Documentation

Mathlib.Order.RelClasses

Unbundled relation classes #

In this file we prove some properties of Is* classes defined in Init.Algebra.Classes. The main difference between these classes and the usual order classes (Preorder etc) is that usual classes extend LE and/or LT while these classes take a relation as an explicit argument.

theorem of_eq {α : Type u} {r : ααProp} [inst : IsRefl α r] {a : α} {b : α} :
a = br a b
theorem comm {α : Type u} {r : ααProp} [inst : IsSymm α r] {a : α} {b : α} :
r a b r b a
theorem antisymm' {α : Type u} {r : ααProp} [inst : IsAntisymm α r] {a : α} {b : α} :
r a br b ab = a
theorem antisymm_iff {α : Type u} {r : ααProp} [inst : IsRefl α r] [inst : IsAntisymm α r] {a : α} {b : α} :
r a b r b a a = b
@[elab_without_expected_type]
theorem antisymm_of {α : Type u} (r : ααProp) [inst : IsAntisymm α r] {a : α} {b : α} :
r a br b aa = b

A version of antisymm with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

@[elab_without_expected_type]
theorem antisymm_of' {α : Type u} (r : ααProp) [inst : IsAntisymm α r] {a : α} {b : α} :
r a br b ab = a

A version of antisymm' with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

theorem comm_of {α : Type u} (r : ααProp) [inst : IsSymm α r] {a : α} {b : α} :
r a b r b a

A version of comm with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

theorem IsRefl.swap {α : Type u} (r : ααProp) [inst : IsRefl α r] :
theorem IsIrrefl.swap {α : Type u} (r : ααProp) [inst : IsIrrefl α r] :
theorem IsTrans.swap {α : Type u} (r : ααProp) [inst : IsTrans α r] :
theorem IsAntisymm.swap {α : Type u} (r : ααProp) [inst : IsAntisymm α r] :
theorem IsAsymm.swap {α : Type u} (r : ααProp) [inst : IsAsymm α r] :
theorem IsTotal.swap {α : Type u} (r : ααProp) [inst : IsTotal α r] :
theorem IsTrichotomous.swap {α : Type u} (r : ααProp) [inst : IsTrichotomous α r] :
theorem IsPreorder.swap {α : Type u} (r : ααProp) [inst : IsPreorder α r] :
theorem IsStrictOrder.swap {α : Type u} (r : ααProp) [inst : IsStrictOrder α r] :
theorem IsPartialOrder.swap {α : Type u} (r : ααProp) [inst : IsPartialOrder α r] :
theorem IsTotalPreorder.swap {α : Type u} (r : ααProp) [inst : IsTotalPreorder α r] :
theorem IsLinearOrder.swap {α : Type u} (r : ααProp) [inst : IsLinearOrder α r] :
theorem IsAsymm.isAntisymm {α : Type u} (r : ααProp) [inst : IsAsymm α r] :
theorem IsAsymm.isIrrefl {α : Type u} {r : ααProp} [inst : IsAsymm α r] :
theorem IsTotal.isTrichotomous {α : Type u} (r : ααProp) [inst : IsTotal α r] :
instance IsTotal.to_isRefl {α : Type u} (r : ααProp) [inst : IsTotal α r] :
IsRefl α r
Equations
theorem ne_of_irrefl {α : Type u} {r : ααProp} [inst : IsIrrefl α r] {x : α} {y : α} :
r x yx y
theorem ne_of_irrefl' {α : Type u} {r : ααProp} [inst : IsIrrefl α r] {x : α} {y : α} :
r x yy x
theorem not_rel_of_subsingleton {α : Type u} (r : ααProp) [inst : IsIrrefl α r] [inst : Subsingleton α] (x : α) (y : α) :
¬r x y
theorem rel_of_subsingleton {α : Type u} (r : ααProp) [inst : IsRefl α r] [inst : Subsingleton α] (x : α) (y : α) :
r x y
@[simp]
theorem empty_relation_apply {α : Type u} (a : α) (b : α) :
theorem eq_empty_relation {α : Type u} (r : ααProp) [inst : IsIrrefl α r] [inst : Subsingleton α] :
r = EmptyRelation
theorem trans_trichotomous_left {α : Type u} {r : ααProp} [inst : IsTrans α r] [inst : IsTrichotomous α r] {a : α} {b : α} {c : α} :
¬r b ar b cr a c
theorem trans_trichotomous_right {α : Type u} {r : ααProp} [inst : IsTrans α r] [inst : IsTrichotomous α r] {a : α} {b : α} {c : α} :
r a b¬r c br a c
theorem transitive_of_trans {α : Type u} (r : ααProp) [inst : IsTrans α r] :
theorem extensional_of_trichotomous_of_irrefl {α : Type u} (r : ααProp) [inst : IsTrichotomous α r] [inst : IsIrrefl α r] {a : α} {b : α} (H : ∀ (x : α), r x a r x b) :
a = b

