Documentation

Mathlib.Algebra.Free

Free constructions #

Main definitions #

inductive FreeAddMagma (α : Type u) :

Free nonabelian additive magma over a given alphabet.

Instances For
    instance instDecidableEqFreeAddMagma :
    {α : Type u_1} → [inst : DecidableEq α] → DecidableEq (FreeAddMagma α)
    Equations
    • instDecidableEqFreeAddMagma = decEqFreeAddMagma✝
    inductive FreeMagma (α : Type u) :

    Free magma over a given alphabet.

    Instances For
      instance instDecidableEqFreeMagma :
      {α : Type u_1} → [inst : DecidableEq α] → DecidableEq (FreeMagma α)
      Equations
      • instDecidableEqFreeMagma = decEqFreeMagma✝
      Equations
      Equations
      • FreeMagma.instInhabitedFreeMagma = { default := FreeMagma.of default }
      Equations
      • FreeAddMagma.instAddFreeAddMagma = { add := FreeAddMagma.add }
      Equations
      • FreeMagma.instMulFreeMagma = { mul := FreeMagma.mul }
      @[simp]
      theorem FreeAddMagma.add_eq {α : Type u} (x : FreeAddMagma α) (y : FreeAddMagma α) :
      @[simp]
      theorem FreeMagma.mul_eq {α : Type u} (x : FreeMagma α) (y : FreeMagma α) :
      FreeMagma.mul x y = x * y
      def FreeAddMagma.recOnAdd {α : Type u} {C : FreeAddMagma αSort l} (x : FreeAddMagma α) (ih1 : (x : α) → C (FreeAddMagma.of x)) (ih2 : (x y : FreeAddMagma α) → C xC yC (x + y)) :
      C x

      Recursor for FreeAddMagma using x + y instead of FreeAddMagma.add x y.

      Equations
      Instances For
        def FreeMagma.recOnMul {α : Type u} {C : FreeMagma αSort l} (x : FreeMagma α) (ih1 : (x : α) → C (FreeMagma.of x)) (ih2 : (x y : FreeMagma α) → C xC yC (x * y)) :
        C x

        Recursor for FreeMagma using x * y instead of FreeMagma.mul x y.

        Equations
        Instances For
          theorem FreeAddMagma.hom_ext {α : Type u} {β : Type v} [Add β] {f : AddHom (FreeAddMagma α) β} {g : AddHom (FreeAddMagma α) β} (h : f FreeAddMagma.of = g FreeAddMagma.of) :
          f = g
          theorem FreeMagma.hom_ext {α : Type u} {β : Type v} [Mul β] {f : FreeMagma α →ₙ* β} {g : FreeMagma α →ₙ* β} (h : f FreeMagma.of = g FreeMagma.of) :
          f = g
          def FreeMagma.liftAux {α : Type u} {β : Type v} [Mul β] (f : αβ) :
          FreeMagma αβ

          Lifts a function α → β to a magma homomorphism FreeMagma α → β given a magma β.

          Equations
          Instances For
            def FreeAddMagma.liftAux {α : Type u} {β : Type v} [Add β] (f : αβ) :
            FreeAddMagma αβ

            Lifts a function α → β to an additive magma homomorphism FreeAddMagma α → β given an additive magma β.

            Equations
            Instances For
              theorem FreeAddMagma.lift.proof_3 {α : Type u_2} {β : Type u_1} [Add β] (F : AddHom (FreeAddMagma α) β) :
              (fun f => { toFun := FreeAddMagma.liftAux f, map_add' := (_ : ∀ (x y : FreeAddMagma α), FreeAddMagma.liftAux f (x + y) = FreeAddMagma.liftAux f (x + y)) }) ((fun F => F FreeAddMagma.of) F) = F
              def FreeAddMagma.lift {α : Type u} {β : Type v} [Add β] :
              (αβ) AddHom (FreeAddMagma α) β

              The universal property of the free additive magma expressing its adjointness.

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                theorem FreeAddMagma.lift.proof_2 {α : Type u_1} {β : Type u_2} [Add β] (f : αβ) :
                (fun F => F FreeAddMagma.of) ((fun f => { toFun := FreeAddMagma.liftAux f, map_add' := (_ : ∀ (x y : FreeAddMagma α), FreeAddMagma.liftAux f (x + y) = FreeAddMagma.liftAux f (x + y)) }) f) = (fun F => F FreeAddMagma.of) ((fun f => { toFun := FreeAddMagma.liftAux f, map_add' := (_ : ∀ (x y : FreeAddMagma α), FreeAddMagma.liftAux f (x + y) = FreeAddMagma.liftAux f (x + y)) }) f)
                theorem FreeAddMagma.lift.proof_1 {α : Type u_1} {β : Type u_2} [Add β] (f : αβ) (x : FreeAddMagma α) (y : FreeAddMagma α) :
                @[simp]
                theorem FreeMagma.lift_symm_apply {α : Type u} {β : Type v} [Mul β] (F : FreeMagma α →ₙ* β) :
                ∀ (a : α), FreeMagma.lift.symm F a = (F FreeMagma.of) a
                @[simp]
                theorem FreeAddMagma.lift_symm_apply {α : Type u} {β : Type v} [Add β] (F : AddHom (FreeAddMagma α) β) :
                ∀ (a : α), FreeAddMagma.lift.symm F a = (F FreeAddMagma.of) a
                def FreeMagma.lift {α : Type u} {β : Type v} [Mul β] :
                (αβ) (FreeMagma α →ₙ* β)

                The universal property of the free magma expressing its adjointness.

