diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Defs.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Defs.sym.json" new file mode 100644--- /dev/null +++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Defs.sym.json" @@ -0,0 +1 @@ +[{"isProp":true,"kind":"theorem","name":["Semiring","toModule","_proof_2"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Defs.1426575160._hygCtx._hyg.6 : Semiring.{u_1} R] (a : R) (b : R) (c : R), Eq.{succ u_1} R (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (MulZeroClass.toMul.{u_1} R (MulZeroOneClass.toMulZeroClass.{u_1} R (MonoidWithZero.toMulZeroOneClass.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Defs.1426575160._hygCtx._hyg.6))))) a (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R 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M._@.Mathlib.Algebra.Module.Defs._hyg.68 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.15.Mathlib.Algebra.Module.Defs._hyg.70))))))) (smul_add._@.Mathlib.Algebra.Module.Defs._hyg.95 : forall (a : R._@.Mathlib.Algebra.Module.Defs._hyg.67) (x : M._@.Mathlib.Algebra.Module.Defs._hyg.68) (y : M._@.Mathlib.Algebra.Module.Defs._hyg.68), Eq.{succ u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (HSMul.hSMul.{u_1, u_2, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHSMul.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (SMul.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 smul._@.Mathlib.Algebra.Module.Defs._hyg.98)) a (HAdd.hAdd.{u_2, u_2, u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHAdd.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddSemigroup.toAdd.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddMonoid.toAddSemigroup.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddCommMonoid.toAddMonoid.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.15.Mathlib.Algebra.Module.Defs._hyg.70)))) x y)) (HAdd.hAdd.{u_2, u_2, u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHAdd.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddSemigroup.toAdd.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddMonoid.toAddSemigroup.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddCommMonoid.toAddMonoid.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.15.Mathlib.Algebra.Module.Defs._hyg.70)))) (HSMul.hSMul.{u_1, u_2, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHSMul.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (SMul.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 smul._@.Mathlib.Algebra.Module.Defs._hyg.98)) a x) (HSMul.hSMul.{u_1, u_2, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHSMul.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (SMul.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 smul._@.Mathlib.Algebra.Module.Defs._hyg.98)) a y))) (add_smul._@.Mathlib.Algebra.Module.Defs._hyg.92 : forall (r : R._@.Mathlib.Algebra.Module.Defs._hyg.67) (s : R._@.Mathlib.Algebra.Module.Defs._hyg.67) (x : M._@.Mathlib.Algebra.Module.Defs._hyg.68), Eq.{succ u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (HSMul.hSMul.{u_1, u_2, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHSMul.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (SMul.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 smul._@.Mathlib.Algebra.Module.Defs._hyg.98)) (HAdd.hAdd.{u_1, u_1, u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 R._@.Mathlib.Algebra.Module.Defs._hyg.67 R._@.Mathlib.Algebra.Module.Defs._hyg.67 (instHAdd.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (Distrib.toAdd.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (NonUnitalNonAssocSemiring.toDistrib.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (Semiring.toNonAssocSemiring.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.12.Mathlib.Algebra.Module.Defs._hyg.69))))) r s) x) (HAdd.hAdd.{u_2, u_2, u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHAdd.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddCommMagma.toAdd.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddCommSemigroup.toAddCommMagma.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddCommMonoid.toAddCommSemigroup.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.15.Mathlib.Algebra.Module.Defs._hyg.70)))) (HSMul.hSMul.{u_1, u_2, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHSMul.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (SMul.