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Found 103 declarations mentioning Module.Relations.G.

• Module.Relations.G 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (self : Module.Relations A) : Type w₀
• Module.Relations.relation 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (self : Module.Relations A) (r : self.R) : self.G →₀ A
• Module.Relations.Solution.var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (self : relations.Solution M) (g : relations.G) : M
• Module.Relations.Solution.ext 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} {inst✝ : Ring A} {relations : Module.Relations A} {M : Type v} {inst✝¹ : AddCommGroup M} {inst✝² : Module A M} {x y : relations.Solution M} (var : x.var = y.var) : x = y
• Module.Relations.Solution.ext_iff 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} {inst✝ : Ring A} {relations : Module.Relations A} {M : Type v} {inst✝¹ : AddCommGroup M} {inst✝² : Module A M} {x y : relations.Solution M} : x = y ↔ x.var = y.var
• Module.Relations.Solution.congr_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution solution' : relations.Solution M} (h : solution = solution') (g : relations.G) : solution.var g = solution'.var g
• Module.Relations.toQuotient 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) : (relations.G →₀ A) →ₗ[A] relations.Quotient
• Module.Relations.Solution.π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : (relations.G →₀ A) →ₗ[A] M
• Module.Relations.map 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) : (relations.R →₀ A) →ₗ[A] relations.G →₀ A
• Module.Relations.Solution.range_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : solution.π.range = Submodule.span A (Set.range solution.var)
• Module.Relations.Solution.IsPresentation.desc_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) {N : Type v'} [AddCommGroup N] [Module A N] (s : relations.Solution N) (g : relations.G) : (h.desc s) (solution.var g) = s.var g
• Module.Relations.Solution.IsPresentationCore.desc_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentationCore) {N : Type w'} [AddCommGroup N] [Module A N] (s : relations.Solution N) (g : relations.G) : (h.desc s) (solution.var g) = s.var g
• Module.Relations.Solution.postcomp_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) {N : Type v'} [AddCommGroup N] [Module A N] (f : M →ₗ[A] N) (g : relations.G) : (solution.postcomp f).var g = f (solution.var g)
• Module.Relations.ker_toQuotient 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) : relations.toQuotient.ker = Submodule.span A (Set.range relations.relation)
• Module.Relations.surjective_toQuotient 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) : Function.Surjective ⇑relations.toQuotient
• Module.Relations.Solution.IsPresentation.surjective_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) : Function.Surjective ⇑solution.π
• Module.Relations.Solution.π_relation 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) (r : relations.R) : solution.π (relations.relation r) = 0
• Module.Relations.Solution.ofQuotient_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) : (Module.Relations.Solution.ofQuotient relations).π = (Submodule.span A (Set.range relations.relation)).mkQ
• Module.Relations.Solution.fromQuotient_comp_toQuotient 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : solution.fromQuotient ∘ₗ relations.toQuotient = solution.π
• Module.Relations.Solution.IsPresentation.ker_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) : solution.π.ker = Submodule.span A (Set.range relations.relation)