                Equations
                • One or more equations did not get rendered due to their size.
                Instances For
                  @[simp]
                  theorem FreeAddMagma.lift_of {α : Type u} {β : Type v} [Add β] (f : αβ) (x : α) :
                  ↑(FreeAddMagma.lift f) (FreeAddMagma.of x) = f x
                  @[simp]
                  theorem FreeMagma.lift_of {α : Type u} {β : Type v} [Mul β] (f : αβ) (x : α) :
                  ↑(FreeMagma.lift f) (FreeMagma.of x) = f x
                  @[simp]
                  theorem FreeAddMagma.lift_comp_of {α : Type u} {β : Type v} [Add β] (f : αβ) :
                  ↑(FreeAddMagma.lift f) FreeAddMagma.of = f
                  @[simp]
                  theorem FreeMagma.lift_comp_of {α : Type u} {β : Type v} [Mul β] (f : αβ) :
                  ↑(FreeMagma.lift f) FreeMagma.of = f
                  @[simp]
                  theorem FreeAddMagma.lift_comp_of' {α : Type u} {β : Type v} [Add β] (f : AddHom (FreeAddMagma α) β) :
                  FreeAddMagma.lift (f FreeAddMagma.of) = f
                  @[simp]
                  theorem FreeMagma.lift_comp_of' {α : Type u} {β : Type v} [Mul β] (f : FreeMagma α →ₙ* β) :
                  FreeMagma.lift (f FreeMagma.of) = f
                  def FreeAddMagma.map {α : Type u} {β : Type v} (f : αβ) :

                  The unique additive magma homomorphism FreeAddMagma α → FreeAddMagma β that sends each of x to of (f x).

                  Equations
                  Instances For
                    def FreeMagma.map {α : Type u} {β : Type v} (f : αβ) :

                    The unique magma homomorphism FreeMagma α →ₙ* FreeMagma β that sends each of x to of (f x).

                    Equations
                    Instances For
                      @[simp]
                      theorem FreeAddMagma.map_of {α : Type u} {β : Type v} (f : αβ) (x : α) :
                      @[simp]
                      theorem FreeMagma.map_of {α : Type u} {β : Type v} (f : αβ) (x : α) :
                      def FreeAddMagma.recOnPure {α : Type u} {C : FreeAddMagma αSort l} (x : FreeAddMagma α) (ih1 : (x : α) → C (pure x)) (ih2 : (x y : FreeAddMagma α) → C xC yC (x + y)) :
                      C x

                      Recursor on FreeAddMagma using pure instead of of.

                      Equations
                      Instances For
                        def FreeMagma.recOnPure {α : Type u} {C : FreeMagma αSort l} (x : FreeMagma α) (ih1 : (x : α) → C (pure x)) (ih2 : (x y : FreeMagma α) → C xC yC (x * y)) :
                        C x

                        Recursor on FreeMagma using pure instead of of.

                        Equations
                        Instances For
                          @[simp]
                          theorem FreeAddMagma.map_pure {α : Type u} {β : Type u} (f : αβ) (x : α) :
                          f <$> pure x = pure (f x)
                          @[simp]
                          theorem FreeMagma.map_pure {α : Type u} {β : Type u} (f : αβ) (x : α) :
                          f <$> pure x = pure (f x)
                          @[simp]
                          theorem FreeAddMagma.map_add' {α : Type u} {β : Type u} (f : αβ) (x : FreeAddMagma α) (y : FreeAddMagma α) :
                          f <$> (x + y) = f <$> x + f <$> y
                          @[simp]
                          theorem FreeMagma.map_mul' {α : Type u} {β : Type u} (f : αβ) (x : FreeMagma α) (y : FreeMagma α) :
                          f <$> (x * y) = f <$> x * f <$> y
                          @[simp]
                          theorem FreeAddMagma.pure_bind {α : Type u} {β : Type u} (f : αFreeAddMagma β) (x : α) :
                          pure x >>= f = f x
                          @[simp]
                          theorem FreeMagma.pure_bind {α : Type u} {β : Type u} (f : αFreeMagma β) (x : α) :
                          pure x >>= f = f x
                          @[simp]
                          theorem FreeAddMagma.add_bind {α : Type u} {β : Type u} (f : αFreeAddMagma β) (x : FreeAddMagma α) (y : FreeAddMagma α) :
                          x + y >>= f = (x >>= f) + (y >>= f)
                          @[simp]
                          theorem FreeMagma.mul_bind {α : Type u} {β : Type u} (f : αFreeMagma β) (x : FreeMagma α) (y : FreeMagma α) :
                          x * y >>= f = (x >>= f) * (y >>= f)
                          @[simp]
                          theorem FreeAddMagma.pure_seq {α : Type u} {β : Type u} {f : αβ} {x : FreeAddMagma α} :
                          (Seq.seq (pure f) fun x => x) = f <$> x
                          @[simp]
                          theorem FreeMagma.pure_seq {α : Type u} {β : Type u} {f : αβ} {x : FreeMagma α} :
                          (Seq.seq (pure f) fun x => x) = f <$> x
                          @[simp]
                          theorem FreeAddMagma.add_seq {α : Type u} {β : Type u} {f : FreeAddMagma (αβ)} {g : FreeAddMagma (αβ)} {x : FreeAddMagma α} :
                          (Seq.seq (f + g) fun x => x) = (Seq.seq f fun x => x) + Seq.seq g fun x => x
                          @[simp]
                          theorem FreeMagma.mul_seq {α : Type u} {β : Type u} {f : FreeMagma (αβ)} {g : FreeMagma (αβ)} {x : FreeMagma α} :
                          (Seq.seq (f * g) fun x => x) = (Seq.seq f fun x => x) * Seq.seq g fun x => x
                          def FreeMagma.traverse {m : Type u → Type u} [Applicative m] {α : Type u} {β : Type u} (F : αm β) :
                          FreeMagma αm (FreeMagma β)

                          FreeMagma is traversable.

                          Equations
                          Instances For
                            def FreeAddMagma.traverse {m : Type u → Type u} [Applicative m] {α : Type u} {β : Type u} (F : αm β) :

                            FreeAddMagma is traversable.

