Add files using upload-large-folder tool

.gitattributes

@@ -95,3 +95,8 @@ data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.MeasureTheory.Function.Sim
 data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Bicategory.Functor.Oplax.sym.json filter=lfs diff=lfs merge=lfs -text
 data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.CStarAlgebra.CStarMatrix.sym.json filter=lfs diff=lfs merge=lfs -text
 data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor.sym.json filter=lfs diff=lfs merge=lfs -text
+data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.LinearAlgebra.TensorProduct.Submodule.sym.json filter=lfs diff=lfs merge=lfs -text
+data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.WithTerminal.Cone.sym.json filter=lfs diff=lfs merge=lfs -text
+data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.AlgebraicGeometry.Modules.Sheaf.sym.json filter=lfs diff=lfs merge=lfs -text
+data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.GuitartExact.Opposite.sym.json filter=lfs diff=lfs merge=lfs -text
+data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.RingTheory.PicardGroup.sym.json filter=lfs diff=lfs merge=lfs -text

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Divisibility.Finite.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.FreeAbelianGroup.UniqueSums.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Commutator.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Subgroup.Finsupp.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GroupWithZero.Nat.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GroupWithZero.Pointwise.Finset.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Group.Unbundled.Int.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Module.Basic.sym.json

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[inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.22 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.25 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.22)] [inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.28 : Module.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.10) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19)] [inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.32 : SMulPosMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.10))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.10) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19) inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.28)))) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.22) (NegZeroClass.toZero.{u_2} β (SubNegZeroMonoid.toNegZeroClass.{u_2} β (SubtractionMonoid.toSubNegZeroMonoid.{u_2} β (SubtractionCommMonoid.toSubtractionMonoid.{u_2} β (AddCommGroup.toDivisionAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19)))))], (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.13)) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.10)))))))) -> (LE.le.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.22)) b (OfNat.ofNat.{u_2} β 0 (Zero.toOfNat0.{u_2} β (NegZeroClass.toZero.{u_2} β (SubNegZeroMonoid.toNegZeroClass.{u_2} β (SubtractionMonoid.toSubNegZeroMonoid.{u_2} β (SubtractionCommMonoid.toSubtractionMonoid.{u_2} β (AddCommGroup.toDivisionAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19)))))))) -> (Eq.{1} Prop (LE.le.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.22)) (OfNat.ofNat.{u_2} β 0 (Zero.toOfNat0.{u_2} β (NegZeroClass.toZero.{u_2} β (SubNegZeroMonoid.toNegZeroClass.{u_2} β (SubtractionMonoid.toSubNegZeroMonoid.{u_2} β (SubtractionCommMonoid.toSubtractionMonoid.{u_2} β (AddCommGroup.toDivisionAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19))))))) (HSMul.hSMul.{u_1, u_2, u_2} α β β (instHSMul.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.10))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.10) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.19) inst._@.Mathlib.Algebra.Order.Module.Defs.1037484200._hygCtx._hyg.28))))) a b)) True)","typeFull":"∀ {α : Type u_1} {β : Type u_2} {a : α} {b : β} [inst : Ring α] [inst_1 : PartialOrder α] [inst_2 : AddCommGroup β]\n  [inst_3 : PartialOrder β] [IsOrderedAddMonoid β] [inst_5 : Module α β] [SMulPosMono α β],\n  a ≤ 0 → b ≤ 0 → (0 ≤ a • b) = True","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {a : α} {b : β} [inst : Ring α] [inst_1 : PartialOrder α] [inst_2 : AddCommGroup β]\n  [inst_3 : PartialOrder β] [IsOrderedAddMonoid β] [inst_5 : Module α β] [SMulPosMono α β],\n  a ≤ 0 → b ≤ 0 → (0 ≤ a • b) = True","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["PartialOrder"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["SMulPosMono"],["DistribSMul","toSMulZeroClass"],["True"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Ring","toSemiring"],["Module","toDistribMulAction"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Ring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["eq_true"],["SubtractionCommMonoid","toSubtractionMonoid"],["SMulZeroClass","toSMul"],["SubNegZeroMonoid","toNegZeroClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["smul_nonneg_of_nonpos_of_nonpos"],["Ring","toSemiring"],["Module","toDistribMulAction"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["inf_eq_half_smul_add_sub_abs_sub"],"typeFallback":"forall (R : Type.{u_2}) {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.8 : Invertible.{u_2} R (Distrib.toMul.{u_2} R (NonUnitalNonAssocSemiring.toDistrib.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5)))) (AddMonoidWithOne.toOne.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5)))) (OfNat.ofNat.{u_2} R 2 (instOfNatAtLeastTwo.{u_2} R 2 (AddMonoidWithOne.toNatCast.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))] [inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.14 : Lattice.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.20 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.24 : IsOrderedAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17) (PartialOrder.toPreorder.{u_3} M (SemilatticeInf.toPartialOrder.{u_3} M (Lattice.toSemilatticeInf.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.14)))] (x : M) (y : M), Eq.{succ u_3} M (Min.min.{u_3} M (SemilatticeInf.toMin.{u_3} M (Lattice.toSemilatticeInf.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.14)) x y) (HSMul.hSMul.{u_2, u_3, u_3} R M M (instHSMul.{u_2, u_3} R M (SMulZeroClass.toSMul.{u_2, 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.Order.Module.Basic.242405034._hygCtx._hyg.17)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17)))) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5)) (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17))) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.20))))) (Invertible.invOf.{u_2} R (Distrib.toMul.{u_2} R (NonUnitalNonAssocSemiring.toDistrib.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5)))) (AddMonoidWithOne.toOne.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5)))) (OfNat.ofNat.{u_2} R 2 (instOfNatAtLeastTwo.{u_2} R 2 (AddMonoidWithOne.toNatCast.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.5)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.8) (HSub.hSub.{u_3, u_3, u_3} M M M (instHSub.{u_3} M (SubNegMonoid.toSub.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17)))) (HAdd.hAdd.{u_3, u_3, u_3} M M M (instHAdd.{u_3} M (AddCommMagma.toAdd.{u_3} M (AddCommSemigroup.toAddCommMagma.{u_3} M (AddCommMonoid.toAddCommSemigroup.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17))))) x y) (abs.