• Module.Relations.Solution.mk 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (var : relations.G → M) (linearCombination_var_relation : ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0) : relations.Solution M
• Module.Relations.Solution.linearCombination_var_relation 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (self : relations.Solution M) (r : relations.R) : (Finsupp.linearCombination A self.var) (relations.relation r) = 0
• Module.Relations.Solution.π_single 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) (g : relations.G) : (solution.π fun₀ | g => 1) = solution.var g
• Module.Relations.Solution.IsPresentation.uniq_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) {N : Type v'} [AddCommGroup N] [Module A N] {solution' : relations.Solution N} (h' : solution'.IsPresentation) (g : relations.G) : (h.uniq h') (solution.var g) = solution'.var g
• Module.Relations.range_map 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) : relations.map.range = Submodule.span A (Set.range relations.relation)
• Module.Relations.toQuotient_relation 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) (r : relations.R) : relations.toQuotient (relations.relation r) = 0
• Module.Relations.Solution.ofQuotient_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) (g : relations.G) : (Module.Relations.Solution.ofQuotient relations).var g = relations.toQuotient fun₀ | g => 1
• Module.Relations.Solution.IsPresentation.desc_comp_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) {N : Type v'} [AddCommGroup N] [Module A N] (s : relations.Solution N) : h.desc s ∘ₗ solution.π = s.π
• Module.Relations.Solution.surjective_π_iff_span_eq_top 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : Function.Surjective ⇑solution.π ↔ Submodule.span A (Set.range solution.var) = ⊤
• Module.Relations.Solution.ofπ 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : ∀ (r : relations.R), π (relations.relation r) = 0) : relations.Solution M
• Module.Relations.Solution.IsPresentation.uniq_symm_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) {N : Type v'} [AddCommGroup N] [Module A N] {solution' : relations.Solution N} (h' : solution'.IsPresentation) (g : relations.G) : (h.uniq h').symm (solution'.var g) = solution.var g
• Module.Relations.Solution.isPresentation_mk 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) (h₁ : Submodule.span A (Set.range solution.var) = ⊤) (h₂ : solution.π.ker = Submodule.span A (Set.range relations.relation)) : solution.IsPresentation
• Module.Relations.Solution.isPresentation_iff 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : solution.IsPresentation ↔ Submodule.span A (Set.range solution.var) = ⊤ ∧ solution.π.ker = Submodule.span A (Set.range relations.relation)
• Module.Relations.map_single 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) (r : relations.R) : (relations.map fun₀ | r => 1) = relations.relation r
• Module.Relations.Solution.surjective_fromQuotient_iff_surjective_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : Function.Surjective ⇑solution.fromQuotient ↔ Function.Surjective ⇑solution.π
• Module.Relations.Solution.injective_fromQuotient_iff_ker_π_eq_span 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : Function.Injective ⇑solution.fromQuotient ↔ solution.π.ker = Submodule.span A (Set.range relations.relation)
• Module.Relations.Solution.ofπ_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : ∀ (r : relations.R), π (relations.relation r) = 0) : (Module.Relations.Solution.ofπ π hπ).π = π
• Module.Relations.Solution.span_relation_le_ker_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : Submodule.span A (Set.range relations.relation) ≤ solution.π.ker
