diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Presentation.Basic.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Presentation.Basic.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Presentation.Basic.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","π_relation"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.12 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.9)] (solution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.9 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(NegZeroClass.toZero.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4131342894._hygCtx._hyg.9)))))))","typeFull":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (solution : relations.Solution M) (r : relations.R), solution.π (relations.relation r) = 0","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (solution : relations.Solution M) (r : relations.R), solution.π (relations.relation r) = 0","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["LinearMap","instFunLike"],["Module"],["Module","Relations"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["Module","Relations","Solution"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module","Relations","G"],["Zero","toOfNat0"],["Eq"],["Semiring","toModule"],["NonAssocRing","toNonUnitalNonAssocRing"],["Finsupp","module"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommGroup"],["LinearMap"],["Module","Relations","Solution","π"],["OfNat","ofNat"],["Module","Relations","R"],["Ring","toSemiring"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Module","Relations","relation"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Finsupp","instAddCommMonoid"],["Finsupp"],["Ring"]],"valueReferences":[["Module","Relations","Solution","linearCombination_var_relation"]]},{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","mk","injEq"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23)] (var : (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M) (linearCombination_var_relation : forall (r : Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations), Eq.{succ v} M (DFunLike.coe.{max (max (succ v) 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(var_1 : (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M) (linearCombination_var_relation_1 : forall (r : Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations), Eq.{succ v} M (DFunLike.coe.{max (max (succ v) (succ w₀)) (succ u), max (succ w₀) (succ u), succ v} (LinearMap.{u, u, max u w₀, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations r)) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (NegZeroClass.toZero.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23)))))))), Eq.{1} Prop (Eq.{max (succ v) (succ w₀)} (Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26) (Module.Relations.Solution.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var linearCombination_var_relation) (Module.Relations.Solution.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var_1 linearCombination_var_relation_1)) (Eq.{max (succ v) (succ w₀)} ((Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M) var var_1)","typeFull":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (var : relations.G → M)\n  (linearCombination_var_relation : ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0)\n  (var_1 : relations.G → M)\n  (linearCombination_var_relation_1 :\n    ∀ (r : relations.R), 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.937216031._hygCtx._hyg.3))) (Module.Relations.Solution.π.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.937216031._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.937216031._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.937216031._hygCtx._hyg.12 solution)) (Submodule.span.{u, max u w₀} A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.937216031._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.937216031._hygCtx._hyg.3)))))) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.937216031._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A 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{A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.23 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.26 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.23)] {solution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.26} {motive : 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) 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u, v} A A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Finsupp.linearCombination.{w₀, v, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) M A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var') (Module.Relations.relation.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations r)) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (NegZeroClass.toZero.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23)))))))}, (Eq.{max (succ v) (succ w₀)} (Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26) (Module.Relations.Solution.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var linearCombination_var_relation) (Module.Relations.Solution.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var' linearCombination_var_relation')) -> ((HEq.{max (succ v) (succ w₀)} ((Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M) var ((Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M) var') -> P) -> P","typeFull":"{A : Type u} →\n  {inst : Ring A} →\n    {relations : Module.Relations A} →\n      {M : Type v} →\n        {inst_1 : AddCommGroup M} →\n          {inst_2 : Module A M} →\n            {P : Sort u_1} →\n              {var : relations.G → M} →\n                {linearCombination_var_relation :\n                    ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0} →\n                  {var' : relations.G → M} →\n                    {linearCombination_var_relation' :\n                        ∀ (r : relations.R), (Finsupp.linearCombination A var') (relations.relation r) = 0} →\n                      { var := var, linearCombination_var_relation := linearCombination_var_relation } =\n                          { var := var', linearCombination_var_relation := linearCombination_var_relation' } →\n                        (var ≍ var' → P) → P","typeReadable":"{A : Type u} →\n  {inst : Ring A} →\n    {relations : Module.Relations A} →\n      {M : Type v} →\n        {inst_1 : AddCommGroup M} →\n          {inst_2 : Module A M} →\n            {P : Sort u_1} →\n              {var : relations.G → M} →\n                {linearCombination_var_relation :\n                    ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0} →\n                  {var' : relations.G → M} →\n                    {linearCombination_var_relation' :\n                        ∀ (r : relations.R), (Finsupp.linearCombination A var') (relations.relation r) = 0} →\n                      { var := var, linearCombination_var_relation := linearCombination_var_relation } =\n                          { var := var', linearCombination_var_relation := linearCombination_var_relation' } →\n                        (var ≍ var' → P) → P","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["LinearMap","instFunLike"],["Module"],["Module","Relations"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["Module","Relations","Solution"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module","Relations","G"],["Zero","toOfNat0"],["Eq"],["Semiring","toModule"],["Finsupp","module"],["Module","Relations","Solution","mk"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Finsupp","linearCombination"],["AddCommGroup"],["LinearMap"],["Module","Relations","R"],["OfNat","ofNat"],["Ring","toSemiring"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Module","Relations","relation"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HEq"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Finsupp","instAddCommMonoid"],["Finsupp"],["Ring"]],"valueReferences":[["HEq","refl"],["Module"],["Module","Relations","Solution","mk"],["Module","Relations"],["AddCommGroup"],["heq_of_eq"],["Ring","toSemiring"],["Module","Relations","Solution"],["Eq","refl"],["id"],["AddCommGroup","toAddCommMonoid"],["Module","Relations","Solution","noConfusion"],["Ring"]]},{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","isPresentation_iff"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9)] (solution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12), Iff (Module.Relations.Solution.IsPresentation.