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Mathlib.GroupTheory.Submonoid.Basic

Submonoids: definition and CompleteLattice structure #

This file defines bundled multiplicative and additive submonoids. We also define a CompleteLattice structure on Submonoids, define the closure of a set as the minimal submonoid that includes this set, and prove a few results about extending properties from a dense set (i.e. a set with closure s = ⊤) to the whole monoid, see Submonoid.dense_induction and MonoidHom.ofClosureEqTopLeft/MonoidHom.ofClosureEqTopRight.

Main definitions #

For each of the following definitions in the Submonoid namespace, there is a corresponding definition in the AddSubmonoid namespace.

Implementation notes #

Submonoid inclusion is denoted rather than , although is defined as membership of a submonoid's underlying set.

Note that Submonoid M does not actually require Monoid M, instead requiring only the weaker MulOneClass M.

This file is designed to have very few dependencies. In particular, it should not use natural numbers. Submonoid is implemented by extending Subsemigroup requiring one_mem'.

Tags #

submonoid, submonoids

class OneMemClass (S : Type u_4) (M : Type u_5) [One M] [SetLike S M] :

OneMemClass S M says S is a type of subsets s ≤ M, such that 1 ∈ s for all s.

  • one_mem : ∀ (s : S), 1 s

    By definition, if we have OneMemClass S M, we have 1 ∈ s for all s : S.

Instances
    class ZeroMemClass (S : Type u_4) (M : Type u_5) [Zero M] [SetLike S M] :

    ZeroMemClass S M says S is a type of subsets s ≤ M, such that 0 ∈ s for all s.

    • zero_mem : ∀ (s : S), 0 s

      By definition, if we have ZeroMemClass S M, we have 0 ∈ s for all s : S.

    Instances
      structure Submonoid (M : Type u_4) [MulOneClass M] extends Subsemigroup :
      Type u_4

      A submonoid of a monoid M is a subset containing 1 and closed under multiplication.

      • carrier : Set M
      • mul_mem' : ∀ {a b : M}, a self.carrierb self.carriera * b self.carrier
      • one_mem' : 1 self.carrier

        A submonoid contains 1.

      Instances For
        class SubmonoidClass (S : Type u_4) (M : Type u_5) [MulOneClass M] [SetLike S M] extends MulMemClass , OneMemClass :

        SubmonoidClass S M says S is a type of subsets s ≤ M that contain 1 and are closed under (*)

          Instances
            structure AddSubmonoid (M : Type u_4) [AddZeroClass M] extends AddSubsemigroup :
            Type u_4

            An additive submonoid of an additive monoid M is a subset containing 0 and closed under addition.

            • carrier : Set M
            • add_mem' : ∀ {a b : M}, a self.carrierb self.carriera + b self.carrier
            • zero_mem' : 0 self.carrier

              An additive submonoid contains 0.

            Instances For
              class AddSubmonoidClass (S : Type u_4) (M : Type u_5) [AddZeroClass M] [SetLike S M] extends AddMemClass , ZeroMemClass :

              AddSubmonoidClass S M says S is a type of subsets s ≤ M that contain 0 and are closed under (+)

