Documentation

Mathlib.Algebra.Order.Monoid.Unbundled.Defs

Covariants and contravariants #

This file contains general lemmas and instances to work with the interactions between a relation and an action on a Type.

The intended application is the splitting of the ordering from the algebraic assumptions on the operations in the Ordered[...] hierarchy.

The strategy is to introduce two more flexible typeclasses, CovariantClass and ContravariantClass:

Since Co(ntra)variantClass takes as input the operation (typically (+) or (*)) and the order relation (typically (≤) or (<)), these are the only two typeclasses that I have used.

The general approach is to formulate the lemma that you are interested in and prove it, with the Ordered[...] typeclass of your liking. After that, you convert the single typeclass, say [OrderedCancelMonoid M], into three typeclasses, e.g. [CancelMonoid M] [PartialOrder M] [CovariantClass M M (Function.swap (*)) (≤)] and have a go at seeing if the proof still works!

Note that it is possible to combine several Co(ntra)variantClass assumptions together. Indeed, the usual ordered typeclasses arise from assuming the pair [CovariantClass M M (*) (≤)] [ContravariantClass M M (*) (<)] on top of order/algebraic assumptions.

A formal remark is that normally CovariantClass uses the (≤)-relation, while ContravariantClass uses the (<)-relation. This need not be the case in general, but seems to be the most common usage. In the opposite direction, the implication

[Semigroup α] [PartialOrder α] [ContravariantClass α α (*) (≤)] → LeftCancelSemigroup α

holds -- note the Co*ntra* assumption on the (≤)-relation.

Formalization notes #

We stick to the convention of using Function.swap (*) (or Function.swap (+)), for the typeclass assumptions, since Function.swap is slightly better behaved than flip. However, sometimes as a non-typeclass assumption, we prefer flip (*) (or flip (+)), as it is easier to use.

def Covariant (M : Type u_1) (N : Type u_2) (μ : MNN) (r : NNProp) :

Covariant is useful to formulate succinctly statements about the interactions between an action of a Type on another one and a relation on the acted-upon Type.

See the CovariantClass doc-string for its meaning.

Equations
  • Covariant M N μ r = ∀ (m : M) {n₁ n₂ : N}, r n₁ n₂r (μ m n₁) (μ m n₂)
Instances For
    def Contravariant (M : Type u_1) (N : Type u_2) (μ : MNN) (r : NNProp) :

    Contravariant is useful to formulate succinctly statements about the interactions between an action of a Type on another one and a relation on the acted-upon Type.

    See the ContravariantClass doc-string for its meaning.

    Equations
    • Contravariant M N μ r = ∀ (m : M) {n₁ n₂ : N}, r (μ m n₁) (μ m n₂)r n₁ n₂
    Instances For
      class CovariantClass (M : Type u_1) (N : Type u_2) (μ : MNN) (r : NNProp) :

      Given an action μ of a Type M on a Type N and a relation r on N, informally, the CovariantClass says that "the action μ preserves the relation r."

      More precisely, the CovariantClass is a class taking two Types M N, together with an "action" μ : M → N → N and a relation r : N → N → Prop. Its unique field elim is the assertion that for all m ∈ M and all elements n₁, n₂ ∈ N, if the relation r holds for the pair (n₁, n₂), then, the relation r also holds for the pair (μ m n₁, μ m n₂), obtained from (n₁, n₂) by acting upon it by m.

      If m : M and h : r n₁ n₂, then CovariantClass.elim m h : r (μ m n₁) (μ m n₂).

      • elim : Covariant M N μ r

        For all m ∈ M and all elements n₁, n₂ ∈ N, if the relation r holds for the pair (n₁, n₂), then, the relation r also holds for the pair (μ m n₁, μ m n₂)

      Instances
        theorem CovariantClass.elim {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} [self : CovariantClass M N μ r] :
        Covariant M N μ r

        For all m ∈ M and all elements n₁, n₂ ∈ N, if the relation r holds for the pair (n₁, n₂), then, the relation r also holds for the pair (μ m n₁, μ m n₂)

        class ContravariantClass (M : Type u_1) (N : Type u_2) (μ : MNN) (r : NNProp) :

        Given an action μ of a Type M on a Type N and a relation r on N, informally, the ContravariantClass says that "if the result of the action μ on a pair satisfies the relation r, then the initial pair satisfied the relation r."