                            Equations
                            Instances For
                              @[simp]
                              theorem FreeAddMagma.traverse_pure {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : α) :
                              traverse F (pure x) = pure <$> F x
                              @[simp]
                              theorem FreeMagma.traverse_pure {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : α) :
                              traverse F (pure x) = pure <$> F x
                              @[simp]
                              theorem FreeAddMagma.traverse_pure' {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) :
                              traverse F pure = fun x => pure <$> F x
                              @[simp]
                              theorem FreeMagma.traverse_pure' {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) :
                              traverse F pure = fun x => pure <$> F x
                              @[simp]
                              theorem FreeAddMagma.traverse_add {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : FreeAddMagma α) (y : FreeAddMagma α) :
                              traverse F (x + y) = Seq.seq ((fun x x_1 => x + x_1) <$> traverse F x) fun x => traverse F y
                              @[simp]
                              theorem FreeMagma.traverse_mul {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : FreeMagma α) (y : FreeMagma α) :
                              traverse F (x * y) = Seq.seq ((fun x x_1 => x * x_1) <$> traverse F x) fun x => traverse F y
                              @[simp]
                              theorem FreeAddMagma.traverse_add' {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) :
                              Function.comp (traverse F) Add.add = fun x y => Seq.seq ((fun x x_1 => x + x_1) <$> traverse F x) fun x => traverse F y
                              @[simp]
                              theorem FreeMagma.traverse_mul' {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) :
                              Function.comp (traverse F) Mul.mul = fun x y => Seq.seq ((fun x x_1 => x * x_1) <$> traverse F x) fun x => traverse F y
                              @[simp]
                              theorem FreeAddMagma.traverse_eq {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : FreeAddMagma α) :
                              @[simp]
                              theorem FreeMagma.traverse_eq {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : FreeMagma α) :
                              @[simp]
                              theorem FreeAddMagma.add_map_seq {α : Type u} (x : FreeAddMagma α) (y : FreeAddMagma α) :
                              (Seq.seq ((fun x x_1 => x + x_1) <$> x) fun x => y) = x + y
                              @[simp]
                              theorem FreeMagma.mul_map_seq {α : Type u} (x : FreeMagma α) (y : FreeMagma α) :
                              (Seq.seq ((fun x x_1 => x * x_1) <$> x) fun x => y) = x * y
                              def FreeMagma.repr {α : Type u} [Repr α] :

                              Representation of an element of a free magma.

                              Equations
                              Instances For

                                Representation of an element of a free additive magma.

                                Equations
                                Instances For
                                  instance instReprFreeAddMagma {α : Type u} [Repr α] :
                                  Equations
                                  instance instReprFreeMagma {α : Type u} [Repr α] :
                                  Equations
                                  def FreeMagma.length {α : Type u} :
                                  FreeMagma α

                                  Length of an element of a free magma.

                                  Equations
                                  Instances For

                                    Length of an element of a free additive magma.

                                    Equations
                                    Instances For
                                      inductive AddMagma.AssocRel (α : Type u) [Add α] :
                                      ααProp

                                      Associativity relations for an additive magma.

                                      Instances For
                                        inductive Magma.AssocRel (α : Type u) [Mul α] :
                                        ααProp

                                        Associativity relations for a magma.

                                        Instances For
                                          def AddMagma.FreeAddSemigroup (α : Type u) [Add α] :

                                          Additive semigroup quotient of an additive magma.

                                          Equations
                                          Instances For
                                            def Magma.AssocQuotient (α : Type u) [Mul α] :

                                            Semigroup quotient of a magma.

                                            Equations
                                            Instances For
                                              theorem AddMagma.FreeAddSemigroup.quot_mk_assoc {α : Type u} [Add α] (x : α) (y : α) (z : α) :
                                              Quot.mk (AddMagma.AssocRel α) (x + y + z) = Quot.mk (AddMagma.AssocRel α) (x + (y + z))
                                              theorem Magma.AssocQuotient.quot_mk_assoc {α : Type u} [Mul α] (x : α) (y : α) (z : α) :
                                              Quot.mk (Magma.AssocRel α) (x * y * z) = Quot.mk (Magma.AssocRel α) (x * (y * z))
                                              theorem AddMagma.FreeAddSemigroup.quot_mk_assoc_left {α : Type u} [Add α] (x : α) (y : α) (z : α) (w : α) :
                                              Quot.mk (AddMagma.AssocRel α) (x + (y + z + w)) = Quot.mk (AddMagma.AssocRel α) (x + (y + (z + w)))
                                              theorem Magma.AssocQuotient.quot_mk_assoc_left {α : Type u} [Mul α] (x : α) (y : α) (z : α) (w : α) :
                                              Quot.mk (Magma.AssocRel α) (x * (y * z * w)) = Quot.mk (Magma.AssocRel α) (x * (y * (z * w)))
                                              theorem AddMagma.FreeAddSemigroup.instAddSemigroupAssocQuotient.proof_2 {α : Type u_1} [Add α] (a₁ : α) (a₂ : α) (b : α) :
                                              AddMagma.AssocRel α a₁ a₂(fun x y => Quot.mk (AddMagma.AssocRel α) (x + y)) a₁ b = (fun x y => Quot.mk (AddMagma.AssocRel α) (x + y)) a₂ b
                                              Equations
                                              theorem AddMagma.FreeAddSemigroup.instAddSemigroupAssocQuotient.proof_1 {α : Type u_1} [Add α] (a : α) (b₁ : α) (b₂ : α) :
                                              AddMagma.AssocRel α b₁ b₂(fun x y => Quot.mk (AddMagma.AssocRel α) (x + y)) a b₁ = (fun x y => Quot.mk (AddMagma.AssocRel α) (x + y)) a b₂
                                              Equations

                                              Embedding from additive magma to its free additive semigroup.

                                              Equations
                                              Instances For
                                                theorem AddMagma.FreeAddSemigroup.of.proof_1 {α : Type u_1} [Add α] (_x : α) (_y : α) :

                                                Embedding from magma to its free semigroup.