• Module.Relations.Solution.π_comp_map 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : solution.π ∘ₗ relations.map = 0
• Module.Relations.Solution.IsPresentation.π_desc_apply 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) {N : Type v'} [AddCommGroup N] [Module A N] (s : relations.Solution N) (x : relations.G →₀ A) : (h.desc s) (solution.π x) = s.π x
• Module.Relations.Solution.π_comp_map_apply 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) (x : relations.R →₀ A) : solution.π (relations.map x) = 0
• Module.Relations.Solution.ofπ_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : ∀ (r : relations.R), π (relations.relation r) = 0) (g : relations.G) : (Module.Relations.Solution.ofπ π hπ).var g = π fun₀ | g => 1
• Module.Relations.toQuotient_map 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) : relations.toQuotient ∘ₗ relations.map = 0
• Module.Relations.Solution.IsPresentation.exact 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) : Function.Exact ⇑relations.map ⇑solution.π
• Module.Relations.Solution.fromQuotient_toQuotient 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) (x : relations.G →₀ A) : solution.fromQuotient (relations.toQuotient x) = solution.π x
• Module.Relations.Solution.ofπ' 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : π ∘ₗ relations.map = 0) : relations.Solution M
• Module.Relations.toQuotient_map_apply 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) (x : relations.R →₀ A) : relations.toQuotient (relations.map x) = 0
• Module.Relations.Solution.IsPresentation.linearEquiv_symm_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {solution : relations.Solution M} (h : solution.IsPresentation) (g : relations.G) : h.linearEquiv.symm (solution.var g) = relations.toQuotient fun₀ | g => 1
• Module.Relations.Solution.ofπ'_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : π ∘ₗ relations.map = 0) : (Module.Relations.Solution.ofπ' π hπ).π = π
• Module.Relations.Solution.ofπ'_var 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : π ∘ₗ relations.map = 0) (g : relations.G) : (Module.Relations.Solution.ofπ' π hπ).var g = π fun₀ | g => 1
• Module.Relations.Quotient.linearMap_ext 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {f f' : relations.Quotient →ₗ[A] M} (h : ∀ (g : relations.G), f (relations.toQuotient fun₀ | g => 1) = f' (relations.toQuotient fun₀ | g => 1)) : f = f'
• Module.Relations.Quotient.linearMap_ext_iff 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] {f f' : relations.Quotient →ₗ[A] M} : f = f' ↔ ∀ (g : relations.G), f (relations.toQuotient fun₀ | g => 1) = f' (relations.toQuotient fun₀ | g => 1)
• exteriorPower.presentation.relations_G 📋 Mathlib.LinearAlgebra.ExteriorPower.Basic
  (R : Type u) [CommRing R] (ι : Type u_4) [DecidableEq ι] (M : Type u_5) [AddCommGroup M] [Module R M] : (exteriorPower.presentation.relations R ι M).G = (ι → M)
• exteriorPower.presentation.relationsSolutionEquiv_symm_apply_var 📋 Mathlib.LinearAlgebra.ExteriorPower.Basic
  {R : Type u} [CommRing R] {N : Type u_2} [AddCommGroup N] [Module R N] {ι : Type u_4} [DecidableEq ι] {M : Type u_5} [AddCommGroup M] [Module R M] (f : M [⋀^ι]→ₗ[R] N) (m : (exteriorPower.presentation.relations R ι M).G) : (exteriorPower.presentation.relationsSolutionEquiv.symm f).var m = f m
• exteriorPower.presentation_G 📋 Mathlib.LinearAlgebra.ExteriorPower.Basic
  (R : Type u) [CommRing R] (n : ℕ) (M : Type u_1) [AddCommGroup M] [Module R M] : (exteriorPower.presentation R n M).G = (Fin n → M)