        More precisely, the ContravariantClass is a class taking two Types M N, together with an "action" μ : M → N → N and a relation r : N → N → Prop. Its unique field elim is the assertion that for all m ∈ M and all elements n₁, n₂ ∈ N, if the relation r holds for the pair (μ m n₁, μ m n₂) obtained from (n₁, n₂) by acting upon it by m, then, the relation r also holds for the pair (n₁, n₂).

        If m : M and h : r (μ m n₁) (μ m n₂), then ContravariantClass.elim m h : r n₁ n₂.

        • elim : Contravariant M N μ r

          For all m ∈ M and all elements n₁, n₂ ∈ N, if the relation r holds for the pair (μ m n₁, μ m n₂) obtained from (n₁, n₂) by acting upon it by m, then, the relation r also holds for the pair (n₁, n₂).

        Instances
          theorem ContravariantClass.elim {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} [self : ContravariantClass M N μ r] :
          Contravariant M N μ r

          For all m ∈ M and all elements n₁, n₂ ∈ N, if the relation r holds for the pair (μ m n₁, μ m n₂) obtained from (n₁, n₂) by acting upon it by m, then, the relation r also holds for the pair (n₁, n₂).

          @[reducible, inline]
          abbrev MulLeftMono (M : Type u_1) [Mul M] [LE M] :

          Typeclass for monotonicity of multiplication on the left, namely b₁ ≤ b₂ → a * b₁ ≤ a * b₂.

          You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCommMonoid.

          Equations
          Instances For
            @[reducible, inline]
            abbrev MulRightMono (M : Type u_1) [Mul M] [LE M] :

            Typeclass for monotonicity of multiplication on the right, namely a₁ ≤ a₂ → a₁ * b ≤ a₂ * b.

            You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCommMonoid.

            Equations
            Instances For
              @[reducible, inline]
              abbrev AddLeftMono (M : Type u_1) [Add M] [LE M] :

              Typeclass for monotonicity of addition on the left, namely b₁ ≤ b₂ → a + b₁ ≤ a + b₂.

              You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedAddCommMonoid.

              Equations
              Instances For
                @[reducible, inline]
                abbrev AddRightMono (M : Type u_1) [Add M] [LE M] :

                Typeclass for monotonicity of addition on the right, namely a₁ ≤ a₂ → a₁ + b ≤ a₂ + b.

                You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedAddCommMonoid.

                Equations
                Instances For
                  @[reducible, inline]
                  abbrev MulLeftStrictMono (M : Type u_1) [Mul M] [LT M] :

                  Typeclass for monotonicity of multiplication on the left, namely b₁ < b₂ → a * b₁ < a * b₂.

                  You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCommGroup.

                  Equations
                  Instances For
                    @[reducible, inline]
                    abbrev MulRightStrictMono (M : Type u_1) [Mul M] [LT M] :

                    Typeclass for monotonicity of multiplication on the right, namely a₁ < a₂ → a₁ * b < a₂ * b.

                    You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCommGroup.

                    Equations
                    Instances For
                      @[reducible, inline]
                      abbrev AddLeftStrictMono (M : Type u_1) [Add M] [LT M] :

                      Typeclass for monotonicity of addition on the left, namely b₁ < b₂ → a + b₁ < a + b₂.

                      You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedAddCommGroup.

                      Equations
                      Instances For
                        @[reducible, inline]
                        abbrev AddRightStrictMono (M : Type u_1) [Add M] [LT M] :

                        Typeclass for monotonicity of addition on the right, namely a₁ < a₂ → a₁ + b < a₂ + b.

                        You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedAddCommGroup.

                        Equations
                        Instances For
                          @[reducible, inline]
                          abbrev MulLeftReflectLT (M : Type u_1) [Mul M] [LT M] :

                          Typeclass for strict reverse monotonicity of multiplication on the left, namely a * b₁ < a * b₂ → b₁ < b₂.

                          You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCommGroup.

                          Equations
                          Instances For
                            @[reducible, inline]
                            abbrev MulRightReflectLT (M : Type u_1) [Mul M] [LT M] :

                            Typeclass for strict reverse monotonicity of multiplication on the right, namely a₁ * b < a₂ * b → a₁ < a₂.

                            You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCommGroup.

                            Equations
                            Instances For
                              @[reducible, inline]
                              abbrev AddLeftReflectLT (M : Type u_1) [Add M] [LT M] :

                              Typeclass for strict reverse monotonicity of addition on the left, namely a + b₁ < a + b₂ → b₁ < b₂.