In a trichotomous irreflexive order, every element is determined by the set of predecessors.

def partialOrderOfSO {α : Type u} (r : ααProp) [inst : IsStrictOrder α r] :

Construct a partial order from a isStrictOrder relation.

See note [reducible non-instances].

Equations
def linearOrderOfSTO {α : Type u} (r : ααProp) [inst : IsStrictTotalOrder α r] [inst : (x y : α) → Decidable ¬r x y] :

Construct a linear order from an IsStrictTotalOrder relation.

See note [reducible non-instances].

Equations
  • One or more equations did not get rendered due to their size.
theorem IsStrictTotalOrder.swap {α : Type u} (r : ααProp) [inst : IsStrictTotalOrder α r] :

Order connection #

class IsOrderConnected (α : Type u) (lt : ααProp) :
  • A connected order is one satisfying the condition a < c → a < b ∨ b < c→ a < b ∨ b < c∨ b < c.

    conn : ∀ (a b c : α), lt a clt a b lt b c

A connected order is one satisfying the condition a < c → a < b ∨ b < c→ a < b ∨ b < c∨ b < c. This is recognizable as an intuitionistic substitute for a ≤ b ∨ b ≤ a≤ b ∨ b ≤ a∨ b ≤ a≤ a on the constructive reals, and is also known as negative transitivity, since the contrapositive asserts transitivity of the relation ¬ a < b¬ a < b.

Instances
    theorem IsOrderConnected.neg_trans {α : Type u} {r : ααProp} [inst : IsOrderConnected α r] {a : α} {b : α} {c : α} (h₁ : ¬r a b) (h₂ : ¬r b c) :
    ¬r a c
    theorem isStrictWeakOrder_of_isOrderConnected {α : Type u} {r : ααProp} [inst : IsAsymm α r] [inst : IsOrderConnected α r] :

    Well-order #

    theorem IsWellFounded_iff (α : Type u) (r : ααProp) :
    class IsWellFounded (α : Type u) (r : ααProp) :

    A well-founded relation. Not to be confused with isWellOrder.

    Instances
      theorem WellFounded.prod_lex {α : Type u} {β : Type v} {ra : ααProp} {rb : ββProp} (ha : WellFounded ra) (hb : WellFounded rb) :
      theorem IsWellFounded.induction {α : Type u} (r : ααProp) [inst : IsWellFounded α r] {C : αProp} (a : α) :
      ((x : α) → ((y : α) → r y xC y) → C x) → C a

      Induction on a well-founded relation.

      theorem IsWellFounded.apply {α : Type u} (r : ααProp) [inst : IsWellFounded α r] (a : α) :
      Acc r a

      All values are accessible under the well-founded relation.

      noncomputable def IsWellFounded.fix {α : Type u} (r : ααProp) [inst : IsWellFounded α r] {C : αSort u_1} :
      ((x : α) → ((y : α) → r y xC y) → C x) → (x : α) → C x

      Creates data, given a way to generate a value from all that compare as less under a well-founded relation. See also IsWellFounded.fix_eq.