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12 solution) (And (Eq.{succ v} (Submodule.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12) (Submodule.span.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12 (Set.range.{v, succ w₀} M (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) (Module.Relations.Solution.var.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12 solution))) (Top.top.{v} (Submodule.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12) (Submodule.instTop.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12))) (Eq.{max (succ u) (succ w₀)} (Submodule.{u, max u w₀} A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3)))))) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3))))) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3)))) (LinearMap.ker.{u, u, max u w₀, v} A A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3)))))) M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12 (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3))) (Module.Relations.Solution.π.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.12 solution)) (Submodule.span.{u, max u w₀} A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3)))))) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3))))) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3))) (Set.range.{max u w₀, succ w₁} (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3)))))) (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations) (Module.Relations.relation.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.4165828814._hygCtx._hyg.3 relations)))))","typeFull":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (solution : relations.Solution M),\n  solution.IsPresentation ↔\n    Submodule.span A (Set.range solution.var) = ⊤ ∧ solution.π.ker = Submodule.span A (Set.range relations.relation)","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (solution : relations.Solution M),\n  solution.IsPresentation ↔\n    Submodule.span A (Set.range solution.var) = ⊤ ∧ solution.π.ker = Submodule.span A (Set.range 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{A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.12 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.9)] (solution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.12), LE.le.{max u w₀} (Submodule.{u, max u w₀} A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))))) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3))))) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))) (Preorder.toLE.{max u w₀} (Submodule.{u, max u w₀} A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))))) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3))))) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))) (PartialOrder.toPreorder.{max u w₀} (Submodule.{u, max u w₀} A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))))) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3))))) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1746160624._hygCtx._hyg.3)))) (Submodule.instPartialOrder.{u, max u w₀} A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.12 (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))) π (Finsupp.single.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))))) g (OfNat.ofNat.{u} A 1 (One.toOfNat1.{u} A (AddMonoidWithOne.toOne.{u} A (AddGroupWithOne.toAddMonoidWithOne.{u} A (Ring.toAddGroupWithOne.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))))))","typeFull":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : π ∘ₗ relations.map = 0) (g : relations.G),\n  (Module.Relations.Solution.ofπ' π hπ).var g = π (Finsupp.single g 1)","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : π ∘ₗ relations.map = 0) (g : 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{A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.12 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.9)] (solution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.423787974._hygCtx._hyg.12), Iff (Function.Surjective.{max (succ u) (succ w₀), succ v} 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: Ring A] →\n    {relations : Module.Relations A} →\n      {M : Type v} →\n        [inst_1 : AddCommGroup M] →\n          [inst_2 : Module A M] →\n            {motive : relations.Solution M → Sort u_1} →\n              (t : relations.Solution M) →\n                ((var : relations.G → M) →\n                    (linearCombination_var_relation :\n                        ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0) →\n                      motive { var := var, linearCombination_var_relation := linearCombination_var_relation }) →\n                  motive t","typeReadable":"{A : Type u} →\n  [inst : Ring A] →\n    {relations : Module.Relations A} →\n      {M : Type v} →\n        [inst_1 : AddCommGroup M] →\n          [inst_2 : Module A M] →\n            {motive : relations.Solution M → Sort u_1} →\n              (t : relations.Solution M) →\n                ((var : relations.G → M) →\n                    (linearCombination_var_relation :\n                        ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0) →\n                      motive { var := var, linearCombination_var_relation := linearCombination_var_relation }) →\n                  motive 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      (t : relations.Solution M) → motive 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.75) f f')), motive (Module.Relations.Solution.IsPresentationCore.mk.{w', w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.26 solution desc postcomp_desc postcomp_injective)) -> (motive t)","typeFull":"{A : Type u} →\n  [inst : Ring A] →\n    {relations : Module.Relations A} →\n      {M : Type v} →\n        [inst_1 : AddCommGroup M] →\n          [inst_2 : Module A M] →\n            {solution : relations.Solution M} →\n              {motive : solution.IsPresentationCore → Sort u_1} →\n                (t : solution.IsPresentationCore) →\n                  ((desc :\n                        {N : Type w'} →\n                          [inst_3 : AddCommGroup N] → [inst_4 : Module A N] → relations.Solution N → M →ₗ[A] N) →\n                      (postcomp_desc :\n                          ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] (s : relations.Solution N),\n                            solution.postcomp (desc s) = s) →\n                        (postcomp_injective :\n                            ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] {f f' : M →ₗ[A] N},\n                              solution.postcomp f = solution.postcomp f' → f = f') →\n                          motive\n                            { desc := desc, postcomp_desc := postcomp_desc,\n                              postcomp_injective := postcomp_injective }) →\n                    motive t","typeReadable":"{A : Type u} →\n  [inst : Ring A] →\n    {relations : Module.Relations A} →\n      {M : Type v} →\n        [inst_1 : AddCommGroup M] →\n          [inst_2 : Module A M] →\n            {solution : relations.Solution M} →\n              {motive : solution.IsPresentationCore → Sort u_1} →\n                (t : solution.IsPresentationCore) →\n                  ((desc :\n                        {N : Type w'} →\n                          [inst_3 : AddCommGroup N] → [inst_4 : Module A N] → relations.Solution N → M →ₗ[A] N) →\n                      (postcomp_desc :\n                          ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] (s : relations.Solution N),\n                            solution.postcomp (desc s) = s) →\n                        (postcomp_injective :\n                            ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] {f f' : M →ₗ[A] N},\n                              solution.postcomp f = solution.postcomp f' → f = f') →\n                          motive\n                            { desc := desc, postcomp_desc := postcomp_desc,\n                              postcomp_injective := postcomp_injective }) →\n                    motive t","typeReferences":[["Module","Relations","Solution","postcomp"],["Module"],["Module","Relations"],["AddCommGroup"],["LinearMap"],["Module","Relations","Solution","IsPresentationCore"],["Module","Relations","Solution","IsPresentationCore","mk"],["Ring","toSemiring"],["Module","Relations","Solution"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["AddCommGroup","toAddCommMonoid"],["Eq"],["Ring"]],"valueReferences":[["Module","Relations","Solution","IsPresentationCore","rec"]]},{"isProp":false,"kind":"definition","name":["Module","Relations","Solution","ofQuotient"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3 : Ring.