Documentation Verification Report

Slope

📁 Source: Mathlib/Analysis/Convex/Slope.lean

Statistics

Metric	Count
Definitions	0
Theorems `slope_anti_adjacent`, `strictAntiOn`, `strictMonoOn`, `secant_mono`, `secant_mono_aux1`, `secant_mono_aux2`, `secant_mono_aux3`, `slope_mono_adjacent`, `strictAntiOn`, `strictMonoOn`, `strict_mono_of_lt`, `secant_strict_mono`, `slope_anti_adjacent`, `secant_strict_mono`, `secant_strict_mono_aux1`, `secant_strict_mono_aux2`, `secant_strict_mono_aux3`, `slope_strict_mono_adjacent`, `concaveOn_iff_slope_anti_adjacent`, `concaveOn_of_slope_anti_adjacent`, `convexOn_iff_slope_mono_adjacent`, `convexOn_of_slope_mono_adjacent`, `strictConcaveOn_iff_slope_strict_anti_adjacent`, `strictConcaveOn_of_slope_strict_anti_adjacent`, `strictConvexOn_iff_slope_strict_mono_adjacent`, `strictConvexOn_of_slope_strict_mono_adjacent`	26
Total	26

ConcaveOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`slope_anti_adjacent` 📖	mathematical	`ConcaveOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`Preorder.toLE` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`ConvexOn.slope_mono_adjacent` `neg` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.LinearCombination.le_of_le` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.LinearCombination.le_rearrange` `Mathlib.Tactic.Ring.le_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.neg_congr` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.isInt_add` `le_refl` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero`
`strictAntiOn` 📖	mathematical	`ConcaveOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`StrictAntiOn` `Set.instInter` `Set.Ici`	—	`neg_neg` `StrictMonoOn.neg` `IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono` `AddLeftCancelSemigroup.toIsLeftCancelAdd` `IsOrderedAddMonoid.toAddLeftMono` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `IsRightCancelAdd.addRightStrictMono_of_addRightMono` `AddRightCancelSemigroup.toIsRightCancelAdd` `covariant_swap_add_of_covariant_add` `ConvexOn.strictMonoOn` `neg_convexOn_iff` `neg_lt_neg`
`strictMonoOn` 📖	mathematical	`ConcaveOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`StrictMonoOn` `Set.instInter` `Set.Iic`	—	`neg_neg` `StrictAntiOn.neg` `IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono` `AddLeftCancelSemigroup.toIsLeftCancelAdd` `IsOrderedAddMonoid.toAddLeftMono` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `IsRightCancelAdd.addRightStrictMono_of_addRightMono` `AddRightCancelSemigroup.toIsRightCancelAdd` `covariant_swap_add_of_covariant_add` `ConvexOn.strictAntiOn` `neg_convexOn_iff` `neg_lt_neg`

ConvexOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`secant_mono` 📖	mathematical	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLE` `PartialOrder.toPreorder`	`DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`eq_or_lt_of_le` `lt_or_gt_of_ne` `neg_div_neg_eq` `neg_sub` `secant_mono_aux3` `slope_mono_adjacent` `secant_mono_aux2`
`secant_mono_aux1` 📖	mathematical	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`Preorder.toLE` `Distrib.toMul` `NonUnitalNonAssocSemiring.toDistrib` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing` `Field.toDivisionRing` `Distrib.toAdd`	—	`lt_of_not_ge` `Mathlib.Tactic.Linarith.lt_irrefl` `Mathlib.Tactic.Ring.of_eq` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Linarith.add_lt_of_neg_of_le` `Mathlib.Tactic.Linarith.sub_neg_of_lt` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.Linarith.sub_nonpos_of_le` `Mathlib.Tactic.Linarith.add_neg` `le_of_lt` `div_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq` `IsCancelMulZero.toIsLeftCancelMulZero` `instIsCancelMulZero` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.subst_add` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₁` `Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃` `mul_one` `Mathlib.Tactic.FieldSimp.NF.one_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `ne_of_gt` `Mathlib.Tactic.FieldSimp.NF.cons_ne_zero` `one_ne_zero` `NeZero.one` `GroupWithZero.toNontrivial` `Mathlib.Tactic.LinearCombination.le_of_le` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.FieldSimp.le_eq_cancel_le` `PosMulStrictMono.toPosMulMono` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval` `Mathlib.Tactic.FieldSimp.NF.inv_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₂` `Mathlib.Tactic.FieldSimp.NF.eval_cons_eq_eval_of_eq_of_eq` `Mathlib.Tactic.FieldSimp.NF.eval_cons_of_pow_eq_zero` `LT.lt.ne'` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.RingNF.nat_rawCast_1` `pow_one` `Mathlib.Tactic.RingNF.int_rawCast_neg` `Mathlib.Tactic.RingNF.mul_neg` `add_zero` `Mathlib.Tactic.RingNF.add_neg` `Mathlib.Tactic.RingNF.mul_assoc_rev` `Mathlib.Tactic.RingNF.add_assoc_rev` `Mathlib.Tactic.LinearCombination.le_rearrange` `IsOrderedRing.toIsOrderedAddMonoid` `Mathlib.Tactic.Ring.le_congr` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `le_refl`
`secant_mono_aux2` 📖	mathematical	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`Preorder.toLE` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`lt_of_not_ge` `Mathlib.Tactic.Linarith.lt_irrefl` `Mathlib.Tactic.Ring.of_eq` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Linarith.add_lt_of_neg_of_le` `Mathlib.Tactic.Linarith.sub_neg_of_lt` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.Linarith.sub_nonpos_of_le` `Mathlib.Tactic.Linarith.add_neg` `Mathlib.Tactic.FieldSimp.le_eq_cancel_le` `PosMulStrictMono.toPosMulMono` `IsStrictOrderedRing.toPosMulStrictMono` `PosMulStrictMono.toPosMulReflectLE` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `LT.lt.ne'` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `PosMulReflectLE.toPosMulReflectLT` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `NeZero.one` `GroupWithZero.toNontrivial` `le_of_not_gt` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Ring.zero_mul` `secant_mono_aux1`