      Equations
      theorem IsWellFounded.fix_eq {α : Type u} (r : ααProp) [inst : IsWellFounded α r] {C : αSort u_1} (F : (x : α) → ((y : α) → r y xC y) → C x) (x : α) :
      IsWellFounded.fix r F x = F x fun y x => IsWellFounded.fix r F y

      The value from IsWellFounded.fix is built from the previous ones as specified.

      def IsWellFounded.toWellFoundedRelation {α : Type u} (r : ααProp) [inst : IsWellFounded α r] :

      Derive a WellFoundedRelation instance from an isWellFounded instance.

      Equations
      theorem WellFounded.asymmetric {α : Sort u_1} {r : ααProp} (h : WellFounded r) (a : α) (b : α) :
      r a b¬r b a
      instance instIsAsymm {α : Type u} (r : ααProp) [inst : IsWellFounded α r] :
      IsAsymm α r
      Equations
      instance instIsIrrefl {α : Type u} (r : ααProp) [inst : IsWellFounded α r] :
      Equations
      def WellFoundedLT (α : Type u_1) [inst : LT α] :

      A class for a well founded relation <.

      Equations
      def WellFoundedGT (α : Type u_1) [inst : LT α] :

      A class for a well founded relation >.

      Equations
      class IsWellOrder (α : Type u) (r : ααProp) extends IsTrichotomous , IsTrans , IsWellFounded :

        A well order is a well-founded linear order.

        Instances
          instance instIsTrichotomous {α : Type u_1} (r : ααProp) [inst : IsWellOrder α r] :
          Equations
          instance instIsTrans_1 {α : Type u_1} (r : ααProp) [inst : IsWellOrder α r] :
          IsTrans α r
          Equations
          instance instIsIrrefl_1 {α : Type u_1} (r : ααProp) [inst : IsWellOrder α r] :
          Equations
          instance instIsAsymm_1 {α : Type u_1} (r : ααProp) [inst : IsWellOrder α r] :
          IsAsymm α r
          Equations
          theorem WellFoundedLT.induction {α : Type u} [inst : LT α] [inst : WellFoundedLT α] {C : αProp} (a : α) :
          ((x : α) → ((y : α) → y < xC y) → C x) → C a

          Inducts on a well-founded < relation.

          theorem WellFoundedLT.apply {α : Type u} [inst : LT α] [inst : WellFoundedLT α] (a : α) :
          Acc (fun x x_1 => x < x_1) a

          All values are accessible under the well-founded <.

          noncomputable def WellFoundedLT.fix {α : Type u} [inst : LT α] [inst : WellFoundedLT α] {C : αSort u_1} :
          ((x : α) → ((y : α) → y < xC y) → C x) → (x : α) → C x

          Creates data, given a way to generate a value from all that compare as lesser. See also WellFoundedLT.fix_eq.

          Equations
          theorem WellFoundedLT.fix_eq {α : Type u} [inst : LT α] [inst : WellFoundedLT α] {C : αSort u_1} (F : (x : α) → ((y : α) → y < xC y) → C x) (x : α) :
          WellFoundedLT.fix F x = F x fun y x => WellFoundedLT.fix F y

          The value from WellFoundedLT.fix is built from the previous ones as specified.

          Derive a WellFoundedRelation instance from a WellFoundedLT instance.

          Equations
          theorem WellFoundedGT.induction {α : Type u} [inst : LT α] [inst : WellFoundedGT α] {C : αProp} (a : α) :
          ((x : α) → ((y : α) → x < yC y) → C x) → C a

          Inducts on a well-founded > relation.

          theorem WellFoundedGT.apply {α : Type u} [inst : LT α] [inst : WellFoundedGT α] (a : α) :
          Acc (fun x x_1 => x > x_1) a

          All values are accessible under the well-founded >.

          noncomputable def WellFoundedGT.fix {α : Type u} [inst : LT α] [inst : WellFoundedGT α] {C : αSort u_1} :
          ((x : α) → ((y : α) → x < yC y) → C x) → (x : α) → C x

          Creates data, given a way to generate a value from all that compare as greater. See also WellFoundedGT.fix_eq.