{u} A] (relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3), Module.Relations.Solution.{w₀, w₁, max u w₀, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3 relations (Module.Relations.Quotient.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3 relations) (Module.Relations.instAddCommGroupQuotient.{u, w₀, w₁} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3 relations) (Module.Relations.instQuotient.{u, w₀, w₁} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3 relations)","typeFull":"{A : Type u} → [inst : Ring A] → (relations : Module.Relations A) → relations.Solution relations.Quotient","typeReadable":"{A : Type u} → [inst : Ring A] → (relations : Module.Relations A) → relations.Solution relations.Quotient","typeReferences":[["Module","Relations","instAddCommGroupQuotient"],["Module","Relations","Quotient"],["Module","Relations"],["Module","Relations","instQuotient"],["Module","Relations","Solution"],["Ring"]],"valueReferences":[["Module","Relations","Solution","ofQuotient","_proof_1"],["Module","Relations","instAddCommGroupQuotient"],["Module","Relations","Quotient"],["Module","Relations","instQuotient"],["Module","Relations","toQuotient"],["Module","Relations","Solution","ofπ"]]},{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","ofπ","eq_1"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.9 : AddCommGroup.{v} M] 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.12) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))))) => M) (LinearMap.instFunLike.{u, u, max u w₀, v} A A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))))) M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.9) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.12 (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))) π (Finsupp.single.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3))))) g (OfNat.ofNat.{u} A 1 (One.toOfNat1.{u} A (AddMonoidWithOne.toOne.{u} A (AddGroupWithOne.toAddMonoidWithOne.{u} A (Ring.toAddGroupWithOne.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3))))))) (fun (r : Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) => Module.Relations.Solution.ofπ._proof_1.{v, w₀, u, w₁} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.12 π hπ r))","typeFull":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : ∀ (r : relations.R), π (relations.relation r) = 0),\n  Module.Relations.Solution.ofπ π hπ = { var := fun g => π (Finsupp.single g 1), linearCombination_var_relation := ⋯ }","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : ∀ (r : relations.R), π (relations.relation r) = 0),\n  Module.Relations.Solution.ofπ π hπ = { var := fun g => π (Finsupp.single g 1), linearCombination_var_relation := ⋯ }","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["LinearMap","instFunLike"],["Module"],["Module","Relations"],["AddGroupWithOne","toAddMonoidWithOne"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["Module","Relations","Solution"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Finsupp","single"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["RingHom","id"],["Module","Relations","G"],["Zero","toOfNat0"],["Module","Relations","Solution","ofπ","_proof_1"],["Semiring","toModule"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Module","Relations","Solution","ofπ"],["Finsupp","module"],["Module","Relations","Solution","mk"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommGroup"],["LinearMap"],["OfNat","ofNat"],["Module","Relations","R"],["Ring","toSemiring"],["One","toOfNat1"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Module","Relations","relation"],["AddMonoidWithOne","toOne"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Finsupp","instAddCommMonoid"],["Finsupp"],["Ring"]],"valueReferences":[["Eq","refl"],["Module","Relations","Solution","ofπ"],["Module","Relations","Solution"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Module","Presentation","Basic",0,"Module","Relations","Solution","ext","match_1"],"typeFallback":"forall {A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 : Type.{u_4}} {inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 : Ring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61} {relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63 : Module.Relations.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62} {M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 : Type.{u_3}} {inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65 : AddCommGroup.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64} {inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.26.Mathlib.Algebra.Module.Presentation.Basic._hyg.66 : Module.{u_4, u_3} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62) (AddCommGroup.toAddCommMonoid.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65)} (motive : (Module.Relations.Solution.{u_1, u_2, u_3, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63 M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.26.Mathlib.Algebra.Module.Presentation.Basic._hyg.66) -> Prop) (h._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.67.80 : Module.Relations.Solution.{u_1, u_2, u_3, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63 M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.26.Mathlib.Algebra.Module.Presentation.Basic._hyg.66), (forall (var._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.89 : (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) -> M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64) (linearCombination_var_relation._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.90 : forall (r : Module.Relations.R.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63), Eq.{succ u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (DFunLike.coe.{max (max (succ u_3) (succ u_1)) (succ u_4), max (succ u_1) (succ u_4), succ u_3} (LinearMap.{u_4, u_4, max u_4 u_1, u_3} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62) (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62) (RingHom.id.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62))) (Finsupp.{u_1, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (MulZeroClass.toZero.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonUnitalNonAssocSemiring.toMulZeroClass.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62)))))) M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (Finsupp.instAddCommMonoid.{u_1, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62))))) (AddCommGroup.toAddCommMonoid.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65) (Finsupp.module.{u_1, u_4, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62)))) (Semiring.toModule.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.26.Mathlib.Algebra.Module.Presentation.Basic._hyg.66) (Finsupp.{u_1, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (MulZeroClass.toZero.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonUnitalNonAssocSemiring.toMulZeroClass.