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.14 (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17) (HSub.hSub.{u_3, u_3, u_3} M M M (instHSub.{u_3} M (SubNegMonoid.toSub.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.242405034._hygCtx._hyg.17)))) y x))))","typeFull":"∀ (R : Type u_2) {M : Type u_3} [inst : Semiring R] [inst_1 : Invertible 2] [inst_2 : Lattice M]\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [IsOrderedAddMonoid M] (x y : M), x ⊓ y = ⅟2 • (x + y - |y - x|)","typeReadable":"∀ (R : Type u_2) {M : Type u_3} [inst : Semiring R] [inst_1 : Invertible 2] [inst_2 : Lattice M]\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [IsOrderedAddMonoid M] (x y : M), x ⊓ y = ⅟2 • (x + y - |y - 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Module.Basic.2570577961._hygCtx._hyg.5 : Ring.{u_2} R] [inst._@.Mathlib.Algebra.Order.Module.Basic.2570577961._hygCtx._hyg.8 : LinearOrder.{u_2} R] [inst._@.Mathlib.Algebra.Order.Module.Basic.2570577961._hygCtx._hyg.11 : IsOrderedRing.{u_2} R (Ring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.2570577961._hygCtx._hyg.5) (SemilatticeInf.toPartialOrder.{u_2} R (Lattice.toSemilatticeInf.{u_2} R (DistribLattice.toLattice.{u_2} R (instDistribLatticeOfLinearOrder.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.2570577961._hygCtx._hyg.8))))] [inst._@.Mathlib.Algebra.Order.Module.Basic.2570577961._hygCtx._hyg.14 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.2570577961._hygCtx._hyg.17 : LinearOrder.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.2570577961._hygCtx._hyg.20 : IsOrderedAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M 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|a| • 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{α : Type.{u_1}} {β : Type.{u_2}} {a : α} {b : β} [inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.10 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.16 : SMulWithZero.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.20 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.23 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.26 : SMulPosMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.16)) inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13], (LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.20) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.10))) -> (LE.le.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.23) (OfNat.ofNat.{u_2} β 0 (Zero.toOfNat0.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13)) b) -> (Eq.{1} Prop (LE.le.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.23) (HSMul.hSMul.{u_1, u_2, u_2} α β β (instHSMul.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.16))) a b) (OfNat.ofNat.{u_2} β 0 (Zero.toOfNat0.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.1663129321._hygCtx._hyg.13))) True)","typeFull":"∀ {α : Type u_1} {β : Type u_2} {a : α} {b : β} [inst : Zero α] [inst_1 : Zero β] [inst_2 : SMulWithZero α β]\n  [inst_3 : Preorder α] [inst_4 : Preorder β] [SMulPosMono α β], a ≤ 0 → 0 ≤ b → (a • b ≤ 0) = True","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {a : α} {b : β} [inst : Zero α] [inst_1 : Zero β] [inst_2 : SMulWithZero α β]\n  [inst_3 : Preorder α] [inst_4 : Preorder β] [SMulPosMono α β], a ≤ 0 → 0 ≤ b → (a • b ≤ 0) = True","typeReferences":[["True"],["SMulZeroClass","toSMul"],["SMulWithZero","toSMulZeroClass"],["OfNat","ofNat"],["Preorder"],["LE","le"],["HSMul","hSMul"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"],["Eq"],["SMulWithZero"],["SMulPosMono"]],"valueReferences":[["HSMul","hSMul"],["LE","le"],["instHSMul"],["eq_true"],["Zero","toOfNat0"],["SMulZeroClass","toSMul"],["Preorder","toLE"],["smul_nonpos_of_nonpos_of_nonneg"],["SMulWithZero","toSMulZeroClass"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["sup_eq_half_smul_add_add_abs_sub'"],"typeFallback":"forall (𝕜 : Type.{u_1}) {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.5 : DivisionSemiring.{u_1} 𝕜] [inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.8 : NeZero.{u_1} 𝕜 (MulZeroClass.toZero.{u_1} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} 𝕜 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} 𝕜 (Semiring.toNonAssocSemiring.{u_1} 𝕜 (DivisionSemiring.toSemiring.{u_1} 𝕜 inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.5))))) (OfNat.ofNat.{u_1} 𝕜 2 (instOfNatAtLeastTwo.{u_1} 𝕜 2 (AddMonoidWithOne.toNatCast.{u_1} 𝕜 (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} 𝕜 (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} 𝕜 (Semiring.toNonAssocSemiring.{u_1} 𝕜 (DivisionSemiring.toSemiring.{u_1} 𝕜 inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.5))))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))] [inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.14 : Lattice.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.17 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.20 : Module.{u_1, u_3} 𝕜 M (DivisionSemiring.toSemiring.{u_1} 𝕜 inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.5) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.24 : IsOrderedAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.17) (PartialOrder.toPreorder.{u_3} M (SemilatticeInf.toPartialOrder.{u_3} M (Lattice.toSemilatticeInf.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.14)))] (x : M) (y : M), Eq.{succ u_3} M (Max.max.{u_3} M (SemilatticeSup.toMax.{u_3} M (Lattice.toSemilatticeSup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.14)) x y) (HSMul.hSMul.{u_1, u_3, u_3} 𝕜 M M (instHSMul.{u_1, u_3} 𝕜 M (SMulZeroClass.toSMul.{u_1, u_3} 𝕜 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 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inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.20))))) (Inv.inv.{u_1} 𝕜 (InvOneClass.toInv.{u_1} 𝕜 (DivInvOneMonoid.toInvOneClass.{u_1} 𝕜 (DivisionMonoid.toDivInvOneMonoid.{u_1} 𝕜 (GroupWithZero.toDivisionMonoid.{u_1} 𝕜 (DivisionSemiring.toGroupWithZero.{u_1} 𝕜 inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.5))))) (OfNat.ofNat.{u_1} 𝕜 2 (instOfNatAtLeastTwo.{u_1} 𝕜 2 (AddMonoidWithOne.toNatCast.{u_1} 𝕜 (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} 𝕜 (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} 𝕜 (Semiring.toNonAssocSemiring.{u_1} 𝕜 (DivisionSemiring.toSemiring.{u_1} 𝕜 inst._@.Mathlib.Algebra.Order.Module.Basic.56789740._hygCtx._hyg.5))))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))) (HAdd.hAdd.{u_3, u_3, u_3} M M M (instHAdd.{u_3} M (AddCommMagma.toAdd.{u_3} M (AddCommSemigroup.toAddCommMagma.{u_3} M (AddCommMonoid.toAddCommSemigroup.