                Instances
                  abbrev nsmul_mem.match_1 (motive : Prop) :
                  ∀ (x : ), (Unitmotive 0)(∀ (n : ), motive (Nat.succ n))motive x
                  Equations
                  • =
                  Instances For
                    theorem nsmul_mem {M : Type u_4} {A : Type u_5} [AddMonoid M] [SetLike A M] [AddSubmonoidClass A M] {S : A} {x : M} (hx : x S) (n : ) :
                    n x S
                    theorem pow_mem {M : Type u_4} {A : Type u_5} [Monoid M] [SetLike A M] [SubmonoidClass A M] {S : A} {x : M} (hx : x S) (n : ) :
                    x ^ n S
                    theorem AddSubmonoid.instSetLikeAddSubmonoid.proof_1 {M : Type u_1} [AddZeroClass M] (p : AddSubmonoid M) (q : AddSubmonoid M) (h : (fun (s : AddSubmonoid M) => s.carrier) p = (fun (s : AddSubmonoid M) => s.carrier) q) :
                    p = q
                    Equations
                    • AddSubmonoid.instSetLikeAddSubmonoid = { coe := fun (s : AddSubmonoid M) => s.carrier, coe_injective' := }
                    Equations
                    • Submonoid.instSetLikeSubmonoid = { coe := fun (s : Submonoid M) => s.carrier, coe_injective' := }
                    @[simp]
                    theorem AddSubmonoid.mem_toSubsemigroup {M : Type u_1} [AddZeroClass M] {s : AddSubmonoid M} {x : M} :
                    x s.toAddSubsemigroup x s
                    @[simp]
                    theorem Submonoid.mem_toSubsemigroup {M : Type u_1} [MulOneClass M] {s : Submonoid M} {x : M} :
                    x s.toSubsemigroup x s
                    theorem AddSubmonoid.mem_carrier {M : Type u_1} [AddZeroClass M] {s : AddSubmonoid M} {x : M} :
                    x s.carrier x s
                    theorem Submonoid.mem_carrier {M : Type u_1} [MulOneClass M] {s : Submonoid M} {x : M} :
                    x s.carrier x s
                    @[simp]
                    theorem AddSubmonoid.mem_mk {M : Type u_1} [AddZeroClass M] {s : AddSubsemigroup M} {x : M} (h_one : 0 s.carrier) :
                    x { toAddSubsemigroup := s, zero_mem' := h_one } x s
                    @[simp]
                    theorem Submonoid.mem_mk {M : Type u_1} [MulOneClass M] {s : Subsemigroup M} {x : M} (h_one : 1 s.carrier) :
                    x { toSubsemigroup := s, one_mem' := h_one } x s
                    @[simp]
                    theorem AddSubmonoid.coe_set_mk {M : Type u_1} [AddZeroClass M] {s : AddSubsemigroup M} (h_one : 0 s.carrier) :
                    { toAddSubsemigroup := s, zero_mem' := h_one } = s
                    @[simp]
                    theorem Submonoid.coe_set_mk {M : Type u_1} [MulOneClass M] {s : Subsemigroup M} (h_one : 1 s.carrier) :
                    { toSubsemigroup := s, one_mem' := h_one } = s
                    @[simp]
                    theorem AddSubmonoid.mk_le_mk {M : Type u_1} [AddZeroClass M] {s : AddSubsemigroup M} {t : AddSubsemigroup M} (h_one : 0 s.carrier) (h_one' : 0 t.carrier) :
                    { toAddSubsemigroup := s, zero_mem' := h_one } { toAddSubsemigroup := t, zero_mem' := h_one' } s t
                    @[simp]
                    theorem Submonoid.mk_le_mk {M : Type u_1} [MulOneClass M] {s : Subsemigroup M} {t : Subsemigroup M} (h_one : 1 s.carrier) (h_one' : 1 t.carrier) :
                    { toSubsemigroup := s, one_mem' := h_one } { toSubsemigroup := t, one_mem' := h_one' } s t
                    theorem AddSubmonoid.ext {M : Type u_1} [AddZeroClass M] {S : AddSubmonoid M} {T : AddSubmonoid M} (h : ∀ (x : M), x S x T) :
                    S = T

                    Two AddSubmonoids are equal if they have the same elements.

                    theorem Submonoid.ext {M : Type u_1} [MulOneClass M] {S : Submonoid M} {T : Submonoid M} (h : ∀ (x : M), x S x T) :
                    S = T

                    Two submonoids are equal if they have the same elements.

                    def AddSubmonoid.copy {M : Type u_1} [AddZeroClass M] (S : AddSubmonoid M) (s : Set M) (hs : s = S) :

                    Copy an additive submonoid replacing carrier with a set that is equal to it.

                    Equations
                    • AddSubmonoid.copy S s hs = { toAddSubsemigroup := { carrier := s, add_mem' := }, zero_mem' := }
                    Instances For
                      theorem AddSubmonoid.copy.proof_1 {M : Type u_1} [AddZeroClass M] (S : AddSubmonoid M) (s : Set M) (hs : s = S) :
                      ∀ {a b : M}, a sb sa + b s
                      theorem AddSubmonoid.copy.proof_2 {M : Type u_1} [AddZeroClass M] (S : AddSubmonoid M) (s : Set M) (hs : s = S) :
                      0 s
                      def Submonoid.copy {M : Type u_1} [MulOneClass M] (S : Submonoid M) (s : Set M) (hs : s = S) :

                      Copy a submonoid replacing carrier with a set that is equal to it.

