\n\n/-!\n# The R-algebra structure on families of R-algebras\n\nThe R-algebra structure on `∀ i : I, A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nuniverse u v w\n\nnamespace Pi\n\n-- The indexing type\nvariable {I : Type u}\n\n-- The scalar type\nvariable {R : Type*}\n\n-- The family of types already equipped with instances\nvariable {f : I → Type v}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nvariable (I f)\n\ninstance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; simp [Algebra.commutes]\n    smul_def' := fun a f => by ext; simp [Algebra.smul_def] }\n#align pi.algebra Pi.algebra\n\ntheorem algebraMap_def {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]\n    (a : R) : algebraMap R (∀ i, f i) a = fun i => algebraMap R (f i) a :=\n  rfl\n#align pi.algebra_map_def Pi.algebraMap_def\n\n@[simp]\ntheorem algebraMap_apply {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]\n    (a : R) (i : I) : algebraMap R (∀ i, f i) a i = algebraMap R (f i) a :=\n  rfl\n#align pi.algebra_map_apply Pi.algebraMap_apply\n\n-- One could also build a `∀ i, R i`-algebra structure on `∀ i, A i`,\n-- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.\nvariable {I} (R)\n\n/-- `Function.eval` as an `AlgHom`. The name matches `Pi.evalRingHom`, `Pi.evalMonoidHom`,\netc. -/\n@[simps]\ndef evalAlgHom {_ : CommSemiring R} [∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] (i : I) :\n    (∀ i, f i) →ₐ[R] f i :=\n  { Pi.evalRingHom f i with\n    toFun := fun f => f i\n    commutes' := fun _ => rfl }\n#align pi.eval_alg_hom Pi.evalAlgHom\n\nvariable (A B : Type*) [CommSemiring R] [Semiring B] [Algebra R B]\n\n/-- `Function.const` as an `AlgHom`. The name matches `Pi.constRingHom`, `Pi.constMonoidHom`,\netc. -/\n@[simps]\ndef constAlgHom : B →ₐ[R] A → B :=\n  { Pi.constRingHom A B with\n    toFun := Function.const _\n    commutes' := fun _ => rfl }\n#align pi.const_alg_hom Pi.constAlgHom\n\n/-- When `R` is commutative and permits an `algebraMap`, `Pi.constRingHom` is equal to that\nmap. -/\n@[simp]\ntheorem constRingHom_eq_algebraMap : constRingHom A R = algebraMap R (A → R) :=\n  rfl\n#align pi.const_ring_hom_eq_algebra_map Pi.constRingHom_eq_algebraMap\n\n@[simp]\ntheorem constAlgHom_eq_algebra_ofId : constAlgHom R A R = Algebra.ofId R (A → R) :=\n  rfl\n#align pi.const_alg_hom_eq_algebra_of_id Pi.constAlgHom_eq_algebra_ofId\n\nend Pi\n\n/-- A special case of `Pi.algebra` for non-dependent types. Lean struggles to elaborate\ndefinitions elsewhere in the library without this, -/\ninstance Function.algebra {R : Type*} (I : Type*) (A : Type*) [CommSemiring R] [Semiring A]\n    [Algebra R A] : Algebra R (I → A) :=\n  Pi.algebra _ _\n#align function.algebra Function.algebra\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {I : Type*}\n\nvariable [CommSemiring R] [Semiring A] [Semiring B]\n\nvariable [Algebra R A] [Algebra R B]\n\n/-- `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps]\nprotected def compLeft (f : A →ₐ[R] B) (I : Type*) : (I → A) →ₐ[R] I → B :=\n  { f.toRingHom.compLeft I with\n    toFun := fun h => f ∘ h\n    commutes' := fun c => by\n      ext\n      ","nextTactic":"exact f.commutes' c","declUpToTactic":"/-- `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps]\nprotected def compLeft (f : A →ₐ[R] B) (I : Type*) : (I → A) ��ₐ[R] I → B :=\n  { f.toRingHom.compLeft I with\n    toFun := fun h => f ∘ h\n    commutes' := fun c => by\n      ext\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Pi.112_0.iEzBlhDeTy24dhL","decl":"/-- `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps]\nprotected def compLeft (f : A →ₐ[R] B) (I : Type*) : (I → A) →ₐ[R] I → B "}
-{"state":"R : Type u_1\nι : Type u_2\nA₁ : ι → Type u_3\nA₂ : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → Semiring (A₁ i)\ninst✝² : (i : ι) → Semiring (A₂ i)\ninst✝¹ : (i : ι) → Algebra R (A₁ i)\ninst✝ : (i : ι) → Algebra R (A₂ i)\ne : (i : ι) → A₁ i ≃ₐ[R] A₂ i\nsrc✝ : ((i : ι) → A₁ i) ≃+* ((i : ι) → A₂ i) := RingEquiv.piCongrRight fun i => toRingEquiv (e i)\nr : R\n⊢ Equiv.toFun\n      { toFun := fun x j => (e j) (x j), invFun := fun x j => (symm (e j)) (x j),\n        left_inv := (_ : Function.LeftInverse src✝.invFun src✝.toFun),\n        right_inv := (_ : Function.RightInverse src✝.invFun src✝.toFun) }\n      ((algebraMap R ((i : ι) → A₁ i)) r) =\n    (algebraMap R ((i : ι) → A₂ i)) r","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Equiv\n\n#align_import algebra.algebra.pi from \"leanprover-community/mathlib\"@\"b16045e4bf61c6fd619a1a68854ab3d605dcd017\"\n\n/-!\n# The R-algebra structure on families of R-algebras\n\nThe R-algebra structure on `∀ i : I, A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nuniverse u v w\n\nnamespace Pi\n\n-- The indexing type\nvariable {I : Type u}\n\n-- The scalar type\nvariable {R : Type*}\n\n-- The family of types already equipped with instances\nvariable {f : I → Type v}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nvariable (I f)\n\ninstance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; simp [Algebra.commutes]\n    smul_def' := fun a f => by ext; simp [Algebra.smul_def] }\n#align pi.algebra Pi.algebra\n\ntheorem algebraMap_def {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]\n    (a : R) : algebraMap R (∀ i, f i) a = fun i => algebraMap R (f i) a :=\n  rfl\n#align pi.algebra_map_def Pi.algebraMap_def\n\n@[simp]\ntheorem algebraMap_apply {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]\n    (a : R) (i : I) : algebraMap R (∀ i, f i) a i = algebraMap R (f i) a :=\n  rfl\n#align pi.algebra_map_apply Pi.algebraMap_apply\n\n-- One could also build a `∀ i, R i`-algebra structure on `∀ i, A i`,\n-- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.\nvariable {I} (R)\n\n/-- `Function.eval` as an `AlgHom`. The name matches `Pi.evalRingHom`, `Pi.evalMonoidHom`,\netc. -/\n@[simps]\ndef evalAlgHom {_ : CommSemiring R} [∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] (i : I) :\n    (∀ i, f i) →ₐ[R] f i :=\n  { Pi.evalRingHom f i with\n    toFun := fun f => f i\n    commutes' := fun _ => rfl }\n#align pi.eval_alg_hom Pi.evalAlgHom\n\nvariable (A B : Type*) [CommSemiring R] [Semiring B] [Algebra R B]\n\n/-- `Function.const` as an `AlgHom`. The name matches `Pi.constRingHom`, `Pi.constMonoidHom`,\netc. -/\n@[simps]\ndef constAlgHom : B →ₐ[R] A → B :=\n  { Pi.constRingHom A B with\n    toFun := Function.const _\n    commutes' := fun _ => rfl }\n#align pi.const_alg_hom Pi.constAlgHom\n\n/-- When `R` is commutative and permits an `algebraMap`, `Pi.constRingHom` is equal to that\nmap. -/\n@[simp]\ntheorem constRingHom_eq_algebraMap : constRingHom A R = algebraMap R (A → R) :=\n  rfl\n#align pi.const_ring_hom_eq_algebra_map Pi.constRingHom_eq_algebraMap\n\n@[simp]\ntheorem constAlgHom_eq_algebra_ofId : constAlgHom R A R = Algebra.ofId R (A → R) :=\n  rfl\n#align pi.const_alg_hom_eq_algebra_of_id Pi.constAlgHom_eq_algebra_ofId\n\nend Pi\n\n/-- A special case of `Pi.algebra` for non-dependent types. Lean struggles to elaborate\ndefinitions elsewhere in the library without this, -/\ninstance Function.algebra {R : Type*} (I : Type*) (A : Type*) [CommSemiring R] [Semiring A]\n    [Algebra R A] : Algebra R (I → A) :=\n  Pi.algebra _ _\n#align function.algebra Function.algebra\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {I : Type*}\n\nvariable [CommSemiring R] [Semiring A] [Semiring B]\n\nvariable [Algebra R A] [Algebra R B]\n\n/-- `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps]\nprotected def compLeft (f : A →ₐ[R] B) (I : Type*) : (I → A) →ₐ[R] I → B :=\n  { f.toRingHom.compLeft I with\n    toFun := fun h => f ∘ h\n    commutes' := fun c => by\n      ext\n      exact f.commutes' c }\n#align alg_hom.comp_left AlgHom.compLeft\n\nend AlgHom\n\nnamespace AlgEquiv\n\n/-- A family of algebra equivalences `∀ i, (A₁ i ≃ₐ A₂ i)` generates a\nmultiplicative equivalence between `∀ i, A₁ i` and `∀ i, A₂ i`.\n\nThis is the `AlgEquiv` version of `Equiv.piCongrRight`, and the dependent version of\n`AlgEquiv.arrowCongr`.\n-/\n@[simps apply]\ndef piCongrRight {R ι : Type*} {A₁ A₂ : ι → Type*} [CommSemiring R] [∀ i, Semiring (A₁ i)]\n    [∀ i, Semiring (A₂ i)] [∀ i, Algebra R (A₁ i)] [∀ i, Algebra R (A₂ i)]\n    (e : ∀ i, A₁ i ≃ₐ[R] A₂ i) : (∀ i, A₁ i) ≃ₐ[R] ∀ i, A₂ i :=\n  { @RingEquiv.piCongrRight ι A₁ A₂ _ _ fun i => (e i).toRingEquiv with\n    toFun := fun x j => e j (x j)\n    invFun := fun x j => (e j).symm (x j)\n    commutes' := fun r => by\n      ","nextTactic":"ext i","declUpToTactic":"/-- A family of algebra equivalences `∀ i, (A₁ i ≃ₐ A₂ i)` generates a\nmultiplicative equivalence between `∀ i, A₁ i` and `∀ i, A₂ i`.\n\nThis is the `AlgEquiv` version of `Equiv.piCongrRight`, and the dependent version of\n`AlgEquiv.arrowCongr`.\n-/\n@[simps apply]\ndef piCongrRight {R ι : Type*} {A₁ A₂ : ι → Type*} [CommSemiring R] [∀ i, Semiring (A₁ i)]\n    [∀ i, Semiring (A₂ i)] [∀ i, Algebra R (A₁ i)] [∀ i, Algebra R (A₂ i)]\n    (e : ∀ i, A₁ i ≃ₐ[R] A₂ i) : (∀ i, A₁ i) ≃ₐ[R] ∀ i, A₂ i :=\n  { @RingEquiv.piCongrRight ι A₁ A₂ _ _ fun i => (e i).toRingEquiv with\n    toFun := fun x j => e j (x j)\n    invFun := fun x j => (e j).symm (x j)\n    commutes' := fun r => by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Pi.127_0.iEzBlhDeTy24dhL","decl":"/-- A family of algebra equivalences `∀ i, (A₁ i ≃ₐ A₂ i)` generates a\nmultiplicative equivalence between `∀ i, A₁ i` and `∀ i, A₂ i`.\n\nThis is the `AlgEquiv` version of `Equiv.piCongrRight`, and the dependent version of\n`AlgEquiv.arrowCongr`.\n-/\n@[simps apply]\ndef piCongrRight {R ι : Type*} {A₁ A₂ : ι → Type*} [CommSemiring R] [∀ i, Semiring (A₁ i)]\n    [∀ i, Semiring (A₂ i)] [∀ i, Algebra R (A₁ i)] [∀ i, Algebra R (A₂ i)]\n    (e : ∀ i, A₁ i ≃ₐ[R] A₂ i) : (∀ i, A₁ i) ≃ₐ[R] ∀ i, A₂ i "}
-{"state":"case h\nR : Type u_1\nι : Type u_2\nA₁ : ι → Type u_3\nA₂ : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → Semiring (A₁ i)\ninst✝² : (i : ι) → Semiring (A₂ i)\ninst✝¹ : (i : ι) → Algebra R (A₁ i)\ninst✝ : (i : ι) → Algebra R (A₂ i)\ne : (i : ι) → A₁ i ≃ₐ[R] A₂ i\nsrc✝ : ((i : ι) → A₁ i) ≃+* ((i : ι) → A₂ i) := RingEquiv.piCongrRight fun i => toRingEquiv (e i)\nr : R\ni : ι\n⊢ Equiv.toFun\n      { toFun := fun x j => (e j) (x j), invFun := fun x j => (symm (e j)) (x j),\n        left_inv := (_ : Function.LeftInverse src✝.invFun src✝.toFun),\n        right_inv := (_ : Function.RightInverse src✝.invFun src✝.toFun) }\n      ((algebraMap R ((i : ι) → A₁ i)) r) i =\n    (algebraMap R ((i : ι) → A₂ i)) r i","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Equiv\n\n#align_import algebra.algebra.pi from \"leanprover-community/mathlib\"@\"b16045e4bf61c6fd619a1a68854ab3d605dcd017\"\n\n/-!\n# The R-algebra structure on families of R-algebras\n\nThe R-algebra structure on `∀ i : I, A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nuniverse u v w\n\nnamespace Pi\n\n-- The indexing type\nvariable {I : Type u}\n\n-- The scalar type\nvariable {R : Type*}\n\n-- The family of types already equipped with instances\nvariable {f : I → Type v}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nvariable (I f)\n\ninstance algebra {r : CommSemiring R} [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :\n    Algebra R (∀ i : I, f i) :=\n  { (Pi.ringHom fun i => algebraMap R (f i) : R →+* ∀ i : I, f i) with\n    commutes' := fun a f => by ext; simp [Algebra.commutes]\n    smul_def' := fun a f => by ext; simp [Algebra.smul_def] }\n#align pi.algebra Pi.algebra\n\ntheorem algebraMap_def {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]\n    (a : R) : algebraMap R (∀ i, f i) a = fun i => algebraMap R (f i) a :=\n  rfl\n#align pi.algebra_map_def Pi.algebraMap_def\n\n@[simp]\ntheorem algebraMap_apply {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]\n    (a : R) (i : I) : algebraMap R (∀ i, f i) a i = algebraMap R (f i) a :=\n  rfl\n#align pi.algebra_map_apply Pi.algebraMap_apply\n\n-- One could also build a `∀ i, R i`-algebra structure on `∀ i, A i`,\n-- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.\nvariable {I} (R)\n\n/-- `Function.eval` as an `AlgHom`. The name matches `Pi.evalRingHom`, `Pi.evalMonoidHom`,\netc. -/\n@[simps]\ndef evalAlgHom {_ : CommSemiring R} [∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] (i : I) :\n    (∀ i, f i) →ₐ[R] f i :=\n  { Pi.evalRingHom f i with\n    toFun := fun f => f i\n    commutes' := fun _ => rfl }\n#align pi.eval_alg_hom Pi.evalAlgHom\n\nvariable (A B : Type*) [CommSemiring R] [Semiring B] [Algebra R B]\n\n/-- `Function.const` as an `AlgHom`. The name matches `Pi.constRingHom`, `Pi.constMonoidHom`,\netc. -/\n@[simps]\ndef constAlgHom : B →ₐ[R] A → B :=\n  { Pi.constRingHom A B with\n    toFun := Function.const _\n    commutes' := fun _ => rfl }\n#align pi.const_alg_hom Pi.constAlgHom\n\n/-- When `R` is commutative and permits an `algebraMap`, `Pi.constRingHom` is equal to that\nmap. -/\n@[simp]\ntheorem constRingHom_eq_algebraMap : constRingHom A R = algebraMap R (A → R) :=\n  rfl\n#align pi.const_ring_hom_eq_algebra_map Pi.constRingHom_eq_algebraMap\n\n@[simp]\ntheorem constAlgHom_eq_algebra_ofId : constAlgHom R A R = Algebra.ofId R (A → R) :=\n  rfl\n#align pi.const_alg_hom_eq_algebra_of_id Pi.constAlgHom_eq_algebra_ofId\n\nend Pi\n\n/-- A special case of `Pi.algebra` for non-dependent types. Lean struggles to elaborate\ndefinitions elsewhere in the library without this, -/\ninstance Function.algebra {R : Type*} (I : Type*) (A : Type*) [CommSemiring R] [Semiring A]\n    [Algebra R A] : Algebra R (I → A) :=\n  Pi.algebra _ _\n#align function.algebra Function.algebra\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {I : Type*}\n\nvariable [CommSemiring R] [Semiring A] [Semiring B]\n\nvariable [Algebra R A] [Algebra R B]\n\n/-- `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps]\nprotected def compLeft (f : A →ₐ[R] B) (I : Type*) : (I → A) →ₐ[R] I → B :=\n  { f.toRingHom.compLeft I with\n    toFun := fun h => f ∘ h\n    commutes' := fun c => by\n      ext\n      exact f.commutes' c }\n#align alg_hom.comp_left AlgHom.compLeft\n\nend AlgHom\n\nnamespace AlgEquiv\n\n/-- A family of algebra equivalences `∀ i, (A₁ i ≃ₐ A₂ i)` generates a\nmultiplicative equivalence between `∀ i, A₁ i` and `∀ i, A₂ i`.\n\nThis is the `AlgEquiv` version of `Equiv.piCongrRight`, and the dependent version of\n`AlgEquiv.arrowCongr`.\n-/\n@[simps apply]\ndef piCongrRight {R ι : Type*} {A₁ A₂ : ι → Type*} [CommSemiring R] [∀ i, Semiring (A₁ i)]\n    [∀ i, Semiring (A₂ i)] [∀ i, Algebra R (A₁ i)] [∀ i, Algebra R (A₂ i)]\n    (e : ∀ i, A₁ i ≃ₐ[R] A₂ i) : (∀ i, A₁ i) ≃ₐ[R] ∀ i, A₂ i :=\n  { @RingEquiv.piCongrRight ι A₁ A₂ _ _ fun i => (e i).toRingEquiv with\n    toFun := fun x j => e j (x j)\n    invFun := fun x j => (e j).symm (x j)\n    commutes' := fun r => by\n      ext i\n      ","nextTactic":"simp","declUpToTactic":"/-- A family of algebra equivalences `∀ i, (A₁ i ≃ₐ A₂ i)` generates a\nmultiplicative equivalence between `∀ i, A₁ i` and `∀ i, A₂ i`.\n\nThis is the `AlgEquiv` version of `Equiv.piCongrRight`, and the dependent version of\n`AlgEquiv.arrowCongr`.\n-/\n@[simps apply]\ndef piCongrRight {R ι : Type*} {A₁ A₂ : ι → Type*} [CommSemiring R] [∀ i, Semiring (A₁ i)]\n    [∀ i, Semiring (A₂ i)] [∀ i, Algebra R (A₁ i)] [∀ i, Algebra R (A₂ i)]\n    (e : ∀ i, A₁ i ≃ₐ[R] A₂ i) : (∀ i, A₁ i) ≃ₐ[R] ∀ i, A₂ i :=\n  { @RingEquiv.piCongrRight ι A₁ A₂ _ _ fun i => (e i).toRingEquiv with\n    toFun := fun x j => e j (x j)\n    invFun := fun x j => (e j).symm (x j)\n    commutes' := fun r => by\n      ext i\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Pi.127_0.iEzBlhDeTy24dhL","decl":"/-- A family of algebra equivalences `∀ i, (A₁ i ≃ₐ A₂ i)` generates a\nmultiplicative equivalence between `∀ i, A₁ i` and `∀ i, A₂ i`.\n\nThis is the `AlgEquiv` version of `Equiv.piCongrRight`, and the dependent version of\n`AlgEquiv.arrowCongr`.\n-/\n@[simps apply]\ndef piCongrRight {R ι : Type*} {A₁ A₂ : ι → Type*} [CommSemiring R] [∀ i, Semiring (A₁ i)]\n    [∀ i, Semiring (A₂ i)] [∀ i, Algebra R (A₁ i)] [∀ i, Algebra R (A₂ i)]\n    (e : ∀ i, A₁ i ≃ₐ[R] A₂ i) : (∀ i, A₁ i) ≃ₐ[R] ∀ i, A₂ i "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_Prod.jsonl