{u_3} M 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(x + y + |y - x|)","typeReadable":"∀ (𝕜 : Type u_1) {M : Type u_3} [inst : DivisionSemiring 𝕜] [NeZero 2] [inst_2 : Lattice M] [inst_3 : AddCommGroup M]\n  [inst_4 : Module 𝕜 M] [IsOrderedAddMonoid M] (x y : M), x ⊔ y = 2⁻¹ • (x + y + |y - x|)","typeReferences":[["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Semiring","toNonAssocSemiring"],["SubNegMonoid","toSub"],["IsOrderedAddMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["abs"],["NonAssocSemiring","toAddCommMonoidWithOne"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["InvOneClass","toInv"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["AddZeroClass","toAddZero"],["Nat"],["AddMonoidWithOne","toNatCast"],["Lattice"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["AddMonoid","toAddZeroClass"],["GroupWithZero","toDivisionMonoid"],["Module"],["Nat","instNeZeroSucc"],["NeZero"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["Inv","inv"],["Lattice","toSemilatticeInf"],["instHAdd"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring"],["OfNat","ofNat"],["HAdd","hAdd"],["Max","max"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["SemilatticeSup","toMax"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["instOfNatAtLeastTwo"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["two_ne_zero'"],["OfNat","ofNat"],["Nat","instNeZeroSucc"],["sup_eq_half_smul_add_add_abs_sub"],["invertibleOfNonzero"],["Nat"],["AddMonoidWithOne","toNatCast"],["Semiring","toNonAssocSemiring"],["MulZeroClass","toZero"],["instOfNatNat"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Nat","instAtLeastTwoHAddOfNat"]]},{"isProp":true,"kind":"theorem","name":["inf_eq_half_smul_add_sub_abs_sub'"],"typeFallback":"forall (𝕜 : Type.{u_1}) {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.5 : DivisionSemiring.{u_1} 𝕜] [inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.8 : NeZero.{u_1} 𝕜 (MulZeroClass.toZero.{u_1} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} 𝕜 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} 𝕜 (Semiring.toNonAssocSemiring.{u_1} 𝕜 (DivisionSemiring.toSemiring.{u_1} 𝕜 inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.5))))) (OfNat.ofNat.{u_1} 𝕜 2 (instOfNatAtLeastTwo.{u_1} 𝕜 2 (AddMonoidWithOne.toNatCast.{u_1} 𝕜 (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} 𝕜 (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} 𝕜 (Semiring.toNonAssocSemiring.{u_1} 𝕜 (DivisionSemiring.toSemiring.{u_1} 𝕜 inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.5))))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))] [inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.14 : Lattice.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.17 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.20 : Module.{u_1, u_3} 𝕜 M (DivisionSemiring.toSemiring.{u_1} 𝕜 inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.5) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.24 : IsOrderedAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.17) (PartialOrder.toPreorder.{u_3} M (SemilatticeInf.toPartialOrder.{u_3} M (Lattice.toSemilatticeInf.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.14)))] (x : M) (y : M), Eq.{succ u_3} M (Min.min.{u_3} M (SemilatticeInf.toMin.{u_3} M (Lattice.toSemilatticeInf.{u_3} M 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inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.5))))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))) (HSub.hSub.{u_3, u_3, u_3} M M M (instHSub.{u_3} M (SubNegMonoid.toSub.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.17)))) (HAdd.hAdd.{u_3, u_3, u_3} M M M (instHAdd.{u_3} M (AddCommMagma.toAdd.{u_3} M (AddCommSemigroup.toAddCommMagma.{u_3} M (AddCommMonoid.toAddCommSemigroup.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.17))))) x y) (abs.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.14 (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.2708070848._hygCtx._hyg.17) (HSub.hSub.{u_3, u_3, u_3} M M M (instHSub.{u_3} M (SubNegMonoid.toSub.{u_3} M 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x|)","typeReferences":[["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Semiring","toNonAssocSemiring"],["SubNegMonoid","toSub"],["IsOrderedAddMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["abs"],["NonAssocSemiring","toAddCommMonoidWithOne"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["InvOneClass","toInv"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["AddZeroClass","toAddZero"],["Nat"],["AddMonoidWithOne","toNatCast"],["Lattice"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["AddMonoid","toAddZeroClass"],["GroupWithZero","toDivisionMonoid"],["SemilatticeInf","toMin"],["Module"],["Nat","instNeZeroSucc"],["NeZero"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["Inv","inv"],["Lattice","toSemilatticeInf"],["instHAdd"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring"],["OfNat","ofNat"],["HAdd","hAdd"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Min","min"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["inf_eq_half_smul_add_sub_abs_sub"],["instOfNatAtLeastTwo"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["two_ne_zero'"],["OfNat","ofNat"],["Nat","instNeZeroSucc"],["invertibleOfNonzero"],["Nat"],["AddMonoidWithOne","toNatCast"],["Semiring","toNonAssocSemiring"],["MulZeroClass","toZero"],["instOfNatNat"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Nat","instAtLeastTwoHAddOfNat"]]},{"isProp":true,"kind":"theorem","name":["sup_eq_half_smul_add_add_abs_sub"],"typeFallback":"forall (R : Type.{u_2}) {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.8 : Invertible.{u_2} R (Distrib.toMul.{u_2} R (NonUnitalNonAssocSemiring.toDistrib.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5)))) (AddMonoidWithOne.toOne.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5)))) (OfNat.ofNat.{u_2} R 2 (instOfNatAtLeastTwo.{u_2} R 2 (AddMonoidWithOne.toNatCast.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))] [inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.14 : Lattice.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.17 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.20 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.17)] [inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.24 : IsOrderedAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.17) (PartialOrder.toPreorder.{u_3} M (SemilatticeInf.toPartialOrder.{u_3} M (Lattice.toSemilatticeInf.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.14)))] (x : M) (y : M), Eq.{succ u_3} M (Max.max.{u_3} M (SemilatticeSup.toMax.{u_3} M (Lattice.toSemilatticeSup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.14)) x y) (HSMul.hSMul.{u_2, u_3, u_3} R M M (instHSMul.{u_2, u_3} R M (SMulZeroClass.toSMul.{u_2, 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.Order.Module.Basic.1937207252._hygCtx._hyg.17)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.17)))) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5)) (SubNegMonoid.toAddMonoid.{u_3} M (AddGroup.toSubNegMonoid.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.17))) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.17) inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.20))))) (Invertible.invOf.{u_2} R (Distrib.toMul.{u_2} R (NonUnitalNonAssocSemiring.toDistrib.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5)))) (AddMonoidWithOne.toOne.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5)))) (OfNat.ofNat.{u_2} R 2 (instOfNatAtLeastTwo.{u_2} R 2 (AddMonoidWithOne.toNatCast.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.5)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.8) (HAdd.hAdd.{u_3, u_3, u_3} M M M (instHAdd.{u_3} M (AddCommMagma.toAdd.{u_3} M (AddCommSemigroup.toAddCommMagma.{u_3} M (AddCommMonoid.toAddCommSemigroup.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Order.Module.Basic.1937207252._hygCtx._hyg.17))))) (HAdd.hAdd.{u_3, u_3, u_3} M M M (instHAdd.{u_3} M (AddCommMagma.toAdd.{u_3} M (AddCommSemigroup.toAddCommMagma.{u_3} M (AddCommMonoid.toAddCommSemigroup.