                              You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedAddCommGroup.

                              Equations
                              Instances For
                                @[reducible, inline]
                                abbrev AddRightReflectLT (M : Type u_1) [Add M] [LT M] :

                                Typeclass for strict reverse monotonicity of addition on the right, namely a₁ * b < a₂ * b → a₁ < a₂.

                                You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedAddCommGroup.

                                Equations
                                Instances For
                                  @[reducible, inline]
                                  abbrev MulLeftReflectLE (M : Type u_1) [Mul M] [LE M] :

                                  Typeclass for reverse monotonicity of multiplication on the left, namely a * b₁ ≤ a * b₂ → b₁ ≤ b₂.

                                  You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCancelCommMonoid.

                                  Equations
                                  Instances For
                                    @[reducible, inline]
                                    abbrev MulRightReflectLE (M : Type u_1) [Mul M] [LE M] :

                                    Typeclass for reverse monotonicity of multiplication on the right, namely a₁ * b ≤ a₂ * b → a₁ ≤ a₂.

                                    You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCancelCommMonoid.

                                    Equations
                                    Instances For
                                      @[reducible, inline]
                                      abbrev AddLeftReflectLE (M : Type u_1) [Add M] [LE M] :

                                      Typeclass for reverse monotonicity of addition on the left, namely a + b₁ ≤ a + b₂ → b₁ ≤ b₂.

                                      You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCancelAddCommMonoid.

                                      Equations
                                      Instances For
                                        @[reducible, inline]
                                        abbrev AddRightReflectLE (M : Type u_1) [Add M] [LE M] :

                                        Typeclass for reverse monotonicity of addition on the right, namely a₁ + b ≤ a₂ + b → a₁ ≤ a₂.

                                        You should usually not use this very granular typeclass directly, but rather a typeclass like OrderedCancelAddCommMonoid.