@@ -1,18 +0,0 @@
-{"state":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nsrc✝¹ : Module R (A × B) := instModule\nsrc✝ : R →+* A × B := RingHom.prod (algebraMap R A) (algebraMap R B)\n⊢ ∀ (r : R) (x : A × B),\n    { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n          r *\n        x =\n      x *\n        { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n          r","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      ","nextTactic":"rintro r ⟨a, b⟩","declUpToTactic":"instance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.35_0.SYlFgSZc7uFXKXx","decl":"instance algebra : Algebra R (A × B) "}
-{"state":"case mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nsrc✝¹ : Module R (A × B) := instModule\nsrc✝ : R →+* A × B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n⊢ { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n          map_add' :=\n            (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n        r *\n      (a, b) =\n    (a, b) *\n      { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n          map_add' :=\n            (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n        r","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      ","nextTactic":"dsimp","declUpToTactic":"instance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.35_0.SYlFgSZc7uFXKXx","decl":"instance algebra : Algebra R (A × B) "}
-{"state":"case mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nsrc✝¹ : Module R (A × B) := instModule\nsrc✝ : R →+* A × B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n⊢ ((algebraMap R A) r * a, (algebraMap R B) r * b) = (a * (algebraMap R A) r, b * (algebraMap R B) r)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      ","nextTactic":"rw [commutes r a]","declUpToTactic":"instance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.35_0.SYlFgSZc7uFXKXx","decl":"instance algebra : Algebra R (A × B) "}
-{"state":"case mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nsrc✝¹ : Module R (A × B) := instModule\nsrc✝ : R →+* A × B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n⊢ (a * (algebraMap R A) r, (algebraMap R B) r * b) = (a * (algebraMap R A) r, b * (algebraMap R B) r)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      ","nextTactic":"rw [commutes r b]","declUpToTactic":"instance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.35_0.SYlFgSZc7uFXKXx","decl":"instance algebra : Algebra R (A × B) "}
-{"state":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nsrc✝¹ : Module R (A × B) := instModule\nsrc✝ : R →+* A × B := RingHom.prod (algebraMap R A) (algebraMap R B)\n⊢ ∀ (r : R) (x : A × B),\n    r • x =\n      { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n          r *\n        x","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      ","nextTactic":"rintro r ⟨a, b⟩","declUpToTactic":"instance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.35_0.SYlFgSZc7uFXKXx","decl":"instance algebra : Algebra R (A × B) "}
-{"state":"case mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nsrc✝¹ : Module R (A × B) := instModule\nsrc✝ : R →+* A × B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n⊢ r • (a, b) =\n    { toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n          map_add' :=\n            (_ : ∀ (x y : R), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) }\n        r *\n      (a, b)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      ","nextTactic":"dsimp","declUpToTactic":"instance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.35_0.SYlFgSZc7uFXKXx","decl":"instance algebra : Algebra R (A × B) "}
-{"state":"case mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nsrc✝¹ : Module R (A × B) := instModule\nsrc✝ : R →+* A × B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n⊢ (r • a, r • b) = ((algebraMap R A) r * a, (algebraMap R B) r * b)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      ","nextTactic":"rw [Algebra.smul_def r a, Algebra.smul_def r b]","declUpToTactic":"instance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.35_0.SYlFgSZc7uFXKXx","decl":"instance algebra : Algebra R (A × B) "}
-{"state":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B\ng : A →ₐ[R] C\nsrc✝ : A →+* B × C := RingHom.prod ↑f ↑g\nr : R\n⊢ OneHom.toFun\n      (↑↑{ toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n            map_add' :=\n              (_ : ∀ (x y : A), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) })\n      ((algebraMap R A) r) =\n    (algebraMap R (B × C)) r","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      ","nextTactic":"simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply]","declUpToTactic":"/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.74_0.SYlFgSZc7uFXKXx","decl":"/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C "}
-{"state":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B\ng : A →ₐ[R] C\n⊢ comp (fst R B C) (prod f g) = f","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ","nextTactic":"ext","declUpToTactic":"@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.87_0.SYlFgSZc7uFXKXx","decl":"@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f "}
-{"state":"case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B\ng : A →ₐ[R] C\nx✝ : A\n⊢ (comp (fst R B C) (prod f g)) x✝ = f x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; ","nextTactic":"rfl","declUpToTactic":"@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.87_0.SYlFgSZc7uFXKXx","decl":"@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f "}
-{"state":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B\ng : A →ₐ[R] C\n⊢ comp (snd R B C) (prod f g) = g","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; rfl\n#align alg_hom.fst_prod AlgHom.fst_prod\n\n@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ","nextTactic":"ext","declUpToTactic":"@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.91_0.SYlFgSZc7uFXKXx","decl":"@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g "}
-{"state":"case H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B\ng : A →ₐ[R] C\nx✝ : A\n⊢ (comp (snd R B C) (prod f g)) x✝ = g x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; rfl\n#align alg_hom.fst_prod AlgHom.fst_prod\n\n@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ext; ","nextTactic":"rfl","declUpToTactic":"@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.91_0.SYlFgSZc7uFXKXx","decl":"@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g "}
-{"state":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : (A →ₐ[R] B) × (A →ₐ[R] C)\n⊢ (fun f => (comp (fst R B C) f, comp (snd R B C) f)) ((fun f => prod f.1 f.2) f) = f","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; rfl\n#align alg_hom.fst_prod AlgHom.fst_prod\n\n@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ext; rfl\n#align alg_hom.snd_prod AlgHom.snd_prod\n\n@[simp]\ntheorem prod_fst_snd : prod (fst R A B) (snd R A B) = 1 :=\n  FunLike.coe_injective Pi.prod_fst_snd\n#align alg_hom.prod_fst_snd AlgHom.prod_fst_snd\n\n/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ","nextTactic":"ext","declUpToTactic":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.100_0.SYlFgSZc7uFXKXx","decl":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f "}
-{"state":"case a.H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : (A →ₐ[R] B) × (A →ₐ[R] C)\nx✝ : A\n⊢ ((fun f => (comp (fst R B C) f, comp (snd R B C) f)) ((fun f => prod f.1 f.2) f)).1 x✝ = f.1 x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; rfl\n#align alg_hom.fst_prod AlgHom.fst_prod\n\n@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ext; rfl\n#align alg_hom.snd_prod AlgHom.snd_prod\n\n@[simp]\ntheorem prod_fst_snd : prod (fst R A B) (snd R A B) = 1 :=\n  FunLike.coe_injective Pi.prod_fst_snd\n#align alg_hom.prod_fst_snd AlgHom.prod_fst_snd\n\n/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> ","nextTactic":"rfl","declUpToTactic":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.100_0.SYlFgSZc7uFXKXx","decl":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f "}
-{"state":"case a.H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : (A →ₐ[R] B) × (A →ₐ[R] C)\nx✝ : A\n⊢ ((fun f => (comp (fst R B C) f, comp (snd R B C) f)) ((fun f => prod f.1 f.2) f)).2 x✝ = f.2 x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; rfl\n#align alg_hom.fst_prod AlgHom.fst_prod\n\n@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ext; rfl\n#align alg_hom.snd_prod AlgHom.snd_prod\n\n@[simp]\ntheorem prod_fst_snd : prod (fst R A B) (snd R A B) = 1 :=\n  FunLike.coe_injective Pi.prod_fst_snd\n#align alg_hom.prod_fst_snd AlgHom.prod_fst_snd\n\n/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> ","nextTactic":"rfl","declUpToTactic":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.100_0.SYlFgSZc7uFXKXx","decl":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f "}
-{"state":"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B × C\n⊢ (fun f => prod f.1 f.2) ((fun f => (comp (fst R B C) f, comp (snd R B C) f)) f) = f","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; rfl\n#align alg_hom.fst_prod AlgHom.fst_prod\n\n@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ext; rfl\n#align alg_hom.snd_prod AlgHom.snd_prod\n\n@[simp]\ntheorem prod_fst_snd : prod (fst R A B) (snd R A B) = 1 :=\n  FunLike.coe_injective Pi.prod_fst_snd\n#align alg_hom.prod_fst_snd AlgHom.prod_fst_snd\n\n/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> rfl\n  right_inv f := by ","nextTactic":"ext","declUpToTactic":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> rfl\n  right_inv f := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.100_0.SYlFgSZc7uFXKXx","decl":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f "}
-{"state":"case H.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B × C\nx✝ : A\n⊢ (((fun f => prod f.1 f.2) ((fun f => (comp (fst R B C) f, comp (snd R B C) f)) f)) x✝).1 = (f x✝).1","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; rfl\n#align alg_hom.fst_prod AlgHom.fst_prod\n\n@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ext; rfl\n#align alg_hom.snd_prod AlgHom.snd_prod\n\n@[simp]\ntheorem prod_fst_snd : prod (fst R A B) (snd R A B) = 1 :=\n  FunLike.coe_injective Pi.prod_fst_snd\n#align alg_hom.prod_fst_snd AlgHom.prod_fst_snd\n\n/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> rfl\n  right_inv f := by ext <;> ","nextTactic":"rfl","declUpToTactic":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> rfl\n  right_inv f := by ext <;> ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.100_0.SYlFgSZc7uFXKXx","decl":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f "}
-{"state":"case H.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nf : A →ₐ[R] B × C\nx✝ : A\n⊢ (((fun f => prod f.1 f.2) ((fun f => (comp (fst R B C) f, comp (snd R B C) f)) f)) x✝).2 = (f x✝).2","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Hom\n\n#align_import algebra.algebra.prod from \"leanprover-community/mathlib\"@\"28aa996fc6fb4317f0083c4e6daf79878d81be33\"\n\n/-!\n# The R-algebra structure on products of R-algebras\n\nThe R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.\n\n## Main definitions\n\n* `Pi.algebra`\n* `Pi.evalAlgHom`\n* `Pi.constAlgHom`\n-/\n\n\nvariable {R A B C : Type*}\n\nvariable [CommSemiring R]\n\nvariable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C]\n\nnamespace Prod\n\nvariable (R A B)\n\nopen Algebra\n\ninstance algebra : Algebra R (A × B) :=\n  { Prod.instModule,\n    RingHom.prod (algebraMap R A) (algebraMap R B) with\n    commutes' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [commutes r a]\n      rw [commutes r b]\n    smul_def' := by\n      rintro r ⟨a, b⟩\n      dsimp\n      rw [Algebra.smul_def r a, Algebra.smul_def r b] }\n#align prod.algebra Prod.algebra\n\nvariable {R A B}\n\n@[simp]\ntheorem algebraMap_apply (r : R) : algebraMap R (A × B) r = (algebraMap R A r, algebraMap R B r) :=\n  rfl\n#align prod.algebra_map_apply Prod.algebraMap_apply\n\nend Prod\n\nnamespace AlgHom\n\nvariable (R A B)\n\n/-- First projection as `AlgHom`. -/\ndef fst : A × B →ₐ[R] A :=\n  { RingHom.fst A B with commutes' := fun _r => rfl }\n#align alg_hom.fst AlgHom.fst\n\n/-- Second projection as `AlgHom`. -/\ndef snd : A × B →ₐ[R] B :=\n  { RingHom.snd A B with commutes' := fun _r => rfl }\n#align alg_hom.snd AlgHom.snd\n\nvariable {R A B}\n\n/-- The `Pi.prod` of two morphisms is a morphism. -/\n@[simps!]\ndef prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : A →ₐ[R] B × C :=\n  { f.toRingHom.prod g.toRingHom with\n    commutes' := fun r => by\n      simp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom,\n        commutes, Prod.algebraMap_apply] }\n#align alg_hom.prod AlgHom.prod\n\ntheorem coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = Pi.prod f g :=\n  rfl\n#align alg_hom.coe_prod AlgHom.coe_prod\n\n@[simp]\ntheorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by ext; rfl\n#align alg_hom.fst_prod AlgHom.fst_prod\n\n@[simp]\ntheorem snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (snd R B C).comp (prod f g) = g := by ext; rfl\n#align alg_hom.snd_prod AlgHom.snd_prod\n\n@[simp]\ntheorem prod_fst_snd : prod (fst R A B) (snd R A B) = 1 :=\n  FunLike.coe_injective Pi.prod_fst_snd\n#align alg_hom.prod_fst_snd AlgHom.prod_fst_snd\n\n/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> rfl\n  right_inv f := by ext <;> ","nextTactic":"rfl","declUpToTactic":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f := f.1.prod f.2\n  invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f)\n  left_inv f := by ext <;> rfl\n  right_inv f := by ext <;> ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Prod.100_0.SYlFgSZc7uFXKXx","decl":"/-- Taking the product of two maps with the same domain is equivalent to taking the product of\ntheir codomains. -/\n@[simps]\ndef prodEquiv : (A →ₐ[R] B) × (A →ₐ[R] C) ≃ (A →ₐ[R] B × C)\n    where\n  toFun f "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_RestrictScalars.jsonl