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M 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x|)","typeReferences":[["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Invertible","invOf"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Semiring","toNonAssocSemiring"],["SubNegMonoid","toSub"],["IsOrderedAddMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["abs"],["Invertible"],["NonAssocSemiring","toAddCommMonoidWithOne"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Nat"],["AddMonoidWithOne","toNatCast"],["Lattice"],["AddMonoidWithOne","toOne"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Module"],["Nat","instNeZeroSucc"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["Lattice","toSemilatticeInf"],["instHAdd"],["Distrib","toMul"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["OfNat","ofNat"],["HAdd","hAdd"],["Max","max"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["AddCommMonoid","toAddCommSemigroup"],["SemilatticeSup","toMax"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"],["Semiring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Invertible","invOf"],["AddCommGroup","toAddGroup"],["HMul","hMul"],["SemigroupAction","toSMul"],["two_nsmul_sup_eq_add_add_abs_sub"],["SMulZeroClass","toSMul"],["Semiring","toNonAssocSemiring"],["SubNegMonoid","toSub"],["DistribMulAction","toMulAction"],["HSub","hSub"],["Eq","symm"],["Monoid","toSemigroup"],["AddGroup","toSubNegMonoid"],["abs"],["NonAssocSemiring","toAddCommMonoidWithOne"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["IsOrderedAddMonoid","toAddLeftMono"],["AddZeroClass","toAddZero"],["Nat"],["AddMonoidWithOne","toNatCast"],["AddMonoid","toNSMul"],["Eq","refl"],["one_smul"],["HSMul","hSMul"],["AddMonoidWithOne","toOne"],["AddCommGroup","toAddCommMonoid"],["id"],["instHMul"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["two_smul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Nat","instNeZeroSucc"],["MulOne","toMul"],["Nat","instSemiring"],["instOfNatNat"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddCommMonoid","toNatModule"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["Lattice","toSemilatticeInf"],["instHAdd"],["Distrib","toMul"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["OfNat","ofNat"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Max","max"],["One","toOfNat1"],["SubNegMonoid","toAddMonoid"],["AddCommMonoid","toAddCommSemigroup"],["invOf_mul_self"],["smul_smul"],["SemilatticeSup","toMax"],["instHSub"],["MulAction","toSemigroupAction"],["Nat","instAtLeastTwoHAddOfNat"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Module.Rat.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["PosSMulStrictMono","nnrat_of_rat"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.3 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.6 : MulAction.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))] [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.12 : MulAction.{0, u_1} Rat α Rat.monoid] [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.18 : IsScalarTower.{0, 0, u_1} NNRat Rat α (NNRat.smulDivisionSemiring.{0} Rat (Semifield.toDivisionSemiring.{0} Rat (Field.toSemifield.{0} Rat Rat.instField))) (SemigroupAction.toSMul.{0, u_1} Rat α (Monoid.toSemigroup.{0} Rat Rat.monoid) (MulAction.toSemigroupAction.{0, u_1} Rat α Rat.monoid inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.12)) (SemigroupAction.toSMul.{0, u_1} NNRat α (Monoid.toSemigroup.{0} NNRat (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))) (MulAction.toSemigroupAction.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.27 : PosSMulStrictMono.{0, u_1} Rat α (SemigroupAction.toSMul.{0, u_1} Rat α (Monoid.toSemigroup.{0} Rat Rat.monoid) (MulAction.toSemigroupAction.{0, u_1} Rat α Rat.monoid inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.12)) Rat.instPreorder inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.3 (MulZeroClass.toZero.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing))))))], PosSMulStrictMono.{0, u_1} NNRat α (SemigroupAction.toSMul.{0, u_1} NNRat α (Monoid.toSemigroup.{0} NNRat (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))) (MulAction.toSemigroupAction.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.6)) (PartialOrder.toPreorder.{0} NNRat (SemilatticeInf.toPartialOrder.{0} NNRat (Lattice.toSemilatticeInf.{0} NNRat (DistribLattice.toLattice.{0} NNRat (instDistribLatticeOfLinearOrder.{0} NNRat instLinearOrderNNRat))))) inst._@.Mathlib.Algebra.Order.Module.Rat.2950371126._hygCtx._hyg.3 (MulZeroClass.toZero.{0} NNRat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNRat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] [inst_1 : MulAction ℚ≥0 α] [inst_2 : MulAction ℚ α] [IsScalarTower ℚ≥0 ℚ α]\n  [PosSMulStrictMono ℚ α], PosSMulStrictMono ℚ≥0 α","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] [inst_1 : MulAction ℚ≥0 α] [inst_2 : MulAction ℚ α] [IsScalarTower ℚ≥0 ℚ α]\n  [PosSMulStrictMono ℚ α], PosSMulStrictMono ℚ≥0 α","typeReferences":[["PartialOrder","toPreorder"],["IsScalarTower"],["SemigroupAction","toSMul"],["instLinearOrderNNRat"],["CommRing","toNonUnitalCommRing"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["NNRat"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semifield","toDivisionSemiring"],["Monoid","toSemigroup"],["NNRat","instSemifield"],["SemilatticeInf","toPartialOrder"],["Rat","commRing"],["Rat","instField"],["Rat","instPreorder"],["Lattice","toSemilatticeInf"],["PosSMulStrictMono"],["NNRat","smulDivisionSemiring"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Rat","monoid"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["Preorder"],["DistribLattice","toLattice"],["MulAction"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Rat"],["MulAction","toSemigroupAction"]],"valueReferences":[["DivisionSemiring","toNNRatCast"],["PartialOrder","toPreorder"],["smul_lt_smul_of_pos_left"],["NNRat","cast_smul_eq_nnqsmul"],["Preorder","toLT"],["SemigroupAction","toSMul"],["CommRing","toNonUnitalCommRing"],["instLinearOrderNNRat"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["NNRat"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["Eq","symm"],["NNRat","cast"],["instHSMul"],["Semifield","toDivisionSemiring"],["Monoid","toSemigroup"],["Eq"],["NNRat","instSemifield"],["Rat","commRing"],["SemilatticeInf","toPartialOrder"],["Rat","instField"],["Rat","instPreorder"],["Lattice","toSemilatticeInf"],["PosSMulStrictMono","mk"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["LT","lt"],["DistribLattice","toLattice"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["id"],["Field","toSemifield"],["Rat"],["Eq","mpr"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["PosSMulMono","nnrat_of_rat"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.3 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.6 : MulAction.{0, u_1} Rat α Rat.monoid] [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.12 : MulAction.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))] [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.18 : IsScalarTower.{0, 0, u_1} NNRat Rat α (NNRat.smulDivisionSemiring.{0} Rat (Semifield.toDivisionSemiring.{0} Rat (Field.toSemifield.{0} Rat Rat.instField))) (SemigroupAction.toSMul.{0, u_1} Rat α (Monoid.toSemigroup.{0} Rat Rat.monoid) (MulAction.toSemigroupAction.{0, u_1} Rat α Rat.monoid inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.6)) (SemigroupAction.toSMul.{0, u_1} NNRat α (Monoid.toSemigroup.{0} NNRat (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))) (MulAction.toSemigroupAction.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.27 : PosSMulMono.{0, u_1} Rat α (SemigroupAction.toSMul.