          Equations
          theorem WellFoundedGT.fix_eq {α : Type u} [inst : LT α] [inst : WellFoundedGT α] {C : αSort u_1} (F : (x : α) → ((y : α) → x < yC y) → C x) (x : α) :
          WellFoundedGT.fix F x = F x fun y x => WellFoundedGT.fix F y

          The value from WellFoundedGT.fix is built from the successive ones as specified.

          Derive a WellFoundedRelation instance from a WellFoundedGT instance.

          Equations
          noncomputable def IsWellOrder.linearOrder {α : Type u} (r : ααProp) [inst : IsWellOrder α r] :

          Construct a decidable linear order from a well-founded linear order.

          Equations
          def IsWellOrder.toHasWellFounded {α : Type u} [inst : LT α] [hwo : IsWellOrder α fun x x_1 => x < x_1] :

          Derive a WellFoundedRelation instance from a IsWellOrder instance.

          Equations
          • IsWellOrder.toHasWellFounded = { rel := fun x x_1 => x < x_1, wf := (_ : WellFounded fun x x_1 => x < x_1) }
          theorem Subsingleton.isWellOrder {α : Type u} [inst : Subsingleton α] (r : ααProp) [hr : IsIrrefl α r] :
          instance instIsWellOrder {α : Type u} [inst : IsEmpty α] (r : ααProp) :
          Equations
          instance instIsWellFoundedProdLex {α : Type u} {β : Type v} {r : ααProp} {s : ββProp} [inst : IsWellFounded α r] [inst : IsWellFounded β s] :
          IsWellFounded (α × β) (Prod.Lex r s)
          Equations
          instance instIsWellOrderProdLex {α : Type u} {β : Type v} {r : ααProp} {s : ββProp} [inst : IsWellOrder α r] [inst : IsWellOrder β s] :
          IsWellOrder (α × β) (Prod.Lex r s)
          Equations
          instance instIsWellFoundedInvImage {α : Type u} {β : Type v} (r : ααProp) [inst : IsWellFounded α r] (f : βα) :
          Equations
          theorem Subrelation.isWellFounded {α : Type u} (r : ααProp) [inst : IsWellFounded α r] {s : ααProp} (h : Subrelation s r) :
          def Set.Unbounded {α : Type u} (r : ααProp) (s : Set α) :

          An unbounded or cofinal set.

          Equations
          def Set.Bounded {α : Type u} (r : ααProp) (s : Set α) :

          A bounded or final set. Not to be confused with Metric.bounded.

          Equations
          @[simp]
          theorem Set.not_bounded_iff {α : Type u} {r : ααProp} (s : Set α) :
          @[simp]
          theorem Set.not_unbounded_iff {α : Type u} {r : ααProp} (s : Set α) :
          theorem Set.unbounded_of_isEmpty {α : Type u} [inst : IsEmpty α] {r : ααProp} (s : Set α) :
          instance Prod.isRefl_preimage_fst {α : Type u} {r : ααProp} [inst : IsRefl α r] :
          IsRefl (α × α) (Prod.fst ⁻¹'o r)
          Equations
          instance Prod.isRefl_preimage_snd {α : Type u} {r : ααProp} [inst : IsRefl α r] :
          IsRefl (α × α) (Prod.snd ⁻¹'o r)
          Equations
          instance Prod.isTrans_preimage_fst {α : Type u} {r : ααProp} [inst : IsTrans α r] :
          IsTrans (α × α) (Prod.fst ⁻¹'o r)
          Equations
          instance Prod.isTrans_preimage_snd {α : Type u} {r : ααProp} [inst : IsTrans α r] :
          IsTrans (α × α) (Prod.snd ⁻¹'o r)
          Equations

          Strict-non strict relations #

          class IsNonstrictStrictOrder (α : Type u_1) (r : ααProp) (s : ααProp) :
          • The relation r is the nonstrict relation corresponding to the strict relation s.

            right_iff_left_not_left : ∀ (a b : α), s a b r a b ¬r b a

          An unbundled relation class stating that r is the nonstrict relation corresponding to the strict relation s. Compare Preorder.lt_iff_le_not_le. This is mostly meant to provide dot notation on (⊆)⊆) and (⊂)⊂).