@@ -1,6 +0,0 @@
-{"state":"R : Type u_1\nS✝ : Type u_2\nM✝ : Type u_3\nA : Type u_4\ninst✝⁴ : Semiring S✝\ninst✝³ : AddCommMonoid M✝\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R S✝\ninst✝ : Module S✝ M✝\nr : R\nS : S✝\nM : RestrictScalars R S✝ M✝\n⊢ (r • S) • M = r • S • M","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.restrict_scalars from \"leanprover-community/mathlib\"@\"c310cfdc40da4d99a10a58c33a95360ef9e6e0bf\"\n\n/-!\n\n# The `RestrictScalars` type alias\n\nSee the documentation attached to the `RestrictScalars` definition for advice on how and when to\nuse this type alias. As described there, it is often a better choice to use the `IsScalarTower`\ntypeclass instead.\n\n## Main definitions\n\n* `RestrictScalars R S M`: the `S`-module `M` viewed as an `R` module when `S` is an `R`-algebra.\n  Note that by default we do *not* have a `Module S (RestrictScalars R S M)` instance\n  for the original action.\n  This is available as a def `RestrictScalars.moduleOrig` if really needed.\n* `RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M`: the additive equivalence\n  between the restricted and original space (in fact, they are definitionally equal,\n  but sometimes it is helpful to avoid using this fact, to keep instances from leaking).\n* `RestrictScalars.ringEquiv : RestrictScalars R S A ≃+* A`: the ring equivalence\n   between the restricted and original space when the module is an algebra.\n\n## See also\n\nThere are many similarly-named definitions elsewhere which do not refer to this type alias. These\nrefer to restricting the scalar type in a bundled type, such as from `A →ₗ[R] B` to `A →ₗ[S] B`:\n\n* `LinearMap.restrictScalars`\n* `LinearEquiv.restrictScalars`\n* `AlgHom.restrictScalars`\n* `AlgEquiv.restrictScalars`\n* `Submodule.restrictScalars`\n* `Subalgebra.restrictScalars`\n-/\n\n\nvariable (R S M A : Type*)\n\n/-- If we put an `R`-algebra structure on a semiring `S`, we get a natural equivalence from the\ncategory of `S`-modules to the category of representations of the algebra `S` (over `R`). The type\nsynonym `RestrictScalars` is essentially this equivalence.\n\nWarning: use this type synonym judiciously! Consider an example where we want to construct an\n`R`-linear map from `M` to `S`, given:\n```lean\nvariable (R S M : Type*)\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module S M]\n```\nWith the assumptions above we can't directly state our map as we have no `Module R M` structure, but\n`RestrictScalars` permits it to be written as:\n```lean\n-- an `R`-module structure on `M` is provided by `RestrictScalars` which is compatible\nexample : RestrictScalars R S M →ₗ[R] S := sorry\n```\nHowever, it is usually better just to add this extra structure as an argument:\n```lean\n-- an `R`-module structure on `M` and proof of its compatibility is provided by the user\nexample [Module R M] [IsScalarTower R S M] : M →ₗ[R] S := sorry\n```\nThe advantage of the second approach is that it defers the duty of providing the missing typeclasses\n`[Module R M] [IsScalarTower R S M]`. If some concrete `M` naturally carries these (as is often\nthe case) then we have avoided `RestrictScalars` entirely. If not, we can pass\n`RestrictScalars R S M` later on instead of `M`.\n\nNote that this means we almost always want to state definitions and lemmas in the language of\n`IsScalarTower` rather than `RestrictScalars`.\n\nAn example of when one might want to use `RestrictScalars` would be if one has a vector space\nover a field of characteristic zero and wishes to make use of the `ℚ`-algebra structure. -/\n@[nolint unusedArguments]\ndef RestrictScalars (_R _S M : Type*) : Type _ := M\n#align restrict_scalars RestrictScalars\n\ninstance [I : Inhabited M] : Inhabited (RestrictScalars R S M) := I\n\ninstance [I : AddCommMonoid M] : AddCommMonoid (RestrictScalars R S M) := I\n\ninstance [I : AddCommGroup M] : AddCommGroup (RestrictScalars R S M) := I\n\nsection Module\n\nsection\n\nvariable [Semiring S] [AddCommMonoid M]\n\n/-- We temporarily install an action of the original ring on `RestrictScalars R S M`. -/\ndef RestrictScalars.moduleOrig [I : Module S M] : Module S (RestrictScalars R S M) := I\n#align restrict_scalars.module_orig RestrictScalars.moduleOrig\n\nvariable [CommSemiring R] [Algebra R S]\n\nsection\n\nattribute [local instance] RestrictScalars.moduleOrig\n\n/-- When `M` is a module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`.\n\nThe preferred way of setting this up is `[Module R M] [Module S M] [IsScalarTower R S M]`.\n-/\ninstance RestrictScalars.module [Module S M] : Module R (RestrictScalars R S M) :=\n  Module.compHom M (algebraMap R S)\n\n/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    ","nextTactic":"rw [Algebra.smul_def]","declUpToTactic":"/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_RestrictScalars.111_0.FyKmAB3YCMLCVBR","decl":"/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) "}
-{"state":"R : Type u_1\nS✝ : Type u_2\nM✝ : Type u_3\nA : Type u_4\ninst✝⁴ : Semiring S✝\ninst✝³ : AddCommMonoid M✝\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R S✝\ninst✝ : Module S✝ M✝\nr : R\nS : S✝\nM : RestrictScalars R S✝ M✝\n⊢ ((algebraMap R S✝) r * S) • M = r • S • M","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.restrict_scalars from \"leanprover-community/mathlib\"@\"c310cfdc40da4d99a10a58c33a95360ef9e6e0bf\"\n\n/-!\n\n# The `RestrictScalars` type alias\n\nSee the documentation attached to the `RestrictScalars` definition for advice on how and when to\nuse this type alias. As described there, it is often a better choice to use the `IsScalarTower`\ntypeclass instead.\n\n## Main definitions\n\n* `RestrictScalars R S M`: the `S`-module `M` viewed as an `R` module when `S` is an `R`-algebra.\n  Note that by default we do *not* have a `Module S (RestrictScalars R S M)` instance\n  for the original action.\n  This is available as a def `RestrictScalars.moduleOrig` if really needed.\n* `RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M`: the additive equivalence\n  between the restricted and original space (in fact, they are definitionally equal,\n  but sometimes it is helpful to avoid using this fact, to keep instances from leaking).\n* `RestrictScalars.ringEquiv : RestrictScalars R S A ≃+* A`: the ring equivalence\n   between the restricted and original space when the module is an algebra.\n\n## See also\n\nThere are many similarly-named definitions elsewhere which do not refer to this type alias. These\nrefer to restricting the scalar type in a bundled type, such as from `A →ₗ[R] B` to `A →ₗ[S] B`:\n\n* `LinearMap.restrictScalars`\n* `LinearEquiv.restrictScalars`\n* `AlgHom.restrictScalars`\n* `AlgEquiv.restrictScalars`\n* `Submodule.restrictScalars`\n* `Subalgebra.restrictScalars`\n-/\n\n\nvariable (R S M A : Type*)\n\n/-- If we put an `R`-algebra structure on a semiring `S`, we get a natural equivalence from the\ncategory of `S`-modules to the category of representations of the algebra `S` (over `R`). The type\nsynonym `RestrictScalars` is essentially this equivalence.\n\nWarning: use this type synonym judiciously! Consider an example where we want to construct an\n`R`-linear map from `M` to `S`, given:\n```lean\nvariable (R S M : Type*)\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module S M]\n```\nWith the assumptions above we can't directly state our map as we have no `Module R M` structure, but\n`RestrictScalars` permits it to be written as:\n```lean\n-- an `R`-module structure on `M` is provided by `RestrictScalars` which is compatible\nexample : RestrictScalars R S M →ₗ[R] S := sorry\n```\nHowever, it is usually better just to add this extra structure as an argument:\n```lean\n-- an `R`-module structure on `M` and proof of its compatibility is provided by the user\nexample [Module R M] [IsScalarTower R S M] : M →ₗ[R] S := sorry\n```\nThe advantage of the second approach is that it defers the duty of providing the missing typeclasses\n`[Module R M] [IsScalarTower R S M]`. If some concrete `M` naturally carries these (as is often\nthe case) then we have avoided `RestrictScalars` entirely. If not, we can pass\n`RestrictScalars R S M` later on instead of `M`.\n\nNote that this means we almost always want to state definitions and lemmas in the language of\n`IsScalarTower` rather than `RestrictScalars`.\n\nAn example of when one might want to use `RestrictScalars` would be if one has a vector space\nover a field of characteristic zero and wishes to make use of the `ℚ`-algebra structure. -/\n@[nolint unusedArguments]\ndef RestrictScalars (_R _S M : Type*) : Type _ := M\n#align restrict_scalars RestrictScalars\n\ninstance [I : Inhabited M] : Inhabited (RestrictScalars R S M) := I\n\ninstance [I : AddCommMonoid M] : AddCommMonoid (RestrictScalars R S M) := I\n\ninstance [I : AddCommGroup M] : AddCommGroup (RestrictScalars R S M) := I\n\nsection Module\n\nsection\n\nvariable [Semiring S] [AddCommMonoid M]\n\n/-- We temporarily install an action of the original ring on `RestrictScalars R S M`. -/\ndef RestrictScalars.moduleOrig [I : Module S M] : Module S (RestrictScalars R S M) := I\n#align restrict_scalars.module_orig RestrictScalars.moduleOrig\n\nvariable [CommSemiring R] [Algebra R S]\n\nsection\n\nattribute [local instance] RestrictScalars.moduleOrig\n\n/-- When `M` is a module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`.\n\nThe preferred way of setting this up is `[Module R M] [Module S M] [IsScalarTower R S M]`.\n-/\ninstance RestrictScalars.module [Module S M] : Module R (RestrictScalars R S M) :=\n  Module.compHom M (algebraMap R S)\n\n/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    rw [Algebra.smul_def]\n    ","nextTactic":"rw [mul_smul]","declUpToTactic":"/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    rw [Algebra.smul_def]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_RestrictScalars.111_0.FyKmAB3YCMLCVBR","decl":"/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) "}
-{"state":"R : Type u_1\nS✝ : Type u_2\nM✝ : Type u_3\nA : Type u_4\ninst✝⁴ : Semiring S✝\ninst✝³ : AddCommMonoid M✝\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R S✝\ninst✝ : Module S✝ M✝\nr : R\nS : S✝\nM : RestrictScalars R S✝ M✝\n⊢ (algebraMap R S✝) r • S • M = r • S • M","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.restrict_scalars from \"leanprover-community/mathlib\"@\"c310cfdc40da4d99a10a58c33a95360ef9e6e0bf\"\n\n/-!\n\n# The `RestrictScalars` type alias\n\nSee the documentation attached to the `RestrictScalars` definition for advice on how and when to\nuse this type alias. As described there, it is often a better choice to use the `IsScalarTower`\ntypeclass instead.\n\n## Main definitions\n\n* `RestrictScalars R S M`: the `S`-module `M` viewed as an `R` module when `S` is an `R`-algebra.\n  Note that by default we do *not* have a `Module S (RestrictScalars R S M)` instance\n  for the original action.\n  This is available as a def `RestrictScalars.moduleOrig` if really needed.\n* `RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M`: the additive equivalence\n  between the restricted and original space (in fact, they are definitionally equal,\n  but sometimes it is helpful to avoid using this fact, to keep instances from leaking).\n* `RestrictScalars.ringEquiv : RestrictScalars R S A ≃+* A`: the ring equivalence\n   between the restricted and original space when the module is an algebra.\n\n## See also\n\nThere are many similarly-named definitions elsewhere which do not refer to this type alias. These\nrefer to restricting the scalar type in a bundled type, such as from `A →ₗ[R] B` to `A →ₗ[S] B`:\n\n* `LinearMap.restrictScalars`\n* `LinearEquiv.restrictScalars`\n* `AlgHom.restrictScalars`\n* `AlgEquiv.restrictScalars`\n* `Submodule.restrictScalars`\n* `Subalgebra.restrictScalars`\n-/\n\n\nvariable (R S M A : Type*)\n\n/-- If we put an `R`-algebra structure on a semiring `S`, we get a natural equivalence from the\ncategory of `S`-modules to the category of representations of the algebra `S` (over `R`). The type\nsynonym `RestrictScalars` is essentially this equivalence.\n\nWarning: use this type synonym judiciously! Consider an example where we want to construct an\n`R`-linear map from `M` to `S`, given:\n```lean\nvariable (R S M : Type*)\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module S M]\n```\nWith the assumptions above we can't directly state our map as we have no `Module R M` structure, but\n`RestrictScalars` permits it to be written as:\n```lean\n-- an `R`-module structure on `M` is provided by `RestrictScalars` which is compatible\nexample : RestrictScalars R S M →ₗ[R] S := sorry\n```\nHowever, it is usually better just to add this extra structure as an argument:\n```lean\n-- an `R`-module structure on `M` and proof of its compatibility is provided by the user\nexample [Module R M] [IsScalarTower R S M] : M →ₗ[R] S := sorry\n```\nThe advantage of the second approach is that it defers the duty of providing the missing typeclasses\n`[Module R M] [IsScalarTower R S M]`. If some concrete `M` naturally carries these (as is often\nthe case) then we have avoided `RestrictScalars` entirely. If not, we can pass\n`RestrictScalars R S M` later on instead of `M`.\n\nNote that this means we almost always want to state definitions and lemmas in the language of\n`IsScalarTower` rather than `RestrictScalars`.\n\nAn example of when one might want to use `RestrictScalars` would be if one has a vector space\nover a field of characteristic zero and wishes to make use of the `ℚ`-algebra structure. -/\n@[nolint unusedArguments]\ndef RestrictScalars (_R _S M : Type*) : Type _ := M\n#align restrict_scalars RestrictScalars\n\ninstance [I : Inhabited M] : Inhabited (RestrictScalars R S M) := I\n\ninstance [I : AddCommMonoid M] : AddCommMonoid (RestrictScalars R S M) := I\n\ninstance [I : AddCommGroup M] : AddCommGroup (RestrictScalars R S M) := I\n\nsection Module\n\nsection\n\nvariable [Semiring S] [AddCommMonoid M]\n\n/-- We temporarily install an action of the original ring on `RestrictScalars R S M`. -/\ndef RestrictScalars.moduleOrig [I : Module S M] : Module S (RestrictScalars R S M) := I\n#align restrict_scalars.module_orig RestrictScalars.moduleOrig\n\nvariable [CommSemiring R] [Algebra R S]\n\nsection\n\nattribute [local instance] RestrictScalars.moduleOrig\n\n/-- When `M` is a module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`.\n\nThe preferred way of setting this up is `[Module R M] [Module S M] [IsScalarTower R S M]`.\n-/\ninstance RestrictScalars.module [Module S M] : Module R (RestrictScalars R S M) :=\n  Module.compHom M (algebraMap R S)\n\n/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    rw [Algebra.smul_def]\n    rw [mul_smul]\n    ","nextTactic":"rfl","declUpToTactic":"/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    rw [Algebra.smul_def]\n    rw [mul_smul]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_RestrictScalars.111_0.FyKmAB3YCMLCVBR","decl":"/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) "}
-{"state":"R : Type u_1\nS : Type u_2\nM : Type u_3\nA : Type u_4\ninst✝⁴ : AddCommMonoid M\ninst✝³ : CommSemiring R\ninst✝² : Semiring S\ninst✝¹ : Algebra R S\ninst✝ : Module S M\nr : R\ns : S\nx : M\n⊢ (AddEquiv.symm (addEquiv R S M)) ((r • s) • x) = r • (AddEquiv.symm (addEquiv R S M)) (s • x)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.restrict_scalars from \"leanprover-community/mathlib\"@\"c310cfdc40da4d99a10a58c33a95360ef9e6e0bf\"\n\n/-!\n\n# The `RestrictScalars` type alias\n\nSee the documentation attached to the `RestrictScalars` definition for advice on how and when to\nuse this type alias. As described there, it is often a better choice to use the `IsScalarTower`\ntypeclass instead.\n\n## Main definitions\n\n* `RestrictScalars R S M`: the `S`-module `M` viewed as an `R` module when `S` is an `R`-algebra.\n  Note that by default we do *not* have a `Module S (RestrictScalars R S M)` instance\n  for the original action.\n  This is available as a def `RestrictScalars.moduleOrig` if really needed.\n* `RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M`: the additive equivalence\n  between the restricted and original space (in fact, they are definitionally equal,\n  but sometimes it is helpful to avoid using this fact, to keep instances from leaking).\n* `RestrictScalars.ringEquiv : RestrictScalars R S A ≃+* A`: the ring equivalence\n   between the restricted and original space when the module is an algebra.\n\n## See also\n\nThere are many similarly-named definitions elsewhere which do not refer to this type alias. These\nrefer to restricting the scalar type in a bundled type, such as from `A →ₗ[R] B` to `A →ₗ[S] B`:\n\n* `LinearMap.restrictScalars`\n* `LinearEquiv.restrictScalars`\n* `AlgHom.restrictScalars`\n* `AlgEquiv.restrictScalars`\n* `Submodule.restrictScalars`\n* `Subalgebra.restrictScalars`\n-/\n\n\nvariable (R S M A : Type*)\n\n/-- If we put an `R`-algebra structure on a semiring `S`, we get a natural equivalence from the\ncategory of `S`-modules to the category of representations of the algebra `S` (over `R`). The type\nsynonym `RestrictScalars` is essentially this equivalence.\n\nWarning: use this type synonym judiciously! Consider an example where we want to construct an\n`R`-linear map from `M` to `S`, given:\n```lean\nvariable (R S M : Type*)\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module S M]\n```\nWith the assumptions above we can't directly state our map as we have no `Module R M` structure, but\n`RestrictScalars` permits it to be written as:\n```lean\n-- an `R`-module structure on `M` is provided by `RestrictScalars` which is compatible\nexample : RestrictScalars R S M →ₗ[R] S := sorry\n```\nHowever, it is usually better just to add this extra structure as an argument:\n```lean\n-- an `R`-module structure on `M` and proof of its compatibility is provided by the user\nexample [Module R M] [IsScalarTower R S M] : M →ₗ[R] S := sorry\n```\nThe advantage of the second approach is that it defers the duty of providing the missing typeclasses\n`[Module R M] [IsScalarTower R S M]`. If some concrete `M` naturally carries these (as is often\nthe case) then we have avoided `RestrictScalars` entirely. If not, we can pass\n`RestrictScalars R S M` later on instead of `M`.\n\nNote that this means we almost always want to state definitions and lemmas in the language of\n`IsScalarTower` rather than `RestrictScalars`.\n\nAn example of when one might want to use `RestrictScalars` would be if one has a vector space\nover a field of characteristic zero and wishes to make use of the `ℚ`-algebra structure. -/\n@[nolint unusedArguments]\ndef RestrictScalars (_R _S M : Type*) : Type _ := M\n#align restrict_scalars RestrictScalars\n\ninstance [I : Inhabited M] : Inhabited (RestrictScalars R S M) := I\n\ninstance [I : AddCommMonoid M] : AddCommMonoid (RestrictScalars R S M) := I\n\ninstance [I : AddCommGroup M] : AddCommGroup (RestrictScalars R S M) := I\n\nsection Module\n\nsection\n\nvariable [Semiring S] [AddCommMonoid M]\n\n/-- We temporarily install an action of the original ring on `RestrictScalars R S M`. -/\ndef RestrictScalars.moduleOrig [I : Module S M] : Module S (RestrictScalars R S M) := I\n#align restrict_scalars.module_orig RestrictScalars.moduleOrig\n\nvariable [CommSemiring R] [Algebra R S]\n\nsection\n\nattribute [local instance] RestrictScalars.moduleOrig\n\n/-- When `M` is a module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`.\n\nThe preferred way of setting this up is `[Module R M] [Module S M] [IsScalarTower R S M]`.\n-/\ninstance RestrictScalars.module [Module S M] : Module R (RestrictScalars R S M) :=\n  Module.compHom M (algebraMap R S)\n\n/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    rw [Algebra.smul_def]\n    rw [mul_smul]\n    rfl⟩\n#align restrict_scalars.is_scalar_tower RestrictScalars.isScalarTower\n\nend\n\n/-- When `M` is a right-module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nright-module structure over `R`.\nThe preferred way of setting this up is\n`[Module Rᵐᵒᵖ M] [Module Sᵐᵒᵖ M] [IsScalarTower Rᵐᵒᵖ Sᵐᵒᵖ M]`.\n-/\ninstance RestrictScalars.opModule [Module Sᵐᵒᵖ M] : Module Rᵐᵒᵖ (RestrictScalars R S M) :=\n  letI : Module Sᵐᵒᵖ (RestrictScalars R S M) := ‹Module Sᵐᵒᵖ M›\n  Module.compHom M (RingHom.op $ algebraMap R S)\n#align restrict_scalars.op_module RestrictScalars.opModule\n\ninstance RestrictScalars.isCentralScalar [Module S M] [Module Sᵐᵒᵖ M] [IsCentralScalar S M] :\n    IsCentralScalar R (RestrictScalars R S M) where\n  op_smul_eq_smul r _x := (op_smul_eq_smul (algebraMap R S r) (_ : M) : _)\n#align restrict_scalars.is_central_scalar RestrictScalars.isCentralScalar\n\n/-- The `R`-algebra homomorphism from the original coefficient algebra `S` to endomorphisms\nof `RestrictScalars R S M`.\n-/\ndef RestrictScalars.lsmul [Module S M] : S →ₐ[R] Module.End R (RestrictScalars R S M) :=\n  -- We use `RestrictScalars.moduleOrig` in the implementation,\n  -- but not in the type.\n  letI : Module S (RestrictScalars R S M) := RestrictScalars.moduleOrig R S M\n  Algebra.lsmul R R (RestrictScalars R S M)\n#align restrict_scalars.lsmul RestrictScalars.lsmul\n\nend\n\nvariable [AddCommMonoid M]\n\n/-- `RestrictScalars.addEquiv` is the additive equivalence with the original module. -/\ndef RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M :=\n  AddEquiv.refl M\n#align restrict_scalars.add_equiv RestrictScalars.addEquiv\n\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [Module S M]\n\ntheorem RestrictScalars.smul_def (c : R) (x : RestrictScalars R S M) :\n    c • x = (RestrictScalars.addEquiv R S M).symm\n      (algebraMap R S c • RestrictScalars.addEquiv R S M x) :=\n  rfl\n#align restrict_scalars.smul_def RestrictScalars.smul_def\n\n@[simp]\ntheorem RestrictScalars.addEquiv_map_smul (c : R) (x : RestrictScalars R S M) :\n    RestrictScalars.addEquiv R S M (c • x) = algebraMap R S c • RestrictScalars.addEquiv R S M x :=\n  rfl\n#align restrict_scalars.add_equiv_map_smul RestrictScalars.addEquiv_map_smul\n\ntheorem RestrictScalars.addEquiv_symm_map_algebraMap_smul (r : R) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm (algebraMap R S r • x) =\n      r • (RestrictScalars.addEquiv R S M).symm x :=\n  rfl\n#align restrict_scalars.add_equiv_symm_map_algebra_map_smul RestrictScalars.addEquiv_symm_map_algebraMap_smul\n\ntheorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) := by\n  ","nextTactic":"rw [Algebra.smul_def]","declUpToTactic":"theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_RestrictScalars.176_0.FyKmAB3YCMLCVBR","decl":"theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) "}
-{"state":"R : Type u_1\nS : Type u_2\nM : Type u_3\nA : Type u_4\ninst✝⁴ : AddCommMonoid M\ninst✝³ : CommSemiring R\ninst✝² : Semiring S\ninst✝¹ : Algebra R S\ninst✝ : Module S M\nr : R\ns : S\nx : M\n⊢ (AddEquiv.symm (addEquiv R S M)) (((algebraMap R S) r * s) • x) = r • (AddEquiv.symm (addEquiv R S M)) (s • x)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.restrict_scalars from \"leanprover-community/mathlib\"@\"c310cfdc40da4d99a10a58c33a95360ef9e6e0bf\"\n\n/-!\n\n# The `RestrictScalars` type alias\n\nSee the documentation attached to the `RestrictScalars` definition for advice on how and when to\nuse this type alias. As described there, it is often a better choice to use the `IsScalarTower`\ntypeclass instead.\n\n## Main definitions\n\n* `RestrictScalars R S M`: the `S`-module `M` viewed as an `R` module when `S` is an `R`-algebra.\n  Note that by default we do *not* have a `Module S (RestrictScalars R S M)` instance\n  for the original action.\n  This is available as a def `RestrictScalars.moduleOrig` if really needed.\n* `RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M`: the additive equivalence\n  between the restricted and original space (in fact, they are definitionally equal,\n  but sometimes it is helpful to avoid using this fact, to keep instances from leaking).\n* `RestrictScalars.ringEquiv : RestrictScalars R S A ≃+* A`: the ring equivalence\n   between the restricted and original space when the module is an algebra.\n\n## See also\n\nThere are many similarly-named definitions elsewhere which do not refer to this type alias. These\nrefer to restricting the scalar type in a bundled type, such as from `A →ₗ[R] B` to `A →ₗ[S] B`:\n\n* `LinearMap.restrictScalars`\n* `LinearEquiv.restrictScalars`\n* `AlgHom.restrictScalars`\n* `AlgEquiv.restrictScalars`\n* `Submodule.restrictScalars`\n* `Subalgebra.restrictScalars`\n-/\n\n\nvariable (R S M A : Type*)\n\n/-- If we put an `R`-algebra structure on a semiring `S`, we get a natural equivalence from the\ncategory of `S`-modules to the category of representations of the algebra `S` (over `R`). The type\nsynonym `RestrictScalars` is essentially this equivalence.\n\nWarning: use this type synonym judiciously! Consider an example where we want to construct an\n`R`-linear map from `M` to `S`, given:\n```lean\nvariable (R S M : Type*)\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module S M]\n```\nWith the assumptions above we can't directly state our map as we have no `Module R M` structure, but\n`RestrictScalars` permits it to be written as:\n```lean\n-- an `R`-module structure on `M` is provided by `RestrictScalars` which is compatible\nexample : RestrictScalars R S M →ₗ[R] S := sorry\n```\nHowever, it is usually better just to add this extra structure as an argument:\n```lean\n-- an `R`-module structure on `M` and proof of its compatibility is provided by the user\nexample [Module R M] [IsScalarTower R S M] : M →ₗ[R] S := sorry\n```\nThe advantage of the second approach is that it defers the duty of providing the missing typeclasses\n`[Module R M] [IsScalarTower R S M]`. If some concrete `M` naturally carries these (as is often\nthe case) then we have avoided `RestrictScalars` entirely. If not, we can pass\n`RestrictScalars R S M` later on instead of `M`.\n\nNote that this means we almost always want to state definitions and lemmas in the language of\n`IsScalarTower` rather than `RestrictScalars`.\n\nAn example of when one might want to use `RestrictScalars` would be if one has a vector space\nover a field of characteristic zero and wishes to make use of the `ℚ`-algebra structure. -/\n@[nolint unusedArguments]\ndef RestrictScalars (_R _S M : Type*) : Type _ := M\n#align restrict_scalars RestrictScalars\n\ninstance [I : Inhabited M] : Inhabited (RestrictScalars R S M) := I\n\ninstance [I : AddCommMonoid M] : AddCommMonoid (RestrictScalars R S M) := I\n\ninstance [I : AddCommGroup M] : AddCommGroup (RestrictScalars R S M) := I\n\nsection Module\n\nsection\n\nvariable [Semiring S] [AddCommMonoid M]\n\n/-- We temporarily install an action of the original ring on `RestrictScalars R S M`. -/\ndef RestrictScalars.moduleOrig [I : Module S M] : Module S (RestrictScalars R S M) := I\n#align restrict_scalars.module_orig RestrictScalars.moduleOrig\n\nvariable [CommSemiring R] [Algebra R S]\n\nsection\n\nattribute [local instance] RestrictScalars.moduleOrig\n\n/-- When `M` is a module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`.\n\nThe preferred way of setting this up is `[Module R M] [Module S M] [IsScalarTower R S M]`.\n-/\ninstance RestrictScalars.module [Module S M] : Module R (RestrictScalars R S M) :=\n  Module.compHom M (algebraMap R S)\n\n/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    rw [Algebra.smul_def]\n    rw [mul_smul]\n    rfl⟩\n#align restrict_scalars.is_scalar_tower RestrictScalars.isScalarTower\n\nend\n\n/-- When `M` is a right-module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nright-module structure over `R`.\nThe preferred way of setting this up is\n`[Module Rᵐᵒᵖ M] [Module Sᵐᵒᵖ M] [IsScalarTower Rᵐᵒᵖ Sᵐᵒᵖ M]`.\n-/\ninstance RestrictScalars.opModule [Module Sᵐᵒᵖ M] : Module Rᵐᵒᵖ (RestrictScalars R S M) :=\n  letI : Module Sᵐᵒᵖ (RestrictScalars R S M) := ‹Module Sᵐᵒᵖ M›\n  Module.compHom M (RingHom.op $ algebraMap R S)\n#align restrict_scalars.op_module RestrictScalars.opModule\n\ninstance RestrictScalars.isCentralScalar [Module S M] [Module Sᵐᵒᵖ M] [IsCentralScalar S M] :\n    IsCentralScalar R (RestrictScalars R S M) where\n  op_smul_eq_smul r _x := (op_smul_eq_smul (algebraMap R S r) (_ : M) : _)\n#align restrict_scalars.is_central_scalar RestrictScalars.isCentralScalar\n\n/-- The `R`-algebra homomorphism from the original coefficient algebra `S` to endomorphisms\nof `RestrictScalars R S M`.\n-/\ndef RestrictScalars.lsmul [Module S M] : S →ₐ[R] Module.End R (RestrictScalars R S M) :=\n  -- We use `RestrictScalars.moduleOrig` in the implementation,\n  -- but not in the type.\n  letI : Module S (RestrictScalars R S M) := RestrictScalars.moduleOrig R S M\n  Algebra.lsmul R R (RestrictScalars R S M)\n#align restrict_scalars.lsmul RestrictScalars.lsmul\n\nend\n\nvariable [AddCommMonoid M]\n\n/-- `RestrictScalars.addEquiv` is the additive equivalence with the original module. -/\ndef RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M :=\n  AddEquiv.refl M\n#align restrict_scalars.add_equiv RestrictScalars.addEquiv\n\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [Module S M]\n\ntheorem RestrictScalars.smul_def (c : R) (x : RestrictScalars R S M) :\n    c • x = (RestrictScalars.addEquiv R S M).symm\n      (algebraMap R S c • RestrictScalars.addEquiv R S M x) :=\n  rfl\n#align restrict_scalars.smul_def RestrictScalars.smul_def\n\n@[simp]\ntheorem RestrictScalars.addEquiv_map_smul (c : R) (x : RestrictScalars R S M) :\n    RestrictScalars.addEquiv R S M (c • x) = algebraMap R S c • RestrictScalars.addEquiv R S M x :=\n  rfl\n#align restrict_scalars.add_equiv_map_smul RestrictScalars.addEquiv_map_smul\n\ntheorem RestrictScalars.addEquiv_symm_map_algebraMap_smul (r : R) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm (algebraMap R S r • x) =\n      r • (RestrictScalars.addEquiv R S M).symm x :=\n  rfl\n#align restrict_scalars.add_equiv_symm_map_algebra_map_smul RestrictScalars.addEquiv_symm_map_algebraMap_smul\n\ntheorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) := by\n  rw [Algebra.smul_def]\n  ","nextTactic":"rw [mul_smul]","declUpToTactic":"theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) := by\n  rw [Algebra.smul_def]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_RestrictScalars.176_0.FyKmAB3YCMLCVBR","decl":"theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) "}
-{"state":"R : Type u_1\nS : Type u_2\nM : Type u_3\nA : Type u_4\ninst✝⁴ : AddCommMonoid M\ninst✝³ : CommSemiring R\ninst✝² : Semiring S\ninst✝¹ : Algebra R S\ninst✝ : Module S M\nr : R\ns : S\nx : M\n⊢ (AddEquiv.symm (addEquiv R S M)) ((algebraMap R S) r • s • x) = r • (AddEquiv.symm (addEquiv R S M)) (s • x)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.restrict_scalars from \"leanprover-community/mathlib\"@\"c310cfdc40da4d99a10a58c33a95360ef9e6e0bf\"\n\n/-!\n\n# The `RestrictScalars` type alias\n\nSee the documentation attached to the `RestrictScalars` definition for advice on how and when to\nuse this type alias. As described there, it is often a better choice to use the `IsScalarTower`\ntypeclass instead.\n\n## Main definitions\n\n* `RestrictScalars R S M`: the `S`-module `M` viewed as an `R` module when `S` is an `R`-algebra.\n  Note that by default we do *not* have a `Module S (RestrictScalars R S M)` instance\n  for the original action.\n  This is available as a def `RestrictScalars.moduleOrig` if really needed.\n* `RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M`: the additive equivalence\n  between the restricted and original space (in fact, they are definitionally equal,\n  but sometimes it is helpful to avoid using this fact, to keep instances from leaking).\n* `RestrictScalars.ringEquiv : RestrictScalars R S A ≃+* A`: the ring equivalence\n   between the restricted and original space when the module is an algebra.\n\n## See also\n\nThere are many similarly-named definitions elsewhere which do not refer to this type alias. These\nrefer to restricting the scalar type in a bundled type, such as from `A →ₗ[R] B` to `A →ₗ[S] B`:\n\n* `LinearMap.restrictScalars`\n* `LinearEquiv.restrictScalars`\n* `AlgHom.restrictScalars`\n* `AlgEquiv.restrictScalars`\n* `Submodule.restrictScalars`\n* `Subalgebra.restrictScalars`\n-/\n\n\nvariable (R S M A : Type*)\n\n/-- If we put an `R`-algebra structure on a semiring `S`, we get a natural equivalence from the\ncategory of `S`-modules to the category of representations of the algebra `S` (over `R`). The type\nsynonym `RestrictScalars` is essentially this equivalence.\n\nWarning: use this type synonym judiciously! Consider an example where we want to construct an\n`R`-linear map from `M` to `S`, given:\n```lean\nvariable (R S M : Type*)\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module S M]\n```\nWith the assumptions above we can't directly state our map as we have no `Module R M` structure, but\n`RestrictScalars` permits it to be written as:\n```lean\n-- an `R`-module structure on `M` is provided by `RestrictScalars` which is compatible\nexample : RestrictScalars R S M →ₗ[R] S := sorry\n```\nHowever, it is usually better just to add this extra structure as an argument:\n```lean\n-- an `R`-module structure on `M` and proof of its compatibility is provided by the user\nexample [Module R M] [IsScalarTower R S M] : M →ₗ[R] S := sorry\n```\nThe advantage of the second approach is that it defers the duty of providing the missing typeclasses\n`[Module R M] [IsScalarTower R S M]`. If some concrete `M` naturally carries these (as is often\nthe case) then we have avoided `RestrictScalars` entirely. If not, we can pass\n`RestrictScalars R S M` later on instead of `M`.\n\nNote that this means we almost always want to state definitions and lemmas in the language of\n`IsScalarTower` rather than `RestrictScalars`.\n\nAn example of when one might want to use `RestrictScalars` would be if one has a vector space\nover a field of characteristic zero and wishes to make use of the `ℚ`-algebra structure. -/\n@[nolint unusedArguments]\ndef RestrictScalars (_R _S M : Type*) : Type _ := M\n#align restrict_scalars RestrictScalars\n\ninstance [I : Inhabited M] : Inhabited (RestrictScalars R S M) := I\n\ninstance [I : AddCommMonoid M] : AddCommMonoid (RestrictScalars R S M) := I\n\ninstance [I : AddCommGroup M] : AddCommGroup (RestrictScalars R S M) := I\n\nsection Module\n\nsection\n\nvariable [Semiring S] [AddCommMonoid M]\n\n/-- We temporarily install an action of the original ring on `RestrictScalars R S M`. -/\ndef RestrictScalars.moduleOrig [I : Module S M] : Module S (RestrictScalars R S M) := I\n#align restrict_scalars.module_orig RestrictScalars.moduleOrig\n\nvariable [CommSemiring R] [Algebra R S]\n\nsection\n\nattribute [local instance] RestrictScalars.moduleOrig\n\n/-- When `M` is a module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`.\n\nThe preferred way of setting this up is `[Module R M] [Module S M] [IsScalarTower R S M]`.\n-/\ninstance RestrictScalars.module [Module S M] : Module R (RestrictScalars R S M) :=\n  Module.compHom M (algebraMap R S)\n\n/-- This instance is only relevant when `RestrictScalars.moduleOrig` is available as an instance.\n-/\ninstance RestrictScalars.isScalarTower [Module S M] : IsScalarTower R S (RestrictScalars R S M) :=\n  ⟨fun r S M ↦ by\n    rw [Algebra.smul_def]\n    rw [mul_smul]\n    rfl⟩\n#align restrict_scalars.is_scalar_tower RestrictScalars.isScalarTower\n\nend\n\n/-- When `M` is a right-module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nright-module structure over `R`.\nThe preferred way of setting this up is\n`[Module Rᵐᵒᵖ M] [Module Sᵐᵒᵖ M] [IsScalarTower Rᵐᵒᵖ Sᵐᵒᵖ M]`.\n-/\ninstance RestrictScalars.opModule [Module Sᵐᵒᵖ M] : Module Rᵐᵒᵖ (RestrictScalars R S M) :=\n  letI : Module Sᵐᵒᵖ (RestrictScalars R S M) := ‹Module Sᵐᵒᵖ M›\n  Module.compHom M (RingHom.op $ algebraMap R S)\n#align restrict_scalars.op_module RestrictScalars.opModule\n\ninstance RestrictScalars.isCentralScalar [Module S M] [Module Sᵐᵒᵖ M] [IsCentralScalar S M] :\n    IsCentralScalar R (RestrictScalars R S M) where\n  op_smul_eq_smul r _x := (op_smul_eq_smul (algebraMap R S r) (_ : M) : _)\n#align restrict_scalars.is_central_scalar RestrictScalars.isCentralScalar\n\n/-- The `R`-algebra homomorphism from the original coefficient algebra `S` to endomorphisms\nof `RestrictScalars R S M`.\n-/\ndef RestrictScalars.lsmul [Module S M] : S →ₐ[R] Module.End R (RestrictScalars R S M) :=\n  -- We use `RestrictScalars.moduleOrig` in the implementation,\n  -- but not in the type.\n  letI : Module S (RestrictScalars R S M) := RestrictScalars.moduleOrig R S M\n  Algebra.lsmul R R (RestrictScalars R S M)\n#align restrict_scalars.lsmul RestrictScalars.lsmul\n\nend\n\nvariable [AddCommMonoid M]\n\n/-- `RestrictScalars.addEquiv` is the additive equivalence with the original module. -/\ndef RestrictScalars.addEquiv : RestrictScalars R S M ≃+ M :=\n  AddEquiv.refl M\n#align restrict_scalars.add_equiv RestrictScalars.addEquiv\n\nvariable [CommSemiring R] [Semiring S] [Algebra R S] [Module S M]\n\ntheorem RestrictScalars.smul_def (c : R) (x : RestrictScalars R S M) :\n    c • x = (RestrictScalars.addEquiv R S M).symm\n      (algebraMap R S c • RestrictScalars.addEquiv R S M x) :=\n  rfl\n#align restrict_scalars.smul_def RestrictScalars.smul_def\n\n@[simp]\ntheorem RestrictScalars.addEquiv_map_smul (c : R) (x : RestrictScalars R S M) :\n    RestrictScalars.addEquiv R S M (c • x) = algebraMap R S c • RestrictScalars.addEquiv R S M x :=\n  rfl\n#align restrict_scalars.add_equiv_map_smul RestrictScalars.addEquiv_map_smul\n\ntheorem RestrictScalars.addEquiv_symm_map_algebraMap_smul (r : R) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm (algebraMap R S r • x) =\n      r • (RestrictScalars.addEquiv R S M).symm x :=\n  rfl\n#align restrict_scalars.add_equiv_symm_map_algebra_map_smul RestrictScalars.addEquiv_symm_map_algebraMap_smul\n\ntheorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) := by\n  rw [Algebra.smul_def]\n  rw [mul_smul]\n  ","nextTactic":"rfl","declUpToTactic":"theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) := by\n  rw [Algebra.smul_def]\n  rw [mul_smul]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_RestrictScalars.176_0.FyKmAB3YCMLCVBR","decl":"theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :\n    (RestrictScalars.addEquiv R S M).symm ((r • s) • x) =\n      r • (RestrictScalars.addEquiv R S M).symm (s • x) "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_Subalgebra_Pointwise.jsonl