• exteriorPower.presentation_var 📋 Mathlib.LinearAlgebra.ExteriorPower.Basic
  (R : Type u) [CommRing R] (n : ℕ) (M : Type u_1) [AddCommGroup M] [Module R M] (m : (exteriorPower.presentation.relations R (Fin n) M).G) : (exteriorPower.presentation R n M).var m = (exteriorPower.ιMulti R n) m
• Module.Presentation.CokernelData.lift 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] {pres₂ : Module.Presentation A M₂} {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (self : pres₂.CokernelData f g₁) (i : ι) : pres₂.G →₀ A
• Module.Presentation.cokernelRelations_G 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) : (pres₂.cokernelRelations data).G = pres₂.G
• Module.Presentation.cokernelRelations_relation 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (x✝ : pres₂.R ⊕ ι) : (pres₂.cokernelRelations data).relation x✝ = match x✝ with | Sum.inl r => pres₂.relation r | Sum.inr i => data.lift i
• Module.Presentation.cokernel_G 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) : (pres₂.cokernel data hg₁).G = pres₂.G
• Module.Presentation.CokernelData.ofSection 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) (f : M₁ →ₗ[A] M₂) {ι : Type w₁} (g₁ : ι → M₁) (s : M₂ → pres₂.G →₀ A) (hs : ∀ (m₂ : M₂), pres₂.π (s m₂) = m₂) : pres₂.CokernelData f g₁
• Module.Presentation.cokernel_relation 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) (x✝ : pres₂.R ⊕ ι) : (pres₂.cokernel data hg₁).relation x✝ = match x✝ with | Sum.inl r => pres₂.relation r | Sum.inr i => data.lift i
• Module.Presentation.CokernelData.π_lift 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] {pres₂ : Module.Presentation A M₂} {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (self : pres₂.CokernelData f g₁) (i : ι) : pres₂.π (self.lift i) = f (g₁ i)
• Module.Presentation.CokernelData.mk 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] {pres₂ : Module.Presentation A M₂} {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (lift : ι → pres₂.G →₀ A) (π_lift : ∀ (i : ι), pres₂.π (lift i) = f (g₁ i)) : pres₂.CokernelData f g₁
• Module.Presentation.ofExact_G 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} {M₃ : Type v₃} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] [AddCommGroup M₃] [Module A M₃] {f : M₁ →ₗ[A] M₂} {g : M₂ →ₗ[A] M₃} (pres₂ : Module.Presentation A M₂) {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) : (pres₂.ofExact data hfg hg hg₁).G = pres₂.G
• Module.Presentation.CokernelData.ofSection_lift 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) (f : M₁ →ₗ[A] M₂) {ι : Type w₁} (g₁ : ι → M₁) (s : M₂ → pres₂.G →₀ A) (hs : ∀ (m₂ : M₂), pres₂.π (s m₂) = m₂) (i : ι) : (Module.Presentation.CokernelData.ofSection pres₂ f g₁ s hs).lift i = s (f (g₁ i))
• Module.Presentation.ofExact_relation 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} {M₃ : Type v₃} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] [AddCommGroup M₃] [Module A M₃] {f : M₁ →ₗ[A] M₂} {g : M₂ →ₗ[A] M₃} (pres₂ : Module.Presentation A M₂) {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) (x✝ : pres₂.R ⊕ ι) : (pres₂.ofExact data hfg hg hg₁).relation x✝ = match x✝ with | Sum.inl r => pres₂.relation r | Sum.inr i => data.lift i
• Module.Presentation.ofExact_var 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} {M₃ : Type v₃} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] [AddCommGroup M₃] [Module A M₃] {f : M₁ →ₗ[A] M₂} {g : M₂ →ₗ[A] M₃} (pres₂ : Module.Presentation A M₂) {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) (g✝ : (pres₂.cokernel data hg₁).G) : (pres₂.ofExact data hfg hg hg₁).var g✝ = g (pres₂.var g✝)