{0, u_1} Rat α (Monoid.toSemigroup.{0} Rat Rat.monoid) (MulAction.toSemigroupAction.{0, u_1} Rat α Rat.monoid inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.6)) Rat.instPreorder inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.3 (MulZeroClass.toZero.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing))))))], PosSMulMono.{0, u_1} NNRat α (SemigroupAction.toSMul.{0, u_1} NNRat α (Monoid.toSemigroup.{0} NNRat (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))) (MulAction.toSemigroupAction.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.12)) (PartialOrder.toPreorder.{0} NNRat (SemilatticeInf.toPartialOrder.{0} NNRat (Lattice.toSemilatticeInf.{0} NNRat (DistribLattice.toLattice.{0} NNRat (instDistribLatticeOfLinearOrder.{0} NNRat instLinearOrderNNRat))))) inst._@.Mathlib.Algebra.Order.Module.Rat.2950371125._hygCtx._hyg.3 (MulZeroClass.toZero.{0} NNRat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNRat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] [inst_1 : MulAction ℚ α] [inst_2 : MulAction ℚ≥0 α] [IsScalarTower ℚ≥0 ℚ α]\n  [PosSMulMono ℚ α], PosSMulMono ℚ≥0 α","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] [inst_1 : MulAction ℚ α] [inst_2 : MulAction ℚ≥0 α] [IsScalarTower ℚ≥0 ℚ α]\n  [PosSMulMono ℚ α], PosSMulMono ℚ≥0 α","typeReferences":[["PartialOrder","toPreorder"],["IsScalarTower"],["SemigroupAction","toSMul"],["instLinearOrderNNRat"],["CommRing","toNonUnitalCommRing"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["NNRat"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semifield","toDivisionSemiring"],["Monoid","toSemigroup"],["NNRat","instSemifield"],["SemilatticeInf","toPartialOrder"],["Rat","commRing"],["Rat","instField"],["Rat","instPreorder"],["Lattice","toSemilatticeInf"],["NNRat","smulDivisionSemiring"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Rat","monoid"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["Preorder"],["DistribLattice","toLattice"],["MulAction"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Rat"],["MulAction","toSemigroupAction"]],"valueReferences":[["DivisionSemiring","toNNRatCast"],["PartialOrder","toPreorder"],["NNRat","cast_smul_eq_nnqsmul"],["SemigroupAction","toSMul"],["CommRing","toNonUnitalCommRing"],["instLinearOrderNNRat"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["NNRat"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["Eq","symm"],["NNRat","cast"],["instHSMul"],["Semifield","toDivisionSemiring"],["Monoid","toSemigroup"],["Eq"],["Preorder","toLE"],["NNRat","instSemifield"],["Rat","commRing"],["SemilatticeInf","toPartialOrder"],["Rat","instField"],["Rat","instPreorder"],["Lattice","toSemilatticeInf"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["PosSMulMono","mk"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["smul_le_smul_of_nonneg_left"],["DivisionSemiring","toSemiring"],["DistribLattice","toLattice"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["HSMul","hSMul"],["id"],["Field","toSemifield"],["Rat"],["Eq","mpr"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["LinearOrderedField","toPosSMulStrictMono_rat"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Rat.4136574846._hygCtx._hyg.3 : Field.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Rat.4136574846._hygCtx._hyg.6 : LinearOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Rat.4136574846._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.4136574846._hygCtx._hyg.3))) (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.4136574846._hygCtx._hyg.6))))], PosSMulStrictMono.{0, u_1} Rat α (Rat.smulDivisionRing.{u_1} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.4136574846._hygCtx._hyg.3)) Rat.instPreorder (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.4136574846._hygCtx._hyg.6))))) (MulZeroClass.toZero.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing))))))","typeFull":"∀ {α : Type u_1} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α], PosSMulStrictMono ℚ α","typeReadable":"∀ {α : Type u_1} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α], PosSMulStrictMono ℚ α","typeReferences":[["Rat","instPreorder"],["PartialOrder","toPreorder"],["IsStrictOrderedRing"],["Lattice","toSemilatticeInf"],["PosSMulStrictMono"],["Field"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["LinearOrder"],["Field","toDivisionRing"],["CommRing","toNonUnitalCommRing"],["DivisionSemiring","toSemiring"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["MulZeroClass","toZero"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Rat"],["Semifield","toDivisionSemiring"],["Rat","smulDivisionRing"],["Rat","commRing"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Rat","cast_pos"],["Preorder","toLT"],["mul_lt_mul_of_pos_left"],["HMul","hMul"],["CommRing","toNonUnitalCommRing"],["Rat","instOfNat"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["Rat","smul_def"],["Rat","cast"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["instHSMul"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Rat","instLT"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Rat","smulDivisionRing"],["SemilatticeInf","toPartialOrder"],["Rat","commRing"],["IsStrictOrderedRing","toPosMulStrictMono"],["Rat","instPreorder"],["Field","toCommRing"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toDistrib"],["PosSMulStrictMono","mk"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivisionRing","toRing"],["Distrib","toMul"],["Field","toDivisionRing"],["DivisionSemiring","toSemiring"],["OfNat","ofNat"],["DivisionRing","toRatCast"],["LT","lt"],["DistribLattice","toLattice"],["MulZeroClass","toZero"],["Iff","mpr"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["Field","toSemifield"],["id"],["Rat"],["instHMul"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Order","Module","Rat",0,"abs_nnqsmul","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {a : α} {b₁ : β} [inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.10 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.16 : SMulZeroClass.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.20 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.23 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.16) inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.10], (LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.20) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.10)) a) -> (LE.le.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.23) (OfNat.ofNat.{u_2} β 0 (Zero.toOfNat0.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.13)) b₁) -> (Eq.{1} Prop (LE.le.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.23) (OfNat.ofNat.{u_2} β 0 (Zero.toOfNat0.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.13)) (HSMul.hSMul.{u_1, u_2, u_2} α β β (instHSMul.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Defs.2127240519._hygCtx._hyg.16)) a b₁)) True)","typeFull":"∀ {α : Type u_1} {β : Type u_2} {a : α} {b₁ : β} [inst : Zero α] [inst_1 : Zero β] [inst_2 : SMulZeroClass α β]\n  [inst_3 : Preorder α] [inst_4 : Preorder β] [PosSMulMono α β], 0 ≤ a → 0 ≤ b₁ → (0 ≤ a • b₁) = True","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {a : α} {b₁ : β} [inst : Zero α] [inst_1 : Zero β] [inst_2 : SMulZeroClass α β]\n  [inst_3 : Preorder α] [inst_4 : Preorder β] [PosSMulMono α β], 0 ≤ a → 0 ≤ b₁ → (0 ≤ a • b₁) = True","typeReferences":[["True"],["SMulZeroClass","toSMul"],["OfNat","ofNat"],["PosSMulMono"],["Preorder"],["SMulZeroClass"],["LE","le"],["HSMul","hSMul"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"],["Eq"]],"valueReferences":[["HSMul","hSMul"],["LE","le"],["instHSMul"],["eq_true"],["SMulZeroClass","toSMul"],["Zero","toOfNat0"],["Preorder","toLE"],["smul_nonneg"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["abs_nnqsmul"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3 : AddCommGroup.