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 smul._@.Mathlib.Algebra.Module.Defs._hyg.98)) r x) (HSMul.hSMul.{u_1, u_2, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHSMul.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (SMul.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 smul._@.Mathlib.Algebra.Module.Defs._hyg.98)) s x))) (zero_smul._@.Mathlib.Algebra.Module.Defs._hyg.93 : forall (x : M._@.Mathlib.Algebra.Module.Defs._hyg.68), Eq.{succ u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (HSMul.hSMul.{u_1, u_2, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (instHSMul.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (SMul.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 smul._@.Mathlib.Algebra.Module.Defs._hyg.98)) (OfNat.ofNat.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 0 (Zero.toOfNat0.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (MulZeroClass.toZero.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (Semiring.toNonAssocSemiring.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.12.Mathlib.Algebra.Module.Defs._hyg.69)))))) x) (OfNat.ofNat.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 0 (Zero.toOfNat0.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddZero.toZero.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddZeroClass.toAddZero.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddMonoid.toAddZeroClass.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 (AddCommMonoid.toAddMonoid.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.15.Mathlib.Algebra.Module.Defs._hyg.70))))))), motive (Module.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.12.Mathlib.Algebra.Module.Defs._hyg.69 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.15.Mathlib.Algebra.Module.Defs._hyg.70 (DistribMulAction.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (MonoidWithZero.toMonoid.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (Semiring.toMonoidWithZero.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.12.Mathlib.Algebra.Module.Defs._hyg.69)) (AddCommMonoid.toAddMonoid.{u_2} M._@.Mathlib.Algebra.Module.Defs._hyg.68 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.15.Mathlib.Algebra.Module.Defs._hyg.70) (MulAction.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (MonoidWithZero.toMonoid.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (Semiring.toMonoidWithZero.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.12.Mathlib.Algebra.Module.Defs._hyg.69)) (SemigroupAction.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 (Monoid.toSemigroup.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (MonoidWithZero.toMonoid.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 (Semiring.toMonoidWithZero.{u_1} R._@.Mathlib.Algebra.Module.Defs._hyg.67 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx.12.Mathlib.Algebra.Module.Defs._hyg.69))) (SMul.mk.{u_1, u_2} R._@.Mathlib.Algebra.Module.Defs._hyg.67 M._@.Mathlib.Algebra.Module.Defs._hyg.68 smul._@.Mathlib.Algebra.Module.Defs._hyg.98) mul_smul._@.Mathlib.Algebra.Module.Defs._hyg.97) one_smul._@.Mathlib.Algebra.Module.Defs._hyg.96) smul_zero._@.Mathlib.Algebra.Module.Defs._hyg.94 smul_add._@.Mathlib.Algebra.Module.Defs._hyg.95) add_smul._@.Mathlib.Algebra.Module.Defs._hyg.92 zero_smul._@.Mathlib.Algebra.Module.Defs._hyg.93)) -> (motive h._@.Mathlib.Algebra.Module.Defs._hyg.71.83)","typeFull":"∀ {R : Type u_1} {M : Type u_2} {inst : Semiring R} {inst_1 : AddCommMonoid M} (motive : Module R M → Prop)\n (h : Module R M),\n (∀ (smul : R → M → M) (mul_smul : ∀ (x y : R) (b : M), (x * y) • b = x • y • b) (one_smul : ∀ (b : M), 1 • b = b)\n (smul_zero : ∀ (a : R), a • 0 = 0) (smul_add : ∀ (a : R) (x y : M), a • (x + y) = a • x + a • y)\n (add_smul : ∀ (r s : R) (x : M), (r + s) • x = r • x + s • x) (zero_smul : ∀ (x : M), 0 • x = 0),\n motive\n { smul := smul, mul_smul := mul_smul, one_smul := one_smul, smul_zero := smul_zero, smul_add := smul_add,\n add_smul := add_smul, zero_smul := zero_smul }) →\n motive h","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {inst : Semiring R} {inst_1 : AddCommMonoid M} (motive : Module R M → Prop)\n (h : Module R M),\n (∀ (smul : R → M → M) (mul_smul : ∀ (x y : R) (b : M), (x * y) • b = x • y • b) (one_smul : ∀ (b : M), 1 • b = b)\n (smul_zero : ∀ (a : R), a • 0 = 0) (smul_add : ∀ (a : R) (x