                                                Equations
                                                Instances For
                                                  Equations
                                                  • AddMagma.FreeAddSemigroup.instInhabitedAssocQuotient = { default := AddMagma.FreeAddSemigroup.of default }
                                                  Equations
                                                  • Magma.AssocQuotient.instInhabitedAssocQuotient = { default := Magma.AssocQuotient.of default }
                                                  theorem AddMagma.FreeAddSemigroup.induction_on {α : Type u} [Add α] {C : AddMagma.FreeAddSemigroup αProp} (x : AddMagma.FreeAddSemigroup α) (ih : (x : α) → C (AddMagma.FreeAddSemigroup.of x)) :
                                                  C x
                                                  theorem Magma.AssocQuotient.induction_on {α : Type u} [Mul α] {C : Magma.AssocQuotient αProp} (x : Magma.AssocQuotient α) (ih : (x : α) → C (Magma.AssocQuotient.of x)) :
                                                  C x
                                                  theorem AddMagma.FreeAddSemigroup.hom_ext {α : Type u} [Add α] {β : Type v} [AddSemigroup β] {f : AddHom (AddMagma.FreeAddSemigroup α) β} {g : AddHom (AddMagma.FreeAddSemigroup α) β} (h : AddHom.comp f AddMagma.FreeAddSemigroup.of = AddHom.comp g AddMagma.FreeAddSemigroup.of) :
                                                  f = g
                                                  theorem Magma.AssocQuotient.hom_ext {α : Type u} [Mul α] {β : Type v} [Semigroup β] {f : Magma.AssocQuotient α →ₙ* β} {g : Magma.AssocQuotient α →ₙ* β} (h : MulHom.comp f Magma.AssocQuotient.of = MulHom.comp g Magma.AssocQuotient.of) :
                                                  f = g
                                                  theorem AddMagma.FreeAddSemigroup.lift.proof_1 {α : Type u_1} [Add α] {β : Type u_2} [AddSemigroup β] (f : AddHom α β) (a : α) (b : α) :
                                                  AddMagma.AssocRel α a bf a = f b
                                                  theorem AddMagma.FreeAddSemigroup.lift.proof_2 {α : Type u_1} [Add α] {β : Type u_2} [AddSemigroup β] (f : AddHom α β) (x : AddMagma.FreeAddSemigroup α) (y : AddMagma.FreeAddSemigroup α) :
                                                  (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) (x + y) = (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) x + (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) y
                                                  theorem AddMagma.FreeAddSemigroup.lift.proof_4 {α : Type u_2} [Add α] {β : Type u_1} [AddSemigroup β] (f : AddHom (AddMagma.FreeAddSemigroup α) β) :
                                                  (fun f => { toFun := fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b), map_add' := (_ : ∀ (x y : AddMagma.FreeAddSemigroup α), (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) (x + y) = (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) x + (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) y) }) ((fun f => AddHom.comp f AddMagma.FreeAddSemigroup.of) f) = f
                                                  theorem AddMagma.FreeAddSemigroup.lift.proof_3 {α : Type u_2} [Add α] {β : Type u_1} [AddSemigroup β] (f : AddHom α β) :
                                                  (fun f => AddHom.comp f AddMagma.FreeAddSemigroup.of) ((fun f => { toFun := fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b), map_add' := (_ : ∀ (x y : AddMagma.FreeAddSemigroup α), (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) (x + y) = (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) x + (fun x => Quot.liftOn x f (_ : ∀ (a b : α), AddMagma.AssocRel α a bf a = f b)) y) }) f) = f

                                                  Lifts an additive magma homomorphism α → β to an additive semigroup homomorphism AddMagma.AssocQuotient α → β given an additive semigroup β.

                                                  Equations
                                                  • One or more equations did not get rendered due to their size.
                                                  Instances For
                                                    @[simp]
                                                    theorem Magma.AssocQuotient.lift_symm_apply {α : Type u} [Mul α] {β : Type v} [Semigroup β] (f : Magma.AssocQuotient α →ₙ* β) :
                                                    Magma.AssocQuotient.lift.symm f = MulHom.comp f Magma.AssocQuotient.of
                                                    @[simp]
                                                    theorem AddMagma.FreeAddSemigroup.lift_symm_apply {α : Type u} [Add α] {β : Type v} [AddSemigroup β] (f : AddHom (AddMagma.FreeAddSemigroup α) β) :
                                                    AddMagma.FreeAddSemigroup.lift.symm f = AddHom.comp f AddMagma.FreeAddSemigroup.of
                                                    def Magma.AssocQuotient.lift {α : Type u} [Mul α] {β : Type v} [Semigroup β] :

                                                    Lifts a magma homomorphism α → β to a semigroup homomorphism Magma.AssocQuotient α → β given a semigroup β.

                                                    Equations
                                                    • One or more equations did not get rendered due to their size.
                                                    Instances For
                                                      @[simp]
                                                      theorem AddMagma.FreeAddSemigroup.lift_of {α : Type u} [Add α] {β : Type v} [AddSemigroup β] (f : AddHom α β) (x : α) :
                                                      ↑(AddMagma.FreeAddSemigroup.lift f) (AddMagma.FreeAddSemigroup.of x) = f x
                                                      @[simp]
                                                      theorem Magma.AssocQuotient.lift_of {α : Type u} [Mul α] {β : Type v} [Semigroup β] (f : α →ₙ* β) (x : α) :
                                                      ↑(Magma.AssocQuotient.lift f) (Magma.AssocQuotient.of x) = f x
                                                      @[simp]
                                                      theorem AddMagma.FreeAddSemigroup.lift_comp_of {α : Type u} [Add α] {β : Type v} [AddSemigroup β] (f : AddHom α β) :
                                                      AddHom.comp (AddMagma.FreeAddSemigroup.lift f) AddMagma.FreeAddSemigroup.of = f
                                                      @[simp]
                                                      theorem Magma.AssocQuotient.lift_comp_of {α : Type u} [Mul α] {β : Type v} [Semigroup β] (f : α →ₙ* β) :
                                                      MulHom.comp (Magma.AssocQuotient.lift f) Magma.AssocQuotient.of = f
                                                      @[simp]
                                                      theorem AddMagma.FreeAddSemigroup.lift_comp_of' {α : Type u} [Add α] {β : Type v} [AddSemigroup β] (f : AddHom (AddMagma.FreeAddSemigroup α) β) :
                                                      AddMagma.FreeAddSemigroup.lift (AddHom.comp f AddMagma.FreeAddSemigroup.of) = f
                                                      @[simp]
                                                      theorem Magma.AssocQuotient.lift_comp_of' {α : Type u} [Mul α] {β : Type v} [Semigroup β] (f : Magma.AssocQuotient α →ₙ* β) :
                                                      Magma.AssocQuotient.lift (MulHom.comp f Magma.AssocQuotient.of) = f

                                                      From an additive magma homomorphism α → β to an additive semigroup homomorphism AddMagma.AssocQuotient α → AddMagma.AssocQuotient β.

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                                                        From a magma homomorphism α →ₙ* β to a semigroup homomorphism Magma.AssocQuotient α →ₙ* Magma.AssocQuotient β.