                      Equations
                      • Submonoid.copy S s hs = { toSubsemigroup := { carrier := s, mul_mem' := }, one_mem' := }
                      Instances For
                        @[simp]
                        theorem AddSubmonoid.coe_copy {M : Type u_1} [AddZeroClass M] {S : AddSubmonoid M} {s : Set M} (hs : s = S) :
                        (AddSubmonoid.copy S s hs) = s
                        @[simp]
                        theorem Submonoid.coe_copy {M : Type u_1} [MulOneClass M] {S : Submonoid M} {s : Set M} (hs : s = S) :
                        (Submonoid.copy S s hs) = s
                        theorem AddSubmonoid.copy_eq {M : Type u_1} [AddZeroClass M] {S : AddSubmonoid M} {s : Set M} (hs : s = S) :
                        theorem Submonoid.copy_eq {M : Type u_1} [MulOneClass M] {S : Submonoid M} {s : Set M} (hs : s = S) :
                        Submonoid.copy S s hs = S
                        theorem AddSubmonoid.zero_mem {M : Type u_1} [AddZeroClass M] (S : AddSubmonoid M) :
                        0 S

                        An AddSubmonoid contains the monoid's 0.

                        theorem Submonoid.one_mem {M : Type u_1} [MulOneClass M] (S : Submonoid M) :
                        1 S

                        A submonoid contains the monoid's 1.

                        theorem AddSubmonoid.add_mem {M : Type u_1} [AddZeroClass M] (S : AddSubmonoid M) {x : M} {y : M} :
                        x Sy Sx + y S

                        An AddSubmonoid is closed under addition.

                        theorem Submonoid.mul_mem {M : Type u_1} [MulOneClass M] (S : Submonoid M) {x : M} {y : M} :
                        x Sy Sx * y S

                        A submonoid is closed under multiplication.

                        The additive submonoid M of the AddMonoid M.

                        Equations
                        • AddSubmonoid.instTopAddSubmonoid = { top := { toAddSubsemigroup := { carrier := Set.univ, add_mem' := }, zero_mem' := } }
                        theorem AddSubmonoid.instTopAddSubmonoid.proof_1 {M : Type u_1} [AddZeroClass M] :
                        ∀ {a b : M}, a Set.univb Set.univa + b Set.univ

                        The submonoid M of the monoid M.

                        Equations
                        • Submonoid.instTopSubmonoid = { top := { toSubsemigroup := { carrier := Set.univ, mul_mem' := }, one_mem' := } }

                        The trivial AddSubmonoid {0} of an AddMonoid M.

                        Equations
                        • AddSubmonoid.instBotAddSubmonoid = { bot := { toAddSubsemigroup := { carrier := {0}, add_mem' := }, zero_mem' := } }
                        theorem AddSubmonoid.instBotAddSubmonoid.proof_1 {M : Type u_1} [AddZeroClass M] :
                        ∀ {a b : M}, a {0}b {0}a + b {0}

                        The trivial submonoid {1} of a monoid M.

                        Equations
                        • Submonoid.instBotSubmonoid = { bot := { toSubsemigroup := { carrier := {1}, mul_mem' := }, one_mem' := } }
                        Equations
                        • AddSubmonoid.instInhabitedAddSubmonoid = { default := }
                        Equations
                        • Submonoid.instInhabitedSubmonoid = { default := }
                        @[simp]
                        theorem AddSubmonoid.mem_bot {M : Type u_1} [AddZeroClass M] {x : M} :
                        x x = 0
                        @[simp]
                        theorem Submonoid.mem_bot {M : Type u_1} [MulOneClass M] {x : M} :
                        x x = 1
                        @[simp]
                        theorem AddSubmonoid.mem_top {M : Type u_1} [AddZeroClass M] (x : M) :
                        @[simp]
                        theorem Submonoid.mem_top {M : Type u_1} [MulOneClass M] (x : M) :
                        @[simp]
                        theorem AddSubmonoid.coe_top {M : Type u_1} [AddZeroClass M] :
                        = Set.univ
                        @[simp]
                        theorem Submonoid.coe_top {M : Type u_1} [MulOneClass M] :
                        = Set.univ
                        @[simp]
                        theorem AddSubmonoid.coe_bot {M : Type u_1} [AddZeroClass M] :
                        = {0}
                        @[simp]
                        theorem Submonoid.coe_bot {M : Type u_1} [MulOneClass M] :
                        = {1}

                        The inf of two AddSubmonoids is their intersection.