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62)))))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Finsupp.{u_1, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (MulZeroClass.toZero.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonUnitalNonAssocSemiring.toMulZeroClass.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62)))))) => M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64) (LinearMap.instFunLike.{u_4, u_4, max u_1 u_4, u_3} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Finsupp.{u_1, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (MulZeroClass.toZero.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonUnitalNonAssocSemiring.toMulZeroClass.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62)))))) M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62) (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62) (Finsupp.instAddCommMonoid.{u_1, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62))))) (AddCommGroup.toAddCommMonoid.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65) (Finsupp.module.{u_1, u_4, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62)))) (Semiring.toModule.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.26.Mathlib.Algebra.Module.Presentation.Basic._hyg.66 (RingHom.id.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Semiring.toNonAssocSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62)))) (Finsupp.linearCombination.{u_1, u_3, u_4} (Module.Relations.G.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63) M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 (Ring.toSemiring.{u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62) (AddCommGroup.toAddCommMonoid.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.26.Mathlib.Algebra.Module.Presentation.Basic._hyg.66 var._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.89) (Module.Relations.relation.{u_1, u_2, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63 r)) (OfNat.ofNat.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 0 (Zero.toOfNat0.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (NegZeroClass.toZero.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (SubNegZeroMonoid.toNegZeroClass.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (SubtractionMonoid.toSubNegZeroMonoid.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (SubtractionCommMonoid.toSubtractionMonoid.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 (AddCommGroup.toDivisionAddCommMonoid.{u_3} M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65)))))))), motive (Module.Relations.Solution.mk.{u_1, u_2, u_3, u_4} A._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.61 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.17.Mathlib.Algebra.Module.Presentation.Basic._hyg.62 relations._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.63 M._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.64 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.23.Mathlib.Algebra.Module.Presentation.Basic._hyg.65 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx.26.Mathlib.Algebra.Module.Presentation.Basic._hyg.66 var._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.89 linearCombination_var_relation._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.90)) -> (motive h._@.Mathlib.Algebra.Module.Presentation.Basic._hyg.67.80)","typeFull":"∀ {A : Type u_4} {inst : Ring A} {relations : Module.Relations A} {M : Type u_3} {inst_1 : AddCommGroup M}\n  {inst_2 : Module A M} (motive : relations.Solution M → Prop) (h : relations.Solution M),\n  (∀ (var : relations.G → M)\n      (linearCombination_var_relation :\n        ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0),\n      motive { var := var, linearCombination_var_relation := linearCombination_var_relation }) →\n    motive h","typeReadable":"∀ {A : Type u_4} {inst : Ring A} {relations : Module.Relations A} {M : Type u_3} {inst_1 : AddCommGroup M}\n  {inst_2 : Module A M} (motive : relations.Solution M → Prop) (h : relations.Solution M),\n  (∀ (var : relations.G → M)\n      (linearCombination_var_relation :\n        ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0),\n      motive { var := var, linearCombination_var_relation := linearCombination_var_relation }) →\n    motive h","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["LinearMap","instFunLike"],["Module"],["Module","Relations"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["Module","Relations","Solution"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module","Relations","G"],["Zero","toOfNat0"],["Eq"],["Semiring","toModule"],["Finsupp","module"],["Module","Relations","Solution","mk"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Finsupp","linearCombination"],["AddCommGroup"],["LinearMap"],["OfNat","ofNat"],["Module","Relations","R"],["Ring","toSemiring"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Module","Relations","relation"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Finsupp","instAddCommMonoid"],["Finsupp"],["Ring"]],"valueReferences":[["Module","Relations","Solution","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","ofπ'","eq_1"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.12 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.9)] (π : LinearMap.{u, u, max u w₀, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))))) M (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.9) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.12) (hπ : Eq.{max (max (succ u) (succ v)) (succ w₁)} (LinearMap.{u, u, max u w₁, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) (Finsupp.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))))) M (Finsupp.instAddCommMonoid.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.9) (Finsupp.module.{w₁, u, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.12) (LinearMap.comp.{u, u, u, max u w₁, max u w₀, v} A A A (Finsupp.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))))) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))))) M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))))) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.9) (Finsupp.module.{w₁, u, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.12 (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) => M) (LinearMap.instFunLike.{u, u, max w₀ u, v} A A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Finsupp.linearCombination.{w₀, v, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) M A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var) (Module.Relations.relation.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations r)) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (NegZeroClass.toZero.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23)))))))), Eq.{max (succ v) (succ w₀)} (Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26) (Module.Relations.Solution.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var linearCombination_var_relation) (Module.Relations.Solution.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var_1 (Eq.ndrec.{0, max (succ v) (succ w₀)} ((Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M) var (fun (var : (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M) => forall (r : Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations), Eq.{succ v} M (DFunLike.coe.{max (max (succ v) (succ w₀)) (succ u), max (succ w₀) (succ u), succ v} (LinearMap.{u, u, max u w₀, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) M (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) => M) (LinearMap.instFunLike.{u, u, max w₀ u, v} A A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Finsupp.linearCombination.{w₀, v, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) M A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var) (Module.Relations.relation.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations r)) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (NegZeroClass.toZero.