@@ -1,22 +0,0 @@
-{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ toSubmodule S * toSubmodule T ≤ toSubmodule (S ⊔ T)","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  ","nextTactic":"rw [Submodule.mul_le]","declUpToTactic":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.27_0.A8LWG0bRLHJclKD","decl":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ toSubmodule S, ∀ n ∈ toSubmodule T, m * n ∈ toSubmodule (S ⊔ T)","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  ","nextTactic":"intro y hy z hz","declUpToTactic":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.27_0.A8LWG0bRLHJclKD","decl":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ toSubmodule S\nz : A\nhz : z ∈ toSubmodule T\n⊢ y * z ∈ toSubmodule (S ⊔ T)","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  ","nextTactic":"show y * z ∈ S ⊔ T","declUpToTactic":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.27_0.A8LWG0bRLHJclKD","decl":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ toSubmodule S\nz : A\nhz : z ∈ toSubmodule T\n⊢ y * z ∈ S ⊔ T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  ","nextTactic":"exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)","declUpToTactic":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.27_0.A8LWG0bRLHJclKD","decl":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ toSubmodule S * toSubmodule S = toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  ","nextTactic":"apply le_antisymm","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) "}
-{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ toSubmodule S * toSubmodule S ≤ toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · ","nextTactic":"refine' (mul_toSubmodule_le _ _).trans_eq _","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) "}
-{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ toSubmodule (S ⊔ S) = toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    ","nextTactic":"rw [sup_idem]","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) "}
-{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ toSubmodule S ≤ toSubmodule S * toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · ","nextTactic":"intro x hx1","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) "}
-{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ toSubmodule S\n⊢ x ∈ toSubmodule S * toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    ","nextTactic":"rw [← mul_one x]","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) "}
-{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ toSubmodule S\n⊢ x * 1 ∈ toSubmodule S * toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    ","nextTactic":"exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) "}
-{"state":"R✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ toSubmodule S * toSubmodule T = toSubmodule (S ⊔ T)","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  ","nextTactic":"refine' le_antisymm (mul_toSubmodule_le _ _) _","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"R✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ toSubmodule (S ⊔ T) ≤ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  ","nextTactic":"rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"R✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)\n⊢ x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  ","nextTactic":"refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_1\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhx : x ∈ ↑S ∪ ↑T\n⊢ x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · ","nextTactic":"cases' hx with hxS hxT","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_1.inl\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhxS : x ∈ ↑S\n⊢ x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · ","nextTactic":"rw [← mul_one x]","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_1.inl\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhxS : x ∈ ↑S\n⊢ x * 1 ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      ","nextTactic":"exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_1.inr\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhxT : x ∈ ↑T\n⊢ x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · ","nextTactic":"rw [← one_mul x]","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_1.inr\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhxT : x ∈ ↑T\n⊢ 1 * x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      ","nextTactic":"exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_2\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)\nr : R\n⊢ (algebraMap R A) r ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n  · ","nextTactic":"rw [← one_mul (algebraMap _ _ _)]","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_2\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)\nr : R\n⊢ 1 * (algebraMap R A) r ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n  · rw [← one_mul (algebraMap _ _ _)]\n    ","nextTactic":"exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n  · rw [← one_mul (algebraMap _ _ _)]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_3\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx y : A\nhx : x ∈ toSubmodule S * toSubmodule T\nhy : y ∈ toSubmodule S * toSubmodule T\n⊢ x * y ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n  · rw [← one_mul (algebraMap _ _ _)]\n    exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)\n  ","nextTactic":"have := Submodule.mul_mem_mul hx hy","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n  · rw [← one_mul (algebraMap _ _ _)]\n    exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}
-{"state":"case refine'_3\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx y : A\nhx : x ∈ toSubmodule S * toSubmodule T\nhy : y ∈ toSubmodule S * toSubmodule T\nthis : x * y ∈ toSubmodule S * toSubmodule T * (toSubmodule S * toSubmodule T)\n⊢ x * y ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n  rw [Submodule.mul_le]\n  intro y hy z hz\n  show y * z ∈ S ⊔ T\n  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n    = (Subalgebra.toSubmodule S) := by\n  apply le_antisymm\n  · refine' (mul_toSubmodule_le _ _).trans_eq _\n    rw [sup_idem]\n  · intro x hx1\n    rw [← mul_one x]\n    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n  · rw [← one_mul (algebraMap _ _ _)]\n    exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)\n  have := Submodule.mul_mem_mul hx hy\n  ","nextTactic":"rwa [mul_assoc, mul_comm _ (Subalgebra.toSubmodule T), ← mul_assoc _ _ (Subalgebra.toSubmodule S),\n    mul_self, mul_comm (Subalgebra.toSubmodule T), ← mul_assoc, mul_self] at this","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) := by\n  refine' le_antisymm (mul_toSubmodule_le _ _) _\n  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n  refine'\n    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n      fun x y hx hy => _\n  · cases' hx with hxS hxT\n    · rw [← mul_one x]\n      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n    · rw [← one_mul x]\n      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n  · rw [← one_mul (algebraMap _ _ _)]\n    exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)\n  have := Submodule.mul_mem_mul hx hy\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n        = Subalgebra.toSubmodule (S ⊔ T) "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_Subalgebra_Tower.jsonl

@@ -1,11 +0,0 @@
-{"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\nU : Subalgebra S A\nx : R\n⊢ (algebraMap R A) x ∈ U.carrier","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      ","nextTactic":"rw [algebraMap_apply R S A]","declUpToTactic":"/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.87_0.Zq8PWcMlDFAlf8P","decl":"/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A "}
-{"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\nU : Subalgebra S A\nx : R\n⊢ (algebraMap S A) ((algebraMap R S) x) ∈ U.carrier","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      ","nextTactic":"exact U.algebraMap_mem _","declUpToTactic":"/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.87_0.Zq8PWcMlDFAlf8P","decl":"/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A "}
-{"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\n⊢ ↑(restrictScalars R ⊤) = ↑⊤","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by ","nextTactic":"dsimp","declUpToTactic":"@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.101_0.Zq8PWcMlDFAlf8P","decl":"@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ "}
-{"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\nU V : Subalgebra S A\nH : restrictScalars R U = restrictScalars R V\nx : A\n⊢ x ∈ U ↔ x ∈ V","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by ","nextTactic":"rw [← mem_restrictScalars R, H, mem_restrictScalars]","declUpToTactic":"theorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.117_0.Zq8PWcMlDFAlf8P","decl":"theorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) "}
-{"state":"R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ LinearMap.range (toAlgHom R (↥S) A) = toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n  f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ","nextTactic":"ext","declUpToTactic":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.134_0.Zq8PWcMlDFAlf8P","decl":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S "}
-{"state":"case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx✝ : A\n⊢ x✝ ∈ LinearMap.range (toAlgHom R (↥S) A) ↔ x✝ ∈ toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n  f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ext\n  ","nextTactic":"simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n    IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]","declUpToTactic":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ext\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.134_0.Zq8PWcMlDFAlf8P","decl":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S "}
-{"state":"case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx✝ : A\n⊢ (∃ y, (algebraMap (↥S) A) y = x✝) ↔ ∃ y, (Submodule.subtype (toSubmodule S)) y = x✝","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n  f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ext\n  simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n    IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n  ","nextTactic":"rfl","declUpToTactic":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ext\n  simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n    IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.134_0.Zq8PWcMlDFAlf8P","decl":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S "}
-{"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nt : Set A\nz : A\n⊢ z ∈ Subsemiring.closure (Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) ∪ t) ↔\n    z ∈ Subsemiring.closure (Set.range ⇑(algebraMap S A) ∪ t)","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n  f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ext\n  simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n    IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n  rfl\n\nend CommSemiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\ntheorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z ↦\n    show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n         z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n      ","nextTactic":"suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n        rw [this]","declUpToTactic":"theorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z ↦\n    show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n         z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.155_0.Zq8PWcMlDFAlf8P","decl":"theorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R "}
-{"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nt : Set A\nz : A\nthis : Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) = Set.range ⇑(algebraMap S A)\n⊢ z ∈ Subsemiring.closure (Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) ∪ t) ↔\n    z ∈ Subsemiring.closure (Set.range ⇑(algebraMap S A) ∪ t)","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n  f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ext\n  simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n    IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n  rfl\n\nend CommSemiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\ntheorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z ↦\n    show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n         z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n      suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n        ","nextTactic":"rw [this]","declUpToTactic":"theorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z ↦\n    show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n         z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n      suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.155_0.Zq8PWcMlDFAlf8P","decl":"theorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R "}
-{"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nt : Set A\nz : A\n⊢ Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) = Set.range ⇑(algebraMap S A)","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n  f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ext\n  simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n    IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n  rfl\n\nend CommSemiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\ntheorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z ↦\n    show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n         z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n      suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n        rw [this]\n      ","nextTactic":"ext z","declUpToTactic":"theorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z ↦\n    show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n         z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n      suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n        rw [this]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.155_0.Zq8PWcMlDFAlf8P","decl":"theorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R "}
-{"state":"case h\nR : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nt : Set A\nz✝ z : A\n⊢ z ∈ Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) ↔ z ∈ Set.range ⇑(algebraMap S A)","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x ↦ by\n      rw [algebraMap_apply R S A]\n      exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n  ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n  f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n    [Algebra R A] (S : Subalgebra R A) :\n    LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n  ext\n  simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n    IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n  rfl\n\nend CommSemiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\ntheorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z ↦\n    show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n         z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n      suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n        rw [this]\n      ext z\n      ","nextTactic":"exact ⟨fun ⟨⟨_, y, h1⟩, h2⟩ ↦ ⟨y, h2 ▸ h1⟩, fun ⟨y, hy⟩ ↦ ⟨⟨z, y, hy⟩, rfl⟩⟩","declUpToTactic":"theorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z ↦\n    show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n         z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n      suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n        rw [this]\n      ext z\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.155_0.Zq8PWcMlDFAlf8P","decl":"theorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_AlgebraicCard.jsonl

@@ -1,21 +0,0 @@
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, v} #{ x // IsAlgebraic R x } ≤ lift.{v, u} #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  ","nextTactic":"rw [← mk_uLift]","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ #(ULift.{u, v} { x // IsAlgebraic R x }) ≤ lift.{v, u} #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  ","nextTactic":"rw [← mk_uLift]","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ #(ULift.{u, v} { x // IsAlgebraic R x }) ≤ #(ULift.{v, u} R[X]) * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  ","nextTactic":"choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\n⊢ #(ULift.{u, v} { x // IsAlgebraic R x }) ≤ #(ULift.{v, u} R[X]) * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  ","nextTactic":"refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ lift.{u, v} #↑(g ⁻¹' {f}) ≤ ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  ","nextTactic":"rw [lift_le_aleph0]","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ #↑(g ⁻¹' {f}) ≤ ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  ","nextTactic":"rw [le_aleph0_iff_set_countable]","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ Set.Countable (g ⁻¹' {f})","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  ","nextTactic":"suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\nthis : MapsTo Subtype.val (g ⁻¹' {f}) (rootSet f A)\n⊢ Set.Countable (g ⁻¹' {f})\ncase this\nR : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ MapsTo Subtype.val (g ⁻¹' {f}) (rootSet f A)","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  ","nextTactic":"exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"case this\nR : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nf : R[X]\n⊢ MapsTo Subtype.val (g ⁻¹' {f}) (rootSet f A)","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  ","nextTactic":"rintro x (rfl : g x = f)","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"case this\nR : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\ng : ↑{x | IsAlgebraic R x} → R[X]\nhg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0\nhg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0\nx : ↑{x | IsAlgebraic R x}\n⊢ ↑x ∈ rootSet (g x) A","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  ","nextTactic":"exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩","declUpToTactic":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.45_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{v, u} (max #R ℵ₀) * ℵ₀ ≤ max (lift.{v, u} #R) ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by ","nextTactic":"simp","declUpToTactic":"theorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.59_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ "}
-{"state":"R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Algebra R A\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : Countable R\n⊢ Set.Countable {x | IsAlgebraic R x}","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  ","nextTactic":"rw [← le_aleph0_iff_set_countable]","declUpToTactic":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.75_0.jqoHMWuzh6xnVhi","decl":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } "}
-{"state":"R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Algebra R A\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : Countable R\n⊢ #↑{x | IsAlgebraic R x} ≤ ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  ","nextTactic":"rw [← lift_le]","declUpToTactic":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.75_0.jqoHMWuzh6xnVhi","decl":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } "}
-{"state":"R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst��³ : IsDomain A\ninst✝² : Algebra R A\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : Countable R\n⊢ lift.{?u.51811, v} #↑{x | IsAlgebraic R x} ≤ lift.{?u.51811, v} ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  ","nextTactic":"apply (cardinal_mk_lift_le_max R A).trans","declUpToTactic":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.75_0.jqoHMWuzh6xnVhi","decl":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } "}
-{"state":"R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Algebra R A\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : Countable R\n⊢ max (lift.{v, u} #R) ℵ₀ ≤ lift.{u, v} ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  apply (cardinal_mk_lift_le_max R A).trans\n  ","nextTactic":"simp","declUpToTactic":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  apply (cardinal_mk_lift_le_max R A).trans\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.75_0.jqoHMWuzh6xnVhi","decl":"@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } "}
-{"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ #{ x // IsAlgebraic R x } ≤ #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  apply (cardinal_mk_lift_le_max R A).trans\n  simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n    #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n  (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  ","nextTactic":"rw [← lift_id #_]","declUpToTactic":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.96_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ "}
-{"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  apply (cardinal_mk_lift_le_max R A).trans\n  simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n    #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n  (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  rw [← lift_id #_]\n  ","nextTactic":"rw [← lift_id #R[X]]","declUpToTactic":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  rw [← lift_id #_]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.96_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ "}
-{"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ lift.{u, u} #R[X] * ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  apply (cardinal_mk_lift_le_max R A).trans\n  simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n    #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n  (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  rw [← lift_id #_]\n  rw [← lift_id #R[X]]\n  ","nextTactic":"exact cardinal_mk_lift_le_mul R A","declUpToTactic":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  rw [← lift_id #_]\n  rw [← lift_id #R[X]]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.96_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ "}
-{"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ #{ x // IsAlgebraic R x } ≤ max #R ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  apply (cardinal_mk_lift_le_max R A).trans\n  simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n    #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n  (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  rw [← lift_id #_]\n  rw [← lift_id #R[X]]\n  exact cardinal_mk_lift_le_mul R A\n#align algebraic.cardinal_mk_le_mul Algebraic.cardinal_mk_le_mul\n\ntheorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n  ","nextTactic":"rw [← lift_id #_]","declUpToTactic":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.102_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ "}
-{"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  apply (cardinal_mk_lift_le_max R A).trans\n  simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n    #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n  (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  rw [← lift_id #_]\n  rw [← lift_id #R[X]]\n  exact cardinal_mk_lift_le_mul R A\n#align algebraic.cardinal_mk_le_mul Algebraic.cardinal_mk_le_mul\n\ntheorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n  rw [← lift_id #_]\n  ","nextTactic":"rw [← lift_id #R]","declUpToTactic":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n  rw [← lift_id #_]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.102_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ "}
-{"state":"R A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀","srcUpToTactic":"/-\nCopyright (c) 2022 Violeta Hernández Palacios. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Violeta Hernández Palacios\n-/\nimport Mathlib.Data.Polynomial.Cardinal\nimport Mathlib.RingTheory.Algebraic\n\n#align_import algebra.algebraic_card from \"leanprover-community/mathlib\"@\"40494fe75ecbd6d2ec61711baa630cf0a7b7d064\"\n\n/-!\n### Cardinality of algebraic numbers\n\nIn this file, we prove variants of the following result: the cardinality of algebraic numbers under\nan R-algebra is at most `# R[X] * ℵ₀`.\n\nAlthough this can be used to prove that real or complex transcendental numbers exist, a more direct\nproof is given by `Liouville.is_transcendental`.\n-/\n\n\nuniverse u v\n\nopen Cardinal Polynomial Set\n\nopen Cardinal Polynomial\n\nnamespace Algebraic\n\ntheorem infinite_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A] [Algebra R A]\n    [CharZero A] : { x : A | IsAlgebraic R x }.Infinite :=\n  infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat\n#align algebraic.infinite_of_char_zero Algebraic.infinite_of_charZero\n\ntheorem aleph0_le_cardinal_mk_of_charZero (R A : Type*) [CommRing R] [IsDomain R] [Ring A]\n    [Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=\n  infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)\n#align algebraic.aleph_0_le_cardinal_mk_of_char_zero Algebraic.aleph0_le_cardinal_mk_of_charZero\n\nsection lift\n\nvariable (R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_lift_le_mul :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by\n  rw [← mk_uLift]\n  rw [← mk_uLift]\n  choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop\n  refine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n  rw [lift_le_aleph0]\n  rw [le_aleph0_iff_set_countable]\n  suffices : MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A)\n  exact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n  rintro x (rfl : g x = f)\n  exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩\n#align algebraic.cardinal_mk_lift_le_mul Algebraic.cardinal_mk_lift_le_mul\n\ntheorem cardinal_mk_lift_le_max :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=\n  (cardinal_mk_lift_le_mul R A).trans <|\n    (mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans <| by simp\n#align algebraic.cardinal_mk_lift_le_max Algebraic.cardinal_mk_lift_le_max\n\n@[simp]\ntheorem cardinal_mk_lift_of_infinite [Infinite R] :\n    Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=\n  ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|\n    lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>\n      NoZeroSMulDivisors.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩\n#align algebraic.cardinal_mk_lift_of_infinite Algebraic.cardinal_mk_lift_of_infinite\n\nvariable [Countable R]\n\n@[simp]\nprotected theorem countable : Set.Countable { x : A | IsAlgebraic R x } := by\n  rw [← le_aleph0_iff_set_countable]\n  rw [← lift_le]\n  apply (cardinal_mk_lift_le_max R A).trans\n  simp\n#align algebraic.countable Algebraic.countable\n\n@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n    #{ x : A // IsAlgebraic R x } = ℵ₀ :=\n  (Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinal_mk_of_charZero R A)\n#align algebraic.cardinal_mk_of_countble_of_char_zero Algebraic.cardinal_mk_of_countable_of_charZero\n\nend lift\n\nsection NonLift\n\nvariable (R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]\n  [NoZeroSMulDivisors R A]\n\ntheorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by\n  rw [← lift_id #_]\n  rw [← lift_id #R[X]]\n  exact cardinal_mk_lift_le_mul R A\n#align algebraic.cardinal_mk_le_mul Algebraic.cardinal_mk_le_mul\n\ntheorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n  rw [← lift_id #_]\n  rw [← lift_id #R]\n  ","nextTactic":"exact cardinal_mk_lift_le_max R A","declUpToTactic":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by\n  rw [← lift_id #_]\n  rw [← lift_id #R]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_AlgebraicCard.102_0.jqoHMWuzh6xnVhi","decl":"theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Associated.jsonl

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Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Basic.jsonl

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Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Finprod.jsonl

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Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Multiset_Lemmas.jsonl

@@ -1,10 +0,0 @@
-{"state":"α : Type u_1\ninst✝ : CommMonoid α\ns : Multiset α\na✝ : α\nl : List α\na : α\nh : a ∈ ⟦l⟧\n⊢ a ∣ prod ⟦l⟧","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by ","nextTactic":"simpa using List.dvd_prod h","declUpToTactic":"theorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.18_0.kGoXX9k6n37QCh9","decl":"theorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod "}
-{"state":"α : Type u_1\ninst✝ : CanonicallyOrderedCommMonoid α\nm : Multiset α\nl : List α\n⊢ prod ⟦l⟧ = 1 ↔ ∀ x ∈ ⟦l⟧, x = 1","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by ","nextTactic":"simpa using List.prod_eq_one_iff l","declUpToTactic":"@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.22_0.kGoXX9k6n37QCh9","decl":"@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) "}
-{"state":"α : Type u_1\nR : Type u_2\ninst✝¹ : CanonicallyOrderedCommSemiring R\ninst✝ : Nontrivial R\nm : Multiset R\n⊢ 0 < Multiset.prod m ↔ ∀ x ∈ m, 0 < x","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  ","nextTactic":"rcases m with ⟨l⟩","declUpToTactic":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.31_0.kGoXX9k6n37QCh9","decl":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) "}
-{"state":"case mk\nα : Type u_1\nR : Type u_2\ninst✝¹ : CanonicallyOrderedCommSemiring R\ninst✝ : Nontrivial R\nm : Multiset R\nl : List R\n⊢ 0 < Multiset.prod (Quot.mk Setoid.r l) ↔ ∀ x ∈ Quot.mk Setoid.r l, 0 < x","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  ","nextTactic":"rw [Multiset.quot_mk_to_coe'']","declUpToTactic":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.31_0.kGoXX9k6n37QCh9","decl":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) "}
-{"state":"case mk\nα : Type u_1\nR : Type u_2\ninst✝¹ : CanonicallyOrderedCommSemiring R\ninst✝ : Nontrivial R\nm : Multiset R\nl : List R\n⊢ 0 < Multiset.prod ↑l ↔ ∀ x ∈ ↑l, 0 < x","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  ","nextTactic":"rw [Multiset.coe_prod]","declUpToTactic":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.31_0.kGoXX9k6n37QCh9","decl":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) "}
-{"state":"case mk\nα : Type u_1\nR : Type u_2\ninst✝¹ : CanonicallyOrderedCommSemiring R\ninst✝ : Nontrivial R\nm : Multiset R\nl : List R\n⊢ 0 < List.prod l ↔ ∀ x ∈ ↑l, 0 < x","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  ","nextTactic":"exact CanonicallyOrderedCommSemiring.list_prod_pos","declUpToTactic":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.31_0.kGoXX9k6n37QCh9","decl":"@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) "}
-{"state":"α : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\nh : ∀ b ∈ s, Commute a b\n⊢ Commute a (sum s)","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  exact CanonicallyOrderedCommSemiring.list_prod_pos\n\nopen Multiset\n\nnamespace Commute\n\nvariable [NonUnitalNonAssocSemiring α] (s : Multiset α)\n\ntheorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  ","nextTactic":"induction s using Quotient.inductionOn","declUpToTactic":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.45_0.kGoXX9k6n37QCh9","decl":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum "}
-{"state":"case h\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a (sum ⟦a✝⟧)","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  exact CanonicallyOrderedCommSemiring.list_prod_pos\n\nopen Multiset\n\nnamespace Commute\n\nvariable [NonUnitalNonAssocSemiring α] (s : Multiset α)\n\ntheorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  ","nextTactic":"rw [quot_mk_to_coe]","declUpToTactic":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.45_0.kGoXX9k6n37QCh9","decl":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum "}
-{"state":"case h\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a (sum ↑a✝)","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  exact CanonicallyOrderedCommSemiring.list_prod_pos\n\nopen Multiset\n\nnamespace Commute\n\nvariable [NonUnitalNonAssocSemiring α] (s : Multiset α)\n\ntheorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  rw [quot_mk_to_coe]\n  ","nextTactic":"rw [coe_sum]","declUpToTactic":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  rw [quot_mk_to_coe]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.45_0.kGoXX9k6n37QCh9","decl":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum "}
-{"state":"case h\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ns : Multiset α\na : α\na✝ : List α\nh : ∀ b ∈ ⟦a✝⟧, Commute a b\n⊢ Commute a (List.sum a✝)","srcUpToTactic":"/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Bhavik Mehta, Eric Wieser\n-/\nimport Mathlib.Data.List.BigOperators.Lemmas\nimport Mathlib.Algebra.BigOperators.Multiset.Basic\n\n#align_import algebra.big_operators.multiset.lemmas from \"leanprover-community/mathlib\"@\"0a0ec35061ed9960bf0e7ffb0335f44447b58977\"\n\n/-! # Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports -/\n\n\nvariable {α : Type*}\n\nnamespace Multiset\n\ntheorem dvd_prod [CommMonoid α] {s : Multiset α} {a : α} : a ∈ s → a ∣ s.prod :=\n  Quotient.inductionOn s (fun l a h => by simpa using List.dvd_prod h) a\n#align multiset.dvd_prod Multiset.dvd_prod\n\n@[to_additive]\ntheorem prod_eq_one_iff [CanonicallyOrderedCommMonoid α] {m : Multiset α} :\n    m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) :=\n  Quotient.inductionOn m fun l => by simpa using List.prod_eq_one_iff l\n#align multiset.prod_eq_one_iff Multiset.prod_eq_one_iff\n#align multiset.sum_eq_zero_iff Multiset.sum_eq_zero_iff\n\nend Multiset\n\n@[simp]\nlemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]\n    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) := by\n  rcases m with ⟨l⟩\n  rw [Multiset.quot_mk_to_coe'']\n  rw [Multiset.coe_prod]\n  exact CanonicallyOrderedCommSemiring.list_prod_pos\n\nopen Multiset\n\nnamespace Commute\n\nvariable [NonUnitalNonAssocSemiring α] (s : Multiset α)\n\ntheorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  rw [quot_mk_to_coe]\n  rw [coe_sum]\n  ","nextTactic":"exact Commute.list_sum_right _ _ h","declUpToTactic":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by\n  induction s using Quotient.inductionOn\n  rw [quot_mk_to_coe]\n  rw [coe_sum]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Multiset_Lemmas.45_0.kGoXX9k6n37QCh9","decl":"theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_NatAntidiagonal.jsonl

@@ -1,11 +0,0 @@
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal (n + 1), f p = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  ","nextTactic":"rw [antidiagonal_succ, prod_cons, prod_map]","declUpToTactic":"theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.25_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ f (0, n + 1) *\n      ∏ x in antidiagonal n,\n        f\n          ((Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Function.Embedding.refl ℕ))\n            x) =\n    f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; ","nextTactic":"rfl","declUpToTactic":"theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.25_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal n, f (Prod.swap p) = ∏ p in antidiagonal n, f p","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  ","nextTactic":"conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.36_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n| ∏ p in antidiagonal n, f (Prod.swap p)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","nextTactic":"rw [← map_swap_antidiagonal, Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.36_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n| ∏ p in antidiagonal n, f (Prod.swap p)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","nextTactic":"rw [← map_swap_antidiagonal, Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.36_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n| ∏ p in antidiagonal n, f (Prod.swap p)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","nextTactic":"rw [← map_swap_antidiagonal, Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.36_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal (n + 1), f p = f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  ","nextTactic":"rw [← prod_antidiagonal_swap]","declUpToTactic":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.43_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal (n + 1), f (Prod.swap p) = f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  ","nextTactic":"rw [prod_antidiagonal_succ]","declUpToTactic":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.43_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ f (Prod.swap (0, n + 1)) * ∏ p in antidiagonal n, f (Prod.swap (p.1 + 1, p.2)) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  ","nextTactic":"rw [← prod_antidiagonal_swap]","declUpToTactic":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.43_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ f (Prod.swap (0, n + 1)) * ∏ p in antidiagonal n, f (Prod.swap ((Prod.swap p).1 + 1, (Prod.swap p).2)) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  rw [← prod_antidiagonal_swap]\n  ","nextTactic":"rfl","declUpToTactic":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  rw [← prod_antidiagonal_swap]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.43_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) "}
-{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → ℕ → M\np : ℕ × ℕ\nhp : p ∈ antidiagonal n\n⊢ f p n = f p (p.1 + p.2)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  rw [← prod_antidiagonal_swap]\n  rfl\n#align finset.nat.prod_antidiagonal_succ' Finset.Nat.prod_antidiagonal_succ'\n\ntheorem sum_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (n + 1, 0) + ∑ p in antidiagonal n, f (p.1, p.2 + 1) :=\n  @prod_antidiagonal_succ' (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ' Finset.Nat.sum_antidiagonal_succ'\n\n@[to_additive]\ntheorem prod_antidiagonal_subst {n : ℕ} {f : ℕ × ℕ → ℕ → M} :\n    ∏ p in antidiagonal n, f p n = ∏ p in antidiagonal n, f p (p.1 + p.2) :=\n  prod_congr rfl fun p hp ↦ by ","nextTactic":"rw [mem_antidiagonal.mp hp]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_subst {n : ℕ} {f : ℕ × ℕ → ℕ → M} :\n    ∏ p in antidiagonal n, f p n = ∏ p in antidiagonal n, f p (p.1 + p.2) :=\n  prod_congr rfl fun p hp ↦ by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.56_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_subst {n : ℕ} {f : ℕ × ℕ → ℕ → M} :\n    ∏ p in antidiagonal n, f p n = ∏ p in antidiagonal n, f p (p.1 + p.2) "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Option.jsonl