                                        Equations
                                        Instances For
                                          theorem rel_iff_cov (M : Type u_1) (N : Type u_2) (μ : MNN) (r : NNProp) [CovariantClass M N μ r] [ContravariantClass M N μ r] (m : M) {a : N} {b : N} :
                                          r (μ m a) (μ m b) r a b
                                          theorem Covariant.flip {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} (h : Covariant M N μ r) :
                                          Covariant M N μ (flip r)
                                          theorem Contravariant.flip {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} (h : Contravariant M N μ r) :
                                          Contravariant M N μ (flip r)
                                          theorem act_rel_act_of_rel {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} [CovariantClass M N μ r] (m : M) {a : N} {b : N} (ab : r a b) :
                                          r (μ m a) (μ m b)
                                          theorem AddGroup.covariant_iff_contravariant {N : Type u_2} {r : NNProp} [AddGroup N] :
                                          Covariant N N (fun (x x_1 : N) => x + x_1) r Contravariant N N (fun (x x_1 : N) => x + x_1) r
                                          theorem Group.covariant_iff_contravariant {N : Type u_2} {r : NNProp} [Group N] :
                                          Covariant N N (fun (x x_1 : N) => x * x_1) r Contravariant N N (fun (x x_1 : N) => x * x_1) r
                                          @[instance 100]
                                          instance AddGroup.covconv {N : Type u_2} {r : NNProp} [AddGroup N] [CovariantClass N N (fun (x x_1 : N) => x + x_1) r] :
                                          ContravariantClass N N (fun (x x_1 : N) => x + x_1) r
                                          Equations
                                          • =
                                          @[instance 100]
                                          instance Group.covconv {N : Type u_2} {r : NNProp} [Group N] [CovariantClass N N (fun (x x_1 : N) => x * x_1) r] :
                                          ContravariantClass N N (fun (x x_1 : N) => x * x_1) r
                                          Equations
                                          • =
                                          theorem AddGroup.covariant_swap_iff_contravariant_swap {N : Type u_2} {r : NNProp} [AddGroup N] :
                                          Covariant N N (Function.swap fun (x x_1 : N) => x + x_1) r Contravariant N N (Function.swap fun (x x_1 : N) => x + x_1) r
                                          theorem Group.covariant_swap_iff_contravariant_swap {N : Type u_2} {r : NNProp} [Group N] :
                                          Covariant N N (Function.swap fun (x x_1 : N) => x * x_1) r Contravariant N N (Function.swap fun (x x_1 : N) => x * x_1) r
                                          @[instance 100]
                                          instance AddGroup.covconv_swap {N : Type u_2} {r : NNProp} [AddGroup N] [CovariantClass N N (Function.swap fun (x x_1 : N) => x + x_1) r] :
                                          ContravariantClass N N (Function.swap fun (x x_1 : N) => x + x_1) r
                                          Equations
                                          • =
                                          @[instance 100]
                                          instance Group.covconv_swap {N : Type u_2} {r : NNProp} [Group N] [CovariantClass N N (Function.swap fun (x x_1 : N) => x * x_1) r] :
                                          ContravariantClass N N (Function.swap fun (x x_1 : N) => x * x_1) r
                                          Equations
                                          • =
                                          theorem act_rel_of_rel_of_act_rel {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} [CovariantClass M N μ r] [IsTrans N r] (m : M) {a : N} {b : N} {c : N} (ab : r a b) (rl : r (μ m b) c) :
                                          r (μ m a) c
                                          theorem rel_act_of_rel_of_rel_act {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} [CovariantClass M N μ r] [IsTrans N r] (m : M) {a : N} {b : N} {c : N} (ab : r a b) (rr : r c (μ m a)) :
                                          r c (μ m b)
                                          theorem act_rel_act_of_rel_of_rel {N : Type u_2} {r : NNProp} {mu : NNN} [IsTrans N r] [i : CovariantClass N N mu r] [i' : CovariantClass N N (Function.swap mu) r] {a : N} {b : N} {c : N} {d : N} (ab : r a b) (cd : r c d) :
                                          r (mu a c) (mu b d)
                                          theorem rel_of_act_rel_act {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} [ContravariantClass M N μ r] (m : M) {a : N} {b : N} (ab : r (μ m a) (μ m b)) :
                                          r a b
                                          theorem act_rel_of_act_rel_of_rel_act_rel {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} [ContravariantClass M N μ r] [IsTrans N r] (m : M) {a : N} {b : N} {c : N} (ab : r (μ m a) b) (rl : r (μ m b) (μ m c)) :
                                          r (μ m a) c
                                          theorem rel_act_of_act_rel_act_of_rel_act {M : Type u_1} {N : Type u_2} {μ : MNN} {r : NNProp} [ContravariantClass M N μ r] [IsTrans N r] (m : M) {a : N} {b : N} {c : N} (ab : r (μ m a) (μ m b)) (rr : r b (μ m c)) :
                                          r a (μ m c)
                                          theorem Covariant.monotone_of_const {M : Type u_1} {N : Type u_2} {μ : MNN} [Preorder N] [CovariantClass M N μ fun (x x_1 : N) => x x_1] (m : M) :
                                          Monotone (μ m)

                                          The partial application of a constant to a covariant operator is monotone.

                                          theorem Monotone.covariant_of_const {M : Type u_1} {N : Type u_2} {μ : MNN} {α : Type u_3} [Preorder α] [Preorder N] {f : Nα} [CovariantClass M N μ fun (x x_1 : N) => x x_1] (hf : Monotone f) (m : M) :
                                          Monotone fun (x : N) => f (μ m x)

                                          A monotone function remains monotone when composed with the partial application of a covariant operator. E.g., ∀ (m : ℕ), Monotone f → Monotone (fun n ↦ f (m + n)).

                                          theorem Monotone.covariant_of_const' {N : Type u_2} {α : Type u_3} [Preorder α] [Preorder N] {f : Nα} {μ : NNN} [CovariantClass N N (Function.swap μ) fun (x x_1 : N) => x x_1] (hf : Monotone f) (m : N) :
                                          Monotone fun (x : N) => f (μ x m)

                                          Same as Monotone.covariant_of_const, but with the constant on the other side of the operator. E.g., ∀ (m : ℕ), Monotone f → Monotone (fun n ↦ f (n + m)).

                                          theorem Antitone.covariant_of_const {M : Type u_1} {N : Type u_2} {μ : MNN} {α : Type u_3} [Preorder α] [Preorder N] {f : Nα} [CovariantClass M N μ fun (x x_1 : N) => x x_1] (hf : Antitone f) (m : M) :
                                          Antitone fun (x : N) => f (μ m x)

                                          Dual of Monotone.covariant_of_const

                                          theorem Antitone.covariant_of_const' {N : Type u_2} {α : Type u_3} [Preorder α] [Preorder N] {f : Nα} {μ : NNN} [CovariantClass N N (Function.swap μ) fun (x x_1 : N) => x x_1] (hf : Antitone f) (m : N) :
                                          Antitone fun (x : N) => f (μ x m)