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                                                          @[simp]
                                                          theorem AddMagma.FreeAddSemigroup.map_of {α : Type u} [Add α] {β : Type v} [Add β] (f : AddHom α β) (x : α) :
                                                          ↑(AddMagma.FreeAddSemigroup.map f) (AddMagma.FreeAddSemigroup.of x) = AddMagma.FreeAddSemigroup.of (f x)
                                                          @[simp]
                                                          theorem Magma.AssocQuotient.map_of {α : Type u} [Mul α] {β : Type v} [Mul β] (f : α →ₙ* β) (x : α) :
                                                          ↑(Magma.AssocQuotient.map f) (Magma.AssocQuotient.of x) = Magma.AssocQuotient.of (f x)
                                                          structure FreeAddSemigroup (α : Type u) :
                                                          • head : α

                                                            The head of the element

                                                          • tail : List α

                                                            The tail of the element

                                                          Free additive semigroup over a given alphabet.

                                                          Instances For
                                                            theorem FreeAddSemigroup.ext_iff {α : Type u} (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) :
                                                            x = y x.head = y.head x.tail = y.tail
                                                            theorem FreeAddSemigroup.ext {α : Type u} (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) (head : x.head = y.head) (tail : x.tail = y.tail) :
                                                            x = y
                                                            theorem FreeSemigroup.ext {α : Type u} (x : FreeSemigroup α) (y : FreeSemigroup α) (head : x.head = y.head) (tail : x.tail = y.tail) :
                                                            x = y
                                                            theorem FreeSemigroup.ext_iff {α : Type u} (x : FreeSemigroup α) (y : FreeSemigroup α) :
                                                            x = y x.head = y.head x.tail = y.tail
                                                            structure FreeSemigroup (α : Type u) :
                                                            • head : α

                                                              The head of the element

                                                            • tail : List α

                                                              The tail of the element

                                                            Free semigroup over a given alphabet.

                                                            Instances For
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                                                              theorem FreeAddSemigroup.instAddSemigroupFreeAddSemigroup.proof_1 {α : Type u_1} (_L1 : FreeAddSemigroup α) (_L2 : FreeAddSemigroup α) (_L3 : FreeAddSemigroup α) :
                                                              _L1 + _L2 + _L3 = _L1 + (_L2 + _L3)
                                                              Equations
                                                              @[simp]
                                                              theorem FreeAddSemigroup.head_add {α : Type u} (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) :
                                                              (x + y).head = x.head
                                                              @[simp]
                                                              theorem FreeSemigroup.head_mul {α : Type u} (x : FreeSemigroup α) (y : FreeSemigroup α) :
                                                              (x * y).head = x.head
                                                              @[simp]
                                                              theorem FreeAddSemigroup.tail_add {α : Type u} (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) :
                                                              (x + y).tail = x.tail ++ y.head :: y.tail
                                                              @[simp]
                                                              theorem FreeSemigroup.tail_mul {α : Type u} (x : FreeSemigroup α) (y : FreeSemigroup α) :
                                                              (x * y).tail = x.tail ++ y.head :: y.tail
                                                              @[simp]
                                                              theorem FreeAddSemigroup.mk_add_mk {α : Type u} (x : α) (y : α) (L1 : List α) (L2 : List α) :
                                                              { head := x, tail := L1 } + { head := y, tail := L2 } = { head := x, tail := L1 ++ y :: L2 }
                                                              @[simp]
                                                              theorem FreeSemigroup.mk_mul_mk {α : Type u} (x : α) (y : α) (L1 : List α) (L2 : List α) :
                                                              { head := x, tail := L1 } * { head := y, tail := L2 } = { head := x, tail := L1 ++ y :: L2 }
                                                              def FreeAddSemigroup.of {α : Type u} (x : α) :

                                                              The embedding α → FreeAddSemigroup α.

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                                                                @[simp]
                                                                theorem FreeAddSemigroup.of_head {α : Type u} (x : α) :
                                                                @[simp]
                                                                theorem FreeSemigroup.of_head {α : Type u} (x : α) :
                                                                (FreeSemigroup.of x).head = x
                                                                @[simp]
                                                                theorem FreeAddSemigroup.of_tail {α : Type u} (x : α) :
                                                                @[simp]
                                                                theorem FreeSemigroup.of_tail {α : Type u} (x : α) :
                                                                (FreeSemigroup.of x).tail = []
                                                                def FreeSemigroup.of {α : Type u} (x : α) :

                                                                The embedding α → FreeSemigroup α.

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                                                                  Length of an element of free additive semigroup

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                                                                    Length of an element of free semigroup.

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                                                                      Equations
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                                                                      def FreeAddSemigroup.recOnAdd {α : Type u} {C : FreeAddSemigroup αSort l} (x : FreeAddSemigroup α) (ih1 : (x : α) → C (FreeAddSemigroup.of x)) (ih2 : (x : α) → (y : FreeAddSemigroup α) → C (FreeAddSemigroup.of x)C yC (FreeAddSemigroup.of x + y)) :
                                                                      C x

                                                                      Recursor for free additive semigroup using of and +.

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                                                                      • One or more equations did not get rendered due to their size.
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                                                                        def FreeSemigroup.recOnMul {α : Type u} {C : FreeSemigroup αSort l} (x : FreeSemigroup α) (ih1 : (x : α) → C (FreeSemigroup.of x)) (ih2 : (x : α) → (y : FreeSemigroup α) → C (FreeSemigroup.of x)C yC (FreeSemigroup.of x * y)) :
                                                                        C x

                                                                        Recursor for free semigroup using of and *.

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                                                                        • One or more equations did not get rendered due to their size.
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                                                                          theorem FreeAddSemigroup.hom_ext {α : Type u} {β : Type v} [Add β] {f : AddHom (FreeAddSemigroup α) β} {g : AddHom (FreeAddSemigroup α) β} (h : f FreeAddSemigroup.of = g FreeAddSemigroup.of) :
                                                                          f = g
                                                                          theorem FreeSemigroup.hom_ext {α : Type u} {β : Type v} [Mul β] {f : FreeSemigroup α →ₙ* β} {g : FreeSemigroup α →ₙ* β} (h : f FreeSemigroup.of = g FreeSemigroup.of) :
                                                                          f = g
                                                                          def FreeAddSemigroup.lift {α : Type u} {β : Type v} [AddSemigroup β] :
                                                                          (αβ) AddHom (FreeAddSemigroup α) β

                                                                          Lifts a function α → β to an additive semigroup homomorphism FreeAddSemigroup α → β given an additive semigroup β.