          Instances
            theorem right_iff_left_not_left {α : Type u} {r : ααProp} {s : ααProp} [inst : IsNonstrictStrictOrder α r s] {a : α} {b : α} :
            s a b r a b ¬r b a
            theorem right_iff_left_not_left_of {α : Type u} (r : ααProp) (s : ααProp) [inst : IsNonstrictStrictOrder α r s] {a : α} {b : α} :
            s a b r a b ¬r b a

            A version of right_iff_left_not_left with explicit r and s.

            instance instIsIrrefl_2 {α : Type u} {r : ααProp} {s : ααProp} [inst : IsNonstrictStrictOrder α r s] :
            Equations

            ⊆⊆ and ⊂⊂ #

            theorem subset_of_eq_of_subset {α : Type u} [inst : HasSubset α] {a : α} {b : α} {c : α} (hab : a = b) (hbc : b c) :
            a c
            theorem subset_of_subset_of_eq {α : Type u} [inst : HasSubset α] {a : α} {b : α} {c : α} (hab : a b) (hbc : b = c) :
            a c
            theorem subset_refl {α : Type u} [inst : HasSubset α] [inst : IsRefl α fun x x_1 => x x_1] (a : α) :
            a a
            theorem subset_rfl {α : Type u} [inst : HasSubset α] {a : α} [inst : IsRefl α fun x x_1 => x x_1] :
            a a
            theorem subset_of_eq {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] :
            a = ba b
            theorem superset_of_eq {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] :
            a = bb a
            theorem ne_of_not_subset {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] :
            ¬a ba b
            theorem ne_of_not_superset {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] :
            ¬a bb a
            theorem subset_trans {α : Type u} [inst : HasSubset α] [inst : IsTrans α fun x x_1 => x x_1] {a : α} {b : α} {c : α} :
            a bb ca c
            theorem subset_antisymm {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] :
            a bb aa = b
            theorem superset_antisymm {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] :
            a bb ab = a
            theorem Eq.trans_subset {α : Type u} [inst : HasSubset α] {a : α} {b : α} {c : α} (hab : a = b) (hbc : b c) :
            a c

            Alias of subset_of_eq_of_subset.

            theorem HasSubset.subset.trans_eq {α : Type u} [inst : HasSubset α] {a : α} {b : α} {c : α} (hab : a b) (hbc : b = c) :
            a c

            Alias of subset_of_subset_of_eq.

            theorem Eq.subset' {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] :
            a = ba b

            Alias of subset_of_eq.

            theorem Eq.superset {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] :
            a = bb a

            Alias of superset_of_eq.

            theorem HasSubset.Subset.trans {α : Type u} [inst : HasSubset α] [inst : IsTrans α fun x x_1 => x x_1] {a : α} {b : α} {c : α} :
            a bb ca c

            Alias of subset_trans.

            theorem HasSubset.Subset.antisymm {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] :
            a bb aa = b

            Alias of subset_antisymm.

            theorem HasSubset.Subset.antisymm' {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] :
            a bb ab = a

            Alias of superset_antisymm.

            theorem subset_antisymm_iff {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] [inst : IsAntisymm α fun x x_1 => x x_1] :
            a = b a b b a
            theorem superset_antisymm_iff {α : Type u} [inst : HasSubset α] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] [inst : IsAntisymm α fun x x_1 => x x_1] :
            a = b b a a b
            theorem ssubset_of_eq_of_ssubset {α : Type u} [inst : HasSSubset α] {a : α} {b : α} {c : α} (hab : a = b) (hbc : b c) :
            a c
            theorem ssubset_of_ssubset_of_eq {α : Type u} [inst : HasSSubset α] {a : α} {b : α} {c : α} (hab : a b) (hbc : b = c) :
            a c
            theorem ssubset_irrefl {α : Type u} [inst : HasSSubset α] [inst : IsIrrefl α fun x x_1 => x x_1] (a : α) :
            ¬a a
            theorem ssubset_irrfl {α : Type u} [inst : HasSSubset α] [inst : IsIrrefl α fun x x_1 => x x_1] {a : α} :
            ¬a a
            theorem ne_of_ssubset {α : Type u} [inst : HasSSubset α] [inst : IsIrrefl α fun x x_1 => x x_1] {a : α} {b : α} :
            a ba b
            theorem ne_of_ssuperset {α : Type u} [inst : HasSSubset α] [inst : IsIrrefl α fun x x_1 => x x_1] {a : α} {b : α} :
            a bb a
            theorem ssubset_trans {α : Type u} [inst : HasSSubset α] [inst : IsTrans α fun x x_1 => x x_1] {a : α} {b : α} {c : α} :
            a bb ca c
            theorem ssubset_asymm {α : Type u} [inst : HasSSubset α] [inst : IsAsymm α fun x x_1 => x x_1] {a : α} {b : α} :
            a b¬b a
            theorem Eq.trans_ssubset {α : Type u} [inst : HasSSubset α] {a : α} {b : α} {c : α} (hab : a = b) (hbc : b c) :
            a c