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23)))))))) linearCombination_var_relation var_1 e_var))","typeFull":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (var var_1 : relations.G → M) (e_var : var = var_1)\n  (linearCombination_var_relation : ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0),\n  { var := var, linearCombination_var_relation := linearCombination_var_relation } =\n    { var := var_1, linearCombination_var_relation := ⋯ }","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (var var_1 : relations.G → M) (e_var : var = var_1)\n  (linearCombination_var_relation : ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0),\n  { var := var, linearCombination_var_relation := linearCombination_var_relation } =\n    { var := var_1, linearCombination_var_relation := ⋯ }","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["LinearMap","instFunLike"],["Module"],["Module","Relations"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["Module","Relations","Solution"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module","Relations","G"],["Zero","toOfNat0"],["Eq","ndrec"],["Eq"],["Semiring","toModule"],["Finsupp","module"],["Module","Relations","Solution","mk"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Finsupp","linearCombination"],["AddCommGroup"],["LinearMap"],["Module","Relations","R"],["OfNat","ofNat"],["Ring","toSemiring"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Module","Relations","relation"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Finsupp","instAddCommMonoid"],["Finsupp"],["Ring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["LinearMap","instFunLike"],["SubtractionCommMonoid","toSubtractionMonoid"],["DFunLike","coe"],["SubNegZeroMonoid","toNegZeroClass"],["Module","Relations","Solution"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module","Relations","G"],["Zero","toOfNat0"],["Eq","ndrec"],["Eq"],["Eq","rec"],["Semiring","toModule"],["Finsupp","module"],["Module","Relations","Solution","mk"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Finsupp","linearCombination"],["LinearMap"],["Module","Relations","R"],["OfNat","ofNat"],["Ring","toSemiring"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Eq","refl"],["Module","Relations","relation"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["Finsupp","instAddCommMonoid"],["Finsupp"]]},{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","IsPresentation","desc_var"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.12 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.9)] {solution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.12} (h : Module.Relations.Solution.IsPresentation.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.12 solution) {N : Type.{v'}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.20 : AddCommGroup.{v'} N] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.23 : Module.{u, v'} A N (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v'} N inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.20)] (s : Module.Relations.Solution.{w₀, w₁, v', u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.3 relations N inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2730720690._hygCtx._hyg.23) (g : 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Finsupp.linearCombination.{w₀, v, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) M A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var) (Module.Relations.relation.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations r)) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (NegZeroClass.toZero.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23)))))))} {var_1 : (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M} {linearCombination_var_relation_1 : forall (r : Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations), Eq.{succ v} M (DFunLike.coe.{max (max (succ v) (succ w₀)) (succ u), max (succ w₀) (succ u), succ v} (LinearMap.{u, u, max u w₀, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) M (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17)))) (Finsupp.linearCombination.{w₀, v, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) M A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var_1) (Module.Relations.relation.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations r)) (OfNat.ofNat.{v} M 0 (Zero.toOfNat0.{v} M (NegZeroClass.toZero.{v} M (SubNegZeroMonoid.toNegZeroClass.{v} M (SubtractionMonoid.toSubNegZeroMonoid.{v} M (SubtractionCommMonoid.toSubtractionMonoid.{v} M (AddCommGroup.toDivisionAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23)))))))}, (Eq.{max (succ v) (succ w₀)} (Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26) (Module.Relations.Solution.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var linearCombination_var_relation) (Module.Relations.Solution.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.26 var_1 linearCombination_var_relation_1)) -> (Eq.{max (succ v) (succ w₀)} ((Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 relations) -> M) var var_1)","typeFull":"∀ {A : Type u} {inst : Ring A} {relations : Module.Relations A} {M : Type v} {inst_1 : AddCommGroup M}\n  {inst_2 : Module A M} {var : relations.G → M}\n  {linearCombination_var_relation : ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0}\n  {var_1 : relations.G → M}\n  {linearCombination_var_relation_1 :\n    ∀ (r : relations.R), (Finsupp.linearCombination A var_1) (relations.relation r) = 0},\n  { var := var, linearCombination_var_relation := linearCombination_var_relation } =\n      { var := var_1, linearCombination_var_relation := linearCombination_var_relation_1 } →\n    var = var_1","typeReadable":"∀ {A : Type u} {inst : Ring A} {relations : Module.Relations A} {M : Type v} {inst_1 : AddCommGroup M}\n  {inst_2 : Module A M} {var : relations.G → M}\n  {linearCombination_var_relation : ∀ (r : relations.R), (Finsupp.linearCombination A var) (relations.relation r) = 0}\n  {var_1 : relations.G → M}\n  {linearCombination_var_relation_1 :\n    ∀ (r : relations.R), (Finsupp.linearCombination A var_1) (relations.relation r) = 0},\n  { var := var, linearCombination_var_relation := linearCombination_var_relation } =\n      { var := var_1, linearCombination_var_relation := linearCombination_var_relation_1 } →\n    var = 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 {inst'_2 : Module A' M'} →\n                            {t' : relations'.Solution M'} →\n                              A = A' →\n                                inst ≍ inst' →\n                                  relations ≍ relations' →\n                                    M = M' →\n                                      inst_1 ≍ inst'_1 →\n                                        inst_2 ≍ inst'_2 → t ≍ t' → Module.Relations.Solution.noConfusionType P t t'","typeReadable":"{P : Sort u_1} →\n  {A : Type u} →\n    {inst : Ring A} →\n      {relations : Module.Relations A} →\n        {M : Type v} →\n          {inst_1 : AddCommGroup M} →\n            {inst_2 : Module A M} →\n              {t : relations.Solution M} →\n                {A' : Type u} →\n                  {inst' : Ring A'} →\n                    {relations' : Module.Relations A'} →\n                      {M' : Type v} →\n                        {inst'_1 : AddCommGroup M'} →\n                          {inst'_2 : Module A' M'} →\n  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inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))))))","typeFull":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : ∀ (r : relations.R), π (relations.relation r) = 0)\n  (g : relations.G), (Module.Relations.Solution.ofπ π hπ).var g = π (Finsupp.single g 1)","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (π : (relations.G →₀ A) →ₗ[A] M) (hπ : ∀ (r : relations.R), π (relations.relation r) = 0)\n  (g : relations.G), (Module.Relations.Solution.ofπ π hπ).var g = π (Finsupp.single g 1)","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["LinearMap","instFunLike"],["Module"],["Module","Relations"],["AddGroupWithOne","toAddMonoidWithOne"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Finsupp","single"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["RingHom","id"],["Module","Relations","G"],["Zero","toOfNat0"],["Semiring","toModule"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Module","Relations","Solution","ofπ"],["Finsupp","module"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommGroup"],["LinearMap"],["OfNat","ofNat"],["Module","Relations","R"],["Ring","toSemiring"],["Module","Relations","Solution","var"],["One","toOfNat1"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Module","Relations","relation"],["AddMonoidWithOne","toOne"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Finsupp","instAddCommMonoid"],["Finsupp"],["Ring"]],"valueReferences":[["Module","Relations","Solution","var"],["Eq","refl"],["Module","Relations","Solution","ofπ"]]},{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","ofπ_π"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.12 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.9)] (π : LinearMap.