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                                                                          • One or more equations did not get rendered due to their size.
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                                                                            theorem FreeAddSemigroup.lift.proof_2 {α : Type u_1} {β : Type u_2} [AddSemigroup β] (f : αβ) :
                                                                            (fun f => f FreeAddSemigroup.of) ((fun f => { toFun := fun x => List.foldl (fun a b => a + f b) (f x.head) x.tail, map_add' := (_ : ∀ (x y : FreeAddSemigroup α), List.foldl (fun a b => a + f b) (f x.head) (x.tail ++ y.head :: y.tail) = List.foldl (fun x y => x + f y) (f x.head) x.tail + List.foldl (fun x y => x + f y) (f y.head) y.tail) }) f) = (fun f => f FreeAddSemigroup.of) ((fun f => { toFun := fun x => List.foldl (fun a b => a + f b) (f x.head) x.tail, map_add' := (_ : ∀ (x y : FreeAddSemigroup α), List.foldl (fun a b => a + f b) (f x.head) (x.tail ++ y.head :: y.tail) = List.foldl (fun x y => x + f y) (f x.head) x.tail + List.foldl (fun x y => x + f y) (f y.head) y.tail) }) f)
                                                                            theorem FreeAddSemigroup.lift.proof_1 {α : Type u_1} {β : Type u_2} [AddSemigroup β] (f : αβ) (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) :
                                                                            List.foldl (fun a b => a + f b) (f x.head) (x.tail ++ y.head :: y.tail) = List.foldl (fun x y => x + f y) (f x.head) x.tail + List.foldl (fun x y => x + f y) (f y.head) y.tail
                                                                            theorem FreeAddSemigroup.lift.proof_3 {α : Type u_2} {β : Type u_1} [AddSemigroup β] (f : AddHom (FreeAddSemigroup α) β) :
                                                                            (fun f => { toFun := fun x => List.foldl (fun a b => a + f b) (f x.head) x.tail, map_add' := (_ : ∀ (x y : FreeAddSemigroup α), List.foldl (fun a b => a + f b) (f x.head) (x.tail ++ y.head :: y.tail) = List.foldl (fun x y => x + f y) (f x.head) x.tail + List.foldl (fun x y => x + f y) (f y.head) y.tail) }) ((fun f => f FreeAddSemigroup.of) f) = f
                                                                            @[simp]
                                                                            theorem FreeSemigroup.lift_symm_apply {α : Type u} {β : Type v} [Semigroup β] (f : FreeSemigroup α →ₙ* β) :
                                                                            ∀ (a : α), FreeSemigroup.lift.symm f a = (f FreeSemigroup.of) a
                                                                            @[simp]
                                                                            theorem FreeAddSemigroup.lift_symm_apply {α : Type u} {β : Type v} [AddSemigroup β] (f : AddHom (FreeAddSemigroup α) β) :
                                                                            ∀ (a : α), FreeAddSemigroup.lift.symm f a = (f FreeAddSemigroup.of) a
                                                                            def FreeSemigroup.lift {α : Type u} {β : Type v} [Semigroup β] :
                                                                            (αβ) (FreeSemigroup α →ₙ* β)

                                                                            Lifts a function α → β to a semigroup homomorphism FreeSemigroup α → β given a semigroup β.

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                                                                            • One or more equations did not get rendered due to their size.
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                                                                              @[simp]
                                                                              theorem FreeAddSemigroup.lift_of {α : Type u} {β : Type v} [AddSemigroup β] (f : αβ) (x : α) :
                                                                              ↑(FreeAddSemigroup.lift f) (FreeAddSemigroup.of x) = f x
                                                                              @[simp]
                                                                              theorem FreeSemigroup.lift_of {α : Type u} {β : Type v} [Semigroup β] (f : αβ) (x : α) :
                                                                              ↑(FreeSemigroup.lift f) (FreeSemigroup.of x) = f x
                                                                              @[simp]
                                                                              theorem FreeAddSemigroup.lift_comp_of {α : Type u} {β : Type v} [AddSemigroup β] (f : αβ) :
                                                                              ↑(FreeAddSemigroup.lift f) FreeAddSemigroup.of = f
                                                                              @[simp]
                                                                              theorem FreeSemigroup.lift_comp_of {α : Type u} {β : Type v} [Semigroup β] (f : αβ) :
                                                                              ↑(FreeSemigroup.lift f) FreeSemigroup.of = f
                                                                              @[simp]
                                                                              theorem FreeAddSemigroup.lift_comp_of' {α : Type u} {β : Type v} [AddSemigroup β] (f : AddHom (FreeAddSemigroup α) β) :
                                                                              FreeAddSemigroup.lift (f FreeAddSemigroup.of) = f
                                                                              @[simp]
                                                                              theorem FreeSemigroup.lift_comp_of' {α : Type u} {β : Type v} [Semigroup β] (f : FreeSemigroup α →ₙ* β) :
                                                                              FreeSemigroup.lift (f FreeSemigroup.of) = f
                                                                              theorem FreeAddSemigroup.lift_of_add {α : Type u} {β : Type v} [AddSemigroup β] (f : αβ) (x : α) (y : FreeAddSemigroup α) :
                                                                              ↑(FreeAddSemigroup.lift f) (FreeAddSemigroup.of x + y) = f x + ↑(FreeAddSemigroup.lift f) y
                                                                              theorem FreeSemigroup.lift_of_mul {α : Type u} {β : Type v} [Semigroup β] (f : αβ) (x : α) (y : FreeSemigroup α) :
                                                                              ↑(FreeSemigroup.lift f) (FreeSemigroup.of x * y) = f x * ↑(FreeSemigroup.lift f) y
                                                                              def FreeAddSemigroup.map {α : Type u} {β : Type v} (f : αβ) :

                                                                              The unique additive semigroup homomorphism that sends of x to of (f x).

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                                                                                def FreeSemigroup.map {α : Type u} {β : Type v} (f : αβ) :

                                                                                The unique semigroup homomorphism that sends of x to of (f x).