                                          Dual of Monotone.covariant_of_const'

                                          theorem covariant_le_of_covariant_lt (M : Type u_1) (N : Type u_2) (μ : MNN) [PartialOrder N] :
                                          (Covariant M N μ fun (x x_1 : N) => x < x_1)Covariant M N μ fun (x x_1 : N) => x x_1
                                          theorem covariantClass_le_of_lt (M : Type u_1) (N : Type u_2) (μ : MNN) [PartialOrder N] [CovariantClass M N μ fun (x x_1 : N) => x < x_1] :
                                          CovariantClass M N μ fun (x x_1 : N) => x x_1
                                          theorem contravariant_le_iff_contravariant_lt_and_eq (M : Type u_1) (N : Type u_2) (μ : MNN) [PartialOrder N] :
                                          (Contravariant M N μ fun (x x_1 : N) => x x_1) (Contravariant M N μ fun (x x_1 : N) => x < x_1) Contravariant M N μ fun (x x_1 : N) => x = x_1
                                          theorem contravariant_lt_of_contravariant_le (M : Type u_1) (N : Type u_2) (μ : MNN) [PartialOrder N] :
                                          (Contravariant M N μ fun (x x_1 : N) => x x_1)Contravariant M N μ fun (x x_1 : N) => x < x_1
                                          theorem covariant_le_iff_contravariant_lt (M : Type u_1) (N : Type u_2) (μ : MNN) [LinearOrder N] :
                                          (Covariant M N μ fun (x x_1 : N) => x x_1) Contravariant M N μ fun (x x_1 : N) => x < x_1
                                          theorem covariant_lt_iff_contravariant_le (M : Type u_1) (N : Type u_2) (μ : MNN) [LinearOrder N] :
                                          (Covariant M N μ fun (x x_1 : N) => x < x_1) Contravariant M N μ fun (x x_1 : N) => x x_1
                                          theorem covariant_flip_iff (N : Type u_2) (r : NNProp) (mu : NNN) [IsSymmOp N N mu] :
                                          Covariant N N (flip mu) r Covariant N N mu r
                                          theorem contravariant_flip_iff (N : Type u_2) (r : NNProp) (mu : NNN) [IsSymmOp N N mu] :
                                          Contravariant N N (flip mu) r Contravariant N N mu r
                                          instance contravariant_lt_of_covariant_le (N : Type u_2) (mu : NNN) [LinearOrder N] [CovariantClass N N mu fun (x x_1 : N) => x x_1] :
                                          ContravariantClass N N mu fun (x x_1 : N) => x < x_1
                                          Equations
                                          • =
                                          instance covariant_lt_of_contravariant_le (N : Type u_2) (mu : NNN) [LinearOrder N] [ContravariantClass N N mu fun (x x_1 : N) => x x_1] :
                                          CovariantClass N N mu fun (x x_1 : N) => x < x_1
                                          Equations
                                          • =
                                          instance covariant_swap_add_of_covariant_add (N : Type u_2) (r : NNProp) [AddCommSemigroup N] [CovariantClass N N (fun (x x_1 : N) => x + x_1) r] :
                                          CovariantClass N N (Function.swap fun (x x_1 : N) => x + x_1) r
                                          Equations
                                          • =
                                          instance covariant_swap_mul_of_covariant_mul (N : Type u_2) (r : NNProp) [CommSemigroup N] [CovariantClass N N (fun (x x_1 : N) => x * x_1) r] :
                                          CovariantClass N N (Function.swap fun (x x_1 : N) => x * x_1) r
                                          Equations
                                          • =
                                          instance contravariant_swap_add_of_contravariant_add (N : Type u_2) (r : NNProp) [AddCommSemigroup N] [ContravariantClass N N (fun (x x_1 : N) => x + x_1) r] :
                                          ContravariantClass N N (Function.swap fun (x x_1 : N) => x + x_1) r
                                          Equations
                                          • =
                                          instance contravariant_swap_mul_of_contravariant_mul (N : Type u_2) (r : NNProp) [CommSemigroup N] [ContravariantClass N N (fun (x x_1 : N) => x * x_1) r] :
                                          ContravariantClass N N (Function.swap fun (x x_1 : N) => x * x_1) r
                                          Equations
                                          • =