{u, u, max u w₀, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.9) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.12) (hπ : forall (r : Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3426846455._hygCtx._hyg.3 relations), Eq.{succ v} M (DFunLike.coe.{max (max (succ u) (succ v)) (succ w₀), 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.12) 0 (Zero.toOfNat0.{max (max u v) w₁} (LinearMap.{u, u, max u w₁, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.9) (Finsupp.module.{w₁, u, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.12) (LinearMap.instZero.{u, u, max u w₁, v} A A (Finsupp.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3)))))) M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.846070317._hygCtx._hyg.9) 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[inst_2 : Module A M] {f f' : relations.Quotient →ₗ[A] M},\n  (∀ (g : relations.G),\n      f (relations.toQuotient (Finsupp.single g 1)) = f' (relations.toQuotient (Finsupp.single g 1))) →\n    f = f'","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] {f f' : relations.Quotient →ₗ[A] M},\n  (∀ (g : relations.G),\n      f (relations.toQuotient (Finsupp.single g 1)) = f' (relations.toQuotient (Finsupp.single g 1))) →\n    f = 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relations N inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.37) -> (LinearMap.{u, u, v, w'} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17))) M N (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.23) (AddCommGroup.toAddCommMonoid.{w'} N inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.34) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.37)) desc desc_1)","typeFull":"∀ {A : Type u} {inst : Ring A} {relations : Module.Relations A} {M : Type v} {inst_1 : AddCommGroup M}\n  {inst_2 : Module A M} {solution : relations.Solution M}\n  {desc : {N : Type w'} → [inst_3 : AddCommGroup N] → [inst_4 : Module A N] → relations.Solution N → M →ₗ[A] N}\n  {postcomp_desc :\n    ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] (s : relations.Solution N),\n      solution.postcomp (desc s) = s}\n  {postcomp_injective :\n    ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] {f f' : M →ₗ[A] N},\n      solution.postcomp f = solution.postcomp f' → f = f'}\n  {desc_1 : {N : Type w'} → [inst_3 : AddCommGroup N] → [inst_4 : Module A N] → relations.Solution N → M →ₗ[A] N}\n  {postcomp_desc_1 :\n    ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] (s : relations.Solution N),\n      solution.postcomp (desc_1 s) = s}\n  {postcomp_injective_1 :\n    ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] {f f' : M →ₗ[A] N},\n      solution.postcomp f = solution.postcomp f' → f = f'},\n  { desc := desc, postcomp_desc := postcomp_desc, postcomp_injective := postcomp_injective } =\n      { desc := desc_1, postcomp_desc := postcomp_desc_1, postcomp_injective := postcomp_injective_1 } →\n    desc = desc_1","typeReadable":"∀ {A : Type u} {inst : Ring A} {relations : Module.Relations A} {M : Type v} {inst_1 : AddCommGroup M}\n  {inst_2 : Module A M} {solution : relations.Solution M}\n  {desc : {N : Type w'} → [inst_3 : AddCommGroup N] → [inst_4 : Module A N] → relations.Solution N → M →ₗ[A] N}\n  {postcomp_desc :\n    ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] (s : relations.Solution N),\n      solution.postcomp (desc s) = s}\n  {postcomp_injective :\n    ∀ {N : Type w'} [inst_3 : AddCommGroup N] [inst_4 : Module A N] {f f' : M →ₗ[A] N},\n      solution.postcomp f = solution.postcomp f' → f = f'}\n  {desc_1 : {N : Type w'} → [inst_3 : AddCommGroup N] → [inst_4 : Module A N] → relations.Solution N → M →ₗ[A] N}\n  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inst._@.Mathlib.Algebra.Module.Presentation.Basic.1492669931._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1492669931._hygCtx._hyg.3)))))) (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1492669931._hygCtx._hyg.3 relations) (Module.Relations.relation.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.1492669931._hygCtx._hyg.3 relations)))","typeFull":"∀ {A : Type u} [inst : Ring A] (relations : Module.Relations A),\n  relations.map.range = Submodule.span A (Set.range relations.relation)","typeReadable":"∀ {A : Type u} [inst : Ring A] (relations : Module.Relations A),\n  relations.map.range = Submodule.span A (Set.range 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{A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 : Ring.{u} A] {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19)] (toRelations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15) (toSolution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22) (toIsPresentation : Module.Relations.Solution.IsPresentation.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 toSolution) (toRelations_1 : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15) (toSolution_1 : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations_1 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22) (toIsPresentation_1 : Module.Relations.Solution.IsPresentation.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations_1 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 toSolution_1), Eq.{1} Prop (Eq.{max (max (max (succ u) (succ v)) (succ (succ w₀))) (succ (succ w₁))} (Module.Presentation.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22) (Module.Presentation.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 toRelations toSolution toIsPresentation) (Module.Presentation.mk.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 toRelations_1 toSolution_1 toIsPresentation_1)) (And (Eq.{max (max (succ u) (succ (succ w₀))) (succ (succ w₁))} (Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15) toRelations toRelations_1) (HEq.