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                                                                                  @[simp]
                                                                                  theorem FreeAddSemigroup.map_of {α : Type u} {β : Type v} (f : αβ) (x : α) :
                                                                                  @[simp]
                                                                                  theorem FreeSemigroup.map_of {α : Type u} {β : Type v} (f : αβ) (x : α) :
                                                                                  @[simp]
                                                                                  theorem FreeSemigroup.length_map {α : Type u} {β : Type v} (f : αβ) (x : FreeSemigroup α) :
                                                                                  def FreeAddSemigroup.recOnPure {α : Type u} {C : FreeAddSemigroup αSort l} (x : FreeAddSemigroup α) (ih1 : (x : α) → C (pure x)) (ih2 : (x : α) → (y : FreeAddSemigroup α) → C (pure x)C yC (pure x + y)) :
                                                                                  C x

                                                                                  Recursor that uses pure instead of of.

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                                                                                    def FreeSemigroup.recOnPure {α : Type u} {C : FreeSemigroup αSort l} (x : FreeSemigroup α) (ih1 : (x : α) → C (pure x)) (ih2 : (x : α) → (y : FreeSemigroup α) → C (pure x)C yC (pure x * y)) :
                                                                                    C x

                                                                                    Recursor that uses pure instead of of.

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                                                                                      @[simp]
                                                                                      theorem FreeAddSemigroup.map_pure {α : Type u} {β : Type u} (f : αβ) (x : α) :
                                                                                      f <$> pure x = pure (f x)
                                                                                      @[simp]
                                                                                      theorem FreeSemigroup.map_pure {α : Type u} {β : Type u} (f : αβ) (x : α) :
                                                                                      f <$> pure x = pure (f x)
                                                                                      @[simp]
                                                                                      theorem FreeAddSemigroup.map_add' {α : Type u} {β : Type u} (f : αβ) (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) :
                                                                                      f <$> (x + y) = f <$> x + f <$> y
                                                                                      @[simp]
                                                                                      theorem FreeSemigroup.map_mul' {α : Type u} {β : Type u} (f : αβ) (x : FreeSemigroup α) (y : FreeSemigroup α) :
                                                                                      f <$> (x * y) = f <$> x * f <$> y
                                                                                      @[simp]
                                                                                      theorem FreeAddSemigroup.pure_bind {α : Type u} {β : Type u} (f : αFreeAddSemigroup β) (x : α) :
                                                                                      pure x >>= f = f x
                                                                                      @[simp]
                                                                                      theorem FreeSemigroup.pure_bind {α : Type u} {β : Type u} (f : αFreeSemigroup β) (x : α) :
                                                                                      pure x >>= f = f x
                                                                                      @[simp]
                                                                                      theorem FreeAddSemigroup.add_bind {α : Type u} {β : Type u} (f : αFreeAddSemigroup β) (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) :
                                                                                      x + y >>= f = (x >>= f) + (y >>= f)
                                                                                      @[simp]
                                                                                      theorem FreeSemigroup.mul_bind {α : Type u} {β : Type u} (f : αFreeSemigroup β) (x : FreeSemigroup α) (y : FreeSemigroup α) :
                                                                                      x * y >>= f = (x >>= f) * (y >>= f)
                                                                                      @[simp]
                                                                                      theorem FreeAddSemigroup.pure_seq {α : Type u} {β : Type u} {f : αβ} {x : FreeAddSemigroup α} :
                                                                                      (Seq.seq (pure f) fun x => x) = f <$> x
                                                                                      @[simp]
                                                                                      theorem FreeSemigroup.pure_seq {α : Type u} {β : Type u} {f : αβ} {x : FreeSemigroup α} :
                                                                                      (Seq.seq (pure f) fun x => x) = f <$> x
                                                                                      @[simp]
                                                                                      theorem FreeAddSemigroup.add_seq {α : Type u} {β : Type u} {f : FreeAddSemigroup (αβ)} {g : FreeAddSemigroup (αβ)} {x : FreeAddSemigroup α} :
                                                                                      (Seq.seq (f + g) fun x => x) = (Seq.seq f fun x => x) + Seq.seq g fun x => x
                                                                                      @[simp]
                                                                                      theorem FreeSemigroup.mul_seq {α : Type u} {β : Type u} {f : FreeSemigroup (αβ)} {g : FreeSemigroup (αβ)} {x : FreeSemigroup α} :
                                                                                      (Seq.seq (f * g) fun x => x) = (Seq.seq f fun x => x) * Seq.seq g fun x => x
                                                                                      def FreeAddSemigroup.traverse {m : Type u → Type u} [Applicative m] {α : Type u} {β : Type u} (F : αm β) (x : FreeAddSemigroup α) :

                                                                                      FreeAddSemigroup is traversable.

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                                                                                        def FreeSemigroup.traverse {m : Type u → Type u} [Applicative m] {α : Type u} {β : Type u} (F : αm β) (x : FreeSemigroup α) :

                                                                                        FreeSemigroup is traversable.