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.6 : LinearOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} α (AddCommGroup.toAddCommMonoid.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.6)))))] [inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.12 : DistribMulAction.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3)))] [inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.18 : PosSMulMono.{0, u_1} NNRat α (SMulZeroClass.toSMul.{0, u_1} NNRat α (AddZero.toZero.{u_1} α (AddZeroClass.toAddZero.{u_1} α (AddMonoid.toAddZeroClass.{u_1} α (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} NNRat α (AddMonoid.toAddZeroClass.{u_1} α (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.12))) (PartialOrder.toPreorder.{0} NNRat (SemilatticeInf.toPartialOrder.{0} NNRat (Lattice.toSemilatticeInf.{0} NNRat (DistribLattice.toLattice.{0} NNRat (instDistribLatticeOfLinearOrder.{0} NNRat instLinearOrderNNRat))))) (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.6))))) (MulZeroClass.toZero.{0} NNRat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNRat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield))))))] (q : NNRat) (a : α), Eq.{succ u_1} α (abs.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.6)) (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3) (HSMul.hSMul.{0, u_1, u_1} NNRat α α (instHSMul.{0, u_1} NNRat α (SMulZeroClass.toSMul.{0, u_1} NNRat α (AddZero.toZero.{u_1} α (AddZeroClass.toAddZero.{u_1} α (AddMonoid.toAddZeroClass.{u_1} α (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} NNRat α (AddMonoid.toAddZeroClass.{u_1} α (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.12)))) q a)) (HSMul.hSMul.{0, u_1, u_1} NNRat α α (instHSMul.{0, u_1} NNRat α (SMulZeroClass.toSMul.{0, u_1} NNRat α (AddZero.toZero.{u_1} α (AddZeroClass.toAddZero.{u_1} α (AddMonoid.toAddZeroClass.{u_1} α (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} NNRat α (AddMonoid.toAddZeroClass.{u_1} α (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} NNRat α (MonoidWithZero.toMonoid.{0} NNRat (Semiring.toMonoidWithZero.{0} NNRat (DivisionSemiring.toSemiring.{0} NNRat (Semifield.toDivisionSemiring.{0} NNRat NNRat.instSemifield)))) (SubNegMonoid.toAddMonoid.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3))) inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.12)))) q (abs.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.6)) (AddCommGroup.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Rat.1588690266._hygCtx._hyg.3) a))","typeFull":"∀ {α : Type u_1} [inst : AddCommGroup α] [inst_1 : LinearOrder α] [IsOrderedAddMonoid α]\n  [inst_3 : DistribMulAction ℚ≥0 α] [PosSMulMono ℚ≥0 α] (q : ℚ≥0) (a : α), |q • a| = q • |a|","typeReadable":"∀ {α : Type u_1} [inst : AddCommGroup α] [inst_1 : LinearOrder α] [IsOrderedAddMonoid α]\n  [inst_3 : DistribMulAction ℚ≥0 α] [PosSMulMono ℚ≥0 α] (q : ℚ≥0) (a : α), |q • a| = q • 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{α : Type.{u_1}} {β : Type.{u_2}} {a : α} {b : β} [inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.10 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.16 : SMulZeroClass.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.20 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.23 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.16) inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Order.Module.Defs.2636034251._hygCtx._hyg.23 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Tropical.Lattice.sym.json

@@ -0,0 +1 @@
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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Tropical.Lattice.3921894153._hygCtx._hyg.4 : InfSet.{u_1} R], InfSet.{u_1} (Tropical.{u_1} R)","typeFull":"{R : Type u_1} → [InfSet R] → InfSet (Tropical R)","typeReadable":"{R : Type u_1} → [InfSet R] → InfSet (Tropical R)","typeReferences":[["Tropical"],["InfSet"]],"valueReferences":[["Set","image"],["Tropical"],["InfSet","sInf"],["InfSet","mk"],["Tropical","untrop"],["Tropical","trop"]]},{"isProp":false,"kind":"definition","name":["instSemilatticeInfTropical"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Tropical.Lattice.2213668161._hygCtx._hyg.4 : SemilatticeInf.{u_1} R], SemilatticeInf.{u_1} (Tropical.{u_1} R)","typeFull":"{R : Type u_1} → [SemilatticeInf R] → SemilatticeInf (Tropical R)","typeReadable":"{R : Type u_1} → [SemilatticeInf R] → SemilatticeInf (Tropical 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Calculus.TangentCone.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Complex.ExponentialBounds.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.sym.json

@@ -0,0 +1 @@
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Type u_1} {R : Type u_3} [inst : AddGroup G] [inst_1 : Fintype G] [inst_2 : Semifield R] [IsDomain R]\n  [inst_4 : CharZero R] (ψ : AddChar G R), (Finset.univ.expect fun a => ψ a) = if ψ = 0 then 1 else 0","typeReadable":"∀ {G : Type u_1} {R : Type u_3} [inst : AddGroup G] [inst_1 : Fintype G] [inst_2 : Semifield R] [IsDomain R]\n  [inst_4 : CharZero R] (ψ : AddChar G R), (Finset.univ.expect fun a => ψ a) = if ψ = 0 then 1 else 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{G : Type.{u_1}} {R : Type.{u_3}} [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.2464149917._hygCtx._hyg.5 : AddGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.2464149917._hygCtx._hyg.8 : Fintype.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.2464149917._hygCtx._hyg.11 : Semifield.{u_3} R] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.2464149917._hygCtx._hyg.14 : IsDomain.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.2464149917._hygCtx._hyg.11))] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.2464149917._hygCtx._hyg.17 : CharZero.{u_3} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_3} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R 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{R : Type u_3} [inst : AddGroup G] [inst_1 : Fintype G] [inst_2 : Semifield R] [IsDomain R]\n  [inst_4 : CharZero R] {ψ : AddChar G R}, (Finset.univ.expect fun x => ψ x) ≠ 0 ↔ ψ = 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(G : Type.{u_1}) (R : Type.{u_3}) [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.5 : AddCommGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.8 : RCLike.{u_3} R] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.17 : Fintype.{u_1} G], Eq.{1} Prop (LE.le.{0} Nat instLENat (Fintype.card.{max u_3 u_1} (AddChar.{u_1, u_3} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.5))) R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R (Field.toSemifield.{u_3} R (NormedField.toField.{u_3} R (DenselyNormedField.toNormedField.{u_3} R (RCLike.toDenselyNormedField.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.8))))))))) (AddChar.instFintype.{u_1, u_3} G R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.5 inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.8 (Finite.of_fintype.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.17))) (Fintype.card.