            Alias of ssubset_of_eq_of_ssubset.

            theorem HasSSubset.SSubset.trans_eq {α : Type u} [inst : HasSSubset α] {a : α} {b : α} {c : α} (hab : a b) (hbc : b = c) :
            a c

            Alias of ssubset_of_ssubset_of_eq.

            theorem HasSSubset.SSubset.false {α : Type u} [inst : HasSSubset α] [inst : IsIrrefl α fun x x_1 => x x_1] {a : α} :
            ¬a a

            Alias of ssubset_irrfl.

            theorem HasSSubset.SSubset.ne {α : Type u} [inst : HasSSubset α] [inst : IsIrrefl α fun x x_1 => x x_1] {a : α} {b : α} :
            a ba b

            Alias of ne_of_ssubset.

            theorem HasSSubset.SSubset.ne' {α : Type u} [inst : HasSSubset α] [inst : IsIrrefl α fun x x_1 => x x_1] {a : α} {b : α} :
            a bb a

            Alias of ne_of_ssuperset.

            theorem HasSSubset.SSubset.trans {α : Type u} [inst : HasSSubset α] [inst : IsTrans α fun x x_1 => x x_1] {a : α} {b : α} {c : α} :
            a bb ca c

            Alias of ssubset_trans.

            theorem HasSSubset.SSubset.asymm {α : Type u} [inst : HasSSubset α] [inst : IsAsymm α fun x x_1 => x x_1] {a : α} {b : α} :
            a b¬b a

            Alias of ssubset_asymm.

            theorem ssubset_iff_subset_not_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} :
            a b a b ¬b a
            theorem subset_of_ssubset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} (h : a b) :
            a b
            theorem not_subset_of_ssubset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} (h : a b) :
            ¬b a
            theorem not_ssubset_of_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} (h : a b) :
            ¬b a
            theorem ssubset_of_subset_not_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} (h₁ : a b) (h₂ : ¬b a) :
            a b
            theorem HasSSubset.SSubset.subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} (h : a b) :
            a b

            Alias of subset_of_ssubset.

            theorem HasSSubset.SSubset.not_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} (h : a b) :
            ¬b a

            Alias of not_subset_of_ssubset.

            theorem HasSubset.Subset.not_ssubset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} (h : a b) :
            ¬b a

            Alias of not_ssubset_of_subset.

            theorem HasSubset.Subset.ssubset_of_not_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} (h₁ : a b) (h₂ : ¬b a) :
            a b

            Alias of ssubset_of_subset_not_subset.

            theorem ssubset_of_subset_of_ssubset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} {c : α} [inst : IsTrans α fun x x_1 => x x_1] (h₁ : a b) (h₂ : b c) :
            a c
            theorem ssubset_of_ssubset_of_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} {c : α} [inst : IsTrans α fun x x_1 => x x_1] (h₁ : a b) (h₂ : b c) :
            a c
            theorem ssubset_of_subset_of_ne {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] (h₁ : a b) (h₂ : a b) :
            a b
            theorem ssubset_of_ne_of_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] (h₁ : a b) (h₂ : a b) :
            a b
            theorem eq_or_ssubset_of_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] (h : a b) :
            a = b a b
            theorem ssubset_or_eq_of_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] (h : a b) :
            a b a = b
            theorem HasSubset.Subset.trans_ssubset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} {c : α} [inst : IsTrans α fun x x_1 => x x_1] (h₁ : a b) (h₂ : b c) :
            a c