`secant_mono_aux3` 📖	mathematical	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`Preorder.toLE` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`lt_of_not_ge` `Mathlib.Tactic.Linarith.lt_irrefl` `Mathlib.Tactic.Ring.of_eq` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Linarith.add_lt_of_neg_of_le` `Mathlib.Tactic.Linarith.sub_neg_of_lt` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.Linarith.sub_nonpos_of_le` `Mathlib.Tactic.Linarith.add_neg` `Mathlib.Tactic.FieldSimp.le_eq_cancel_le` `PosMulStrictMono.toPosMulMono` `IsStrictOrderedRing.toPosMulStrictMono` `PosMulStrictMono.toPosMulReflectLE` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `LT.lt.ne'` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `PosMulReflectLE.toPosMulReflectLT` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `NeZero.one` `GroupWithZero.toNontrivial` `le_of_not_gt` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Ring.zero_mul` `secant_mono_aux1`
`slope_mono_adjacent` 📖	mathematical	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`Preorder.toLE` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`LT.lt.trans` `le_of_lt` `div_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `sub_pos` `IsRightCancelAdd.addRightStrictMono_of_addRightMono` `AddRightCancelSemigroup.toIsRightCancelAdd` `covariant_swap_add_of_covariant_add` `IsOrderedAddMonoid.toAddLeftMono` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq` `IsCancelMulZero.toIsLeftCancelMulZero` `instIsCancelMulZero` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.subst_add` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₁` `Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃` `mul_one` `Mathlib.Tactic.FieldSimp.NF.one_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `ne_of_gt` `Mathlib.Tactic.FieldSimp.NF.cons_ne_zero` `one_ne_zero` `NeZero.one` `GroupWithZero.toNontrivial` `Mathlib.Tactic.Ring.of_eq` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.FieldSimp.le_eq_cancel_le` `PosMulStrictMono.toPosMulMono` `LT.lt.ne'` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `le_of_not_gt` `Mathlib.Tactic.Linarith.lt_irrefl` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero` `Mathlib.Tactic.Linarith.add_lt_of_le_of_neg` `Mathlib.Tactic.Linarith.sub_nonpos_of_le` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval` `Mathlib.Tactic.FieldSimp.NF.inv_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₂` `Mathlib.Tactic.FieldSimp.NF.eval_cons_eq_eval_of_eq_of_eq` `Mathlib.Tactic.FieldSimp.NF.eval_cons_of_pow_eq_zero` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.RingNF.nat_rawCast_1` `pow_one` `Mathlib.Tactic.RingNF.int_rawCast_neg` `Mathlib.Tactic.RingNF.mul_neg` `add_zero` `Mathlib.Tactic.RingNF.add_neg` `Mathlib.Tactic.RingNF.mul_assoc_rev` `Mathlib.Tactic.RingNF.add_assoc_rev` `Mathlib.Tactic.Linarith.sub_neg_of_lt`
`strictAntiOn` 📖	mathematical	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`StrictAntiOn` `Set.instInter` `Set.Iic`	—	`strictMonoOn` `comp_affineMap` `neg_neg` `neg_lt_neg` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `StrictMonoOn.comp_strictAntiOn` `StrictMonoOn.neg` `IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono` `AddLeftCancelSemigroup.toIsLeftCancelAdd` `IsOrderedAddMonoid.toAddLeftMono` `IsRightCancelAdd.addRightStrictMono_of_addRightMono` `AddRightCancelSemigroup.toIsRightCancelAdd` `covariant_swap_add_of_covariant_add` `strictMonoOn_id`
`strictMonoOn` 📖	mathematical	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`StrictMonoOn` `Set.instInter` `Set.Ici`	—	`lt_right_of_left_lt` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `IsStrictOrderedModule.toPosSMulStrictMono` `IsStrictOrderedRing.toIsStrictOrderedModule` `openSegment_eq_Ioo` `LT.lt.trans` `eq_or_lt_of_le` `Convex.segment_subset` `segment_eq_Icc` `LE.le.trans` `LT.lt.le`
`strict_mono_of_lt` 📖	mathematical	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`StrictMonoOn` `Set.instInter` `Set.Ici`	—	`strictMonoOn`

StrictConcaveOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`secant_strict_mono` 📖	mathematical	`StrictConcaveOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`StrictConvexOn.secant_strict_mono` `neg` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.LinearCombination.lt_of_lt` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.LinearCombination.le_rearrange` `Mathlib.Tactic.Ring.le_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.neg_congr` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `le_refl` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero`
`slope_anti_adjacent` 📖	mathematical	`StrictConcaveOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`StrictConvexOn.slope_strict_mono_adjacent` `neg` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.LinearCombination.lt_of_lt` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.LinearCombination.le_rearrange` `Mathlib.Tactic.Ring.le_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.neg_congr` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.isInt_add` `le_refl` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero`

StrictConvexOn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`secant_strict_mono` 📖	mathematical	`StrictConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`lt_or_gt_of_ne` `neg_div_neg_eq` `neg_sub` `secant_strict_mono_aux3` `slope_strict_mono_adjacent` `secant_strict_mono_aux2`
`secant_strict_mono_aux1` 📖	mathematical	`StrictConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`Distrib.toMul` `NonUnitalNonAssocSemiring.toDistrib` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing` `Field.toDivisionRing` `Distrib.toAdd`	—	`lt_of_not_ge` `Mathlib.Tactic.Linarith.lt_irrefl` `Mathlib.Tactic.Ring.of_eq` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Linarith.add_lt_of_neg_of_le` `Mathlib.Tactic.Linarith.sub_neg_of_lt` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.Linarith.sub_nonpos_of_le` `Mathlib.Tactic.Linarith.add_neg` `div_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Mathlib.Tactic.Ring.neg_congr` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Linarith.lt_of_lt_of_eq` `neg_eq_zero` `sub_eq_zero_of_eq` `Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq` `IsCancelMulZero.toIsLeftCancelMulZero` `instIsCancelMulZero` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.subst_add` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₁` `Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃` `mul_one` `Mathlib.Tactic.FieldSimp.NF.one_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `ne_of_gt` `Mathlib.Tactic.FieldSimp.NF.cons_ne_zero` `one_ne_zero` `NeZero.one` `GroupWithZero.toNontrivial` `Mathlib.Tactic.LinearCombination.lt_of_lt` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.FieldSimp.lt_eq_cancel_lt` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval` `Mathlib.Tactic.FieldSimp.NF.inv_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₂` `Mathlib.Tactic.FieldSimp.NF.eval_cons_eq_eval_of_eq_of_eq` `Mathlib.Tactic.FieldSimp.NF.eval_cons_of_pow_eq_zero` `LT.lt.ne'` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.RingNF.nat_rawCast_1` `pow_one` `Mathlib.Tactic.RingNF.int_rawCast_neg` `Mathlib.Tactic.RingNF.mul_neg` `add_zero` `Mathlib.Tactic.RingNF.add_neg` `Mathlib.Tactic.RingNF.mul_assoc_rev` `Mathlib.Tactic.RingNF.add_assoc_rev` `Mathlib.Tactic.LinearCombination.le_rearrange` `IsOrderedRing.toIsOrderedAddMonoid` `Mathlib.Tactic.Ring.le_congr` `le_refl`
`secant_strict_mono_aux2` 📖	mathematical	`StrictConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`lt_of_not_ge` `Mathlib.Tactic.Linarith.lt_irrefl` `Mathlib.Tactic.Ring.of_eq` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Linarith.add_lt_of_neg_of_le` `Mathlib.Tactic.Linarith.sub_neg_of_lt` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.Linarith.sub_nonpos_of_le` `Mathlib.Tactic.Linarith.add_neg` `Mathlib.Tactic.FieldSimp.lt_eq_cancel_lt` `IsStrictOrderedRing.toPosMulStrictMono` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `LT.lt.ne'` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `NeZero.one` `GroupWithZero.toNontrivial` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Linarith.add_lt_of_le_of_neg` `secant_strict_mono_aux1`
`secant_strict_mono_aux3` 📖	mathematical	`StrictConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`lt_of_not_ge` `Mathlib.Tactic.Linarith.lt_irrefl` `Mathlib.Tactic.Ring.of_eq` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Linarith.add_lt_of_neg_of_le` `Mathlib.Tactic.Linarith.sub_neg_of_lt` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.Linarith.sub_nonpos_of_le` `Mathlib.Tactic.Linarith.add_neg` `Mathlib.Tactic.FieldSimp.lt_eq_cancel_lt` `IsStrictOrderedRing.toPosMulStrictMono` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `LT.lt.ne'` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `NeZero.one` `GroupWithZero.toNontrivial` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Linarith.add_lt_of_le_of_neg` `secant_strict_mono_aux1`
`slope_strict_mono_adjacent` 📖	mathematical	`StrictConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Set` `Set.instMembership` `Preorder.toLT` `PartialOrder.toPreorder`	`DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`LT.lt.trans` `LT.lt.ne` `div_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `sub_pos` `IsRightCancelAdd.addRightStrictMono_of_addRightMono` `AddRightCancelSemigroup.toIsRightCancelAdd` `covariant_swap_add_of_covariant_add` `IsOrderedAddMonoid.toAddLeftMono` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq` `IsCancelMulZero.toIsLeftCancelMulZero` `instIsCancelMulZero` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.subst_add` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₁` `Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃` `mul_one` `Mathlib.Tactic.FieldSimp.NF.one_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `ne_of_gt` `Mathlib.Tactic.FieldSimp.NF.cons_ne_zero` `one_ne_zero` `NeZero.one` `GroupWithZero.toNontrivial` `Mathlib.Tactic.Ring.of_eq` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.FieldSimp.lt_eq_cancel_lt` `LT.lt.ne'` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `lt_of_not_ge` `Mathlib.Tactic.Linarith.lt_irrefl` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero` `Mathlib.Tactic.Linarith.add_lt_of_neg_of_le` `Mathlib.Tactic.Linarith.sub_neg_of_lt` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval` `Mathlib.Tactic.FieldSimp.NF.inv_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₂` `Mathlib.Tactic.FieldSimp.NF.eval_cons_eq_eval_of_eq_of_eq` `Mathlib.Tactic.FieldSimp.NF.eval_cons_of_pow_eq_zero` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.RingNF.nat_rawCast_1` `pow_one` `Mathlib.Tactic.RingNF.int_rawCast_neg` `Mathlib.Tactic.RingNF.mul_neg` `add_zero` `Mathlib.Tactic.RingNF.add_neg` `Mathlib.Tactic.RingNF.mul_assoc_rev` `Mathlib.Tactic.RingNF.add_assoc_rev` `Mathlib.Tactic.Linarith.sub_nonpos_of_le`

