Documentation

Mathlib.Algebra.Order.Monoid.Basic

Ordered monoids #

This file develops some additional material on ordered monoids.

theorem Function.Injective.orderedAddCommMonoid.proof_1 {α : Type u_2} [OrderedAddCommMonoid α] {β : Type u_1} [Add β] (f : βα) (hf : Function.Injective f) (mul : ∀ (x y : β), f (x + y) = f x + f y) (a : β) (b : β) (ab : a b) (c : β) :
f (c + a) f (c + b)
@[reducible]
def Function.Injective.orderedAddCommMonoid {α : Type u} [OrderedAddCommMonoid α] {β : Type u_2} [Zero β] [Add β] [SMul β] (f : βα) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ), f (n x) = n f x) :

Pullback an OrderedAddCommMonoid under an injective map.

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    @[reducible]
    def Function.Injective.orderedCommMonoid {α : Type u} [OrderedCommMonoid α] {β : Type u_2} [One β] [Mul β] [Pow β ] (f : βα) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) :

    Pullback an OrderedCommMonoid under an injective map. See note [reducible non-instances].

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      @[reducible]
      def Function.Injective.linearOrderedAddCommMonoid {α : Type u} [LinearOrderedAddCommMonoid α] {β : Type u_2} [Zero β] [Add β] [SMul β] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ), f (n x) = n f x) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

      Pullback an OrderedAddCommMonoid under an injective map.

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        @[reducible]
        def Function.Injective.linearOrderedCommMonoid {α : Type u} [LinearOrderedCommMonoid α] {β : Type u_2} [One β] [Mul β] [Pow β ] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

        Pullback a LinearOrderedCommMonoid under an injective map. See note [reducible non-instances].

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          @[reducible]
          def Function.Injective.orderedCancelAddCommMonoid {α : Type u} {β : Type u_1} [OrderedCancelAddCommMonoid α] [Zero β] [Add β] [SMul β] (f : βα) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ), f (n x) = n f x) :

          Pullback an OrderedCancelAddCommMonoid under an injective map.

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            theorem Function.Injective.orderedCancelAddCommMonoid.proof_1 {α : Type u_1} {β : Type u_2} [OrderedCancelAddCommMonoid α] [Add β] (f : βα) (mul : ∀ (x y : β), f (x + y) = f x + f y) (a : β) (b : β) (c : β) (bc : f (a + b) f (a + c)) :
            f b f c
            @[reducible]
            def Function.Injective.orderedCancelCommMonoid {α : Type u} {β : Type u_1} [OrderedCancelCommMonoid α] [One β] [Mul β] [Pow β ] (f : βα) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) :

            Pullback an OrderedCancelCommMonoid under an injective map. See note [reducible non-instances].

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              @[reducible]
              def Function.Injective.linearOrderedCancelAddCommMonoid {α : Type u} {β : Type u_1} [LinearOrderedCancelAddCommMonoid α] [Zero β] [Add β] [SMul β] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ), f (n x) = n f x) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

              Pullback a LinearOrderedCancelAddCommMonoid under an injective map.

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                @[reducible]
                def Function.Injective.linearOrderedCancelCommMonoid {α : Type u} {β : Type u_1} [LinearOrderedCancelCommMonoid α] [One β] [Mul β] [Pow β ] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (one : f 1 = 1) (mul : ∀ (x y : β), f (x * y) = f x * f y) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

                Pullback a LinearOrderedCancelCommMonoid under an injective map. See note [reducible non-instances].

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                  theorem OrderEmbedding.addLeft.proof_1 {α : Type u_1} [Add α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] (m : α) :
                  ∀ (x x_1 : α), x < x_1m + x < m + x_1
                  def OrderEmbedding.addLeft {α : Type u_2} [Add α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] (m : α) :
                  α ↪o α

                  The order embedding sending b to a + b, for some fixed a. See also OrderIso.addLeft when working in an additive ordered group.

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                    @[simp]
                    theorem OrderEmbedding.addLeft_apply {α : Type u_2} [Add α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] (m : α) (n : α) :
                    @[simp]
                    theorem OrderEmbedding.mulLeft_apply {α : Type u_2} [Mul α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] (m : α) (n : α) :
                    def OrderEmbedding.mulLeft {α : Type u_2} [Mul α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] (m : α) :
                    α ↪o α

                    The order embedding sending b to a * b, for some fixed a. See also OrderIso.mulLeft when working in an ordered group.

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                      theorem OrderEmbedding.addRight.proof_1 {α : Type u_1} [Add α] [LinearOrder α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] (m : α) :
                      ∀ (x x_1 : α), x < x_1x + m < x_1 + m
                      def OrderEmbedding.addRight {α : Type u_2} [Add α] [LinearOrder α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] (m : α) :
                      α ↪o α

                      The order embedding sending b to b + a, for some fixed a. See also OrderIso.addRight when working in an additive ordered group.

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                        @[simp]
                        theorem OrderEmbedding.addRight_apply {α : Type u_2} [Add α] [LinearOrder α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] (m : α) (n : α) :
                        @[simp]
                        theorem OrderEmbedding.mulRight_apply {α : Type u_2} [Mul α] [LinearOrder α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] (m : α) (n : α) :
                        def OrderEmbedding.mulRight {α : Type u_2} [Mul α] [LinearOrder α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] (m : α) :
                        α ↪o α

                        The order embedding sending b to b * a, for some fixed a. See also OrderIso.mulRight when working in an ordered group.

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