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.17)) True","typeFull":"∀ (G : Type u_1) (R : Type u_3) [inst : AddCommGroup G] [inst_1 : RCLike R] [inst_2 : Fintype G],\n  (Fintype.card (AddChar G R) ≤ Fintype.card G) = True","typeReadable":"∀ (G : Type u_1) (R : Type u_3) [inst : AddCommGroup G] [inst_1 : RCLike R] [inst_2 : Fintype G],\n  (Fintype.card (AddChar G R) ≤ Fintype.card G) = 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u_1) → (R : Type u_3) → [inst : AddCommGroup G] → [inst_1 : RCLike R] → [Finite G] → Fintype (AddChar G R)","typeReadable":"(G : Type u_1) → (R : Type u_3) → [inst : AddCommGroup G] → [inst_1 : RCLike R] → [Finite G] → Fintype (AddChar G 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Type.{u_3}} [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1451397613._hygCtx._hyg.5 : AddGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1451397613._hygCtx._hyg.8 : Fintype.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1451397613._hygCtx._hyg.11 : Semifield.{u_3} R] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1451397613._hygCtx._hyg.14 : IsDomain.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1451397613._hygCtx._hyg.11))] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1451397613._hygCtx._hyg.17 : CharZero.{u_3} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_3} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R 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(G : Type.{u_1}) (R : Type.{u_2}) [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4096832312._hygCtx._hyg.5 : AddCommGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4096832312._hygCtx._hyg.8 : RCLike.{u_2} R] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4096832312._hygCtx._hyg.17 : Finite.{succ u_1} G], Finite.{succ (max u_1 u_2)} (AddChar.{u_1, u_2} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4096832312._hygCtx._hyg.5))) R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (DivisionSemiring.toSemiring.{u_2} R (Semifield.toDivisionSemiring.{u_2} R (Field.toSemifield.{u_2} R (NormedField.toField.{u_2} R (DenselyNormedField.toNormedField.{u_2} R (RCLike.toDenselyNormedField.{u_2} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4096832312._hygCtx._hyg.8)))))))))","typeFull":"∀ (G : 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{K : Type.{u_1}} [inst._@.Mathlib.Analysis.RCLike.Basic.637791388._hygCtx._hyg.4 : RCLike.{u_1} K] {G : Type.{u_3}} [inst._@.Mathlib.Analysis.RCLike.Basic.637791388._hygCtx._hyg.8 : Finite.{succ u_3} G] [inst._@.Mathlib.Analysis.RCLike.Basic.637791388._hygCtx._hyg.11 : AddLeftCancelMonoid.{u_3} G] (ψ : AddChar.{u_3, u_1} G (AddLeftCancelMonoid.toAddMonoid.{u_3} G inst._@.Mathlib.Analysis.RCLike.Basic.637791388._hygCtx._hyg.11) K (MonoidWithZero.toMonoid.{u_1} K (Semiring.toMonoidWithZero.{u_1} K (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K (NormedField.toField.{u_1} K (DenselyNormedField.toNormedField.{u_1} K (RCLike.toDenselyNormedField.{u_1} K inst._@.Mathlib.Analysis.RCLike.Basic.637791388._hygCtx._hyg.4))))))))) (x : G), Eq.{succ u_1} K (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (RingHom.{u_1, u_1} K K (Semiring.toNonAssocSemiring.{u_1} K (CommSemiring.toSemiring.{u_1} K (Semifield.toCommSemiring.{u_1} K 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{G : Type.{u_1}} {R : Type.{u_3}} [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.3857055223._hygCtx._hyg.5 : AddCommGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.3857055223._hygCtx._hyg.8 : RCLike.{u_3} R] {ψ₁ : AddChar.{u_1, u_3} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.3857055223._hygCtx._hyg.5))) R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R (Field.toSemifield.{u_3} R (NormedField.toField.{u_3} R (DenselyNormedField.toNormedField.{u_3} R (RCLike.toDenselyNormedField.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.3857055223._hygCtx._hyg.8))))))))} {ψ₂ : AddChar.{u_1, u_3} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G 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(G : Type.{u_1}) (R : Type.{u_3}) [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.5 : AddCommGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.8 : RCLike.{u_3} R] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.17 : Fintype.{u_1} G], LE.le.{0} Nat instLENat (Fintype.card.{max u_3 u_1} (AddChar.{u_1, u_3} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.5))) R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R (Field.toSemifield.{u_3} R (NormedField.toField.{u_3} R (DenselyNormedField.toNormedField.{u_3} R (RCLike.toDenselyNormedField.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.8))))))))) (AddChar.instFintype.{u_1, u_3} G R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.5 inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.8 (Finite.of_fintype.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.17))) (Fintype.card.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1051699885._hygCtx._hyg.17)","typeFull":"∀ (G : Type u_1) (R : Type u_3) [inst : AddCommGroup G] [inst_1 : RCLike R] [inst_2 : Fintype G],\n  Fintype.card (AddChar G R) ≤ Fintype.card G","typeReadable":"∀ (G : Type u_1) (R : Type u_3) [inst : AddCommGroup G] [inst_1 : RCLike R] [inst_2 : Fintype G],\n  Fintype.card (AddChar G R) ≤ Fintype.card 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{G : Type.{u_1}} {R : Type.{u_3}} [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1825036342._hygCtx._hyg.5 : AddGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1825036342._hygCtx._hyg.8 : RCLike.{u_3} R] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1825036342._hygCtx._hyg.11 : Fintype.{u_1} G] (ψ : AddChar.{u_1, u_3} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1825036342._hygCtx._hyg.5)) R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R (Field.toSemifield.{u_3} R (NormedField.toField.{u_3} R (DenselyNormedField.toNormedField.{u_3} R (RCLike.toDenselyNormedField.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1825036342._hygCtx._hyg.8))))))))), Eq.{succ u_3} R (RCLike.wInner.{u_1, u_3, u_3} G R (fun (a : G) => R) 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{G : Type.{u_1}} {R : Type.{u_3}} [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.5 : AddCommGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.8 : RCLike.{u_3} R] {ψ₁ : AddChar.{u_1, u_3} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.5))) R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R (Field.toSemifield.{u_3} R (NormedField.toField.{u_3} R (DenselyNormedField.toNormedField.{u_3} R (RCLike.toDenselyNormedField.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.8))))))))} {ψ₂ : AddChar.{u_1, u_3} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.5))) R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R (Field.toSemifield.{u_3} R (NormedField.toField.{u_3} R (DenselyNormedField.toNormedField.{u_3} R (RCLike.toDenselyNormedField.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.8))))))))} [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.17 : Fintype.{u_1} G], Iff (Eq.{succ u_3} R (RCLike.wInner.{u_1, u_3, u_3} G R (fun (a : G) => R) inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.17 inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.1946914729._hygCtx._hyg.8 (fun (i : G) => NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u_3} R (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{u_3} R 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{G : Type.{u_1}} {R : Type.{u_3}} [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4141330620._hygCtx._hyg.5 : AddCommGroup.{u_1} G] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4141330620._hygCtx._hyg.8 : RCLike.{u_3} R] [inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4141330620._hygCtx._hyg.17 : Fintype.{u_1} G] (ψ₁ : AddChar.{u_1, u_3} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4141330620._hygCtx._hyg.5))) R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R (DivisionSemiring.toSemiring.{u_3} R (Semifield.toDivisionSemiring.{u_3} R (Field.toSemifield.{u_3} R (NormedField.toField.{u_3} R (DenselyNormedField.toNormedField.{u_3} R (RCLike.toDenselyNormedField.{u_3} R inst._@.Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality.4141330620._hygCtx._hyg.8))))))))) (ψ₂ : AddChar.{u_1, u_3} G 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Normed.Module.Normalize.sym.json