• Module.Presentation.cokernel_var 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) (g : (pres₂.cokernelRelations data).G) : (pres₂.cokernel data hg₁).var g = Submodule.Quotient.mk (pres₂.var g)
• Module.Presentation.cokernelSolution_var 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (g : (pres₂.cokernelRelations data).G) : (pres₂.cokernelSolution data).var g = f.range.mkQ (pres₂.var g)
• Algebra.Presentation.differentialsRelations_G 📋 Mathlib.Algebra.Module.Presentation.Differentials
  {R : Type u} {S : Type v} {ι : Type w} {σ : Type t} [CommRing R] [CommRing S] [Algebra R S] (pres : Algebra.Presentation R S ι σ) : pres.differentialsRelations.G = ι
• Module.Relations.directSum_G 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {ι : Type w} (relations : ι → Module.Relations A) : (Module.Relations.directSum relations).G = ((i : ι) × (relations i).G)
• Module.Presentation.finsupp_G 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {N : Type v} [AddCommGroup N] [Module A N] (pres : Module.Presentation A N) (ι : Type w) [DecidableEq ι] [DecidableEq N] : (pres.finsupp ι).G = ((i : ι) × pres.G)
• Module.Presentation.directSum_G 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {ι : Type w} [DecidableEq ι] {M : ι → Type v} [(i : ι) → AddCommGroup (M i)] [(i : ι) → Module A (M i)] (pres : (i : ι) → Module.Presentation A (M i)) : (Module.Presentation.directSum pres).G = ((i : ι) × (pres i).G)
• Module.Relations.directSum_relation 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {ι : Type w} (relations : ι → Module.Relations A) (x✝ : (i : ι) × (relations i).R) : (Module.Relations.directSum relations).relation x✝ = match x✝ with | ⟨i, r⟩ => Finsupp.embDomain (Function.Embedding.sigmaMk i) ((relations i).relation r)
• Module.Relations.Solution.directSumEquiv_apply_var 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {ι : Type w} {relations : ι → Module.Relations A} {N : Type v} [AddCommGroup N] [Module A N] (s : (Module.Relations.directSum relations).Solution N) (i : ι) (g : (relations i).G) : (Module.Relations.Solution.directSumEquiv s i).var g = s.var ⟨i, g⟩
• Module.Relations.Solution.directSumEquiv_symm_apply_var 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {ι : Type w} {relations : ι → Module.Relations A} {N : Type v} [AddCommGroup N] [Module A N] (t : (i : ι) → (relations i).Solution N) (x✝ : (Module.Relations.directSum relations).G) : (Module.Relations.Solution.directSumEquiv.symm t).var x✝ = match x✝ with | ⟨i, g⟩ => (t i).var g
• Module.Presentation.finsupp_var 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {N : Type v} [AddCommGroup N] [Module A N] (pres : Module.Presentation A N) (ι : Type w) [DecidableEq ι] [DecidableEq N] (i : ι) (g : pres.G) : (pres.finsupp ι).var ⟨i, g⟩ = fun₀ | i => pres.var g
• Module.Relations.Solution.directSum_var 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {ι : Type w} [DecidableEq ι] {relations : ι → Module.Relations A} {M : ι → Type v} [(i : ι) → AddCommGroup (M i)] [(i : ι) → Module A (M i)] (solution : (i : ι) → (relations i).Solution (M i)) (i : ι) (g : (relations i).G) : (Module.Relations.Solution.directSum solution).var ⟨i, g⟩ = (DirectSum.lof A ι M i) ((solution i).var g)
• Module.Presentation.finsupp_relation 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {N : Type v} [AddCommGroup N] [Module A N] (pres : Module.Presentation A N) (ι : Type w) [DecidableEq ι] [DecidableEq N] (x✝ : (i : ι) × ((fun i => ((fun x => pres) i).toRelations) i).R) : (pres.finsupp ι).relation x✝ = Finsupp.embDomain (Function.Embedding.sigmaMk x✝.fst) (pres.relation x✝.snd)