                        Equations
                        • AddSubmonoid.instInfAddSubmonoid = { inf := fun (S₁ S₂ : AddSubmonoid M) => { toAddSubsemigroup := { carrier := S₁ S₂, add_mem' := }, zero_mem' := } }
                        theorem AddSubmonoid.instInfAddSubmonoid.proof_2 {M : Type u_1} [AddZeroClass M] (S₁ : AddSubmonoid M) (S₂ : AddSubmonoid M) :
                        0 S₁ 0 S₂
                        theorem AddSubmonoid.instInfAddSubmonoid.proof_1 {M : Type u_1} [AddZeroClass M] (S₁ : AddSubmonoid M) (S₂ : AddSubmonoid M) :
                        ∀ {a b : M}, a S₁ S₂b S₁ S₂a + b S₁ S₂
                        abbrev AddSubmonoid.instInfAddSubmonoid.match_1 {M : Type u_1} [AddZeroClass M] (S₁ : AddSubmonoid M) (S₂ : AddSubmonoid M) :
                        ∀ {b : M} (motive : b S₁ S₂Prop) (x : b S₁ S₂), (∀ (hy : b S₁) (hy' : b S₂), motive )motive x
                        Equations
                        • =
                        Instances For

                          The inf of two submonoids is their intersection.

                          Equations
                          • Submonoid.instInfSubmonoid = { inf := fun (S₁ S₂ : Submonoid M) => { toSubsemigroup := { carrier := S₁ S₂, mul_mem' := }, one_mem' := } }
                          @[simp]
                          theorem AddSubmonoid.coe_inf {M : Type u_1} [AddZeroClass M] (p : AddSubmonoid M) (p' : AddSubmonoid M) :
                          (p p') = p p'
                          @[simp]
                          theorem Submonoid.coe_inf {M : Type u_1} [MulOneClass M] (p : Submonoid M) (p' : Submonoid M) :
                          (p p') = p p'
                          @[simp]
                          theorem AddSubmonoid.mem_inf {M : Type u_1} [AddZeroClass M] {p : AddSubmonoid M} {p' : AddSubmonoid M} {x : M} :
                          x p p' x p x p'
                          @[simp]
                          theorem Submonoid.mem_inf {M : Type u_1} [MulOneClass M] {p : Submonoid M} {p' : Submonoid M} {x : M} :
                          x p p' x p x p'
                          theorem AddSubmonoid.instInfSetAddSubmonoid.proof_1 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) :
                          ∀ {a b : M}, a ⋂ t ∈ s, tb ⋂ t ∈ s, ta + b ⋂ x ∈ s, x
                          Equations
                          • AddSubmonoid.instInfSetAddSubmonoid = { sInf := fun (s : Set (AddSubmonoid M)) => { toAddSubsemigroup := { carrier := ⋂ t ∈ s, t, add_mem' := }, zero_mem' := } }
                          theorem AddSubmonoid.instInfSetAddSubmonoid.proof_2 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) :
                          0 ⋂ x ∈ s, x
                          Equations
                          • Submonoid.instInfSetSubmonoid = { sInf := fun (s : Set (Submonoid M)) => { toSubsemigroup := { carrier := ⋂ t ∈ s, t, mul_mem' := }, one_mem' := } }
                          @[simp]
                          theorem AddSubmonoid.coe_sInf {M : Type u_1} [AddZeroClass M] (S : Set (AddSubmonoid M)) :
                          (sInf S) = ⋂ s ∈ S, s
                          @[simp]
                          theorem Submonoid.coe_sInf {M : Type u_1} [MulOneClass M] (S : Set (Submonoid M)) :
                          (sInf S) = ⋂ s ∈ S, s
                          theorem AddSubmonoid.mem_sInf {M : Type u_1} [AddZeroClass M] {S : Set (AddSubmonoid M)} {x : M} :
                          x sInf S pS, x p
                          theorem Submonoid.mem_sInf {M : Type u_1} [MulOneClass M] {S : Set (Submonoid M)} {x : M} :
                          x sInf S pS, x p
                          theorem AddSubmonoid.mem_iInf {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} {S : ιAddSubmonoid M} {x : M} :
                          x ⨅ (i : ι), S i ∀ (i : ι), x S i
                          theorem Submonoid.mem_iInf {M : Type u_1} [MulOneClass M] {ι : Sort u_4} {S : ιSubmonoid M} {x : M} :
                          x ⨅ (i : ι), S i ∀ (i : ι), x S i
                          @[simp]
                          theorem AddSubmonoid.coe_iInf {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} {S : ιAddSubmonoid M} :
                          (⨅ (i : ι), S i) = ⋂ (i : ι), (S i)
                          @[simp]
                          theorem Submonoid.coe_iInf {M : Type u_1} [MulOneClass M] {ι : Sort u_4} {S : ιSubmonoid M} :
                          (⨅ (i : ι), S i) = ⋂ (i : ι), (S i)

                          The AddSubmonoids of an AddMonoid form a complete lattice.