@@ -1,4 +0,0 @@
-{"state":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : Option α → M\ns : Finset α\n⊢ ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x)","srcUpToTactic":"/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Data.Finset.Option\n\n#align_import algebra.big_operators.option from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Lemmas about products and sums over finite sets in `Option α`\n\nIn this file we prove formulas for products and sums over `Finset.insertNone s` and\n`Finset.eraseNone s`.\n-/\n\nopen BigOperators\n\nopen Function\n\nnamespace Finset\n\nvariable {α M : Type*} [CommMonoid M]\n\n@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by ","nextTactic":"simp [insertNone]","declUpToTactic":"@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Option.26_0.TnTVWDXfJXrB22D","decl":"@[to_additive (attr "}
-{"state":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x","srcUpToTactic":"/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Data.Finset.Option\n\n#align_import algebra.big_operators.option from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Lemmas about products and sums over finite sets in `Option α`\n\nIn this file we prove formulas for products and sums over `Finset.insertNone s` and\n`Finset.eraseNone s`.\n-/\n\nopen BigOperators\n\nopen Function\n\nnamespace Finset\n\nvariable {α M : Type*} [CommMonoid M]\n\n@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by simp [insertNone]\n#align finset.prod_insert_none Finset.prod_insertNone\n#align finset.sum_insert_none Finset.sum_insertNone\n\n@[to_additive]\ntheorem mul_prod_eq_prod_insertNone (f : α → M) (x : M) (s : Finset α) :\n    x * ∏ i in s, f i = ∏ i in insertNone s, i.elim x f :=\n  (prod_insertNone (fun i => i.elim x f) _).symm\n\n@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  ","nextTactic":"classical calc\n      ∏ x in eraseNone s, f x = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x in s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n      _ = ∏ x in s, Option.elim' 1 f x := prod_erase _ rfl","declUpToTactic":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Option.37_0.TnTVWDXfJXrB22D","decl":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x "}
-{"state":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x","srcUpToTactic":"/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Data.Finset.Option\n\n#align_import algebra.big_operators.option from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Lemmas about products and sums over finite sets in `Option α`\n\nIn this file we prove formulas for products and sums over `Finset.insertNone s` and\n`Finset.eraseNone s`.\n-/\n\nopen BigOperators\n\nopen Function\n\nnamespace Finset\n\nvariable {α M : Type*} [CommMonoid M]\n\n@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by simp [insertNone]\n#align finset.prod_insert_none Finset.prod_insertNone\n#align finset.sum_insert_none Finset.sum_insertNone\n\n@[to_additive]\ntheorem mul_prod_eq_prod_insertNone (f : α → M) (x : M) (s : Finset α) :\n    x * ∏ i in s, f i = ∏ i in insertNone s, i.elim x f :=\n  (prod_insertNone (fun i => i.elim x f) _).symm\n\n@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  classical ","nextTactic":"calc\n      ∏ x in eraseNone s, f x = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x in s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n      _ = ∏ x in s, Option.elim' 1 f x := prod_erase _ rfl","declUpToTactic":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  classical ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Option.37_0.TnTVWDXfJXrB22D","decl":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x "}
-{"state":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in map Embedding.some (eraseNone s), Option.elim' 1 f x = ∏ x in erase s none, Option.elim' 1 f x","srcUpToTactic":"/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Data.Finset.Option\n\n#align_import algebra.big_operators.option from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Lemmas about products and sums over finite sets in `Option α`\n\nIn this file we prove formulas for products and sums over `Finset.insertNone s` and\n`Finset.eraseNone s`.\n-/\n\nopen BigOperators\n\nopen Function\n\nnamespace Finset\n\nvariable {α M : Type*} [CommMonoid M]\n\n@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by simp [insertNone]\n#align finset.prod_insert_none Finset.prod_insertNone\n#align finset.sum_insert_none Finset.sum_insertNone\n\n@[to_additive]\ntheorem mul_prod_eq_prod_insertNone (f : α → M) (x : M) (s : Finset α) :\n    x * ∏ i in s, f i = ∏ i in insertNone s, i.elim x f :=\n  (prod_insertNone (fun i => i.elim x f) _).symm\n\n@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  classical calc\n      ∏ x in eraseNone s, f x = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x in s.erase none, Option.elim' 1 f x := by ","nextTactic":"rw [map_some_eraseNone]","declUpToTactic":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  classical calc\n      ∏ x in eraseNone s, f x = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x in s.erase none, Option.elim' 1 f x := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Option.37_0.TnTVWDXfJXrB22D","decl":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Pi.jsonl

@@ -1,11 +0,0 @@
-{"state":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns : Finset γ\nf : γ → α\ng : γ → β\nthis : DecidableEq γ\n⊢ ∀ ⦃a : γ⦄ {s : Finset γ},\n    a ∉ s →\n      (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) →\n        (∏ x in insert a s, f x, ∏ x in insert a s, g x) = ∏ x in insert a s, (f x, g x)","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by ","nextTactic":"simp (config := { contextual := true }) [Prod.ext_iff]","declUpToTactic":"@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.63_0.f38FU5HrzKIlXWd","decl":"@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) "}
-{"state":"ι : Type u_1\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\nR : Type u_2\ninst✝ : Semiring R\nx : ι → R\n⊢ x = ∑ i : ι, x i • fun j => if i = j then 1 else 0","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ","nextTactic":"ext","declUpToTactic":"/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.71_0.f38FU5HrzKIlXWd","decl":"/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 "}
-{"state":"case h\nι : Type u_1\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\nR : Type u_2\ninst✝ : Semiring R\nx : ι → R\nx✝ : ι\n⊢ x x✝ = Finset.sum Finset.univ (fun i => x i • fun j => if i = j then 1 else 0) x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  ","nextTactic":"simp","declUpToTactic":"/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.71_0.f38FU5HrzKIlXWd","decl":"/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 "}
-{"state":"I : Type u_1\ninst✝² : DecidableEq I\nZ : I → Type u_2\ninst✝¹ : (i : I) → CommMonoid (Z i)\ninst✝ : Fintype I\nf : (i : I) → Z i\n⊢ ∏ i : I, Pi.mulSingle i (f i) = f","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ","nextTactic":"ext a","declUpToTactic":"@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.84_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f "}
-{"state":"case h\nI : Type u_1\ninst✝² : DecidableEq I\nZ : I → Type u_2\ninst✝¹ : (i : I) → CommMonoid (Z i)\ninst✝ : Fintype I\nf : (i : I) → Z i\na : I\n⊢ Finset.prod univ (fun i => Pi.mulSingle i (f i)) a = f a","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ext a\n  ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ext a\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.84_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f "}
-{"state":"I : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\n⊢ g = h","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ext a\n  simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  ","nextTactic":"cases nonempty_fintype I","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "}
-{"state":"case intro\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\n⊢ g = h","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ext a\n  simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ","nextTactic":"ext k","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "}
-{"state":"case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ g k = h k","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ext a\n  simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ext k\n  ","nextTactic":"rw [← Finset.univ_prod_mulSingle k]","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ext k\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "}
-{"state":"case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ g (∏ i : I, Pi.mulSingle i (k i)) = h (∏ i : I, Pi.mulSingle i (k i))","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ext a\n  simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ext k\n  rw [← Finset.univ_prod_mulSingle k]\n  ","nextTactic":"rw [map_prod]","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ext k\n  rw [← Finset.univ_prod_mulSingle k]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "}
-{"state":"case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ ∏ x : I, g (Pi.mulSingle x (k x)) = h (∏ i : I, Pi.mulSingle i (k i))","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ��� a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ext a\n  simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ext k\n  rw [← Finset.univ_prod_mulSingle k]\n  rw [map_prod]\n  ","nextTactic":"rw [map_prod]","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ext k\n  rw [← Finset.univ_prod_mulSingle k]\n  rw [map_prod]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "}
-{"state":"case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ ∏ x : I, g (Pi.mulSingle x (k x)) = ∏ x : I, h (Pi.mulSingle x (k x))","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n    (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n  (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n    (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n  map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n    (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n  funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n    [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n  Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n    (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n  haveI := Classical.decEq γ\n  Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n    (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n  ext\n  simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n    (∏ i, Pi.mulSingle i (f i)) = f := by\n  ext a\n  simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ext k\n  rw [← Finset.univ_prod_mulSingle k]\n  rw [map_prod]\n  rw [map_prod]\n  ","nextTactic":"simp only [H]","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n  cases nonempty_fintype I\n  ext k\n  rw [← Finset.univ_prod_mulSingle k]\n  rw [map_prod]\n  rw [map_prod]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n    (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Bounds.jsonl

@@ -1,2 +0,0 @@
-{"state":"ι : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : ConditionallyCompleteLattice G\ninst✝¹ : CovariantClass G G (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : Nonempty ι\nf : ι → G\nhf : BddAbove (range f)\na : G\n⊢ (⨆ i, f i) / a = ⨆ i, f i / a","srcUpToTactic":"/-\nCopyright (c) 2021 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury G. Kudryashov\n-/\nimport Mathlib.Algebra.Order.Group.OrderIso\nimport Mathlib.Algebra.Order.Monoid.OrderDual\nimport Mathlib.Data.Set.Pointwise.Basic\nimport Mathlib.Order.Bounds.OrderIso\nimport Mathlib.Order.ConditionallyCompleteLattice.Basic\n\n#align_import algebra.bounds from \"leanprover-community/mathlib\"@\"dd71334db81d0bd444af1ee339a29298bef40734\"\n\n/-!\n# Upper/lower bounds in ordered monoids and groups\n\nIn this file we prove a few facts like “`-s` is bounded above iff `s` is bounded below”\n(`bddAbove_neg`).\n-/\n\n\nopen Function Set\n\nopen Pointwise\n\nsection InvNeg\n\nvariable {G : Type*} [Group G] [Preorder G] [CovariantClass G G (· * ·) (· ≤ ·)]\n  [CovariantClass G G (swap (· * ·)) (· ≤ ·)] {s : Set G} {a : G}\n\n@[to_additive (attr := simp)]\ntheorem bddAbove_inv : BddAbove s⁻¹ ↔ BddBelow s :=\n  (OrderIso.inv G).bddAbove_preimage\n#align bdd_above_inv bddAbove_inv\n#align bdd_above_neg bddAbove_neg\n\n@[to_additive (attr := simp)]\ntheorem bddBelow_inv : BddBelow s⁻¹ ↔ BddAbove s :=\n  (OrderIso.inv G).bddBelow_preimage\n#align bdd_below_inv bddBelow_inv\n#align bdd_below_neg bddBelow_neg\n\n@[to_additive]\ntheorem BddAbove.inv (h : BddAbove s) : BddBelow s⁻¹ :=\n  bddBelow_inv.2 h\n#align bdd_above.inv BddAbove.inv\n#align bdd_above.neg BddAbove.neg\n\n@[to_additive]\ntheorem BddBelow.inv (h : BddBelow s) : BddAbove s⁻¹ :=\n  bddAbove_inv.2 h\n#align bdd_below.inv BddBelow.inv\n#align bdd_below.neg BddBelow.neg\n\n@[to_additive (attr := simp)]\ntheorem isLUB_inv : IsLUB s⁻¹ a ↔ IsGLB s a⁻¹ :=\n  (OrderIso.inv G).isLUB_preimage\n#align is_lub_inv isLUB_inv\n#align is_lub_neg isLUB_neg\n\n@[to_additive]\ntheorem isLUB_inv' : IsLUB s⁻¹ a⁻¹ ↔ IsGLB s a :=\n  (OrderIso.inv G).isLUB_preimage'\n#align is_lub_inv' isLUB_inv'\n#align is_lub_neg' isLUB_neg'\n\n@[to_additive]\ntheorem IsGLB.inv (h : IsGLB s a) : IsLUB s⁻¹ a⁻¹ :=\n  isLUB_inv'.2 h\n#align is_glb.inv IsGLB.inv\n#align is_glb.neg IsGLB.neg\n\n@[to_additive (attr := simp)]\ntheorem isGLB_inv : IsGLB s⁻¹ a ↔ IsLUB s a⁻¹ :=\n  (OrderIso.inv G).isGLB_preimage\n#align is_glb_inv isGLB_inv\n#align is_glb_neg isGLB_neg\n\n@[to_additive]\ntheorem isGLB_inv' : IsGLB s⁻¹ a⁻¹ ↔ IsLUB s a :=\n  (OrderIso.inv G).isGLB_preimage'\n#align is_glb_inv' isGLB_inv'\n#align is_glb_neg' isGLB_neg'\n\n@[to_additive]\ntheorem IsLUB.inv (h : IsLUB s a) : IsGLB s⁻¹ a⁻¹ :=\n  isGLB_inv'.2 h\n#align is_lub.inv IsLUB.inv\n#align is_lub.neg IsLUB.neg\n\nend InvNeg\n\nsection mul_add\n\nvariable {M : Type*} [Mul M] [Preorder M] [CovariantClass M M (· * ·) (· ≤ ·)]\n  [CovariantClass M M (swap (· * ·)) (· ≤ ·)]\n\n@[to_additive]\ntheorem mul_mem_upperBounds_mul {s t : Set M} {a b : M} (ha : a ∈ upperBounds s)\n    (hb : b ∈ upperBounds t) : a * b ∈ upperBounds (s * t) :=\n  forall_image2_iff.2 fun _ hx _ hy => mul_le_mul' (ha hx) (hb hy)\n#align mul_mem_upper_bounds_mul mul_mem_upperBounds_mul\n#align add_mem_upper_bounds_add add_mem_upperBounds_add\n\n@[to_additive]\ntheorem subset_upperBounds_mul (s t : Set M) :\n    upperBounds s * upperBounds t ⊆ upperBounds (s * t) :=\n  image2_subset_iff.2 fun _ hx _ hy => mul_mem_upperBounds_mul hx hy\n#align subset_upper_bounds_mul subset_upperBounds_mul\n#align subset_upper_bounds_add subset_upperBounds_add\n\n@[to_additive]\ntheorem mul_mem_lowerBounds_mul {s t : Set M} {a b : M} (ha : a ∈ lowerBounds s)\n    (hb : b ∈ lowerBounds t) : a * b ∈ lowerBounds (s * t) :=\n  mul_mem_upperBounds_mul (M := Mᵒᵈ) ha hb\n#align mul_mem_lower_bounds_mul mul_mem_lowerBounds_mul\n#align add_mem_lower_bounds_add add_mem_lowerBounds_add\n\n@[to_additive]\ntheorem subset_lowerBounds_mul (s t : Set M) :\n    lowerBounds s * lowerBounds t ⊆ lowerBounds (s * t) :=\n  subset_upperBounds_mul (M := Mᵒᵈ) _ _\n#align subset_lower_bounds_mul subset_lowerBounds_mul\n#align subset_lower_bounds_add subset_lowerBounds_add\n\n@[to_additive]\ntheorem BddAbove.mul {s t : Set M} (hs : BddAbove s) (ht : BddAbove t) : BddAbove (s * t) :=\n  (Nonempty.mul hs ht).mono (subset_upperBounds_mul s t)\n#align bdd_above.mul BddAbove.mul\n#align bdd_above.add BddAbove.add\n\n@[to_additive]\ntheorem BddBelow.mul {s t : Set M} (hs : BddBelow s) (ht : BddBelow t) : BddBelow (s * t) :=\n  (Nonempty.mul hs ht).mono (subset_lowerBounds_mul s t)\n#align bdd_below.mul BddBelow.mul\n#align bdd_below.add BddBelow.add\n\nend mul_add\n\nsection ConditionallyCompleteLattice\n\nsection Right\n\nvariable {ι G : Type*} [Group G] [ConditionallyCompleteLattice G]\n  [CovariantClass G G (Function.swap (· * ·)) (· ��� ·)] [Nonempty ι] {f : ι → G}\n\n@[to_additive]\ntheorem ciSup_mul (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) * a = ⨆ i, f i * a :=\n  (OrderIso.mulRight a).map_ciSup hf\n#align csupr_mul ciSup_mul\n#align csupr_add ciSup_add\n\n@[to_additive]\ntheorem ciSup_div (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by\n  ","nextTactic":"simp only [div_eq_mul_inv, ciSup_mul hf]","declUpToTactic":"@[to_additive]\ntheorem ciSup_div (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Bounds.153_0.kwNNY3mkAxBsoBv","decl":"@[to_additive]\ntheorem ciSup_div (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a "}
-{"state":"ι : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : ConditionallyCompleteLattice G\ninst✝¹ : CovariantClass G G (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : Nonempty ι\nf : ι → G\nhf : BddBelow (range f)\na : G\n⊢ (⨅ i, f i) / a = ⨅ i, f i / a","srcUpToTactic":"/-\nCopyright (c) 2021 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury G. Kudryashov\n-/\nimport Mathlib.Algebra.Order.Group.OrderIso\nimport Mathlib.Algebra.Order.Monoid.OrderDual\nimport Mathlib.Data.Set.Pointwise.Basic\nimport Mathlib.Order.Bounds.OrderIso\nimport Mathlib.Order.ConditionallyCompleteLattice.Basic\n\n#align_import algebra.bounds from \"leanprover-community/mathlib\"@\"dd71334db81d0bd444af1ee339a29298bef40734\"\n\n/-!\n# Upper/lower bounds in ordered monoids and groups\n\nIn this file we prove a few facts like “`-s` is bounded above iff `s` is bounded below”\n(`bddAbove_neg`).\n-/\n\n\nopen Function Set\n\nopen Pointwise\n\nsection InvNeg\n\nvariable {G : Type*} [Group G] [Preorder G] [CovariantClass G G (· * ·) (· ≤ ·)]\n  [CovariantClass G G (swap (· * ·)) (· ≤ ·)] {s : Set G} {a : G}\n\n@[to_additive (attr := simp)]\ntheorem bddAbove_inv : BddAbove s⁻¹ ↔ BddBelow s :=\n  (OrderIso.inv G).bddAbove_preimage\n#align bdd_above_inv bddAbove_inv\n#align bdd_above_neg bddAbove_neg\n\n@[to_additive (attr := simp)]\ntheorem bddBelow_inv : BddBelow s⁻¹ ↔ BddAbove s :=\n  (OrderIso.inv G).bddBelow_preimage\n#align bdd_below_inv bddBelow_inv\n#align bdd_below_neg bddBelow_neg\n\n@[to_additive]\ntheorem BddAbove.inv (h : BddAbove s) : BddBelow s⁻¹ :=\n  bddBelow_inv.2 h\n#align bdd_above.inv BddAbove.inv\n#align bdd_above.neg BddAbove.neg\n\n@[to_additive]\ntheorem BddBelow.inv (h : BddBelow s) : BddAbove s⁻¹ :=\n  bddAbove_inv.2 h\n#align bdd_below.inv BddBelow.inv\n#align bdd_below.neg BddBelow.neg\n\n@[to_additive (attr := simp)]\ntheorem isLUB_inv : IsLUB s⁻¹ a ↔ IsGLB s a⁻¹ :=\n  (OrderIso.inv G).isLUB_preimage\n#align is_lub_inv isLUB_inv\n#align is_lub_neg isLUB_neg\n\n@[to_additive]\ntheorem isLUB_inv' : IsLUB s⁻¹ a⁻¹ ↔ IsGLB s a :=\n  (OrderIso.inv G).isLUB_preimage'\n#align is_lub_inv' isLUB_inv'\n#align is_lub_neg' isLUB_neg'\n\n@[to_additive]\ntheorem IsGLB.inv (h : IsGLB s a) : IsLUB s⁻¹ a⁻¹ :=\n  isLUB_inv'.2 h\n#align is_glb.inv IsGLB.inv\n#align is_glb.neg IsGLB.neg\n\n@[to_additive (attr := simp)]\ntheorem isGLB_inv : IsGLB s⁻¹ a ↔ IsLUB s a⁻¹ :=\n  (OrderIso.inv G).isGLB_preimage\n#align is_glb_inv isGLB_inv\n#align is_glb_neg isGLB_neg\n\n@[to_additive]\ntheorem isGLB_inv' : IsGLB s⁻¹ a⁻¹ ↔ IsLUB s a :=\n  (OrderIso.inv G).isGLB_preimage'\n#align is_glb_inv' isGLB_inv'\n#align is_glb_neg' isGLB_neg'\n\n@[to_additive]\ntheorem IsLUB.inv (h : IsLUB s a) : IsGLB s⁻¹ a⁻¹ :=\n  isGLB_inv'.2 h\n#align is_lub.inv IsLUB.inv\n#align is_lub.neg IsLUB.neg\n\nend InvNeg\n\nsection mul_add\n\nvariable {M : Type*} [Mul M] [Preorder M] [CovariantClass M M (· * ·) (· ≤ ·)]\n  [CovariantClass M M (swap (· * ·)) (· ≤ ·)]\n\n@[to_additive]\ntheorem mul_mem_upperBounds_mul {s t : Set M} {a b : M} (ha : a ∈ upperBounds s)\n    (hb : b ∈ upperBounds t) : a * b ∈ upperBounds (s * t) :=\n  forall_image2_iff.2 fun _ hx _ hy => mul_le_mul' (ha hx) (hb hy)\n#align mul_mem_upper_bounds_mul mul_mem_upperBounds_mul\n#align add_mem_upper_bounds_add add_mem_upperBounds_add\n\n@[to_additive]\ntheorem subset_upperBounds_mul (s t : Set M) :\n    upperBounds s * upperBounds t ⊆ upperBounds (s * t) :=\n  image2_subset_iff.2 fun _ hx _ hy => mul_mem_upperBounds_mul hx hy\n#align subset_upper_bounds_mul subset_upperBounds_mul\n#align subset_upper_bounds_add subset_upperBounds_add\n\n@[to_additive]\ntheorem mul_mem_lowerBounds_mul {s t : Set M} {a b : M} (ha : a ∈ lowerBounds s)\n    (hb : b ∈ lowerBounds t) : a * b ∈ lowerBounds (s * t) :=\n  mul_mem_upperBounds_mul (M := Mᵒᵈ) ha hb\n#align mul_mem_lower_bounds_mul mul_mem_lowerBounds_mul\n#align add_mem_lower_bounds_add add_mem_lowerBounds_add\n\n@[to_additive]\ntheorem subset_lowerBounds_mul (s t : Set M) :\n    lowerBounds s * lowerBounds t ⊆ lowerBounds (s * t) :=\n  subset_upperBounds_mul (M := Mᵒᵈ) _ _\n#align subset_lower_bounds_mul subset_lowerBounds_mul\n#align subset_lower_bounds_add subset_lowerBounds_add\n\n@[to_additive]\ntheorem BddAbove.mul {s t : Set M} (hs : BddAbove s) (ht : BddAbove t) : BddAbove (s * t) :=\n  (Nonempty.mul hs ht).mono (subset_upperBounds_mul s t)\n#align bdd_above.mul BddAbove.mul\n#align bdd_above.add BddAbove.add\n\n@[to_additive]\ntheorem BddBelow.mul {s t : Set M} (hs : BddBelow s) (ht : BddBelow t) : BddBelow (s * t) :=\n  (Nonempty.mul hs ht).mono (subset_lowerBounds_mul s t)\n#align bdd_below.mul BddBelow.mul\n#align bdd_below.add BddBelow.add\n\nend mul_add\n\nsection ConditionallyCompleteLattice\n\nsection Right\n\nvariable {ι G : Type*} [Group G] [ConditionallyCompleteLattice G]\n  [CovariantClass G G (Function.swap (· * ·)) (· ≤ ·)] [Nonempty ι] {f : ι → G}\n\n@[to_additive]\ntheorem ciSup_mul (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) * a = ⨆ i, f i * a :=\n  (OrderIso.mulRight a).map_ciSup hf\n#align csupr_mul ciSup_mul\n#align csupr_add ciSup_add\n\n@[to_additive]\ntheorem ciSup_div (hf : BddAbove (Set.range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by\n  simp only [div_eq_mul_inv, ciSup_mul hf]\n#align csupr_div ciSup_div\n#align csupr_sub ciSup_sub\n\n@[to_additive]\ntheorem ciInf_mul (hf : BddBelow (Set.range f)) (a : G) : (⨅ i, f i) * a = ⨅ i, f i * a :=\n  (OrderIso.mulRight a).map_ciInf hf\n\n@[to_additive]\ntheorem ciInf_div (hf : BddBelow (Set.range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by\n  ","nextTactic":"simp only [div_eq_mul_inv, ciInf_mul hf]","declUpToTactic":"@[to_additive]\ntheorem ciInf_div (hf : BddBelow (Set.range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Bounds.163_0.kwNNY3mkAxBsoBv","decl":"@[to_additive]\ntheorem ciInf_div (hf : BddBelow (Set.range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_AlgebraCat_Monoidal.jsonl