• Module.Presentation.directSum_relation 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {ι : Type w} [DecidableEq ι] {M : ι → Type v} [(i : ι) → AddCommGroup (M i)] [(i : ι) → Module A (M i)] (pres : (i : ι) → Module.Presentation A (M i)) (x✝ : (i : ι) × ((fun i => (pres i).toRelations) i).R) : (Module.Presentation.directSum pres).relation x✝ = Finsupp.embDomain (Function.Embedding.sigmaMk x✝.fst) ((pres x✝.fst).relation x✝.snd)
• Module.Presentation.directSum_var 📋 Mathlib.Algebra.Module.Presentation.DirectSum
  {A : Type u} [Ring A] {ι : Type w} [DecidableEq ι] {M : ι → Type v} [(i : ι) → AddCommGroup (M i)] [(i : ι) → Module A (M i)] (pres : (i : ι) → Module.Presentation A (M i)) (i : ι) (g : (pres i).G) : (Module.Presentation.directSum pres).var ⟨i, g⟩ = (DirectSum.lof A ι M i) ((pres i).var g)
• Module.Presentation.finite 📋 Mathlib.Algebra.Module.Presentation.Finite
  {A : Type u} [Ring A] {M : Type v} [AddCommGroup M] [Module A M] (pres : Module.Presentation A M) [Finite pres.G] : Module.Finite A M
• Module.Presentation.finitePresentation 📋 Mathlib.Algebra.Module.Presentation.Finite
  {A : Type u} [Ring A] {M : Type v} [AddCommGroup M] [Module A M] (pres : Module.Presentation A M) [Finite pres.G] [Finite pres.R] : Module.FinitePresentation A M
• Module.finitePresentation_iff_exists_presentation 📋 Mathlib.Algebra.Module.Presentation.Finite
  {A : Type u} [Ring A] {M : Type v} [AddCommGroup M] [Module A M] : Module.FinitePresentation A M ↔ ∃ pres, Finite pres.G ∧ Finite pres.R
• Module.presentationFinsupp_G 📋 Mathlib.Algebra.Module.Presentation.Free
  (A : Type u) [Ring A] (G : Type w₀) : (Module.presentationFinsupp A G).G = G
• Module.Relations.solutionFinsupp 📋 Mathlib.Algebra.Module.Presentation.Free
  {A : Type u} [Ring A] (relations : Module.Relations A) [IsEmpty relations.R] : relations.Solution (relations.G →₀ A)
• Module.Relations.solutionFinsupp_isPresentation 📋 Mathlib.Algebra.Module.Presentation.Free
  {A : Type u} [Ring A] (relations : Module.Relations A) [IsEmpty relations.R] : relations.solutionFinsupp.IsPresentation
• Module.Relations.solutionFinsupp.isPresentationCore 📋 Mathlib.Algebra.Module.Presentation.Free
  {A : Type u} [Ring A] (relations : Module.Relations A) [IsEmpty relations.R] : relations.solutionFinsupp.IsPresentationCore
• Module.Relations.solutionFinsupp_var 📋 Mathlib.Algebra.Module.Presentation.Free
  {A : Type u} [Ring A] (relations : Module.Relations A) [IsEmpty relations.R] (g : relations.G) : relations.solutionFinsupp.var g = fun₀ | g => 1
• Module.presentationFinsupp_var 📋 Mathlib.Algebra.Module.Presentation.Free
  (A : Type u) [Ring A] (G : Type w₀) (g : { G := G, R := PEmpty.{w₁ + 1}, relation := fun r => PEmpty.casesOn (fun x => G →₀ A) r }.G) : (Module.presentationFinsupp A G).var g = fun₀ | g => 1
• Module.Presentation.RestrictScalarsData 📋 Mathlib.Algebra.Module.Presentation.RestrictScalars
  {B : Type u_1} [Ring B] {M : Type u_2} [AddCommGroup M] [Module B M] [DecidableEq B] (presM : Module.Presentation B M) [DecidableEq presM.G] {A : Type u_3} [CommRing A] [Algebra A B] (presB : Module.Presentation A B) : Type (max (max (max u_3 u_6) u_4) u_6 u_5)
• Module.Presentation.restrictScalars 📋 Mathlib.Algebra.Module.Presentation.RestrictScalars
  {B : Type u_1} [Ring B] {M : Type u_2} [AddCommGroup M] [Module B M] [DecidableEq B] (presM : Module.Presentation B M) [DecidableEq presM.G] {A : Type u_3} [CommRing A] [Algebra A B] [Module A M] [IsScalarTower A B M] (presB : Module.Presentation A B) (data : presM.RestrictScalarsData presB) : Module.Presentation A M