                          Equations
                          theorem AddSubmonoid.instCompleteLatticeAddSubmonoid.proof_13 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) (a : AddSubmonoid M) :
                          (bs, b a)sSup s a
                          theorem AddSubmonoid.instCompleteLatticeAddSubmonoid.proof_15 {M : Type u_1} [AddZeroClass M] (s : Set (AddSubmonoid M)) (a : AddSubmonoid M) :
                          (bs, a b)a sInf s
                          theorem AddSubmonoid.instCompleteLatticeAddSubmonoid.proof_9 {M : Type u_1} [AddZeroClass M] :
                          ∀ (x x_1 : AddSubmonoid M) (x_2 : M), x_2 x x_2 x_1x_2 x
                          theorem AddSubmonoid.instCompleteLatticeAddSubmonoid.proof_11 {M : Type u_1} [AddZeroClass M] :
                          ∀ (x x_1 x_2 : AddSubmonoid M), x x_1x x_2x_3x, x_3 x_1 x_3 x_2
                          theorem AddSubmonoid.instCompleteLatticeAddSubmonoid.proof_10 {M : Type u_1} [AddZeroClass M] :
                          ∀ (x x_1 : AddSubmonoid M) (x_2 : M), x_2 x x_2 x_1x_2 x_1

                          Submonoids of a monoid form a complete lattice.

                          Equations
                          Equations
                          • AddSubmonoid.instUniqueAddSubmonoid = { toInhabited := { default := }, uniq := }
                          Equations
                          • Submonoid.instUniqueSubmonoid = { toInhabited := { default := }, uniq := }

                          The add_submonoid generated by a set

                          Equations
                          Instances For
                            def Submonoid.closure {M : Type u_1} [MulOneClass M] (s : Set M) :

                            The Submonoid generated by a set.

                            Equations
                            Instances For
                              theorem AddSubmonoid.mem_closure {M : Type u_1} [AddZeroClass M] {s : Set M} {x : M} :
                              x AddSubmonoid.closure s ∀ (S : AddSubmonoid M), s Sx S
                              theorem Submonoid.mem_closure {M : Type u_1} [MulOneClass M] {s : Set M} {x : M} :
                              x Submonoid.closure s ∀ (S : Submonoid M), s Sx S
                              @[simp]

                              The AddSubmonoid generated by a set includes the set.

                              @[simp]
                              theorem Submonoid.subset_closure {M : Type u_1} [MulOneClass M] {s : Set M} :

                              The submonoid generated by a set includes the set.

                              theorem AddSubmonoid.not_mem_of_not_mem_closure {M : Type u_1} [AddZeroClass M] {s : Set M} {P : M} (hP : PAddSubmonoid.closure s) :
                              Ps
                              theorem Submonoid.not_mem_of_not_mem_closure {M : Type u_1} [MulOneClass M] {s : Set M} {P : M} (hP : PSubmonoid.closure s) :
                              Ps
                              @[simp]
                              theorem AddSubmonoid.closure_le {M : Type u_1} [AddZeroClass M] {s : Set M} {S : AddSubmonoid M} :

                              An additive submonoid S includes closure s if and only if it includes s

                              @[simp]
                              theorem Submonoid.closure_le {M : Type u_1} [MulOneClass M] {s : Set M} {S : Submonoid M} :

                              A submonoid S includes closure s if and only if it includes s.