@@ -1,10 +0,0 @@
-{"state":"R : Type u\ninst✝ : CommRing R\nX Y Z : AlgebraCat R\n⊢ (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n    (α_ ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X) ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Y)\n        ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Z)).hom","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  ","nextTactic":"rfl","declUpToTactic":"theorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.59_0.wM7NNPRVO9JczpO","decl":"theorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom "}
-{"state":"R : Type u\ninst✝ : CommRing R\nX Y Z : AlgebraCat R\n⊢ (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n    (α_ ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X) ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Y)\n        ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Z)).inv","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  ","nextTactic":"rfl","declUpToTactic":"theorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.67_0.wM7NNPRVO9JczpO","decl":"theorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv "}
-{"state":"R : Type u\ninst✝ : CommRing R\nX Y Z : AlgebraCat R\n⊢ (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n    ((((fun X Y =>\n                Iso.refl ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X ⊗ (forget₂ (AlgebraCat R) (ModuleCat R)).obj Y))\n              (X ⊗ Y) Z).symm ≪≫\n          (((fun X Y =>\n                  Iso.refl\n                    ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X ⊗ (forget₂ (AlgebraCat R) (ModuleCat R)).obj Y))\n                X Y).symm ⊗\n            Iso.refl ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Z))) ≪≫\n        α_ ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X) ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Y)\n            ((forget₂ (AlgebraCat R) (ModuleCat R)).obj Z) ≪≫\n          (Iso.refl ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X) ⊗\n              (fun X Y =>\n                  Iso.refl\n                    ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X ⊗ (forget₂ (AlgebraCat R) (ModuleCat R)).obj Y))\n                Y Z) ≪≫\n            (fun X Y =>\n                Iso.refl ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X ⊗ (forget₂ (AlgebraCat R) (ModuleCat R)).obj Y))\n              X (Y ⊗ Z)).hom","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  rfl\n\nset_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        ","nextTactic":"dsimp only [forget₂_module_obj, forget₂_map_associator_hom]","declUpToTactic":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.75_0.wM7NNPRVO9JczpO","decl":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) "}
-{"state":"R : Type u\ninst✝ : CommRing R\nX Y Z : AlgebraCat R\n⊢ (α_ (ModuleCat.of R ↑X) (ModuleCat.of R ↑Y) (ModuleCat.of R ↑Z)).hom =\n    (((Iso.refl (ModuleCat.of R ↑(X ⊗ Y) ⊗ ModuleCat.of R ↑Z)).symm ≪≫\n          ((Iso.refl (ModuleCat.of R ↑X ⊗ ModuleCat.of R ↑Y)).symm ⊗ Iso.refl (ModuleCat.of R ↑Z))) ≪≫\n        α_ (ModuleCat.of R ↑X) (ModuleCat.of R ↑Y) (ModuleCat.of R ↑Z) ≪≫\n          (Iso.refl (ModuleCat.of R ↑X) ⊗ Iso.refl (ModuleCat.of R ↑Y ⊗ ModuleCat.of R ↑Z)) ≪≫\n            Iso.refl (ModuleCat.of R ↑X ⊗ ModuleCat.of R ↑(Y ⊗ Z))).hom","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  rfl\n\nset_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        ","nextTactic":"simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]","declUpToTactic":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.75_0.wM7NNPRVO9JczpO","decl":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) "}
-{"state":"R : Type u\ninst✝ : CommRing R\nX Y Z : AlgebraCat R\n⊢ (α_ (ModuleCat.of R ↑X) (ModuleCat.of R ↑Y) (ModuleCat.of R ↑Z)).hom =\n    (�� (ModuleCat.of R ↑(X ⊗ Y) ⊗ ModuleCat.of R ↑Z) ≫\n        (𝟙 (ModuleCat.of R ↑X ⊗ ModuleCat.of R ↑Y) ⊗ 𝟙 (ModuleCat.of R ↑Z))) ≫\n      (α_ (ModuleCat.of R ↑X) (ModuleCat.of R ↑Y) (ModuleCat.of R ↑Z)).hom ≫\n        𝟙 (ModuleCat.of R ↑X ⊗ ModuleCat.of R ↑Y ⊗ ModuleCat.of R ↑Z) ≫ 𝟙 (ModuleCat.of R ↑X ⊗ ModuleCat.of R ↑(Y ⊗ Z))","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  rfl\n\nset_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        ","nextTactic":"erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]","declUpToTactic":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.75_0.wM7NNPRVO9JczpO","decl":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) "}
-{"state":"R : Type u\ninst✝ : CommRing R\nX : AlgebraCat R\n⊢ (forget₂ (AlgebraCat R) (ModuleCat R)).map (ρ_ X).hom =\n    ((((fun X Y =>\n                Iso.refl ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X ⊗ (forget₂ (AlgebraCat R) (ModuleCat R)).obj Y))\n              X (𝟙_ (AlgebraCat R))).symm ≪≫\n          (Iso.refl ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X) ⊗ (Iso.refl (𝟙_ (ModuleCat R))).symm)) ≪≫\n        ρ_ ((forget₂ (AlgebraCat R) (ModuleCat R)).obj X)).hom","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  rfl\n\nset_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]\n      rightUnitor_eq := fun X => by\n        ","nextTactic":"dsimp","declUpToTactic":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]\n      rightUnitor_eq := fun X => by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.75_0.wM7NNPRVO9JczpO","decl":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) "}
-{"state":"R : Type u\ninst✝ : CommRing R\nX : AlgebraCat R\n⊢ ModuleCat.ofHom (AlgHom.toLinearMap (ρ_ X).hom) =\n    (𝟙 (ModuleCat.of R ↑X ⊗ ModuleCat.of R ↑(𝟙_ (AlgebraCat R))) ≫ (𝟙 (ModuleCat.of R ↑X) ⊗ 𝟙 (𝟙_ (ModuleCat R)))) ≫\n      (ρ_ (ModuleCat.of R ↑X)).hom","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  rfl\n\nset_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]\n      rightUnitor_eq := fun X => by\n        dsimp\n        ","nextTactic":"erw [Category.id_comp, MonoidalCategory.tensor_id, Category.id_comp]","declUpToTactic":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]\n      rightUnitor_eq := fun X => by\n        dsimp\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.75_0.wM7NNPRVO9JczpO","decl":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) "}
-{"state":"R : Type u\ninst✝ : CommRing R\nX : AlgebraCat R\n⊢ ModuleCat.ofHom (AlgHom.toLinearMap (ρ_ X).hom) = (ρ_ (ModuleCat.of R ↑X)).hom","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  rfl\n\nset_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]\n      rightUnitor_eq := fun X => by\n        dsimp\n        erw [Category.id_comp, MonoidalCategory.tensor_id, Category.id_comp]\n        ","nextTactic":"exact congr_arg LinearEquiv.toLinearMap <|\n          TensorProduct.AlgebraTensorModule.rid_eq_rid R X","declUpToTactic":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]\n      rightUnitor_eq := fun X => by\n        dsimp\n        erw [Category.id_comp, MonoidalCategory.tensor_id, Category.id_comp]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.75_0.wM7NNPRVO9JczpO","decl":"set_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) "}
-{"state":"R : Type u\ninst✝ : CommRing R\n⊢ MonoidalFunctor (AlgebraCat R) (ModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  rfl\n\nset_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]\n      rightUnitor_eq := fun X => by\n        dsimp\n        erw [Category.id_comp, MonoidalCategory.tensor_id, Category.id_comp]\n        exact congr_arg LinearEquiv.toLinearMap <|\n          TensorProduct.AlgebraTensorModule.rid_eq_rid R X }\n\nvariable (R) in\n/-- `forget₂ (AlgebraCat R) (ModuleCat R)` as a monoidal functor. -/\ndef toModuleCatMonoidalFunctor : MonoidalFunctor (AlgebraCat.{u} R) (ModuleCat.{u} R) := by\n  ","nextTactic":"unfold instMonoidalCategory","declUpToTactic":"variable (R) in\n/-- `forget₂ (AlgebraCat R) (ModuleCat R)` as a monoidal functor. -/\ndef toModuleCatMonoidalFunctor : MonoidalFunctor (AlgebraCat.{u} R) (ModuleCat.{u} R) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.92_0.wM7NNPRVO9JczpO","decl":"variable (R) in\n/-- `forget₂ (AlgebraCat R) (ModuleCat R)` as a monoidal functor. -/\ndef toModuleCatMonoidalFunctor : MonoidalFunctor (AlgebraCat.{u} R) (ModuleCat.{u} R) "}
-{"state":"R : Type u\ninst✝ : CommRing R\n⊢ MonoidalFunctor (AlgebraCat R) (ModuleCat R)","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.CategoryTheory.Monoidal.Braided\nimport Mathlib.CategoryTheory.Monoidal.Transport\nimport Mathlib.Algebra.Category.AlgebraCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic\nimport Mathlib.RingTheory.TensorProduct\n\n/-!\n# The monoidal category structure on R-algebras\n-/\n\nopen CategoryTheory\nopen scoped MonoidalCategory\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\nnoncomputable section\n\nnamespace instMonoidalCategory\n\nopen scoped TensorProduct\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\n@[simps!]\nnoncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R :=\n  of R (X ⊗[R] Y)\n\n/-- Auxiliary definition used to fight a timeout when building\n`AlgebraCat.instMonoidalCategory`. -/\nnoncomputable abbrev tensorHom {W X Y Z : AlgebraCat.{u} R} (f : W ⟶ X) (g : Y ⟶ Z) :\n    tensorObj W Y ⟶ tensorObj X Z :=\n  Algebra.TensorProduct.map f g\n\nopen MonoidalCategory\n\nend instMonoidalCategory\n\nopen instMonoidalCategory\n\ninstance : MonoidalCategoryStruct (AlgebraCat.{u} R) where\n  tensorObj := instMonoidalCategory.tensorObj\n  whiskerLeft X _ _ f := tensorHom (𝟙 X) f\n  whiskerRight {X₁ X₂} (f : X₁ ⟶ X₂) Y := tensorHom f (𝟙 Y)\n  tensorHom := tensorHom\n  tensorUnit := of R R\n  associator X Y Z := (Algebra.TensorProduct.assoc R X Y Z).toAlgebraIso\n  leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso\n  rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso\n\ntheorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by\n  rfl\n\ntheorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :\n    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n      (α_\n        (forget₂ _ (ModuleCat R) |>.obj X)\n        (forget₂ _ (ModuleCat R) |>.obj Y)\n        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by\n  rfl\n\nset_option maxHeartbeats 800000 in\nnoncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=\n  Monoidal.induced\n    (forget₂ (AlgebraCat R) (ModuleCat R))\n    { μIso := fun X Y => Iso.refl _\n      εIso := Iso.refl _\n      associator_eq := fun X Y Z => by\n        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]\n        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,\n          Iso.refl_hom, MonoidalCategory.tensor_id]\n        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]\n      rightUnitor_eq := fun X => by\n        dsimp\n        erw [Category.id_comp, MonoidalCategory.tensor_id, Category.id_comp]\n        exact congr_arg LinearEquiv.toLinearMap <|\n          TensorProduct.AlgebraTensorModule.rid_eq_rid R X }\n\nvariable (R) in\n/-- `forget₂ (AlgebraCat R) (ModuleCat R)` as a monoidal functor. -/\ndef toModuleCatMonoidalFunctor : MonoidalFunctor (AlgebraCat.{u} R) (ModuleCat.{u} R) := by\n  unfold instMonoidalCategory\n  ","nextTactic":"exact Monoidal.fromInduced (forget₂ (AlgebraCat R) (ModuleCat R)) _","declUpToTactic":"variable (R) in\n/-- `forget₂ (AlgebraCat R) (ModuleCat R)` as a monoidal functor. -/\ndef toModuleCatMonoidalFunctor : MonoidalFunctor (AlgebraCat.{u} R) (ModuleCat.{u} R) := by\n  unfold instMonoidalCategory\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Monoidal.92_0.wM7NNPRVO9JczpO","decl":"variable (R) in\n/-- `forget₂ (AlgebraCat R) (ModuleCat R)` as a monoidal functor. -/\ndef toModuleCatMonoidalFunctor : MonoidalFunctor (AlgebraCat.{u} R) (ModuleCat.{u} R) "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_AlgebraCat_Symmetric.jsonl

@@ -1 +0,0 @@
-{"state":"R : Type u\ninst✝ : CommRing R\n⊢ ∀ (X Y : AlgebraCat R),\n    LaxMonoidalFunctor.μ (toModuleCatMonoidalFunctor R).toLaxMonoidalFunctor X Y ≫\n        (toModuleCatMonoidalFunctor R).toLaxMonoidalFunctor.toFunctor.map\n          ((fun X Y => AlgEquiv.toAlgebraIso (Algebra.TensorProduct.comm R ↑X ↑Y)) X Y).hom =\n      (β_ ((toModuleCatMonoidalFunctor R).toLaxMonoidalFunctor.toFunctor.obj X)\n            ((toModuleCatMonoidalFunctor R).toLaxMonoidalFunctor.toFunctor.obj Y)).hom ≫\n        LaxMonoidalFunctor.μ (toModuleCatMonoidalFunctor R).toLaxMonoidalFunctor Y X","srcUpToTactic":"/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Category.AlgebraCat.Monoidal\nimport Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric\n\n/-!\n# The monoidal structure on `AlgebraCat` is symmetric.\n\nIn this file we show:\n\n* `AlgebraCat.instSymmetricCategory : SymmetricCategory (AlgebraCat.{u} R)`\n-/\nopen CategoryTheory\n\n\nnoncomputable section\n\nuniverse v u\n\nvariable {R : Type u} [CommRing R]\n\nnamespace AlgebraCat\n\ninstance : BraidedCategory (AlgebraCat.{u} R) :=\n  braidedCategoryOfFaithful (toModuleCatMonoidalFunctor R)\n    (fun X Y => (Algebra.TensorProduct.comm R X Y).toAlgebraIso)\n    (by ","nextTactic":"aesop_cat","declUpToTactic":"instance : BraidedCategory (AlgebraCat.{u} R) :=\n  braidedCategoryOfFaithful (toModuleCatMonoidalFunctor R)\n    (fun X Y => (Algebra.TensorProduct.comm R X Y).toAlgebraIso)\n    (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_AlgebraCat_Symmetric.27_0.PmLzvseiTXhH2JS","decl":"instance : BraidedCategory (AlgebraCat.{u} R) "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_BoolRing.jsonl

@@ -1,6 +0,0 @@
-{"state":"⊢ ConcreteCategory BoolRing","srcUpToTactic":"/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport Mathlib.Algebra.Category.Ring.Basic\nimport Mathlib.Algebra.Ring.BooleanRing\nimport Mathlib.Order.Category.BoolAlg\n\n#align_import algebra.category.BoolRing from \"leanprover-community/mathlib\"@\"67779f73e572fd1fec2218648b2078d167d16c0a\"\n\n/-!\n# The category of Boolean rings\n\nThis file defines `BoolRing`, the category of Boolean rings.\n\n## TODO\n\nFinish the equivalence with `BoolAlg`.\n-/\n\nset_option linter.uppercaseLean3 false\n\nuniverse u\n\nopen CategoryTheory Order\n\n/-- The category of Boolean rings. -/\ndef BoolRing :=\n  Bundled BooleanRing\n#align BoolRing BoolRing\n\nnamespace BoolRing\n\ninstance : CoeSort BoolRing (Type*) :=\n  Bundled.coeSort\n\ninstance (X : BoolRing) : BooleanRing X :=\n  X.str\n\n/-- Construct a bundled `BoolRing` from a `BooleanRing`. -/\ndef of (α : Type*) [BooleanRing α] : BoolRing :=\n  Bundled.of α\n#align BoolRing.of BoolRing.of\n\n@[simp]\ntheorem coe_of (α : Type*) [BooleanRing α] : ↥(of α) = α :=\n  rfl\n#align BoolRing.coe_of BoolRing.coe_of\n\ninstance : Inhabited BoolRing :=\n  ⟨of PUnit⟩\n\ninstance : BundledHom.ParentProjection @BooleanRing.toCommRing :=\n  ⟨⟩\n\n-- Porting note: `deriving` `ConcreteCategory` failed, added it manually\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\nderiving instance LargeCategory for BoolRing\n\ninstance : ConcreteCategory BoolRing := by\n  ","nextTactic":"dsimp [BoolRing]","declUpToTactic":"instance : ConcreteCategory BoolRing := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_BoolRing.61_0.P6FSJwyn6wAgFWS","decl":"instance : ConcreteCategory BoolRing "}
-{"state":"⊢ ConcreteCategory (Bundled BooleanRing)","srcUpToTactic":"/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport Mathlib.Algebra.Category.Ring.Basic\nimport Mathlib.Algebra.Ring.BooleanRing\nimport Mathlib.Order.Category.BoolAlg\n\n#align_import algebra.category.BoolRing from \"leanprover-community/mathlib\"@\"67779f73e572fd1fec2218648b2078d167d16c0a\"\n\n/-!\n# The category of Boolean rings\n\nThis file defines `BoolRing`, the category of Boolean rings.\n\n## TODO\n\nFinish the equivalence with `BoolAlg`.\n-/\n\nset_option linter.uppercaseLean3 false\n\nuniverse u\n\nopen CategoryTheory Order\n\n/-- The category of Boolean rings. -/\ndef BoolRing :=\n  Bundled BooleanRing\n#align BoolRing BoolRing\n\nnamespace BoolRing\n\ninstance : CoeSort BoolRing (Type*) :=\n  Bundled.coeSort\n\ninstance (X : BoolRing) : BooleanRing X :=\n  X.str\n\n/-- Construct a bundled `BoolRing` from a `BooleanRing`. -/\ndef of (α : Type*) [BooleanRing α] : BoolRing :=\n  Bundled.of α\n#align BoolRing.of BoolRing.of\n\n@[simp]\ntheorem coe_of (α : Type*) [BooleanRing α] : ↥(of α) = α :=\n  rfl\n#align BoolRing.coe_of BoolRing.coe_of\n\ninstance : Inhabited BoolRing :=\n  ⟨of PUnit⟩\n\ninstance : BundledHom.ParentProjection @BooleanRing.toCommRing :=\n  ⟨⟩\n\n-- Porting note: `deriving` `ConcreteCategory` failed, added it manually\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\nderiving instance LargeCategory for BoolRing\n\ninstance : ConcreteCategory BoolRing := by\n  dsimp [BoolRing]\n  ","nextTactic":"infer_instance","declUpToTactic":"instance : ConcreteCategory BoolRing := by\n  dsimp [BoolRing]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_BoolRing.61_0.P6FSJwyn6wAgFWS","decl":"instance : ConcreteCategory BoolRing "}
-{"state":"α β : BoolRing\ne : ↑α ≃+* ↑β\n⊢ ↑e ≫ ↑(RingEquiv.symm e) = 𝟙 α","srcUpToTactic":"/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport Mathlib.Algebra.Category.Ring.Basic\nimport Mathlib.Algebra.Ring.BooleanRing\nimport Mathlib.Order.Category.BoolAlg\n\n#align_import algebra.category.BoolRing from \"leanprover-community/mathlib\"@\"67779f73e572fd1fec2218648b2078d167d16c0a\"\n\n/-!\n# The category of Boolean rings\n\nThis file defines `BoolRing`, the category of Boolean rings.\n\n## TODO\n\nFinish the equivalence with `BoolAlg`.\n-/\n\nset_option linter.uppercaseLean3 false\n\nuniverse u\n\nopen CategoryTheory Order\n\n/-- The category of Boolean rings. -/\ndef BoolRing :=\n  Bundled BooleanRing\n#align BoolRing BoolRing\n\nnamespace BoolRing\n\ninstance : CoeSort BoolRing (Type*) :=\n  Bundled.coeSort\n\ninstance (X : BoolRing) : BooleanRing X :=\n  X.str\n\n/-- Construct a bundled `BoolRing` from a `BooleanRing`. -/\ndef of (α : Type*) [BooleanRing α] : BoolRing :=\n  Bundled.of α\n#align BoolRing.of BoolRing.of\n\n@[simp]\ntheorem coe_of (α : Type*) [BooleanRing α] : ↥(of α) = α :=\n  rfl\n#align BoolRing.coe_of BoolRing.coe_of\n\ninstance : Inhabited BoolRing :=\n  ⟨of PUnit⟩\n\ninstance : BundledHom.ParentProjection @BooleanRing.toCommRing :=\n  ⟨⟩\n\n-- Porting note: `deriving` `ConcreteCategory` failed, added it manually\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\nderiving instance LargeCategory for BoolRing\n\ninstance : ConcreteCategory BoolRing := by\n  dsimp [BoolRing]\n  infer_instance\n\n-- Porting note: disabled `simps`\n--   Invalid simp lemma BoolRing.hasForgetToCommRing_forget₂_obj_str_add.\n--   The given definition is not a constructor application:\n--     inferInstance.1\n-- @[simps]\ninstance hasForgetToCommRing : HasForget₂ BoolRing CommRingCat :=\n  BundledHom.forget₂ _ _\n#align BoolRing.has_forget_to_CommRing BoolRing.hasForgetToCommRing\n\n/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom := (e : RingHom _ _)\n  inv := (e.symm : RingHom _ _)\n  hom_inv_id := by ","nextTactic":"ext","declUpToTactic":"/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom := (e : RingHom _ _)\n  inv := (e.symm : RingHom _ _)\n  hom_inv_id := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_BoolRing.74_0.P6FSJwyn6wAgFWS","decl":"/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom "}
-{"state":"case w\nα β : BoolRing\ne : ↑α ≃+* ↑β\nx✝ : (forget BoolRing).obj α\n⊢ (↑e ≫ ↑(RingEquiv.symm e)) x✝ = (𝟙 α) x✝","srcUpToTactic":"/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport Mathlib.Algebra.Category.Ring.Basic\nimport Mathlib.Algebra.Ring.BooleanRing\nimport Mathlib.Order.Category.BoolAlg\n\n#align_import algebra.category.BoolRing from \"leanprover-community/mathlib\"@\"67779f73e572fd1fec2218648b2078d167d16c0a\"\n\n/-!\n# The category of Boolean rings\n\nThis file defines `BoolRing`, the category of Boolean rings.\n\n## TODO\n\nFinish the equivalence with `BoolAlg`.\n-/\n\nset_option linter.uppercaseLean3 false\n\nuniverse u\n\nopen CategoryTheory Order\n\n/-- The category of Boolean rings. -/\ndef BoolRing :=\n  Bundled BooleanRing\n#align BoolRing BoolRing\n\nnamespace BoolRing\n\ninstance : CoeSort BoolRing (Type*) :=\n  Bundled.coeSort\n\ninstance (X : BoolRing) : BooleanRing X :=\n  X.str\n\n/-- Construct a bundled `BoolRing` from a `BooleanRing`. -/\ndef of (α : Type*) [BooleanRing α] : BoolRing :=\n  Bundled.of α\n#align BoolRing.of BoolRing.of\n\n@[simp]\ntheorem coe_of (α : Type*) [BooleanRing α] : ↥(of α) = α :=\n  rfl\n#align BoolRing.coe_of BoolRing.coe_of\n\ninstance : Inhabited BoolRing :=\n  ⟨of PUnit⟩\n\ninstance : BundledHom.ParentProjection @BooleanRing.toCommRing :=\n  ⟨⟩\n\n-- Porting note: `deriving` `ConcreteCategory` failed, added it manually\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\nderiving instance LargeCategory for BoolRing\n\ninstance : ConcreteCategory BoolRing := by\n  dsimp [BoolRing]\n  infer_instance\n\n-- Porting note: disabled `simps`\n--   Invalid simp lemma BoolRing.hasForgetToCommRing_forget₂_obj_str_add.\n--   The given definition is not a constructor application:\n--     inferInstance.1\n-- @[simps]\ninstance hasForgetToCommRing : HasForget₂ BoolRing CommRingCat :=\n  BundledHom.forget₂ _ _\n#align BoolRing.has_forget_to_CommRing BoolRing.hasForgetToCommRing\n\n/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom := (e : RingHom _ _)\n  inv := (e.symm : RingHom _ _)\n  hom_inv_id := by ext; ","nextTactic":"exact e.symm_apply_apply _","declUpToTactic":"/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom := (e : RingHom _ _)\n  inv := (e.symm : RingHom _ _)\n  hom_inv_id := by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_BoolRing.74_0.P6FSJwyn6wAgFWS","decl":"/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom "}
-{"state":"α β : BoolRing\ne : ↑α ≃+* ↑β\n⊢ ↑(RingEquiv.symm e) ≫ ↑e = 𝟙 β","srcUpToTactic":"/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport Mathlib.Algebra.Category.Ring.Basic\nimport Mathlib.Algebra.Ring.BooleanRing\nimport Mathlib.Order.Category.BoolAlg\n\n#align_import algebra.category.BoolRing from \"leanprover-community/mathlib\"@\"67779f73e572fd1fec2218648b2078d167d16c0a\"\n\n/-!\n# The category of Boolean rings\n\nThis file defines `BoolRing`, the category of Boolean rings.\n\n## TODO\n\nFinish the equivalence with `BoolAlg`.\n-/\n\nset_option linter.uppercaseLean3 false\n\nuniverse u\n\nopen CategoryTheory Order\n\n/-- The category of Boolean rings. -/\ndef BoolRing :=\n  Bundled BooleanRing\n#align BoolRing BoolRing\n\nnamespace BoolRing\n\ninstance : CoeSort BoolRing (Type*) :=\n  Bundled.coeSort\n\ninstance (X : BoolRing) : BooleanRing X :=\n  X.str\n\n/-- Construct a bundled `BoolRing` from a `BooleanRing`. -/\ndef of (α : Type*) [BooleanRing α] : BoolRing :=\n  Bundled.of α\n#align BoolRing.of BoolRing.of\n\n@[simp]\ntheorem coe_of (α : Type*) [BooleanRing α] : ↥(of α) = α :=\n  rfl\n#align BoolRing.coe_of BoolRing.coe_of\n\ninstance : Inhabited BoolRing :=\n  ⟨of PUnit⟩\n\ninstance : BundledHom.ParentProjection @BooleanRing.toCommRing :=\n  ⟨⟩\n\n-- Porting note: `deriving` `ConcreteCategory` failed, added it manually\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\nderiving instance LargeCategory for BoolRing\n\ninstance : ConcreteCategory BoolRing := by\n  dsimp [BoolRing]\n  infer_instance\n\n-- Porting note: disabled `simps`\n--   Invalid simp lemma BoolRing.hasForgetToCommRing_forget₂_obj_str_add.\n--   The given definition is not a constructor application:\n--     inferInstance.1\n-- @[simps]\ninstance hasForgetToCommRing : HasForget₂ BoolRing CommRingCat :=\n  BundledHom.forget₂ _ _\n#align BoolRing.has_forget_to_CommRing BoolRing.hasForgetToCommRing\n\n/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom := (e : RingHom _ _)\n  inv := (e.symm : RingHom _ _)\n  hom_inv_id := by ext; exact e.symm_apply_apply _\n  inv_hom_id := by ","nextTactic":"ext","declUpToTactic":"/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom := (e : RingHom _ _)\n  inv := (e.symm : RingHom _ _)\n  hom_inv_id := by ext; exact e.symm_apply_apply _\n  inv_hom_id := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_BoolRing.74_0.P6FSJwyn6wAgFWS","decl":"/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom "}
-{"state":"case w\nα β : BoolRing\ne : ↑α ≃+* ↑β\nx✝ : (forget BoolRing).obj β\n⊢ (↑(RingEquiv.symm e) ≫ ↑e) x✝ = (𝟙 β) x✝","srcUpToTactic":"/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport Mathlib.Algebra.Category.Ring.Basic\nimport Mathlib.Algebra.Ring.BooleanRing\nimport Mathlib.Order.Category.BoolAlg\n\n#align_import algebra.category.BoolRing from \"leanprover-community/mathlib\"@\"67779f73e572fd1fec2218648b2078d167d16c0a\"\n\n/-!\n# The category of Boolean rings\n\nThis file defines `BoolRing`, the category of Boolean rings.\n\n## TODO\n\nFinish the equivalence with `BoolAlg`.\n-/\n\nset_option linter.uppercaseLean3 false\n\nuniverse u\n\nopen CategoryTheory Order\n\n/-- The category of Boolean rings. -/\ndef BoolRing :=\n  Bundled BooleanRing\n#align BoolRing BoolRing\n\nnamespace BoolRing\n\ninstance : CoeSort BoolRing (Type*) :=\n  Bundled.coeSort\n\ninstance (X : BoolRing) : BooleanRing X :=\n  X.str\n\n/-- Construct a bundled `BoolRing` from a `BooleanRing`. -/\ndef of (α : Type*) [BooleanRing α] : BoolRing :=\n  Bundled.of α\n#align BoolRing.of BoolRing.of\n\n@[simp]\ntheorem coe_of (α : Type*) [BooleanRing α] : ↥(of α) = α :=\n  rfl\n#align BoolRing.coe_of BoolRing.coe_of\n\ninstance : Inhabited BoolRing :=\n  ⟨of PUnit⟩\n\ninstance : BundledHom.ParentProjection @BooleanRing.toCommRing :=\n  ⟨⟩\n\n-- Porting note: `deriving` `ConcreteCategory` failed, added it manually\n-- see https://github.com/leanprover-community/mathlib4/issues/5020\nderiving instance LargeCategory for BoolRing\n\ninstance : ConcreteCategory BoolRing := by\n  dsimp [BoolRing]\n  infer_instance\n\n-- Porting note: disabled `simps`\n--   Invalid simp lemma BoolRing.hasForgetToCommRing_forget₂_obj_str_add.\n--   The given definition is not a constructor application:\n--     inferInstance.1\n-- @[simps]\ninstance hasForgetToCommRing : HasForget₂ BoolRing CommRingCat :=\n  BundledHom.forget₂ _ _\n#align BoolRing.has_forget_to_CommRing BoolRing.hasForgetToCommRing\n\n/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom := (e : RingHom _ _)\n  inv := (e.symm : RingHom _ _)\n  hom_inv_id := by ext; exact e.symm_apply_apply _\n  inv_hom_id := by ext; ","nextTactic":"exact e.apply_symm_apply _","declUpToTactic":"/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom := (e : RingHom _ _)\n  inv := (e.symm : RingHom _ _)\n  hom_inv_id := by ext; exact e.symm_apply_apply _\n  inv_hom_id := by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_BoolRing.74_0.P6FSJwyn6wAgFWS","decl":"/-- Constructs an isomorphism of Boolean rings from a ring isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β where\n  hom "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_FGModuleCat_Limits.jsonl