@@ -0,0 +1 @@
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NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] {r : ℝ},\n  0 < r → ∀ (x : V), NormedSpace.normalize (r • x) = NormedSpace.normalize x","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["NormedSpace"],["Real","instLT"],["NormedAddCommGroup"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["DistribSMul","toSMulZeroClass"],["ESeminormedAddMonoid","toAddMonoid"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["NormedSpace","normalize"],["Real"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Real","semiring"],["OfNat","ofNat"],["LT","lt"],["Module","toDistribMulAction"],["Real","instZero"],["Real","normedField"],["Real","instMonoid"],["HSMul","hSMul"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["DivInvMonoid","toInv"],["Real","instPreorder"],["Real","instAddGroup"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["SemigroupAction","toSMul"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["SMulZeroClass","toSMul"],["CommGroupWithZero","toDivisionCommMonoid"],["DivisionMonoid","toDivInvMonoid"],["not_false_eq_true"],["DistribMulAction","toMulAction"],["Semifield","toDivisionSemiring"],["Monoid","toSemigroup"],["Real","instAddCommMonoid"],["Real","lattice"],["DistribSMul","toSMulZeroClass"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["NormedSpace","normalize"],["Real"],["Norm","norm"],["Real","instIsOrderedAddMonoid"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["NormedSpace","toNormSMulClass"],["DistribMulAction","toDistribSMul"],["IsOrderedAddMonoid","toAddLeftMono"],["DivisionSemiring","toGroupWithZero"],["AddZeroClass","toAddZero"],["MulZeroOneClass","toMulZeroClass"],["DivInvMonoid","toMonoid"],["eq_false"],["Real","instMonoid"],["HSMul","hSMul"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["instHMul"],["NormedAddCommGroup","toNorm"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["inv_mul_cancel_right₀"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["UniformSpace","toTopologicalSpace"],["Real","norm"],["AddCommMonoid","toAddMonoid"],["Real","instField"],["congrArg"],["LT","lt","ne'"],["MulOne","toMul"],["GroupWithZero","toMonoidWithZero"],["Monoid","toMulOneClass"],["instHSMul"],["Zero","toOfNat0"],["congrFun'"],["Eq"],["DivisionCommMonoid","toDivisionMonoid"],["abs_of_pos"],["Not"],["Real","instMul"],["mul_inv_rev"],["ESeminormedAddMonoid","toAddMonoid"],["Inv","inv"],["True"],["OfNat","ofNat"],["Real","semiring"],["norm_smul"],["eq_self"],["Real","instInv"],["Module","toDistribMulAction"],["Real","instZero"],["of_eq_true"],["Real","normedField"],["MulZeroClass","toZero"],["smul_smul"],["Field","toSemifield"],["False"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["MulAction","toSemigroupAction"],["Semifield","toCommGroupWithZero"]]},{"isProp":true,"kind":"theorem","name":["NormedSpace","normalize_smul"],"typeFallback":"forall {V : Type.{u_1}} [inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.3 : NormedAddCommGroup.{u_1} V] [inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.3)] (r : Real) (x : V), Eq.{succ u_1} V (NormedSpace.normalize.{u_1} V inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.6 (HSMul.hSMul.{0, u_1, u_1} Real V V (instHSMul.{0, u_1} Real V (SMulZeroClass.toSMul.{0, u_1} Real V (AddZero.toZero.{u_1} V (AddZeroClass.toAddZero.{u_1} V (AddMonoid.toAddZeroClass.{u_1} V (ESeminormedAddMonoid.toAddMonoid.{u_1} V (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V 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(NormedAddCommGroup.toENormedAddCommMonoid.{u_1} V inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.3)))) (Module.toDistribMulAction.{0, u_1} Real V Real.semiring (ESeminormedAddCommMonoid.toAddCommMonoid.{u_1} V (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.3)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_1} V (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.3)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u_1} V inst._@.Mathlib.Analysis.Normed.Module.Normalize.2011256697._hygCtx._hyg.3))) (NormedSpace.toModule.{0, u_1} Real V 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Normed.Module.RCLike.Real.sym.json

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{G₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3558189578._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3558189578._hygCtx._hyg.8 : Preorder.{u_3} G₀] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3558189578._hygCtx._hyg.11 : ZeroLEOneClass.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3558189578._hygCtx._hyg.5)))) (InvOneClass.toOne.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3558189578._hygCtx._hyg.5)))) (Preorder.toLE.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3558189578._hygCtx._hyg.8)] (a : G₀), Eq.{1} Prop (LE.le.{u_3} G₀ 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{E : Type.{u_1}} [inst._@.Mathlib.Analysis.Normed.Module.RCLike.Real.1117416928._hygCtx._hyg.3 : SeminormedAddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Normed.Module.RCLike.Real.1117416928._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real E Real.normedField inst._@.Mathlib.Analysis.Normed.Module.RCLike.Real.1117416928._hygCtx._hyg.3] (x : E), Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) (Metric.closedBall.{u_1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} E inst._@.Mathlib.Analysis.Normed.Module.RCLike.Real.1117416928._hygCtx._hyg.3) (OfNat.ofNat.{u_1} E 0 (Zero.toOfNat0.{u_1} E (NegZeroClass.toZero.{u_1} E (SubNegZeroMonoid.toNegZeroClass.{u_1} E (SubtractionMonoid.toSubNegZeroMonoid.{u_1} E (SubtractionCommMonoid.toSubtractionMonoid.{u_1} E (AddCommGroup.toDivisionAddCommMonoid.{u_1} E (SeminormedAddCommGroup.toAddCommGroup.{u_1} E inst._@.Mathlib.Analysis.Normed.Module.RCLike.Real.1117416928._hygCtx._hyg.3)))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne))) (HSMul.hSMul.{0, u_1, u_1} Real E E (instHSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (SeminormedAddGroup.toAddGroup.{u_1} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u_1} E inst._@.Mathlib.Analysis.Normed.Module.RCLike.Real.1117416928._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (SeminormedAddGroup.toAddGroup.{u_1} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u_1} E inst._@.Mathlib.Analysis.Normed.Module.RCLike.Real.1117416928._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (SeminormedAddGroup.toAddGroup.{u_1} E 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Abelian.Monomorphisms.sym.json

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