• Module.Presentation.tautologicalRelations_G 📋 Mathlib.Algebra.Module.Presentation.Tautological
  (A : Type u) [Ring A] (M : Type v) [AddCommGroup M] [Module A M] : (Module.Presentation.tautologicalRelations A M).G = M
• Module.Presentation.tautological_G 📋 Mathlib.Algebra.Module.Presentation.Tautological
  (A : Type u) [Ring A] (M : Type v) [AddCommGroup M] [Module A M] : (Module.Presentation.tautological A M).G = M
• Module.Relations.tensor_G 📋 Mathlib.Algebra.Module.Presentation.Tensor
  {A : Type u} [CommRing A] (relations₁ : Module.Relations A) (relations₂ : Module.Relations A) : (relations₁.tensor relations₂).G = (relations₁.G × relations₂.G)
• Module.Relations.tensor_R 📋 Mathlib.Algebra.Module.Presentation.Tensor
  {A : Type u} [CommRing A] (relations₁ : Module.Relations A) (relations₂ : Module.Relations A) : (relations₁.tensor relations₂).R = (relations₁.R × relations₂.G ⊕ relations₁.G × relations₂.R)
• Module.Presentation.tensor_G 📋 Mathlib.Algebra.Module.Presentation.Tensor
  {A : Type u} [CommRing A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [AddCommGroup M₂] [Module A M₁] [Module A M₂] (pres₁ : Module.Presentation A M₁) (pres₂ : Module.Presentation A M₂) : (pres₁.tensor pres₂).G = (pres₁.G × pres₂.G)
• Module.Presentation.tensor_R 📋 Mathlib.Algebra.Module.Presentation.Tensor
  {A : Type u} [CommRing A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [AddCommGroup M₂] [Module A M₁] [Module A M₂] (pres₁ : Module.Presentation A M₁) (pres₂ : Module.Presentation A M₂) : (pres₁.tensor pres₂).R = (pres₁.R × pres₂.G ⊕ pres₁.G × pres₂.R)
• Module.Relations.Solution.tensor_var 📋 Mathlib.Algebra.Module.Presentation.Tensor
  {A : Type u} [CommRing A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [AddCommGroup M₂] [Module A M₁] [Module A M₂] {relations₁ : Module.Relations A} {relations₂ : Module.Relations A} (solution₁ : relations₁.Solution M₁) (solution₂ : relations₂.Solution M₂) (x✝ : (relations₁.tensor relations₂).G) : (solution₁.tensor solution₂).var x✝ = match x✝ with | (g₁, g₂) => solution₁.var g₁ ⊗ₜ[A] solution₂.var g₂
• Module.Relations.tensor_relation 📋 Mathlib.Algebra.Module.Presentation.Tensor
  {A : Type u} [CommRing A] (relations₁ : Module.Relations A) (relations₂ : Module.Relations A) (x✝ : relations₁.R × relations₂.G ⊕ relations₁.G × relations₂.R) : (relations₁.tensor relations₂).relation x✝ = match x✝ with | Sum.inl (r₁, g₂) => Finsupp.embDomain (Function.Embedding.sectL relations₁.G g₂) (relations₁.relation r₁) | Sum.inr (g₁, r₂) => Finsupp.embDomain (Function.Embedding.sectR g₁ relations₂.G) (relations₂.relation r₂)
• Module.Presentation.tensor_var 📋 Mathlib.Algebra.Module.Presentation.Tensor
  {A : Type u} [CommRing A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [AddCommGroup M₂] [Module A M₁] [Module A M₂] (pres₁ : Module.Presentation A M₁) (pres₂ : Module.Presentation A M₂) (x✝ : (pres₁.tensor pres₂.toRelations).G) : (pres₁.tensor pres₂).var x✝ = match x✝ with | (g₁, g₂) => pres₁.var g₁ ⊗ₜ[A] pres₂.var g₂
• Module.Presentation.tensor_relation 📋 Mathlib.Algebra.Module.Presentation.Tensor
  {A : Type u} [CommRing A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [AddCommGroup M₂] [Module A M₁] [Module A M₂] (pres₁ : Module.Presentation A M₁) (pres₂ : Module.Presentation A M₂) (x✝ : pres₁.R × pres₂.G ⊕ pres₁.G × pres₂.R) : (pres₁.tensor pres₂).relation x✝ = match x✝ with | Sum.inl (r₁, g₂) => Finsupp.embDomain (Function.Embedding.sectL pres₁.G g₂) (pres₁.relation r₁) | Sum.inr (g₁, r₂) => Finsupp.embDomain (Function.Embedding.sectR g₁ pres₂.G) (pres₂.relation r₂)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c