(root)

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`concaveOn_iff_slope_anti_adjacent` 📖	mathematical	—	`ConcaveOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Convex` `Preorder.toLE` `PartialOrder.toPreorder` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`ConcaveOn.slope_anti_adjacent` `concaveOn_of_slope_anti_adjacent`
`concaveOn_of_slope_anti_adjacent` 📖	mathematical	`Convex` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Preorder.toLE` `PartialOrder.toPreorder` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	`ConcaveOn`	—	`neg_convexOn_iff` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `convexOn_of_slope_mono_adjacent` `Mathlib.Tactic.LinearCombination.le_of_le` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.LinearCombination.le_rearrange` `Mathlib.Tactic.Ring.le_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.neg_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `le_refl` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero`
`convexOn_iff_slope_mono_adjacent` 📖	mathematical	—	`ConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Convex` `Preorder.toLE` `PartialOrder.toPreorder` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`ConvexOn.slope_mono_adjacent` `convexOn_of_slope_mono_adjacent`
`convexOn_of_slope_mono_adjacent` 📖	mathematical	`Convex` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Preorder.toLE` `PartialOrder.toPreorder` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	`ConvexOn`	—	`LinearOrder.convexOn_of_lt` `Mathlib.Tactic.LinearCombination.lt_of_lt` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.LinearCombination.add_lt_eq` `Mathlib.Tactic.LinearCombination.mul_const_lt` `Mathlib.Tactic.LinearCombination.mul_const_eq` `Mathlib.Tactic.LinearCombination.le_rearrange` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.Ring.le_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.cast_pos` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_one` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `le_refl` `Mathlib.Tactic.Ring.cast_zero` `Nat.cast_zero` `le_of_mul_le_mul_left` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Mathlib.Tactic.LinearCombination.le_of_le` `Mathlib.Tactic.LinearCombination.add_le_eq` `Mathlib.Tactic.FieldSimp.le_eq_cancel_le` `PosMulStrictMono.toPosMulMono` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.subst_add` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃` `mul_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `LT.lt.ne'` `sub_pos` `IsRightCancelAdd.addRightStrictMono_of_addRightMono` `AddRightCancelSemigroup.toIsRightCancelAdd` `covariant_swap_add_of_covariant_add` `IsOrderedAddMonoid.toAddLeftMono` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `PosMulReflectLE.toPosMulReflectLT` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `NeZero.one` `GroupWithZero.toNontrivial` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Ring.neg_congr`
`strictConcaveOn_iff_slope_strict_anti_adjacent` 📖	mathematical	—	`StrictConcaveOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Convex` `Preorder.toLT` `PartialOrder.toPreorder` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`StrictConcaveOn.slope_anti_adjacent` `strictConcaveOn_of_slope_strict_anti_adjacent`
`strictConcaveOn_of_slope_strict_anti_adjacent` 📖	mathematical	`Convex` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Preorder.toLT` `PartialOrder.toPreorder` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	`StrictConcaveOn`	—	`neg_strictConvexOn_iff` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `strictConvexOn_of_slope_strict_mono_adjacent` `Mathlib.Tactic.LinearCombination.lt_of_lt` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.LinearCombination.le_rearrange` `Mathlib.Tactic.Ring.le_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.div_congr` `Mathlib.Tactic.Ring.neg_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.div_pf` `Mathlib.Tactic.Ring.atom_pf'` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.isInt_add` `Mathlib.Tactic.Ring.add_pf_add_gt` `le_refl` `Mathlib.Tactic.Ring.cast_zero` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_zero`
