{"id":"QuadraticForm","label":"QuadraticForm","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["BilinForm"],"nlp_text":"Let \\(R\\) be a commutative ring, \\(M\\) a \\(R\\)-module. An quadratic form \\(Q\\) over \\(M\\) is a map \\(Q : M \\to R\\), satisfying: \\( Q(a \\bullet x) = a * a * Q(x)\\) for all \\(a \\in R, x \\in M\\). there exists a companion bilinear form \\(B : M \\to M \\to R\\), such that \\(Q(x + y) = Q(x) + Q(y) + B(x, y)\\) In some literatures, the bilinear form is denoted \\(\\Phi \\), and called the polar form associated with the quadratic form \\(Q\\), or simply the polar form of \\(Q\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/QuadraticForm","latex_link":"sec-forms.html#QuadraticForm","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/QuadraticForm","lean_decl_name":"QuadraticForm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean","lean4_code":"instance : CommRing (CliffordAlgebra (0 : QuadraticForm R Unit)) :=\n  { CliffordAlgebra.instRing _ with\n    mul_comm := fun x y => by\n      induction x using CliffordAlgebra.induction with\n      | algebraMap r => apply Algebra.commutes\n      | ι x => simp\n      | add x₁ x₂ hx₁ hx₂ => rw [mul_add, add_mul, hx₁, hx₂]\n      | mul x₁ x₂ hx₁ hx₂ => rw [mul_assoc, hx₂, ← mul_assoc, hx₁, ← mul_assoc] }\n\n","line":77,"end_line":85,"proof_status":"formalized"}
{"id":"LinearMap","label":"LinearMap","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module","RingHom"],"nlp_text":"Let \\(R, S\\) be rings, \\(M\\) an \\(R\\)-module, \\(N\\) an \\(S\\)-module. A linear map from \\(M\\) to \\(N\\) is a function \\(f : M \\to _{l} N\\) over a ring homomorphism \\(\\sigma : R \\to _{+*} S\\), satisfying: \\(f(x + y) = f(x) + f(y)\\) for all \\(x, y \\in M\\). \\(f(r \\bullet x) = \\sigma (r) \\bullet f(x)\\) for all \\(r \\in R\\), \\(x \\in M\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/LinearMap","latex_link":"sec-algebras.html#LinearMap","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/LinearMap","lean_decl_name":"LinearMap","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Algebra/Module/Basic.lean","lean4_code":"theorem LinearMap.continuous_on_pi {ι : Type*} {R : Type*} {M : Type*} [Finite ι] [Semiring R]\n    [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M]\n    [ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous f := by\n  cases nonempty_fintype ι\n  classical\n    -- for the proof, write `f` in the standard basis, and use that each coordinate is a continuous\n    -- function.\n    have : (f : (ι → R) → M) = fun x => ∑ i : ι, x i • f fun j => if i = j then 1 else 0 := by\n      ext x\n      exact f.pi_apply_eq_sum_univ x\n    rw [this]\n    refine continuous_finset_sum _ fun i _ => ?_\n    exact (continuous_apply i).smul continuous_const\n\nend Pi\n\nsection PointwiseLimits\n\nvariable {M₁ M₂ α R S : Type*} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S]\n  [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂]\n\nvariable [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α}\n\n/-- Constructs a bundled linear map from a function and a proof that this function belongs to the\nclosure of the set of linear maps. -/\n","line":245,"end_line":269,"proof_status":"formalized"}
{"id":"Dual","label":"Dual","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["LinearMap"],"nlp_text":"The dual module \\(M^* : M \\to _{l[R]} R\\) is the \\(R\\)-module of all linear maps from \\(M\\) to \\(R\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module.Dual","latex_link":"sec-basics.html#Dual","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module.Dual","lean_decl_name":"Module.Dual","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RepresentationTheory/Basic.lean","lean4_code":"def dual : Representation k G (Module.Dual k V) where\n  toFun g :=\n    { toFun := fun f => f ∘ₗ ρV g⁻¹\n      map_add' := fun f₁ f₂ => by simp only [add_comp]\n      map_smul' := fun r f => by\n        ext\n        simp only [coe_comp, Function.comp_apply, smul_apply, RingHom.id_apply] }\n  map_one' := by ext; simp\n  map_mul' g h := by ext; simp\n\n","line":675,"end_line":684,"proof_status":"formalized"}