                                          theorem covariant_lt_of_covariant_le_of_contravariant_eq (M : Type u_1) (N : Type u_2) (μ : MNN) [ContravariantClass M N μ fun (x x_1 : N) => x = x_1] [PartialOrder N] [CovariantClass M N μ fun (x x_1 : N) => x x_1] :
                                          CovariantClass M N μ fun (x x_1 : N) => x < x_1
                                          theorem contravariant_le_of_contravariant_eq_and_lt (M : Type u_1) (N : Type u_2) (μ : MNN) [PartialOrder N] [ContravariantClass M N μ fun (x x_1 : N) => x = x_1] [ContravariantClass M N μ fun (x x_1 : N) => x < x_1] :
                                          ContravariantClass M N μ fun (x x_1 : N) => x x_1
                                          instance IsLeftCancelAdd.covariant_add_lt_of_covariant_add_le (N : Type u_2) [Add N] [IsLeftCancelAdd N] [PartialOrder N] [CovariantClass N N (fun (x x_1 : N) => x + x_1) fun (x x_1 : N) => x x_1] :
                                          CovariantClass N N (fun (x x_1 : N) => x + x_1) fun (x x_1 : N) => x < x_1
                                          Equations
                                          • =
                                          instance IsLeftCancelMul.covariant_mul_lt_of_covariant_mul_le (N : Type u_2) [Mul N] [IsLeftCancelMul N] [PartialOrder N] [CovariantClass N N (fun (x x_1 : N) => x * x_1) fun (x x_1 : N) => x x_1] :
                                          CovariantClass N N (fun (x x_1 : N) => x * x_1) fun (x x_1 : N) => x < x_1
                                          Equations
                                          • =
                                          instance IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le (N : Type u_2) [Add N] [IsRightCancelAdd N] [PartialOrder N] [CovariantClass N N (Function.swap fun (x x_1 : N) => x + x_1) fun (x x_1 : N) => x x_1] :
                                          CovariantClass N N (Function.swap fun (x x_1 : N) => x + x_1) fun (x x_1 : N) => x < x_1
                                          Equations
                                          • =
                                          instance IsRightCancelMul.covariant_swap_mul_lt_of_covariant_swap_mul_le (N : Type u_2) [Mul N] [IsRightCancelMul N] [PartialOrder N] [CovariantClass N N (Function.swap fun (x x_1 : N) => x * x_1) fun (x x_1 : N) => x x_1] :
                                          CovariantClass N N (Function.swap fun (x x_1 : N) => x * x_1) fun (x x_1 : N) => x < x_1
                                          Equations
                                          • =
                                          instance IsLeftCancelAdd.contravariant_add_le_of_contravariant_add_lt (N : Type u_2) [Add N] [IsLeftCancelAdd N] [PartialOrder N] [ContravariantClass N N (fun (x x_1 : N) => x + x_1) fun (x x_1 : N) => x < x_1] :
                                          ContravariantClass N N (fun (x x_1 : N) => x + x_1) fun (x x_1 : N) => x x_1
                                          Equations
                                          • =
                                          instance IsLeftCancelMul.contravariant_mul_le_of_contravariant_mul_lt (N : Type u_2) [Mul N] [IsLeftCancelMul N] [PartialOrder N] [ContravariantClass N N (fun (x x_1 : N) => x * x_1) fun (x x_1 : N) => x < x_1] :
                                          ContravariantClass N N (fun (x x_1 : N) => x * x_1) fun (x x_1 : N) => x x_1
                                          Equations
                                          • =
                                          instance IsRightCancelAdd.contravariant_swap_add_le_of_contravariant_swap_add_lt (N : Type u_2) [Add N] [IsRightCancelAdd N] [PartialOrder N] [ContravariantClass N N (Function.swap fun (x x_1 : N) => x + x_1) fun (x x_1 : N) => x < x_1] :
                                          ContravariantClass N N (Function.swap fun (x x_1 : N) => x + x_1) fun (x x_1 : N) => x x_1
                                          Equations
                                          • =
                                          instance IsRightCancelMul.contravariant_swap_mul_le_of_contravariant_swap_mul_lt (N : Type u_2) [Mul N] [IsRightCancelMul N] [PartialOrder N] [ContravariantClass N N (Function.swap fun (x x_1 : N) => x * x_1) fun (x x_1 : N) => x < x_1] :
                                          ContravariantClass N N (Function.swap fun (x x_1 : N) => x * x_1) fun (x x_1 : N) => x x_1
                                          Equations
                                          • =