            Alias of ssubset_of_subset_of_ssubset.

            theorem HasSSubset.SSubset.trans_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} {c : α} [inst : IsTrans α fun x x_1 => x x_1] (h₁ : a b) (h₂ : b c) :
            a c

            Alias of ssubset_of_ssubset_of_subset.

            theorem HasSubset.Subset.ssubset_of_ne {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] (h₁ : a b) (h₂ : a b) :
            a b

            Alias of ssubset_of_subset_of_ne.

            theorem Ne.ssubset_of_subset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] (h₁ : a b) (h₂ : a b) :
            a b

            Alias of ssubset_of_ne_of_subset.

            theorem HasSubset.Subset.eq_or_ssubset {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] (h : a b) :
            a = b a b

            Alias of eq_or_ssubset_of_subset.

            theorem HasSubset.Subset.ssubset_or_eq {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] (h : a b) :
            a b a = b

            Alias of ssubset_or_eq_of_subset.

            theorem ssubset_iff_subset_ne {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsAntisymm α fun x x_1 => x x_1] :
            a b a b a b
            theorem subset_iff_ssubset_or_eq {α : Type u} [inst : HasSubset α] [inst : HasSSubset α] [inst : IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x x_1] {a : α} {b : α} [inst : IsRefl α fun x x_1 => x x_1] [inst : IsAntisymm α fun x x_1 => x x_1] :
            a b a b a = b

            Conversion of bundled order typeclasses to unbundled relation typeclasses #

            instance instIsReflLeToLE {α : Type u} [inst : Preorder α] :
            IsRefl α fun x x_1 => x x_1
            Equations
            instance instIsReflGeToLE {α : Type u} [inst : Preorder α] :
            IsRefl α fun x x_1 => x x_1
            Equations
            instance instIsTransLeToLE {α : Type u} [inst : Preorder α] :
            IsTrans α fun x x_1 => x x_1
            Equations
            instance instIsTransGeToLE {α : Type u} [inst : Preorder α] :
            IsTrans α fun x x_1 => x x_1
            Equations
            instance instIsIrreflLtToLT {α : Type u} [inst : Preorder α] :
            IsIrrefl α fun x x_1 => x < x_1
            Equations
            instance instIsIrreflGtToLT {α : Type u} [inst : Preorder α] :
            IsIrrefl α fun x x_1 => x > x_1
            Equations
            instance instIsTransLtToLT {α : Type u} [inst : Preorder α] :
            IsTrans α fun x x_1 => x < x_1
            Equations
            instance instIsTransGtToLT {α : Type u} [inst : Preorder α] :
            IsTrans α fun x x_1 => x > x_1
            Equations
            instance instIsAsymmLtToLT {α : Type u} [inst : Preorder α] :
            IsAsymm α fun x x_1 => x < x_1
            Equations
            instance instIsAsymmGtToLT {α : Type u} [inst : Preorder α] :
            IsAsymm α fun x x_1 => x > x_1
            Equations
            instance instIsNonstrictStrictOrderLeToLELtToLT {α : Type u} [inst : Preorder α] :
            IsNonstrictStrictOrder α (fun x x_1 => x x_1) fun x x_1 => x < x_1
            Equations
            • instIsNonstrictStrictOrderLeToLELtToLT = { right_iff_left_not_left := (_ : ∀ {a b : α}, a < b a b ¬b a) }
            theorem transitive_le {α : Type u} [inst : Preorder α] :
            theorem transitive_lt {α : Type u} [inst : Preorder α] :
            theorem transitive_ge {α : Type u} [inst : Preorder α] :
            theorem transitive_gt {α : Type u} [inst : Preorder α] :
            instance OrderDual.isTotal_le {α : Type u} [inst : LE α] [h : IsTotal α fun x x_1 => x x_1] :
            IsTotal αᵒᵈ fun x x_1 => x x_1
            Equations