{"id":"TensorAlgebra","label":"TensorAlgebra","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["LinearMap","FreeAlgebra"],"nlp_text":"Let \\(A\\) be a free \\(R\\)-algebra generated by module \\(M\\), let \\(\\iota : M \\to A\\) denote the map from \\(M\\) to \\(A\\). An tensor algebra over \\(M\\) (or “of \\(M\\) \") \\(T\\) is the ring quotient of the free \\(R\\)-algebra generated by \\(M\\), by the equivalence relation satisfying: for all \\(a, b\\) in \\(M\\), \\(\\iota (a + b) \\sim \\iota (a) + \\iota (b)\\). for all \\(r\\) in \\(R\\), \\(a\\) in \\(M\\), \\(\\iota (r \\bullet a) \\sim r * \\iota (a)\\). i.e. making the inclusion of \\(M\\) into an \\(R\\)-linear map.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/TensorAlgebra","latex_link":"sec-algebras.html#TensorAlgebra","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/TensorAlgebra","lean_decl_name":"TensorAlgebra","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean","lean4_code":"def TensorAlgebra :=\n  RingQuot (TensorAlgebra.Rel R M)\n\n-- The `Inhabited, Semiring, Algebra` instances should be constructed by a deriving handler.\n-- https://github.com/leanprover-community/mathlib4/issues/380\n","line":59,"end_line":63,"proof_status":"formalized"}
{"id":"Group","label":"Group","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":[],"nlp_text":"A group is a pair \\((G, *)\\), satisfying: \\((a * b) * c = a * (b * c)\\) for all \\(a, b, c \\in G\\) (associativity). there exists \\(1 \\in G\\) such that \\(1 * a = a * 1 = a\\) for all \\(a \\in G\\). for each \\(a \\in G\\) there exists \\(a^{-1} \\in G\\) such that \\(a * a^{-1} = a^{-1} * a = 1\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Group","latex_link":"sec-basics.html#Group","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Group","lean_decl_name":"Group","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/Germ/Basic.lean","lean4_code":"instance instGroup [Group G] : Group (Germ l G) :=\n  { inv_mul_cancel := Quotient.ind' fun _ => congrArg ofFun <| inv_mul_cancel _ }\n\n","line":505,"end_line":507,"proof_status":"formalized"}
{"id":"Ring","label":"Ring","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Group","Monoid"],"nlp_text":"A ring is a triple \\((R, +, *)\\), satisfying: \\(R\\) is a commutative group under \\(+\\). \\(R\\) is a monoid under \\(*\\). for all \\(a, b, c \\in R\\), we have \\(a * (b + c) = a * b + a * c\\) \\((a + b) * c = a * c + b * c\\) (left and right distributivity over \\(+\\)).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Ring","latex_link":"sec-basics.html#Ring","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Ring","lean_decl_name":"Ring","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Tactic/FastInstance.lean","lean4_code":"instance instRing : Ring X := fast_instance% Function.Injective.ring ..\n```\nthis will define `instRing` as a nested constructor application that refers to `instSemiring`\nrather than applications of `Function.Injective.ring` or other non-canonical constructors.\nThe advantage is then that `instRing.toSemiring` unifies almost immediately with `instSemiring`,\nrather than having to break it down into smaller pieces.\n-/\nsyntax (name := fastInstance) \"fast_instance%\" term : term\n\n","line":111,"end_line":119,"proof_status":"formalized"}