y : M), a • (x + y) = a • x + a • y)\n (add_smul : ∀ (r s : R) (x : M), (r + s) • x = r • x + s • x) (zero_smul : ∀ (x : M), 0 • x = 0),\n motive\n { smul := smul, mul_smul := mul_smul, one_smul := one_smul, smul_zero := smul_zero, smul_add := smul_add,\n add_smul := add_smul, zero_smul := zero_smul }) →\n motive h","typeReferences":[["MulOneClass","toMulOne"],["SemigroupAction","mk"],["Module"],["HMul","hMul"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["Module","mk"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["AddSemigroup","toAdd"],["SMul","mk"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["MulOne","toOne"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","mk"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HAdd","hAdd"],["AddCommMonoid","toAddCommSemigroup"],["One","toOfNat1"],["MulZeroClass","toZero"],["AddMonoid","toAddSemigroup"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["instHMul"],["AddZero","toZero"],["MulAction","mk"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["MulOneClass","toMulOne"],["SemigroupAction","mk"],["HMul","hMul"],["SemigroupAction","toSMul"],["Module","casesOn"],["AddCommMonoid","toAddMonoid"],["MulAction","casesOn"],["Semigroup","toMul"],["Module","mk"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["DistribMulAction","casesOn"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["AddSemigroup","toAdd"],["Distrib","toAdd"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","mk"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["SemigroupAction","casesOn"],["OfNat","ofNat"],["HAdd","hAdd"],["One","toOfNat1"],["AddCommMonoid","toAddCommSemigroup"],["SMul","casesOn"],["MulZeroClass","toZero"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["MulAction","mk"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["neg_smul"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Defs.2907391035._hygCtx._hyg.6 : Ring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Defs.2907391035._hygCtx._hyg.9 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Defs.2907391035._hygCtx._hyg.12 : Module.{u_1, u_3} R M (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Defs.2907391035._hygCtx._hyg.6) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.2907391035._hygCtx._hyg.9)] (r : R) (x : M), Eq.{succ u_3} M (HSMul.hSMul.{u_1, u_3, u_3} R M M (instHSMul.{u_1, u_3} R M (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.2907391035._hygCtx._hyg.9)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M 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{R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx._hyg.12 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx._hyg.15 : AddCommMonoid.{v} M] {motive : (Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx._hyg.15) -> Sort.{u_1}} (t : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx._hyg.15), (forall [toDistribMulAction : DistribMulAction.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx._hyg.12)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Defs.2274493915._hygCtx._hyg.15)] (add_smul : forall (r : R) (s : R) (x : M), Eq.{succ v} M (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SemigroupAction.toSMul.{u, v} R M (Monoid.toSemigroup.{u} R 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{inst : Semiring R} →\n {inst_1 : AddCommMonoid M} →\n {P : Sort u_1} →\n {toDistribMulAction : DistribMulAction R M} →\n {add_smul : ∀ (r s : R) (x : M), (r + s) • x = r • x + s • x} →\n {zero_smul : ∀ (x : M), 0 • x = 0} →\n {toDistribMulAction' : DistribMulAction R M} →\n {add_smul' : ∀ (r s : R) (x : M), (r + s) • x = r • x + s • x} →\n {zero_smul' : ∀ (x : M), 0 • x = 0} →\n { toDistribMulAction := toDistribMulAction, add_smul := add_smul, zero_smul := zero_smul } =\n { toDistribMulAction := toDistribMulAction', add_smul := add_smul',\n zero_smul := zero_smul' } →\n (toDistribMulAction ≍ toDistribMulAction' → P) → P","typeReadable":"{R : Type u} →\n {M : Type v} →\n {inst : Semiring R} →\n {inst_1 : AddCommMonoid M} →\n {P : Sort u_1} →\n {toDistribMulAction : DistribMulAction R M} →\n {add_smul : ∀ (r s : R) (x : M), (r + s) • x = r • x + s • x} →\n {zero_smul : ∀ (x : M), 0 • x = 0} →\n {toDistribMulAction' : DistribMulAction R M} →\n {add_smul' : ∀ (r s : R) (x : M), (r + s) • 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