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                                                                                          @[simp]
                                                                                          theorem FreeAddSemigroup.traverse_pure {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : α) :
                                                                                          traverse F (pure x) = pure <$> F x
                                                                                          @[simp]
                                                                                          theorem FreeSemigroup.traverse_pure {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : α) :
                                                                                          traverse F (pure x) = pure <$> F x
                                                                                          @[simp]
                                                                                          theorem FreeAddSemigroup.traverse_pure' {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) :
                                                                                          traverse F pure = fun x => pure <$> F x
                                                                                          @[simp]
                                                                                          theorem FreeSemigroup.traverse_pure' {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) :
                                                                                          traverse F pure = fun x => pure <$> F x
                                                                                          @[simp]
                                                                                          theorem FreeAddSemigroup.traverse_add {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) [LawfulApplicative m] (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) :
                                                                                          traverse F (x + y) = Seq.seq ((fun x x_1 => x + x_1) <$> traverse F x) fun x => traverse F y
                                                                                          @[simp]
                                                                                          theorem FreeSemigroup.traverse_mul {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) [LawfulApplicative m] (x : FreeSemigroup α) (y : FreeSemigroup α) :
                                                                                          traverse F (x * y) = Seq.seq ((fun x x_1 => x * x_1) <$> traverse F x) fun x => traverse F y
                                                                                          @[simp]
                                                                                          theorem FreeAddSemigroup.traverse_add' {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) [LawfulApplicative m] :
                                                                                          Function.comp (traverse F) Add.add = fun x y => Seq.seq ((fun x x_1 => x + x_1) <$> traverse F x) fun x => traverse F y
                                                                                          @[simp]
                                                                                          theorem FreeSemigroup.traverse_mul' {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) [LawfulApplicative m] :
                                                                                          Function.comp (traverse F) Mul.mul = fun x y => Seq.seq ((fun x x_1 => x * x_1) <$> traverse F x) fun x => traverse F y
                                                                                          @[simp]
                                                                                          theorem FreeAddSemigroup.traverse_eq {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : FreeAddSemigroup α) :
                                                                                          @[simp]
                                                                                          theorem FreeSemigroup.traverse_eq {α : Type u} {β : Type u} {m : Type u → Type u} [Applicative m] (F : αm β) (x : FreeSemigroup α) :
                                                                                          @[simp]
                                                                                          theorem FreeAddSemigroup.add_map_seq {α : Type u} (x : FreeAddSemigroup α) (y : FreeAddSemigroup α) :
                                                                                          (Seq.seq ((fun x x_1 => x + x_1) <$> x) fun x => y) = x + y
                                                                                          @[simp]
                                                                                          theorem FreeSemigroup.mul_map_seq {α : Type u} (x : FreeSemigroup α) (y : FreeSemigroup α) :
                                                                                          (Seq.seq ((fun x x_1 => x * x_1) <$> x) fun x => y) = x * y
                                                                                          Equations
                                                                                          Equations

                                                                                          The canonical additive morphism from FreeAddMagma α to FreeAddSemigroup α.

                                                                                          Equations
                                                                                          • FreeAddMagma.toFreeAddSemigroup = FreeAddMagma.lift FreeAddSemigroup.of
                                                                                          Instances For

                                                                                            The canonical multiplicative morphism from FreeMagma α to FreeSemigroup α.

                                                                                            Equations
                                                                                            • FreeMagma.toFreeSemigroup = FreeMagma.lift FreeSemigroup.of
                                                                                            Instances For
                                                                                              @[simp]
                                                                                              theorem FreeAddMagma.toFreeAddSemigroup_of {α : Type u} (x : α) :
                                                                                              FreeAddMagma.toFreeAddSemigroup (FreeAddMagma.of x) = FreeAddSemigroup.of x
                                                                                              @[simp]
                                                                                              theorem FreeMagma.toFreeSemigroup_of {α : Type u} (x : α) :
                                                                                              FreeMagma.toFreeSemigroup (FreeMagma.of x) = FreeSemigroup.of x
                                                                                              @[simp]
                                                                                              theorem FreeAddMagma.toFreeAddSemigroup_comp_of {α : Type u} :
                                                                                              FreeAddMagma.toFreeAddSemigroup FreeAddMagma.of = FreeAddSemigroup.of
                                                                                              @[simp]
                                                                                              theorem FreeMagma.toFreeSemigroup_comp_of {α : Type u} :
                                                                                              FreeMagma.toFreeSemigroup FreeMagma.of = FreeSemigroup.of
                                                                                              theorem FreeAddMagma.toFreeAddSemigroup_comp_map {α : Type u} {β : Type v} (f : αβ) :
                                                                                              AddHom.comp FreeAddMagma.toFreeAddSemigroup (FreeAddMagma.map f) = AddHom.comp (FreeAddSemigroup.map f) FreeAddMagma.toFreeAddSemigroup
                                                                                              theorem FreeMagma.toFreeSemigroup_comp_map {α : Type u} {β : Type v} (f : αβ) :
                                                                                              MulHom.comp FreeMagma.toFreeSemigroup (FreeMagma.map f) = MulHom.comp (FreeSemigroup.map f) FreeMagma.toFreeSemigroup
                                                                                              theorem FreeAddMagma.toFreeAddSemigroup_map {α : Type u} {β : Type v} (f : αβ) (x : FreeAddMagma α) :
                                                                                              FreeAddMagma.toFreeAddSemigroup (↑(FreeAddMagma.map f) x) = ↑(FreeAddSemigroup.map f) (FreeAddMagma.toFreeAddSemigroup x)
                                                                                              theorem FreeMagma.toFreeSemigroup_map {α : Type u} {β : Type v} (f : αβ) (x : FreeMagma α) :
                                                                                              FreeMagma.toFreeSemigroup (↑(FreeMagma.map f) x) = ↑(FreeSemigroup.map f) (FreeMagma.toFreeSemigroup x)
                                                                                              @[simp]
                                                                                              theorem FreeAddMagma.length_toFreeAddSemigroup {α : Type u} (x : FreeAddMagma α) :
                                                                                              FreeAddSemigroup.length (FreeAddMagma.toFreeAddSemigroup x) = FreeAddMagma.length x
                                                                                              @[simp]
                                                                                              theorem FreeMagma.length_toFreeSemigroup {α : Type u} (x : FreeMagma α) :
                                                                                              FreeSemigroup.length (FreeMagma.toFreeSemigroup x) = FreeMagma.length x
                                                                                              theorem FreeAddMagmaAssocQuotientEquiv.proof_2 (α : Type u_1) :
                                                                                              AddHom.comp (AddMagma.FreeAddSemigroup.lift FreeAddMagma.toFreeAddSemigroup) (FreeAddSemigroup.lift (AddMagma.FreeAddSemigroup.of FreeAddMagma.of)) = AddHom.id (FreeAddSemigroup α)

                                                                                              Isomorphism between AddMagma.AssocQuotient (FreeAddMagma α) and FreeAddSemigroup α.

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                                                                                                theorem FreeAddMagmaAssocQuotientEquiv.proof_1 (α : Type u_1) :
                                                                                                AddHom.comp (FreeAddSemigroup.lift (AddMagma.FreeAddSemigroup.of FreeAddMagma.of)) (AddMagma.FreeAddSemigroup.lift FreeAddMagma.toFreeAddSemigroup) = AddHom.id (AddMagma.FreeAddSemigroup (FreeAddMagma α))

                                                                                                Isomorphism between Magma.AssocQuotient (FreeMagma α) and FreeSemigroup α.

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