{"id":"Module","label":"Module","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Ring"],"nlp_text":"Let \\(R\\) be a commutative ring. A module over \\(R\\) (in short \\(R\\)-module) is a pair \\((M, \\bullet )\\), satisfying: M is a group under \\(+\\). \\(\\bullet : R \\to M \\to M\\) is called scalar multiplication, for every \\(a, b \\in R\\), \\(x, y \\in M\\), we have \\(a \\bullet (x + y) = a \\bullet x + b \\bullet y\\) \\((a + b) \\bullet x = a \\bullet x + b \\bullet x\\) \\(a * (b \\bullet x)=(a * b) \\bullet x\\) \\(1_R \\bullet x = x\\)","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module","latex_link":"sec-basics.html#Module","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module","lean_decl_name":"Module","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/Germ/Basic.lean","lean4_code":"instance instModule [Semiring R] [AddCommMonoid M] [Module R M] : Module R (Germ l M) where\n  add_smul c₁ c₂ f :=\n    inductionOn f fun f => by\n      norm_cast\n      simp [add_smul]\n  zero_smul f :=\n    inductionOn f fun f => by\n      norm_cast\n      simp [zero_smul]\n\n","line":652,"end_line":661,"proof_status":"formalized"}
{"id":"DivisionRing","label":"DivisionRing","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Ring"],"nlp_text":"A division ring is a ring \\((R, +, *)\\), satisfying: \\(R\\) contains at least 2 elements. for all \\(a \\neq 0\\) in \\(R\\), there exists a multiplicative inverse \\(a^{-1} \\in R\\) such that \\[ a * a^{-1} = a^{-1} * a = 1 \\]","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/DivisionRing","latex_link":"sec-basics.html#DivisionRing","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/DivisionRing","lean_decl_name":"DivisionRing","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/FilterProduct.lean","lean4_code":"instance instDivisionRing [DivisionRing β] : DivisionRing β* where\n  __ := instRing\n  __ := instDivisionSemiring\n  qsmul := _\n  qsmul_def := fun _ _ => rfl\n\n","line":51,"end_line":56,"proof_status":"formalized"}
{"id":"RingHom","label":"RingHom","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Ring"],"nlp_text":"Let \\((\\alpha , +_\\alpha , *_\\alpha )\\) and \\((\\beta , +_\\beta , *_\\beta )\\) be rings. A ring homomorphism from \\(\\alpha \\) to \\(\\beta \\) is a map \\(\\mathit{1}: \\alpha \\to _{+*}\\beta \\) such that \\(\\mathit{1}(x +_{\\alpha } y) = \\mathit{1}(x) +_{\\beta } \\mathit{1}(y)\\) for each \\(x,y \\in \\alpha \\). \\(\\mathit{1}(x *_{\\alpha } y) = \\mathit{1}(x) *_{\\beta } \\mathit{1}(y)\\) for each \\(x,y \\in \\alpha \\). \\(\\mathit{1}(1_{\\alpha }) = 1_{\\beta }\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/RingHom","latex_link":"sec-algebras.html#RingHom","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/RingHom","lean_decl_name":"RingHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Tactic/Algebraize.lean","lean4_code":"def RingHom.FiniteType (f : A →+* B) : Prop :=\n  @Algebra.FiniteType A B _ _ f.toAlgebra\n```\nAn example when the `Name` is provided (as the `Algebra` does not have the expected name):\n```\n","line":51,"end_line":55,"proof_status":"formalized"}
{"id":"VectorSpace","label":"VectorSpace","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module","DivisionRing"],"nlp_text":"If \\(R\\) is a division ring, then a module \\(M\\) over \\(R\\) is called a vector space.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module","latex_link":"sec-basics.html#VectorSpace","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module","lean_decl_name":"Module","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/Germ/Basic.lean","lean4_code":"instance instModule [Semiring R] [AddCommMonoid M] [Module R M] : Module R (Germ l M) where\n  add_smul c₁ c₂ f :=\n    inductionOn f fun f => by\n      norm_cast\n      simp [add_smul]\n  zero_smul f :=\n    inductionOn f fun f => by\n      norm_cast\n      simp [zero_smul]\n\n","line":652,"end_line":661,"proof_status":"formalized"}