{max (succ v) (succ w₀)} (Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22) toSolution (Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations_1 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22) toSolution_1))","typeFull":"∀ {A : Type u} [inst : Ring A] {M : Type v} [inst_1 : AddCommGroup M] [inst_2 : Module A M]\n  (toRelations : Module.Relations A) (toSolution : toRelations.Solution M)\n  (toIsPresentation : toSolution.IsPresentation) (toRelations_1 : Module.Relations A)\n  (toSolution_1 : toRelations_1.Solution M) (toIsPresentation_1 : toSolution_1.IsPresentation),\n  ({ toRelations := toRelations, toSolution := toSolution, toIsPresentation := toIsPresentation } =\n      { toRelations := toRelations_1, toSolution := toSolution_1, toIsPresentation := toIsPresentation_1 }) =\n    (toRelations = toRelations_1 ∧ toSolution ≍ toSolution_1)","typeReadable":"∀ {A : Type u} [inst : Ring A] {M : Type v} [inst_1 : AddCommGroup M] [inst_2 : Module A M]\n  (toRelations : Module.Relations A) (toSolution : toRelations.Solution M)\n  (toIsPresentation : toSolution.IsPresentation) (toRelations_1 : Module.Relations A)\n  (toSolution_1 : toRelations_1.Solution M) (toIsPresentation_1 : toSolution_1.IsPresentation),\n  ({ toRelations := toRelations, toSolution := toSolution, toIsPresentation := toIsPresentation } =\n      { toRelations := toRelations_1, toSolution := toSolution_1, toIsPresentation := toIsPresentation_1 }) =\n    (toRelations = toRelations_1 ∧ toSolution ≍ toSolution_1)","typeReferences":[["Module","Presentation","mk"],["Module"],["Module","Relations","Solution","IsPresentation"],["Module","Relations"],["And"],["AddCommGroup"],["Ring","toSemiring"],["Module","Relations","Solution"],["Module","Presentation"],["AddCommGroup","toAddCommMonoid"],["HEq"],["Eq"],["Ring"]],"valueReferences":[["Module","Presentation","mk"],["Module","Presentation","mk","inj"],["Module","Relations","Solution","IsPresentation"],["And"],["Module","Relations"],["Lean","injEq_helper"],["Module","Relations","Solution"],["Module","Presentation"],["Eq","propIntro"],["Eq","refl"],["eq_of_heq"],["HEq"],["Eq"],["Eq","ndrec"]]},{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","linearCombination_var_relation"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.905924591._hygCtx._hyg.17 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A 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    [inst' : Ring A'] →\n                      {relations' : Module.Relations A'} →\n                        {M' : Type v} →\n                          [inst'_1 : AddCommGroup M'] →\n                            [inst'_2 : Module A' M'] →\n                              {solution' : relations'.Solution M'} → solution'.IsPresentationCore → Sort u_1","typeReadable":"Sort u_1 →\n  {A : Type u} →\n    [inst : Ring A] →\n      {relations : Module.Relations A} →\n        {M : Type v} →\n          [inst_1 : AddCommGroup M] →\n            [inst_2 : Module A M] →\n              {solution : relations.Solution M} →\n                solution.IsPresentationCore →\n                  {A' : Type u} →\n                    [inst' : Ring A'] →\n                      {relations' : Module.Relations A'} →\n                        {M' : Type v} →\n                          [inst'_1 : AddCommGroup M'] →\n                            [inst'_2 : Module A' M'] →\n                              {solution' : 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: relations.Solution N} (h' : solution'.IsPresentation),\n  solution'.postcomp (h.desc solution' ∘ₗ h'.desc solution) = solution'.postcomp 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{A : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3 : Ring.{u_4} A] {relations : Module.Relations.{u_2, u_3, u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3} {M : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.9 : AddCommGroup.{u_5} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.12 : Module.{u_4, u_5} A M (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.9)] {solution : Module.Relations.Solution.{u_2, u_3, u_5, u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.12} (h : Module.Relations.Solution.IsPresentationCore.{max u_6 u_1, u_2, u_3, u_5, u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.12 solution) {N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46 : AddCommGroup.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47 : Module.{u_4, u_1} A N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46)] (s : Module.Relations.Solution.{u_2, u_3, u_1, u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3 relations N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47), Eq.{max (succ u_1) (succ u_2)} (Module.Relations.Solution.{u_2, u_3, u_1, u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3 relations N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47) (Module.Relations.Solution.postcomp.{u_2, u_3, u_1, u_5, u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.12 solution N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47 (LinearMap.comp.{u_4, u_4, u_4, u_5, max u_6 u_1, u_1} A A A M (ULift.{u_6, u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45) N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.9) (ULift.addCommMonoid.{u_1, u_6} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46)) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.12 (ULift.module'.{u_4, u_1, u_6} A N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47 (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHomCompTriple.ids.{u_4, u_4} A A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3)))) (LinearEquiv.toLinearMap.{u_4, u_4, max u_6 u_1, u_1} A A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHomInvPair.ids.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3)) (RingHomInvPair.ids.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3)) (ULift.{u_6, u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45) N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (ULift.addCommMonoid.{u_1, u_6} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46)) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) (ULift.module'.{u_4, u_1, u_6} A N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47 (ULift.moduleEquiv.{u_4, u_1, u_6} A N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47)) (Module.Relations.Solution.IsPresentationCore.desc.{max u_6 u_1, u_2, u_3, u_5, u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.12 solution h (ULift.{u_6, u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45) (ULift.addCommGroup.{u_1, u_6} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) (ULift.module'.{u_4, u_1, u_6} A N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47) (Module.Relations.Solution.postcomp.{u_2, u_3, max u_1 u_6, u_1, u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3 relations N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47 s (ULift.{u_6, u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45) (ULift.addCommGroup.{u_1, u_6} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) (ULift.module'.{u_4, u_1, u_6} A N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47) (LinearEquiv.toLinearMap.{u_4, u_4, u_1, max u_6 u_1} A A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHomInvPair.ids.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3)) (RingHomInvPair.ids.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3)) N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (ULift.{u_6, u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) (ULift.addCommMonoid.{u_1, u_6} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46)) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47 (ULift.module'.{u_4, u_1, u_6} A N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.53.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.46) inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47) (LinearEquiv.symm.{u_4, u_4, max u_6 u_1, u_1} A A (ULift.{u_6, u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45) N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (ULift.addCommMonoid.{u_1, u_6} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (AddCommGroup.toAddCommMonoid.{u_1} N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx.56.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.47 (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHom.id.{u_4} A (Semiring.toNonAssocSemiring.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3))) (RingHomInvPair.ids.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3)) (RingHomInvPair.ids.{u_4} A (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3)) (ULift.moduleEquiv.{u_4, u_1, u_6} A N._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.45 (Ring.toSemiring.{u_4} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3967909212._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{u_1} 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{inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 : AddCommGroup.