`strictConvexOn_iff_slope_strict_mono_adjacent` 📖	mathematical	—	`StrictConvexOn` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Convex` `Preorder.toLT` `PartialOrder.toPreorder` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	—	`StrictConvexOn.slope_strict_mono_adjacent` `strictConvexOn_of_slope_strict_mono_adjacent`
`strictConvexOn_of_slope_strict_mono_adjacent` 📖	mathematical	`Convex` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `SemilatticeInf.toPartialOrder` `Lattice.toSemilatticeInf` `DistribLattice.toLattice` `instDistribLatticeOfLinearOrder` `NonUnitalNonAssocSemiring.toAddCommMonoid` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `CommRing.toNonUnitalCommRing` `Field.toCommRing` `Algebra.toSMul` `Semifield.toCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `Preorder.toLT` `PartialOrder.toPreorder` `DivInvMonoid.toDiv` `DivisionRing.toDivInvMonoid` `Field.toDivisionRing` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddGroupWithOne.toAddGroup` `Ring.toAddGroupWithOne` `DivisionRing.toRing`	`StrictConvexOn`	—	`LinearOrder.strictConvexOn_of_lt` `Mathlib.Tactic.LinearCombination.lt_of_lt` `IsStrictOrderedRing.toIsOrderedCancelAddMonoid` `Mathlib.Tactic.LinearCombination.add_lt_eq` `Mathlib.Tactic.LinearCombination.mul_const_lt` `Mathlib.Tactic.LinearCombination.mul_const_eq` `Mathlib.Tactic.LinearCombination.le_rearrange` `IsOrderedRing.toIsOrderedAddMonoid` `IsStrictOrderedRing.toIsOrderedRing` `Mathlib.Tactic.Ring.le_congr` `Mathlib.Tactic.Ring.sub_congr` `Mathlib.Tactic.Ring.add_congr` `Mathlib.Tactic.Ring.atom_pf` `Mathlib.Tactic.Ring.mul_congr` `Mathlib.Tactic.Ring.add_mul` `Mathlib.Tactic.Ring.mul_add` `Mathlib.Tactic.Ring.mul_pf_left` `Mathlib.Tactic.Ring.mul_pf_right` `Mathlib.Tactic.Ring.one_mul` `Mathlib.Tactic.Ring.mul_zero` `Mathlib.Tactic.Ring.add_pf_add_zero` `Mathlib.Tactic.Ring.zero_mul` `Mathlib.Tactic.Ring.add_pf_add_gt` `Mathlib.Tactic.Ring.add_pf_add_lt` `Mathlib.Tactic.Ring.add_pf_zero_add` `Mathlib.Tactic.Ring.cast_pos` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_one` `Mathlib.Tactic.Ring.sub_pf` `Mathlib.Tactic.Ring.neg_add` `Mathlib.Tactic.Ring.neg_mul` `Mathlib.Tactic.Ring.neg_one_mul` `Mathlib.Meta.NormNum.IsInt.to_raw_eq` `Mathlib.Meta.NormNum.isInt_mul` `Mathlib.Meta.NormNum.IsInt.of_raw` `Mathlib.Meta.NormNum.IsNat.to_isInt` `Mathlib.Meta.NormNum.IsNat.of_raw` `Mathlib.Tactic.Ring.neg_zero` `Mathlib.Tactic.Ring.add_pf_add_overlap_zero` `Mathlib.Tactic.Ring.add_overlap_pf_zero` `Mathlib.Meta.NormNum.IsInt.to_isNat` `Mathlib.Meta.NormNum.isInt_add` `le_refl` `Mathlib.Tactic.Ring.cast_zero` `Nat.cast_zero` `lt_of_mul_lt_mul_left` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Mathlib.Tactic.FieldSimp.lt_eq_cancel_lt` `Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval` `Mathlib.Tactic.FieldSimp.subst_sub` `Mathlib.Tactic.FieldSimp.NF.atom_eq_eval` `Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons` `one_mul` `Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst` `Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div` `div_one` `Mathlib.Tactic.FieldSimp.NF.eval_cons` `Mathlib.Tactic.FieldSimp.zpow'_one` `Mathlib.Tactic.FieldSimp.subst_add` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval` `Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃` `mul_one` `Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg` `LT.lt.ne'` `sub_pos` `IsRightCancelAdd.addRightStrictMono_of_addRightMono` `AddRightCancelSemigroup.toIsRightCancelAdd` `covariant_swap_add_of_covariant_add` `IsOrderedAddMonoid.toAddLeftMono` `Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval` `Mathlib.Tactic.FieldSimp.NF.cons_pos` `IsStrictOrderedRing.toZeroLEOneClass` `zero_lt_one` `NeZero.one` `GroupWithZero.toNontrivial` `Mathlib.Tactic.Ring.mul_one` `Mathlib.Meta.NormNum.IsNat.to_raw_eq` `Mathlib.Tactic.Ring.neg_congr` `LT.lt.le`

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