{"id":"Algebra","label":"Algebra","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module","RingHom"],"nlp_text":"Let \\(R\\) be a commutative ring. An algebra \\(A\\) over \\(R\\) is a pair \\((A, \\bullet )\\), satisfying: \\(A\\) is a ring under \\(*\\). there exists a ring homomorphism from \\(R\\) to \\(A\\), denoted \\(\\mathit{1}: R \\to _{+*} A\\). \\(\\bullet : R \\to M \\to M\\) is a scalar multiplication for every \\(r \\in R\\), \\(x \\in A\\), we have \\(r * x = x * r\\) (commutativity between \\(R\\) and \\(A\\)) \\(r \\bullet x = r * x\\) (definition of scalar multiplication) where we omitted that the ring homomorphism \\(\\mathit{1}\\) is applied to \\(r\\) before each multiplication.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Algebra","latex_link":"sec-algebras.html#Algebra","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Algebra","lean_decl_name":"Algebra","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/LocallyConstant/Algebra.lean","lean4_code":"instance : Algebra R (LocallyConstant X Y) where\n  algebraMap := constRingHom.comp <| algebraMap R Y\n  commutes' := by\n    intros\n    ext\n    exact Algebra.commutes' _ _\n  smul_def' := by\n    intros\n    ext\n    exact Algebra.smul_def' _ _\n\n","line":250,"end_line":260,"proof_status":"formalized"}
{"id":"RingQuot","label":"RingQuot","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module","RingHom"],"nlp_text":"Let \\(R\\) be a non-commutative ring, \\(r\\) an arbitrary equivalence relation on \\(R\\). The ring quotient of \\(R\\) by \\(r\\) is the quotient of \\(R\\) by the strengthen equivalence relation of \\(r\\) such that for all \\(a, b, c\\) in \\(R\\): \\(a + c \\sim b + c\\) if \\(a \\sim b\\) \\(a * c \\sim b * c\\) if \\(a \\sim b\\) \\(a * b \\sim a * c\\) if \\(b \\sim c\\) i.e. the equivalence relation is compatible with the ring operations \\(+\\) and \\(*\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/RingQuot","latex_link":"sec-algebras.html#RingQuot","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/RingQuot","lean_decl_name":"RingQuot","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean","lean4_code":"instance instInhabited : Inhabited (CliffordAlgebra Q) := RingQuot.instInhabited _\n","line":73,"end_line":73,"proof_status":"formalized"}
{"id":"BilinForm","label":"BilinForm","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module"],"nlp_text":"Let \\(R\\) be a ring, \\(M\\) an \\(R\\)-module. An bilinear form \\(B\\) over \\(M\\) is a map \\(B : M \\to M \\to R\\), satisfying: \\( B(x + y, z) = B(x, z) +B(y, z) \\) \\( B(x, y + z) = B(x, y) +B(x, z) \\) \\( B(a \\bullet x, y) = a * B(x, y)\\) \\( B(x, a \\bullet y) = a * B(x, y)\\) for all \\(a \\in R, x, y, z \\in M\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/BilinForm","latex_link":"sec-forms.html#BilinForm","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/BilinForm","lean_decl_name":"BilinForm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean","lean4_code":"def RootForm : LinearMap.BilinForm R M :=\n  ∑ i, (P.coroot' i).smulRight (P.coroot' i)\n\n/-- An invariant inner product on the coweight space. -/\n","line":80,"end_line":83,"proof_status":"formalized"}
{"id":"FreeAlgebra","label":"FreeAlgebra","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Algebra","RingQuot"],"nlp_text":"Let \\(X\\) be an arbitrary set. An free \\(R\\)-algebra \\(A\\) on \\(X\\) (or “ generated by \\(X\\) \") is the ring quotient of the following inductively constructed set \\(A_{\\mathtt{pre}}\\) for all \\(x\\) in \\(X\\), there exists a map \\(X \\to A_{\\mathtt{pre}}\\). for all \\(r\\) in \\(R\\), there exists a map \\(R \\to A_{\\mathtt{pre}}\\). for all \\(a, b\\) in \\(A_{\\mathtt{pre}}\\), \\(a + b\\) is in \\(A_{\\mathtt{pre}}\\). for all \\(a, b\\) in \\(A_{\\mathtt{pre}}\\), \\(a * b\\) is in \\(A_{\\mathtt{pre}}\\). by that equivalence relation that makes \\(A\\) an \\(R\\)-algebra, namely: there exists a ring homomorphism from \\(R\\) to \\(A_{\\mathtt{pre}}\\), denoted \\(R \\to _{+*} A_{\\mathtt{pre}}\\). \\(A\\) is