{v} M} {inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19)} {toRelations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15} {toSolution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22} {toIsPresentation : Module.Relations.Solution.IsPresentation.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 toSolution} {toRelations_1 : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15} {toSolution_1 : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations_1 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22} {toIsPresentation_1 : Module.Relations.Solution.IsPresentation.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 toRelations_1 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 toSolution_1}, (Eq.{max (max (max (succ u) (succ v)) (succ 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toSolution.IsPresentation} {toRelations_1 : Module.Relations A}\n  {toSolution_1 : toRelations_1.Solution M} {toIsPresentation_1 : toSolution_1.IsPresentation},\n  { toRelations := toRelations, toSolution := toSolution, toIsPresentation := toIsPresentation } =\n      { toRelations := toRelations_1, toSolution := toSolution_1, toIsPresentation := toIsPresentation_1 } →\n    toRelations = toRelations_1 ∧ toSolution ≍ toSolution_1","typeReadable":"∀ {A : Type u} {inst : Ring A} {M : Type v} {inst_1 : AddCommGroup M} {inst_2 : Module A M}\n  {toRelations : Module.Relations A} {toSolution : toRelations.Solution M}\n  {toIsPresentation : toSolution.IsPresentation} {toRelations_1 : Module.Relations A}\n  {toSolution_1 : toRelations_1.Solution M} {toIsPresentation_1 : toSolution_1.IsPresentation},\n  { toRelations := toRelations, toSolution := toSolution, toIsPresentation := toIsPresentation } =\n      { toRelations := toRelations_1, toSolution := toSolution_1, toIsPresentation := toIsPresentation_1 } →\n    toRelations = toRelations_1 ∧ toSolution ≍ toSolution_1","typeReferences":[["Module","Presentation","mk"],["Module"],["Module","Relations","Solution","IsPresentation"],["Module","Relations"],["And"],["AddCommGroup"],["Ring","toSemiring"],["Module","Relations","Solution"],["Module","Presentation"],["AddCommGroup","toAddCommMonoid"],["HEq"],["Eq"],["Ring"]],"valueReferences":[["And","intro"],["Module","Presentation","mk","noConfusion"],["eq_of_heq"],["HEq"],["Module","Relations"],["And"],["Eq"],["Module","Relations","Solution"]]},{"isProp":true,"kind":"theorem","name":["Module","Relations","Solution","ofπ'_π"],"typeFallback":"forall {A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.9 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.12 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.9)] (π : LinearMap.{u, u, max u w₀, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3))) (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3)))))) M (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.9) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.12) (hπ : Eq.{max (max (succ u) (succ v)) (succ w₁)} (LinearMap.{u, u, max u w₁, v} A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3) (RingHom.id.{u} A (Semiring.toNonAssocSemiring.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3))) (Finsupp.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3)))))) M (Finsupp.instAddCommMonoid.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.9) (Finsupp.module.{w₁, u, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.12) (LinearMap.comp.{u, u, u, max u w₁, max u w₀, v} A A A (Finsupp.{w₁, u} (Module.Relations.R.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2221585689._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A 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relations.Solution M), solution.π ∘ₗ relations.map = 0","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (solution : relations.Solution M), solution.π ∘ₗ relations.map = 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{A : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3 : Ring.{u_3} A] (relations : Module.Relations.{u_1, u_2, u_3} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3) (r : Module.Relations.R.{u_1, u_2, u_3} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3 relations), Eq.{succ (max u_3 u_1)} (Module.Relations.Quotient.{u_1, u_2, u_3} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3 relations) (DFunLike.coe.{max (max (succ u_3) (succ (max u_3 u_1))) (succ u_1), max (succ u_3) (succ u_1), succ (max u_3 u_1)} (LinearMap.{u_3, u_3, max u_3 u_1, max u_3 u_1} A A (Ring.toSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3) (Ring.toSemiring.{u_3} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.725361648._hygCtx._hyg.3) (RingHom.id.{u_3} A (Semiring.toNonAssocSemiring.{u_3} A (Ring.toSemiring.{u_3} A 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{A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17 : Ring.{u} A] {relations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.23 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.26 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.23)] {solution : Module.Relations.Solution.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.17 relations M inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Presentation.Basic.188742726._hygCtx._hyg.26} 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{A : Type.{u}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 : Ring.{u} A] {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 : AddCommGroup.{v} M] [inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22 : Module.{u, v} A M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19)] {motive : (Module.Presentation.{w₀, w₁, v, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15 M inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.22) -> Sort.{u_1}}, (forall (toRelations : Module.Relations.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.2151515462._hygCtx._hyg.15) (toSolution : 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M","typeReadable":"{A : Type u} →\n  [inst : Ring A] →\n    {M : Type v} →\n      [inst_1 : AddCommGroup M] →\n        [inst_2 : Module A M] →\n          (G : Type w₀) →\n            (R : Type w₁) →\n              (relation : R → G →₀ A) →\n                (var : G → M) →\n                  (linearCombination_var_relation : ∀ (r : R), (Finsupp.linearCombination A var) (relation r) = 0) →\n                    Function.Bijective\n                        ⇑{ var := var, linearCombination_var_relation := linearCombination_var_relation }.fromQuotient →\n                      Module.Presentation A 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inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))))) => M) (LinearMap.instFunLike.{u, u, max u w₀, v} A A (Finsupp.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A (MulZeroClass.toZero.{u} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))))) M (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3) (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3) (Finsupp.instAddCommMonoid.{w₀, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3))))) (AddCommGroup.toAddCommMonoid.{v} M inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.9) (Finsupp.module.{w₀, u, u} (Module.Relations.G.{w₀, w₁, u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3 relations) A A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} A (NonAssocRing.toNonUnitalNonAssocRing.{u} A (Ring.toNonAssocRing.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3)))) (Semiring.toModule.{u} A (Ring.toSemiring.{u} A inst._@.Mathlib.Algebra.Module.Presentation.Basic.3858320564._hygCtx._hyg.3))) 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= Module.Relations.Solution.ofπ π_1 ⋯","typeReadable":"∀ {A : Type u} [inst : Ring A] {relations : Module.Relations A} {M : Type v} [inst_1 : AddCommGroup M]\n  [inst_2 : Module A M] (π π_1 : (relations.G →₀ A) →ₗ[A] M) (e_π : π = π_1)\n  (hπ : ∀ (r : relations.R), π (relations.relation r) = 0),\n  Module.Relations.Solution.ofπ π hπ = Module.Relations.Solution.ofπ π_1 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