@@ -1,6 +0,0 @@
-{"state":"J✝ : Type\ninst✝⁴ : SmallCategory J✝\ninst✝³ : FinCategory J✝\nk : Type v\ninst✝² : Field k\nJ : Type\ninst✝¹ : Fintype J\nZ : J → ModuleCat k\ninst✝ : ∀ (j : J), FiniteDimensional k ↑(Z j)\n⊢ FiniteDimensional k ↑(ModuleCat.of k ((j : J) → ↑(Z j)))","srcUpToTactic":"/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Category.FGModuleCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Limits\nimport Mathlib.Algebra.Category.ModuleCat.Products\nimport Mathlib.Algebra.Category.ModuleCat.EpiMono\nimport Mathlib.CategoryTheory.Limits.Creates\nimport Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits\nimport Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\n#align_import algebra.category.fgModule.limits from \"leanprover-community/mathlib\"@\"19a70dceb9dff0994b92d2dd049de7d84d28112b\"\n\n/-!\n# `forget₂ (FGModuleCat K) (ModuleCat K)` creates all finite limits.\n\nAnd hence `FGModuleCat K` has all finite limits.\n\n## Future work\nAfter generalising `FGModuleCat` to allow the ring and the module to live in different universes,\ngeneralize this construction so we can take limits over smaller diagrams,\nas is done for the other algebraic categories.\n\nAnalogous constructions for Noetherian modules.\n-/\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nnamespace FGModuleCat\n\nvariable {J : Type} [SmallCategory J] [FinCategory J]\n\nvariable {k : Type v} [Field k]\n\ninstance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by ","nextTactic":"unfold ModuleCat.of","declUpToTactic":"instance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Limits.43_0.lMHchQdGSr2OU7X","decl":"instance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) "}
-{"state":"J✝ : Type\ninst✝⁴ : SmallCategory J✝\ninst✝³ : FinCategory J✝\nk : Type v\ninst✝² : Field k\nJ : Type\ninst✝¹ : Fintype J\nZ : J → ModuleCat k\ninst✝ : ∀ (j : J), FiniteDimensional k ↑(Z j)\n⊢ FiniteDimensional k ↑(ModuleCat.mk ((j : J) → ↑(Z j)))","srcUpToTactic":"/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Category.FGModuleCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Limits\nimport Mathlib.Algebra.Category.ModuleCat.Products\nimport Mathlib.Algebra.Category.ModuleCat.EpiMono\nimport Mathlib.CategoryTheory.Limits.Creates\nimport Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits\nimport Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\n#align_import algebra.category.fgModule.limits from \"leanprover-community/mathlib\"@\"19a70dceb9dff0994b92d2dd049de7d84d28112b\"\n\n/-!\n# `forget₂ (FGModuleCat K) (ModuleCat K)` creates all finite limits.\n\nAnd hence `FGModuleCat K` has all finite limits.\n\n## Future work\nAfter generalising `FGModuleCat` to allow the ring and the module to live in different universes,\ngeneralize this construction so we can take limits over smaller diagrams,\nas is done for the other algebraic categories.\n\nAnalogous constructions for Noetherian modules.\n-/\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nnamespace FGModuleCat\n\nvariable {J : Type} [SmallCategory J] [FinCategory J]\n\nvariable {k : Type v} [Field k]\n\ninstance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by unfold ModuleCat.of; ","nextTactic":"infer_instance","declUpToTactic":"instance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by unfold ModuleCat.of; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Limits.43_0.lMHchQdGSr2OU7X","decl":"instance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) "}
-{"state":"J✝ : Type\ninst✝⁴ : SmallCategory J✝\ninst✝³ : FinCategory J✝\nk : Type v\ninst✝² : Field k\nJ : Type\ninst✝¹ : Fintype J\nZ : J → ModuleCat k\ninst✝ : ∀ (j : J), FiniteDimensional k ↑(Z j)\nthis : FiniteDimensional k ↑(ModuleCat.of k ((j : J) → ↑(Z j)))\n⊢ Mono (ModuleCat.piIsoPi fun j => Z j).hom","srcUpToTactic":"/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Category.FGModuleCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Limits\nimport Mathlib.Algebra.Category.ModuleCat.Products\nimport Mathlib.Algebra.Category.ModuleCat.EpiMono\nimport Mathlib.CategoryTheory.Limits.Creates\nimport Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits\nimport Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\n#align_import algebra.category.fgModule.limits from \"leanprover-community/mathlib\"@\"19a70dceb9dff0994b92d2dd049de7d84d28112b\"\n\n/-!\n# `forget₂ (FGModuleCat K) (ModuleCat K)` creates all finite limits.\n\nAnd hence `FGModuleCat K` has all finite limits.\n\n## Future work\nAfter generalising `FGModuleCat` to allow the ring and the module to live in different universes,\ngeneralize this construction so we can take limits over smaller diagrams,\nas is done for the other algebraic categories.\n\nAnalogous constructions for Noetherian modules.\n-/\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nnamespace FGModuleCat\n\nvariable {J : Type} [SmallCategory J] [FinCategory J]\n\nvariable {k : Type v} [Field k]\n\ninstance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by unfold ModuleCat.of; infer_instance\n  FiniteDimensional.of_injective (ModuleCat.piIsoPi _).hom\n    ((ModuleCat.mono_iff_injective _).1 (by ","nextTactic":"infer_instance","declUpToTactic":"instance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by unfold ModuleCat.of; infer_instance\n  FiniteDimensional.of_injective (ModuleCat.piIsoPi _).hom\n    ((ModuleCat.mono_iff_injective _).1 (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Limits.43_0.lMHchQdGSr2OU7X","decl":"instance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) "}
-{"state":"J : Type\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nk : Type v\ninst✝ : Field k\nF : J ⥤ FGModuleCat k\n⊢ ∀ (j : J), FiniteDimensional k ↑((F ⋙ forget₂ (FGModuleCat k) (ModuleCat k)).obj j)","srcUpToTactic":"/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Category.FGModuleCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Limits\nimport Mathlib.Algebra.Category.ModuleCat.Products\nimport Mathlib.Algebra.Category.ModuleCat.EpiMono\nimport Mathlib.CategoryTheory.Limits.Creates\nimport Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits\nimport Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\n#align_import algebra.category.fgModule.limits from \"leanprover-community/mathlib\"@\"19a70dceb9dff0994b92d2dd049de7d84d28112b\"\n\n/-!\n# `forget₂ (FGModuleCat K) (ModuleCat K)` creates all finite limits.\n\nAnd hence `FGModuleCat K` has all finite limits.\n\n## Future work\nAfter generalising `FGModuleCat` to allow the ring and the module to live in different universes,\ngeneralize this construction so we can take limits over smaller diagrams,\nas is done for the other algebraic categories.\n\nAnalogous constructions for Noetherian modules.\n-/\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nnamespace FGModuleCat\n\nvariable {J : Type} [SmallCategory J] [FinCategory J]\n\nvariable {k : Type v} [Field k]\n\ninstance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by unfold ModuleCat.of; infer_instance\n  FiniteDimensional.of_injective (ModuleCat.piIsoPi _).hom\n    ((ModuleCat.mono_iff_injective _).1 (by infer_instance))\n\n/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) :=\n  haveI : ∀ j, FiniteDimensional k ((F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)).obj j) := by\n    ","nextTactic":"intro j","declUpToTactic":"/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) :=\n  haveI : ∀ j, FiniteDimensional k ((F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)).obj j) := by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Limits.49_0.lMHchQdGSr2OU7X","decl":"/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) "}
-{"state":"J : Type\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nk : Type v\ninst✝ : Field k\nF : J ⥤ FGModuleCat k\nj : J\n⊢ FiniteDimensional k ↑((F ⋙ forget₂ (FGModuleCat k) (ModuleCat k)).obj j)","srcUpToTactic":"/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Category.FGModuleCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Limits\nimport Mathlib.Algebra.Category.ModuleCat.Products\nimport Mathlib.Algebra.Category.ModuleCat.EpiMono\nimport Mathlib.CategoryTheory.Limits.Creates\nimport Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits\nimport Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\n#align_import algebra.category.fgModule.limits from \"leanprover-community/mathlib\"@\"19a70dceb9dff0994b92d2dd049de7d84d28112b\"\n\n/-!\n# `forget₂ (FGModuleCat K) (ModuleCat K)` creates all finite limits.\n\nAnd hence `FGModuleCat K` has all finite limits.\n\n## Future work\nAfter generalising `FGModuleCat` to allow the ring and the module to live in different universes,\ngeneralize this construction so we can take limits over smaller diagrams,\nas is done for the other algebraic categories.\n\nAnalogous constructions for Noetherian modules.\n-/\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nnamespace FGModuleCat\n\nvariable {J : Type} [SmallCategory J] [FinCategory J]\n\nvariable {k : Type v} [Field k]\n\ninstance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by unfold ModuleCat.of; infer_instance\n  FiniteDimensional.of_injective (ModuleCat.piIsoPi _).hom\n    ((ModuleCat.mono_iff_injective _).1 (by infer_instance))\n\n/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) :=\n  haveI : ∀ j, FiniteDimensional k ((F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)).obj j) := by\n    intro j; ","nextTactic":"change FiniteDimensional k (F.obj j)","declUpToTactic":"/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) :=\n  haveI : ∀ j, FiniteDimensional k ((F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)).obj j) := by\n    intro j; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Limits.49_0.lMHchQdGSr2OU7X","decl":"/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) "}
-{"state":"J : Type\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nk : Type v\ninst✝ : Field k\nF : J ⥤ FGModuleCat k\nj : J\n⊢ FiniteDimensional k ↑(F.obj j)","srcUpToTactic":"/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Category.FGModuleCat.Basic\nimport Mathlib.Algebra.Category.ModuleCat.Limits\nimport Mathlib.Algebra.Category.ModuleCat.Products\nimport Mathlib.Algebra.Category.ModuleCat.EpiMono\nimport Mathlib.CategoryTheory.Limits.Creates\nimport Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits\nimport Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\n#align_import algebra.category.fgModule.limits from \"leanprover-community/mathlib\"@\"19a70dceb9dff0994b92d2dd049de7d84d28112b\"\n\n/-!\n# `forget₂ (FGModuleCat K) (ModuleCat K)` creates all finite limits.\n\nAnd hence `FGModuleCat K` has all finite limits.\n\n## Future work\nAfter generalising `FGModuleCat` to allow the ring and the module to live in different universes,\ngeneralize this construction so we can take limits over smaller diagrams,\nas is done for the other algebraic categories.\n\nAnalogous constructions for Noetherian modules.\n-/\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nnamespace FGModuleCat\n\nvariable {J : Type} [SmallCategory J] [FinCategory J]\n\nvariable {k : Type v} [Field k]\n\ninstance {J : Type} [Fintype J] (Z : J → ModuleCat.{v} k) [∀ j, FiniteDimensional k (Z j)] :\n    FiniteDimensional k (∏ fun j => Z j : ModuleCat.{v} k) :=\n  haveI : FiniteDimensional k (ModuleCat.of k (∀ j, Z j)) := by unfold ModuleCat.of; infer_instance\n  FiniteDimensional.of_injective (ModuleCat.piIsoPi _).hom\n    ((ModuleCat.mono_iff_injective _).1 (by infer_instance))\n\n/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) :=\n  haveI : ∀ j, FiniteDimensional k ((F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)).obj j) := by\n    intro j; change FiniteDimensional k (F.obj j); ","nextTactic":"infer_instance","declUpToTactic":"/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) :=\n  haveI : ∀ j, FiniteDimensional k ((F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)).obj j) := by\n    intro j; change FiniteDimensional k (F.obj j); ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_FGModuleCat_Limits.49_0.lMHchQdGSr2OU7X","decl":"/-- Finite limits of finite dimensional vectors spaces are finite dimensional,\nbecause we can realise them as subobjects of a finite product. -/\ninstance (F : J ⥤ FGModuleCat k) :\n    FiniteDimensional k (limit (F ⋙ forget₂ (FGModuleCat k) (ModuleCat.{v} k)) : ModuleCat.{v} k) "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_GroupCat_Abelian.jsonl

@@ -1,22 +0,0 @@
-{"state":"X Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f g ↔ AddMonoidHom.range f = AddMonoidHom.ker g","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  ","nextTactic":"rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]","declUpToTactic":"theorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.51_0.jLTZO2zfNLMKRQk","decl":"theorem exact_iff : Exact f g ↔ f.range = g.ker "}
-{"state":"X Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ f ≫ g = 0 ∧ Fork.ι (kernelCone g) ≫ Cofork.π (cokernelCocone f) = 0 ↔ AddMonoidHom.range f = AddMonoidHom.ker g","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  ","nextTactic":"exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩","declUpToTactic":"theorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.51_0.jLTZO2zfNLMKRQk","decl":"theorem exact_iff : Exact f g ↔ f.range = g.ker "}
-{"state":"X Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\n⊢ PreservesFiniteLimits colim","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  ","nextTactic":"refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_1\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : Exact η γ\n⊢ AddMonoidHom.range (colim.map η) ≤ AddMonoidHom.ker (colim.map γ)\ncase refine_2\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : Exact η γ\n⊢ AddMonoidHom.ker (colim.map γ) ≤ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  ","nextTactic":"all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_1\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : Exact η γ\n⊢ AddMonoidHom.range (colim.map η) ≤ AddMonoidHom.ker (colim.map γ)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals ","nextTactic":"replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : Exact η γ\n⊢ AddMonoidHom.ker (colim.map γ) ≤ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals ","nextTactic":"replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_1\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\n⊢ AddMonoidHom.range (colim.map η) ≤ AddMonoidHom.ker (colim.map γ)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · ","nextTactic":"rw [AddMonoidHom.range_le_ker_iff, ← comp_def]","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_1\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\n⊢ colim.map η ≫ colim.map γ = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    ","nextTactic":"exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"X Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\n⊢ colimit.ι F j ≫ colim.map η ≫ colim.map γ = colimit.ι F j ≫ 0","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by ","nextTactic":"simp [reassoc_of% (h j).w]","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\n⊢ AddMonoidHom.ker (colim.map γ) ≤ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · ","nextTactic":"intro x (hx : _ = _)","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nx : ↑(colim.obj G)\nhx : (colim.map γ) x = 0\n⊢ x ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    ","nextTactic":"rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2.intro.intro\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colim.map γ) ((colimit.ι G j) y) = 0\n⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    ","nextTactic":"erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"X Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = 0\n⊢ ↑(H.obj j) →+ ↑(colimit H)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by ","nextTactic":"exact colimit.ι H j","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2.intro.intro\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\n⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    ","nextTactic":"rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\nk : J\ne₁ e₂ : j ⟶ k\nhk : (H.map e₁) ((γ.app j) y) = (H.map e₂) 0\n⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    ","nextTactic":"rw [map_zero] at hk","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\nk : J\ne₁ e₂ : j ⟶ k\nhk : (H.map e₁) ((γ.app j) y) = 0\n⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    ","nextTactic":"rw [← comp_apply] at hk","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\nk : J\ne₁ e₂ : j ⟶ k\nhk : (γ.app j ≫ H.map e₁) y = 0\n⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    ","nextTactic":"rw [← NatTrans.naturality] at hk","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\nk : J\ne₁ e₂ : j ⟶ k\nhk : (G.map e₁ ≫ γ.app k) y = 0\n⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    ","nextTactic":"rw [comp_apply] at hk","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\nk : J\ne₁ e₂ : j ⟶ k\nhk : (γ.app k) ((G.map e₁) y) = 0\n⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out��\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    rw [comp_apply] at hk\n    ","nextTactic":"rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    rw [comp_apply] at hk\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case refine_2.intro.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\nk : J\ne₁ e₂ : j ⟶ k\nhk : (γ.app k) ((G.map e₁) y) = 0\nt : ↑(F.obj k)\nht : (η.app k) t = (G.map e₁) y\n⊢ (colimit.ι G j) y ∈ AddMonoidHom.range (colim.map η)","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    rw [comp_apply] at hk\n    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩\n    ","nextTactic":"use colimit.ι F k t","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    rw [comp_apply] at hk\n    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case h\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\nk : J\ne₁ e₂ : j ⟶ k\nhk : (γ.app k) ((G.map e₁) y) = 0\nt : ↑(F.obj k)\nht : (η.app k) t = (G.map e₁) y\n⊢ (colim.map η) ((colimit.ι F k) t) = (colimit.ι G j) y","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    rw [comp_apply] at hk\n    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩\n    use colimit.ι F k t\n    ","nextTactic":"erw [← comp_apply, colimit.ι_map, comp_apply, ht]","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    rw [comp_apply] at hk\n    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩\n    use colimit.ι F k t\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}
-{"state":"case h\nX Y Z : AddCommGroupCat\nf : X ⟶ Y\ng : Y ⟶ Z\nJ : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF G H : J ⥤ AddCommGroupCat\nη : F ⟶ G\nγ : G ⟶ H\nh : ∀ (j : J), Exact (η.app j) (γ.app j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : (colimit.ι H j) ((γ.app j) y) = (colimit.ι H j) 0\nk : J\ne₁ e₂ : j ⟶ k\nhk : (γ.app k) ((G.map e₁) y) = 0\nt : ↑(F.obj k)\nht : (η.app k) t = (G.map e₁) y\n⊢ (colimit.ι G k) ((G.map e₁) y) = (colimit.ι G j) y","srcUpToTactic":"/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.Algebra.Category.GroupCat.Colimits\nimport Mathlib.Algebra.Category.GroupCat.FilteredColimits\nimport Mathlib.Algebra.Category.GroupCat.Kernels\nimport Mathlib.Algebra.Category.GroupCat.Limits\nimport Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence\nimport Mathlib.Algebra.Category.ModuleCat.Abelian\nimport Mathlib.CategoryTheory.Abelian.FunctorCategory\nimport Mathlib.CategoryTheory.Limits.ConcreteCategory\n\n#align_import algebra.category.Group.abelian from \"leanprover-community/mathlib\"@\"f7baecbb54bd0f24f228576f97b1752fc3c9b318\"\n\n/-!\n# The category of abelian groups is abelian\n-/\n\nopen CategoryTheory Limits\n\nuniverse u\n\nnoncomputable section\n\nnamespace AddCommGroupCat\n\nvariable {X Y Z : AddCommGroupCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n/-- In the category of abelian groups, every monomorphism is normal. -/\ndef normalMono (_ : Mono f) : NormalMono f :=\n  equivalenceReflectsNormalMono (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalMono _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_mono AddCommGroupCat.normalMono\n\n/-- In the category of abelian groups, every epimorphism is normal. -/\ndef normalEpi (_ : Epi f) : NormalEpi f :=\n  equivalenceReflectsNormalEpi (forget₂ (ModuleCat.{u} ℤ) AddCommGroupCat.{u}).inv <|\n    ModuleCat.normalEpi _ inferInstance\nset_option linter.uppercaseLean3 false in\n#align AddCommGroup.normal_epi AddCommGroupCat.normalEpi\n\n/-- The category of abelian groups is abelian. -/\ninstance : Abelian AddCommGroupCat.{u} where\n  has_finite_products := ⟨HasFiniteProducts.out⟩\n  normalMonoOfMono := normalMono\n  normalEpiOfEpi := normalEpi\n\ntheorem exact_iff : Exact f g ↔ f.range = g.ker := by\n  rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n  exact\n    ⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm\n        ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n      fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,\n          (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le⟩⟩\n\n/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    rw [comp_apply] at hk\n    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩\n    use colimit.ι F k t\n    erw [← comp_apply, colimit.ι_map, comp_apply, ht]\n    ","nextTactic":"exact colimit.w_apply G e₁ y","declUpToTactic":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J := J) (C := AddCommGroupCat.{u}) := by\n  refine Functor.preservesFiniteLimitsOfMapExact _\n    fun F G H η γ h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n  all_goals replace h : ∀ j : J, Exact (η.app j) (γ.app j) :=\n    fun j => Functor.map_exact ((evaluation _ _).obj j) η γ h\n  · rw [AddMonoidHom.range_le_ker_iff, ← comp_def]\n    exact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n  · intro x (hx : _ = _)\n    rcases Concrete.colimit_exists_rep G x with ⟨j, y, rfl⟩\n    erw [← comp_apply, colimit.ι_map, comp_apply,\n      ← map_zero (by exact colimit.ι H j : H.obj j →+ ↑(colimit H))] at hx\n    rcases Concrete.colimit_exists_of_rep_eq H _ _ hx with ⟨k, e₁, e₂, hk : _ = H.map e₂ 0⟩\n    rw [map_zero] at hk\n    rw [← comp_apply] at hk\n    rw [← NatTrans.naturality] at hk\n    rw [comp_apply] at hk\n    rcases ((exact_iff _ _).mp <| h k).ge hk with ⟨t, ht⟩\n    use colimit.ι F k t\n    erw [← comp_apply, colimit.ι_map, comp_apply, ht]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Abelian.59_0.jLTZO2zfNLMKRQk","decl":"/-- The category of abelian groups satisfies Grothedieck's Axiom AB5. -/\ninstance {J : Type u} [SmallCategory J] [IsFiltered J] :\n    PreservesFiniteLimits <| colim (J "}

Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Category_GroupCat_Biproducts.jsonl

@@ -1,15 +0,0 @@
-{"state":"G H : AddCommGroupCat\n⊢ ∀ ⦃X Y : Discrete WalkingPair⦄ (f : X ⟶ Y),\n    ((Functor.const (Discrete WalkingPair)).obj (of (↑G × ↑H))).map f ≫\n        (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H))\n          Y =\n      (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H)) X ≫\n        (pair G H).map f","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by ","nextTactic":"rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩","declUpToTactic":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.38_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone "}
-{"state":"case mk.left.mk.left.up.up.refl\nG H : AddCommGroupCat\n⊢ ((Functor.const (Discrete WalkingPair)).obj (of (↑G × ↑H))).map\n        { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } } ≫\n      (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H))\n        { as := WalkingPair.left } =\n    (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H))\n        { as := WalkingPair.left } ≫\n      (pair G H).map { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } }","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> ","nextTactic":"rfl","declUpToTactic":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.38_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone "}
-{"state":"case mk.right.mk.right.up.up.refl\nG H : AddCommGroupCat\n⊢ ((Functor.const (Discrete WalkingPair)).obj (of (↑G × ↑H))).map\n        { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } } ≫\n      (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H))\n        { as := WalkingPair.right } =\n    (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H))\n        { as := WalkingPair.right } ≫\n      (pair G H).map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } }","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> ","nextTactic":"rfl","declUpToTactic":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.38_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone "}
-{"state":"G H : AddCommGroupCat\n⊢ ∀ (s : Cone (pair G H)) (j : Discrete WalkingPair),\n    (fun s => AddMonoidHom.prod (s.π.app { as := WalkingPair.left }) (s.π.app { as := WalkingPair.right })) s ≫\n        { pt := of (↑G × ↑H),\n                π :=\n                  NatTrans.mk fun j =>\n                    Discrete.casesOn j fun j =>\n                      WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H) }.π.app\n          j =\n      s.π.app j","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by ","nextTactic":"rintro s (⟨⟩ | ⟨⟩)","declUpToTactic":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.38_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone "}
-{"state":"case mk.left\nG H : AddCommGroupCat\ns : Cone (pair G H)\n⊢ (fun s => AddMonoidHom.prod (s.π.app { as := WalkingPair.left }) (s.π.app { as := WalkingPair.right })) s ≫\n      { pt := of (↑G × ↑H),\n              π :=\n                NatTrans.mk fun j =>\n                  Discrete.casesOn j fun j =>\n                    WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H) }.π.app\n        { as := WalkingPair.left } =\n    s.π.app { as := WalkingPair.left }","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.��.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> ","nextTactic":"rfl","declUpToTactic":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.38_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone "}
-{"state":"case mk.right\nG H : AddCommGroupCat\ns : Cone (pair G H)\n⊢ (fun s => AddMonoidHom.prod (s.π.app { as := WalkingPair.left }) (s.π.app { as := WalkingPair.right })) s ≫\n      { pt := of (↑G × ↑H),\n              π :=\n                NatTrans.mk fun j =>\n                  Discrete.casesOn j fun j =>\n                    WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H) }.π.app\n        { as := WalkingPair.right } =\n    s.π.app { as := WalkingPair.right }","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> ","nextTactic":"rfl","declUpToTactic":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.38_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone "}
-{"state":"G H : AddCommGroupCat\ns : Cone (pair G H)\nm :\n  s.pt ⟶\n    { pt := of (↑G × ↑H),\n        π :=\n          NatTrans.mk fun j =>\n            Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H) }.pt\nw :\n  ∀ (j : Discrete WalkingPair),\n    m ≫\n        { pt := of (↑G × ↑H),\n                π :=\n                  NatTrans.mk fun j =>\n                    Discrete.casesOn j fun j =>\n                      WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H) }.π.app\n          j =\n      s.π.app j\n⊢ m = (fun s => AddMonoidHom.prod (s.π.app { as := WalkingPair.left }) (s.π.app { as := WalkingPair.right })) s","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        ","nextTactic":"simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]","declUpToTactic":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.38_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone "}
-{"state":"G H : AddCommGroupCat\ns : Cone (pair G H)\nm :\n  s.pt ⟶\n    { pt := of (↑G × ↑H),\n        π :=\n          NatTrans.mk fun j =>\n            Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H) }.pt\nw :\n  ∀ (j : Discrete WalkingPair),\n    m ≫\n        { pt := of (↑G × ↑H),\n                π :=\n                  NatTrans.mk fun j =>\n                    Discrete.casesOn j fun j =>\n                      WalkingPair.casesOn j (AddMonoidHom.fst ↑G ↑H) (AddMonoidHom.snd ↑G ↑H) }.π.app\n          j =\n      s.π.app j\n⊢ m = AddMonoidHom.prod (m ≫ AddMonoidHom.fst ↑G ↑H) (m ≫ AddMonoidHom.snd ↑G ↑H)","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        ","nextTactic":"rfl","declUpToTactic":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.38_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone "}
-{"state":"J : Type w\nf : J → AddCommGroupCat\ns : Fan f\n⊢ (fun x j => (s.π.app { as := j }) x) 0 = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        rfl }\n#align AddCommGroup.binary_product_limit_cone AddCommGroupCat.binaryProductLimitCone\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_left (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.left⟩ = AddMonoidHom.fst G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_left AddCommGroupCat.binaryProductLimitCone_cone_π_app_left\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_right (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.right⟩ = AddMonoidHom.snd G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_right AddCommGroupCat.binaryProductLimitCone_cone_π_app_right\n\n/-- We verify that the biproduct in `AddCommGroupCat` is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps! hom_apply]\nnoncomputable def biprodIsoProd (G H : AddCommGroupCat.{u}) :\n    (G ⊞ H : AddCommGroupCat) ≅ AddCommGroupCat.of (G × H) :=\n  IsLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit G H) (binaryProductLimitCone G H).isLimit\n#align AddCommGroup.biprod_iso_prod AddCommGroupCat.biprodIsoProd\n\n-- These lemmas have always been bad (#7657), but lean4#2644 made `simp` start noticing\nattribute [nolint simpNF] AddCommGroupCat.biprodIsoProd_hom_apply\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_fst (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.fst = AddMonoidHom.fst G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.left)\n#align AddCommGroup.biprod_iso_prod_inv_comp_fst AddCommGroupCat.biprodIsoProd_inv_comp_fst\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_snd (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.snd = AddMonoidHom.snd G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.right)\n#align AddCommGroup.biprod_iso_prod_inv_comp_snd AddCommGroupCat.biprodIsoProd_inv_comp_snd\n\nnamespace HasLimit\n\nvariable {J : Type w} (f : J → AddCommGroupCat.{max w u})\n\n/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    ","nextTactic":"simp only [Functor.const_obj_obj, map_zero]","declUpToTactic":"/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.98_0.jT6AjHNJZqfPwUM","decl":"/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j "}
-{"state":"J : Type w\nf : J → AddCommGroupCat\ns : Fan f\n⊢ (fun j => 0) = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        rfl }\n#align AddCommGroup.binary_product_limit_cone AddCommGroupCat.binaryProductLimitCone\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_left (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.left⟩ = AddMonoidHom.fst G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_left AddCommGroupCat.binaryProductLimitCone_cone_π_app_left\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_right (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.right⟩ = AddMonoidHom.snd G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_right AddCommGroupCat.binaryProductLimitCone_cone_π_app_right\n\n/-- We verify that the biproduct in `AddCommGroupCat` is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps! hom_apply]\nnoncomputable def biprodIsoProd (G H : AddCommGroupCat.{u}) :\n    (G ⊞ H : AddCommGroupCat) ≅ AddCommGroupCat.of (G × H) :=\n  IsLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit G H) (binaryProductLimitCone G H).isLimit\n#align AddCommGroup.biprod_iso_prod AddCommGroupCat.biprodIsoProd\n\n-- These lemmas have always been bad (#7657), but lean4#2644 made `simp` start noticing\nattribute [nolint simpNF] AddCommGroupCat.biprodIsoProd_hom_apply\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_fst (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.fst = AddMonoidHom.fst G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.left)\n#align AddCommGroup.biprod_iso_prod_inv_comp_fst AddCommGroupCat.biprodIsoProd_inv_comp_fst\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_snd (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.snd = AddMonoidHom.snd G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.right)\n#align AddCommGroup.biprod_iso_prod_inv_comp_snd AddCommGroupCat.biprodIsoProd_inv_comp_snd\n\nnamespace HasLimit\n\nvariable {J : Type w} (f : J → AddCommGroupCat.{max w u})\n\n/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    ","nextTactic":"rfl","declUpToTactic":"/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.98_0.jT6AjHNJZqfPwUM","decl":"/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j "}
-{"state":"J : Type w\nf : J → AddCommGroupCat\ns : Fan f\nx y : ↑s.pt\n⊢ ZeroHom.toFun\n      { toFun := fun x j => (s.π.app { as := j }) x, map_zero' := (_ : (fun x j => (s.π.app { as := j }) x) 0 = 0) }\n      (x + y) =\n    ZeroHom.toFun\n        { toFun := fun x j => (s.π.app { as := j }) x, map_zero' := (_ : (fun x j => (s.π.app { as := j }) x) 0 = 0) }\n        x +\n      ZeroHom.toFun\n        { toFun := fun x j => (s.π.app { as := j }) x, map_zero' := (_ : (fun x j => (s.π.app { as := j }) x) 0 = 0) } y","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        rfl }\n#align AddCommGroup.binary_product_limit_cone AddCommGroupCat.binaryProductLimitCone\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_left (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.left⟩ = AddMonoidHom.fst G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_left AddCommGroupCat.binaryProductLimitCone_cone_π_app_left\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_right (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.right⟩ = AddMonoidHom.snd G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_right AddCommGroupCat.binaryProductLimitCone_cone_π_app_right\n\n/-- We verify that the biproduct in `AddCommGroupCat` is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps! hom_apply]\nnoncomputable def biprodIsoProd (G H : AddCommGroupCat.{u}) :\n    (G ⊞ H : AddCommGroupCat) ≅ AddCommGroupCat.of (G × H) :=\n  IsLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit G H) (binaryProductLimitCone G H).isLimit\n#align AddCommGroup.biprod_iso_prod AddCommGroupCat.biprodIsoProd\n\n-- These lemmas have always been bad (#7657), but lean4#2644 made `simp` start noticing\nattribute [nolint simpNF] AddCommGroupCat.biprodIsoProd_hom_apply\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_fst (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.fst = AddMonoidHom.fst G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.left)\n#align AddCommGroup.biprod_iso_prod_inv_comp_fst AddCommGroupCat.biprodIsoProd_inv_comp_fst\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_snd (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.snd = AddMonoidHom.snd G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.right)\n#align AddCommGroup.biprod_iso_prod_inv_comp_snd AddCommGroupCat.biprodIsoProd_inv_comp_snd\n\nnamespace HasLimit\n\nvariable {J : Type w} (f : J → AddCommGroupCat.{max w u})\n\n/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    rfl\n  map_add' x y := by\n    ","nextTactic":"simp only [Functor.const_obj_obj, map_add]","declUpToTactic":"/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    rfl\n  map_add' x y := by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.98_0.jT6AjHNJZqfPwUM","decl":"/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j "}
-{"state":"J : Type w\nf : J → AddCommGroupCat\ns : Fan f\nx y : ↑s.pt\n⊢ (fun j => (s.π.app { as := j }) x + (s.π.app { as := j }) y) =\n    (fun j => (s.π.app { as := j }) x) + fun j => (s.π.app { as := j }) y","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        rfl }\n#align AddCommGroup.binary_product_limit_cone AddCommGroupCat.binaryProductLimitCone\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_left (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.left⟩ = AddMonoidHom.fst G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_left AddCommGroupCat.binaryProductLimitCone_cone_π_app_left\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_right (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.right⟩ = AddMonoidHom.snd G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_right AddCommGroupCat.binaryProductLimitCone_cone_π_app_right\n\n/-- We verify that the biproduct in `AddCommGroupCat` is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps! hom_apply]\nnoncomputable def biprodIsoProd (G H : AddCommGroupCat.{u}) :\n    (G ⊞ H : AddCommGroupCat) ≅ AddCommGroupCat.of (G × H) :=\n  IsLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit G H) (binaryProductLimitCone G H).isLimit\n#align AddCommGroup.biprod_iso_prod AddCommGroupCat.biprodIsoProd\n\n-- These lemmas have always been bad (#7657), but lean4#2644 made `simp` start noticing\nattribute [nolint simpNF] AddCommGroupCat.biprodIsoProd_hom_apply\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_fst (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.fst = AddMonoidHom.fst G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.left)\n#align AddCommGroup.biprod_iso_prod_inv_comp_fst AddCommGroupCat.biprodIsoProd_inv_comp_fst\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_snd (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.snd = AddMonoidHom.snd G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.right)\n#align AddCommGroup.biprod_iso_prod_inv_comp_snd AddCommGroupCat.biprodIsoProd_inv_comp_snd\n\nnamespace HasLimit\n\nvariable {J : Type w} (f : J → AddCommGroupCat.{max w u})\n\n/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    rfl\n  map_add' x y := by\n    simp only [Functor.const_obj_obj, map_add]\n    ","nextTactic":"rfl","declUpToTactic":"/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    rfl\n  map_add' x y := by\n    simp only [Functor.const_obj_obj, map_add]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.98_0.jT6AjHNJZqfPwUM","decl":"/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j "}
-{"state":"J : Type w\nf : J → AddCommGroupCat\ns : Cone (Discrete.functor f)\nm :\n  s.pt ⟶ { pt := of ((j : J) → ↑(f j)), π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => ↑(f j)) j.as }.pt\nw :\n  ∀ (j : Discrete J),\n    m ≫\n        { pt := of ((j : J) → ↑(f j)),\n                π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => ↑(f j)) j.as }.π.app\n          j =\n      s.π.app j\n⊢ m = lift f s","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        rfl }\n#align AddCommGroup.binary_product_limit_cone AddCommGroupCat.binaryProductLimitCone\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_left (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.left⟩ = AddMonoidHom.fst G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_left AddCommGroupCat.binaryProductLimitCone_cone_π_app_left\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_right (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.right⟩ = AddMonoidHom.snd G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_right AddCommGroupCat.binaryProductLimitCone_cone_π_app_right\n\n/-- We verify that the biproduct in `AddCommGroupCat` is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps! hom_apply]\nnoncomputable def biprodIsoProd (G H : AddCommGroupCat.{u}) :\n    (G ⊞ H : AddCommGroupCat) ≅ AddCommGroupCat.of (G × H) :=\n  IsLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit G H) (binaryProductLimitCone G H).isLimit\n#align AddCommGroup.biprod_iso_prod AddCommGroupCat.biprodIsoProd\n\n-- These lemmas have always been bad (#7657), but lean4#2644 made `simp` start noticing\nattribute [nolint simpNF] AddCommGroupCat.biprodIsoProd_hom_apply\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_fst (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.fst = AddMonoidHom.fst G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.left)\n#align AddCommGroup.biprod_iso_prod_inv_comp_fst AddCommGroupCat.biprodIsoProd_inv_comp_fst\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_snd (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.snd = AddMonoidHom.snd G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.right)\n#align AddCommGroup.biprod_iso_prod_inv_comp_snd AddCommGroupCat.biprodIsoProd_inv_comp_snd\n\nnamespace HasLimit\n\nvariable {J : Type w} (f : J → AddCommGroupCat.{max w u})\n\n/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    rfl\n  map_add' x y := by\n    simp only [Functor.const_obj_obj, map_add]\n    rfl\n#align AddCommGroup.has_limit.lift AddCommGroupCat.HasLimit.lift\n\n/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone :=\n    { pt := AddCommGroupCat.of (∀ j, f j)\n      π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => f j) j.as }\n  isLimit :=\n    { lift := lift.{_, u} f\n      fac := fun s j => rfl\n      uniq := fun s m w => by\n        ","nextTactic":"ext x","declUpToTactic":"/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone :=\n    { pt := AddCommGroupCat.of (∀ j, f j)\n      π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => f j) j.as }\n  isLimit :=\n    { lift := lift.{_, u} f\n      fac := fun s j => rfl\n      uniq := fun s m w => by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.112_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone "}
-{"state":"case w\nJ : Type w\nf : J → AddCommGroupCat\ns : Cone (Discrete.functor f)\nm :\n  s.pt ⟶ { pt := of ((j : J) → ↑(f j)), π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => ↑(f j)) j.as }.pt\nw :\n  ∀ (j : Discrete J),\n    m ≫\n        { pt := of ((j : J) → ↑(f j)),\n                π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => ↑(f j)) j.as }.π.app\n          j =\n      s.π.app j\nx : ↑s.pt\n⊢ m x = (lift f s) x","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        rfl }\n#align AddCommGroup.binary_product_limit_cone AddCommGroupCat.binaryProductLimitCone\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_left (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.left⟩ = AddMonoidHom.fst G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_left AddCommGroupCat.binaryProductLimitCone_cone_π_app_left\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_right (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.right⟩ = AddMonoidHom.snd G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_right AddCommGroupCat.binaryProductLimitCone_cone_π_app_right\n\n/-- We verify that the biproduct in `AddCommGroupCat` is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps! hom_apply]\nnoncomputable def biprodIsoProd (G H : AddCommGroupCat.{u}) :\n    (G ⊞ H : AddCommGroupCat) ≅ AddCommGroupCat.of (G × H) :=\n  IsLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit G H) (binaryProductLimitCone G H).isLimit\n#align AddCommGroup.biprod_iso_prod AddCommGroupCat.biprodIsoProd\n\n-- These lemmas have always been bad (#7657), but lean4#2644 made `simp` start noticing\nattribute [nolint simpNF] AddCommGroupCat.biprodIsoProd_hom_apply\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_fst (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.fst = AddMonoidHom.fst G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.left)\n#align AddCommGroup.biprod_iso_prod_inv_comp_fst AddCommGroupCat.biprodIsoProd_inv_comp_fst\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_snd (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.snd = AddMonoidHom.snd G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.right)\n#align AddCommGroup.biprod_iso_prod_inv_comp_snd AddCommGroupCat.biprodIsoProd_inv_comp_snd\n\nnamespace HasLimit\n\nvariable {J : Type w} (f : J → AddCommGroupCat.{max w u})\n\n/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    rfl\n  map_add' x y := by\n    simp only [Functor.const_obj_obj, map_add]\n    rfl\n#align AddCommGroup.has_limit.lift AddCommGroupCat.HasLimit.lift\n\n/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone :=\n    { pt := AddCommGroupCat.of (∀ j, f j)\n      π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => f j) j.as }\n  isLimit :=\n    { lift := lift.{_, u} f\n      fac := fun s j => rfl\n      uniq := fun s m w => by\n        ext x\n        ","nextTactic":"funext j","declUpToTactic":"/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone :=\n    { pt := AddCommGroupCat.of (∀ j, f j)\n      π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => f j) j.as }\n  isLimit :=\n    { lift := lift.{_, u} f\n      fac := fun s j => rfl\n      uniq := fun s m w => by\n        ext x\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.112_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone "}
-{"state":"case w.h\nJ : Type w\nf : J → AddCommGroupCat\ns : Cone (Discrete.functor f)\nm :\n  s.pt ⟶ { pt := of ((j : J) → ↑(f j)), π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => ↑(f j)) j.as }.pt\nw :\n  ∀ (j : Discrete J),\n    m ≫\n        { pt := of ((j : J) → ↑(f j)),\n                π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => ↑(f j)) j.as }.π.app\n          j =\n      s.π.app j\nx : ↑s.pt\nj : J\n⊢ m x j = (lift f s) x j","srcUpToTactic":"/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.Algebra.Group.Pi\nimport Mathlib.Algebra.Category.GroupCat.Preadditive\nimport Mathlib.CategoryTheory.Preadditive.Biproducts\nimport Mathlib.Algebra.Category.GroupCat.Limits\n\n#align_import algebra.category.Group.biproducts from \"leanprover-community/mathlib\"@\"234ddfeaa5572bc13716dd215c6444410a679a8e\"\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nopen BigOperators\n\nuniverse w u\n\nnamespace AddCommGroupCat\nset_option linter.uppercaseLean3 false -- `AddCommGroup`\n\n-- As `AddCommGroupCat` is preadditive, and has all limits, it automatically has biproducts.\ninstance : HasBinaryBiproducts AddCommGroupCat :=\n  HasBinaryBiproducts.of_hasBinaryProducts\n\ninstance : HasFiniteBiproducts AddCommGroupCat :=\n  HasFiniteBiproducts.of_hasFiniteProducts\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n/-- Construct limit data for a binary product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (G × H)`.\n-/\n@[simps cone_pt isLimit_lift]\ndef binaryProductLimitCone (G H : AddCommGroupCat.{u}) : Limits.LimitCone (pair G H) where\n  cone :=\n    { pt := AddCommGroupCat.of (G × H)\n      π :=\n        { app := fun j =>\n            Discrete.casesOn j fun j =>\n              WalkingPair.casesOn j (AddMonoidHom.fst G H) (AddMonoidHom.snd G H)\n          naturality := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟨⟩⟩⟩ <;> rfl } }\n  isLimit :=\n    { lift := fun s => AddMonoidHom.prod (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩)\n      fac := by rintro s (⟨⟩ | ⟨⟩) <;> rfl\n      uniq := fun s m w => by\n        simp_rw [← w ⟨WalkingPair.left⟩, ← w ⟨WalkingPair.right⟩]\n        rfl }\n#align AddCommGroup.binary_product_limit_cone AddCommGroupCat.binaryProductLimitCone\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_left (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.left⟩ = AddMonoidHom.fst G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_left AddCommGroupCat.binaryProductLimitCone_cone_π_app_left\n\n@[simp]\ntheorem binaryProductLimitCone_cone_π_app_right (G H : AddCommGroupCat.{u}) :\n    (binaryProductLimitCone G H).cone.π.app ⟨WalkingPair.right⟩ = AddMonoidHom.snd G H :=\n  rfl\n#align AddCommGroup.binary_product_limit_cone_cone_π_app_right AddCommGroupCat.binaryProductLimitCone_cone_π_app_right\n\n/-- We verify that the biproduct in `AddCommGroupCat` is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps! hom_apply]\nnoncomputable def biprodIsoProd (G H : AddCommGroupCat.{u}) :\n    (G ⊞ H : AddCommGroupCat) ≅ AddCommGroupCat.of (G × H) :=\n  IsLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit G H) (binaryProductLimitCone G H).isLimit\n#align AddCommGroup.biprod_iso_prod AddCommGroupCat.biprodIsoProd\n\n-- These lemmas have always been bad (#7657), but lean4#2644 made `simp` start noticing\nattribute [nolint simpNF] AddCommGroupCat.biprodIsoProd_hom_apply\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_fst (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.fst = AddMonoidHom.fst G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.left)\n#align AddCommGroup.biprod_iso_prod_inv_comp_fst AddCommGroupCat.biprodIsoProd_inv_comp_fst\n\n@[simp, elementwise]\ntheorem biprodIsoProd_inv_comp_snd (G H : AddCommGroupCat.{u}) :\n    (biprodIsoProd G H).inv ≫ biprod.snd = AddMonoidHom.snd G H :=\n  IsLimit.conePointUniqueUpToIso_inv_comp _ _ (Discrete.mk WalkingPair.right)\n#align AddCommGroup.biprod_iso_prod_inv_comp_snd AddCommGroupCat.biprodIsoProd_inv_comp_snd\n\nnamespace HasLimit\n\nvariable {J : Type w} (f : J → AddCommGroupCat.{max w u})\n\n/-- The map from an arbitrary cone over an indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : Fan f) : s.pt ⟶ AddCommGroupCat.of (∀ j, f j) where\n  toFun x j := s.π.app ⟨j⟩ x\n  map_zero' := by\n    simp only [Functor.const_obj_obj, map_zero]\n    rfl\n  map_add' x y := by\n    simp only [Functor.const_obj_obj, map_add]\n    rfl\n#align AddCommGroup.has_limit.lift AddCommGroupCat.HasLimit.lift\n\n/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone :=\n    { pt := AddCommGroupCat.of (∀ j, f j)\n      π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => f j) j.as }\n  isLimit :=\n    { lift := lift.{_, u} f\n      fac := fun s j => rfl\n      uniq := fun s m w => by\n        ext x\n        funext j\n        ","nextTactic":"exact congr_arg (fun g : s.pt ⟶ f j => (g : s.pt → f j) x) (w ⟨j⟩)","declUpToTactic":"/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone :=\n    { pt := AddCommGroupCat.of (∀ j, f j)\n      π := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => f j) j.as }\n  isLimit :=\n    { lift := lift.{_, u} f\n      fac := fun s j => rfl\n      uniq := fun s m w => by\n        ext x\n        funext j\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Category_GroupCat_Biproducts.112_0.jT6AjHNJZqfPwUM","decl":"/-- Construct limit data for a product in `AddCommGroupCat`, using\n`AddCommGroupCat.of (∀ j, F.obj j)`.\n-/\n@[simps]\ndef productLimitCone : Limits.LimitCone (Discrete.functor f) where\n  cone "}