a commutative group under \\(+\\). \\(A\\) is a monoid under \\(*\\). left and right distributivity under \\(*\\) over \\(+\\). \\(0 * a \\sim a * 0 \\sim 0\\). for all \\(a, b, c\\) in \\(A\\), if \\(a \\sim b\\), we have \\(a + c \\sim b + c\\) \\(c + a \\sim c + b\\) \\(a * c \\sim b * c\\) \\(c * a \\sim c * b\\) (compatibility with the ring operations \\(+\\) and \\(*\\))","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/FreeAlgebra","latex_link":"sec-algebras.html#FreeAlgebra","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/FreeAlgebra","lean_decl_name":"FreeAlgebra","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/FreeAlgebra.lean","lean4_code":"def FreeAlgebra :=\n  Quot (FreeAlgebra.Rel R X)\n\nnamespace FreeAlgebra\n\nattribute [local instance] Pre.hasCoeGenerator Pre.hasCoeSemiring Pre.hasMul Pre.hasAdd\n  Pre.hasZero Pre.hasOne Pre.hasSMul\n\n/-! Define the basic operations -/\n\n","line":157,"end_line":166,"proof_status":"formalized"}
{"id":"AlgHom","label":"AlgHom","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["RingHom"],"nlp_text":"Let \\(A\\) and \\(B\\) be \\(R\\)-algebras. \\(\\mathit{1}_A\\) and \\(\\mathit{1}_B\\) are ring homomorphisms from \\(R\\) to \\(A\\) and \\(B\\), respectively. A algebra homomorphism from \\(A\\) to \\(B\\) is a map \\(f : \\alpha \\to _{a}\\beta \\) such that \\(f\\) is a ring homomorphism \\(f(\\mathit{1}_{A}(r)) = \\mathit{1}_{B}(r)\\) for each \\(r \\in R\\)","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/AlgHom","latex_link":"sec-algebras.html#AlgHom","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/AlgHom","lean_decl_name":"AlgHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean","lean4_code":"theorem AlgHom.closure_ker_inter {F S K A : Type*} [CommRing K] [Ring A] [Algebra K A]\n    [TopologicalSpace K] [T1Space K] [TopologicalSpace A] [ContinuousSub A] [ContinuousSMul K A]\n    [FunLike F A K] [AlgHomClass F K A K] [SetLike S A] [OneMemClass S A] [AddSubgroupClass S A]\n    [SMulMemClass S K A] (φ : F) (hφ : Continuous φ) (s : S) :\n    closure (s ∩ RingHom.ker φ) = closure s ∩ (ker φ : Set A) := by\n  refine subset_antisymm ?_ ?_\n  · simpa only [ker_eq, (isClosed_singleton.preimage hφ).closure_eq]\n      using closure_inter_subset_inter_closure s (ker φ : Set A)\n  · intro x ⟨hxs, (hxφ : φ x = 0)⟩\n    rw [mem_closure_iff_clusterPt, ClusterPt] at hxs\n    have : Tendsto (fun y ↦ y - φ y • 1) (𝓝 x ⊓ 𝓟 s) (𝓝 x) := by\n      conv => congr; rfl; rfl; rw [← sub_zero x, ← zero_smul K 1, ← hxφ]\n      exact Filter.tendsto_inf_left (Continuous.tendsto (by fun_prop) x)\n    refine mem_closure_of_tendsto this <| eventually_inf_principal.mpr ?_\n    filter_upwards [] with g hg using\n      ⟨sub_mem hg (SMulMemClass.smul_mem _ <| one_mem _), by simp [RingHom.mem_ker]⟩\n\n","line":582,"end_line":598,"proof_status":"formalized"}
{"id":"Monoid","label":"Monoid","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":[],"nlp_text":"A monoid is a pair \\((R, *)\\), satisfying: \\((a * b) * c = a * (b * c)\\) for all \\(a, b, c \\in R\\) (associativity). there exists an element \\(1 \\in R\\) such that \\(1 * a = a * 1 = a\\) for all \\(a \\in R\\) i.e. \\(1\\) is the multiplicative identity (neutral element).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Monoid","latex_link":"sec-basics.html#Monoid","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Monoid","lean_decl_name":"Monoid","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/Germ/Basic.lean","lean4_code":"instance instMonoid [Monoid M] : Monoid (Germ l M) :=\n  { Function.Surjective.monoid ofFun Quot.mk_surjective (by rfl)\n      (fun _ _ => by rfl) fun _ _ => by rfl with\n    toSemigroup := instSemigroup\n    toOne := instOne\n    npow := fun n a => a ^ n }\n\n/-- Coercion from functions to germs as a monoid homomorphism. -/\n","line":404,"end_line":411,"proof_status":"formalized"}