                              theorem AddSubmonoid.closure_mono {M : Type u_1} [AddZeroClass M] ⦃s : Set M ⦃t : Set M (h : s t) :

                              Additive submonoid closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t

                              theorem Submonoid.closure_mono {M : Type u_1} [MulOneClass M] ⦃s : Set M ⦃t : Set M (h : s t) :

                              Submonoid closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

                              theorem AddSubmonoid.closure_eq_of_le {M : Type u_1} [AddZeroClass M] {s : Set M} {S : AddSubmonoid M} (h₁ : s S) (h₂ : S AddSubmonoid.closure s) :
                              theorem Submonoid.closure_eq_of_le {M : Type u_1} [MulOneClass M] {s : Set M} {S : Submonoid M} (h₁ : s S) (h₂ : S Submonoid.closure s) :
                              theorem AddSubmonoid.closure_induction {M : Type u_1} [AddZeroClass M] {s : Set M} {p : MProp} {x : M} (h : x AddSubmonoid.closure s) (mem : xs, p x) (one : p 0) (mul : ∀ (x y : M), p xp yp (x + y)) :
                              p x

                              An induction principle for additive closure membership. If p holds for 0 and all elements of s, and is preserved under addition, then p holds for all elements of the additive closure of s.

                              theorem Submonoid.closure_induction {M : Type u_1} [MulOneClass M] {s : Set M} {p : MProp} {x : M} (h : x Submonoid.closure s) (mem : xs, p x) (one : p 1) (mul : ∀ (x y : M), p xp yp (x * y)) :
                              p x

                              An induction principle for closure membership. If p holds for 1 and all elements of s, and is preserved under multiplication, then p holds for all elements of the closure of s.

                              abbrev AddSubmonoid.closure_induction'.match_1 {M : Type u_1} [AddZeroClass M] (s : Set M) {p : (x : M) → x AddSubmonoid.closure sProp} (y : M) (motive : (∃ (x : y AddSubmonoid.closure s), p y x)Prop) :
                              ∀ (x : ∃ (x : y AddSubmonoid.closure s), p y x), (∀ (hy' : y AddSubmonoid.closure s) (hy : p y hy'), motive )motive x
                              Equations
                              • =
                              Instances For
                                theorem AddSubmonoid.closure_induction' {M : Type u_1} [AddZeroClass M] (s : Set M) {p : (x : M) → x AddSubmonoid.closure sProp} (mem : ∀ (x : M) (h : x s), p x ) (one : p 0 ) (mul : ∀ (x : M) (hx : x AddSubmonoid.closure s) (y : M) (hy : y AddSubmonoid.closure s), p x hxp y hyp (x + y) ) {x : M} (hx : x AddSubmonoid.closure s) :
                                p x hx

                                A dependent version of AddSubmonoid.closure_induction.

                                theorem Submonoid.closure_induction' {M : Type u_1} [MulOneClass M] (s : Set M) {p : (x : M) → x Submonoid.closure sProp} (mem : ∀ (x : M) (h : x s), p x ) (one : p 1 ) (mul : ∀ (x : M) (hx : x Submonoid.closure s) (y : M) (hy : y Submonoid.closure s), p x hxp y hyp (x * y) ) {x : M} (hx : x Submonoid.closure s) :
                                p x hx

                                A dependent version of Submonoid.closure_induction.

                                theorem AddSubmonoid.closure_induction₂ {M : Type u_1} [AddZeroClass M] {s : Set M} {p : MMProp} {x : M} {y : M} (hx : x AddSubmonoid.closure s) (hy : y AddSubmonoid.closure s) (Hs : xs, ys, p x y) (H1_left : ∀ (x : M), p 0 x) (H1_right : ∀ (x : M), p x 0) (Hmul_left : ∀ (x y z : M), p x zp y zp (x + y) z) (Hmul_right : ∀ (x y z : M), p z xp z yp z (x + y)) :
                                p x y

                                An induction principle for additive closure membership for predicates with two arguments.

                                theorem Submonoid.closure_induction₂ {M : Type u_1} [MulOneClass M] {s : Set M} {p : MMProp} {x : M} {y : M} (hx : x Submonoid.closure s) (hy : y Submonoid.closure s) (Hs : xs, ys, p x y) (H1_left : ∀ (x : M), p 1 x) (H1_right : ∀ (x : M), p x 1) (Hmul_left : ∀ (x y z : M), p x zp y zp (x * y) z) (Hmul_right : ∀ (x y z : M), p z xp z yp z (x * y)) :
                                p x y

                                An induction principle for closure membership for predicates with two arguments.

                                theorem AddSubmonoid.dense_induction {M : Type u_1} [AddZeroClass M] {p : MProp} (x : M) {s : Set M} (hs : AddSubmonoid.closure s = ) (mem : xs, p x) (one : p 0) (mul : ∀ (x y : M), p xp yp (x + y)) :
                                p x

                                If s is a dense set in an additive monoid M, AddSubmonoid.closure s = ⊤, then in order to prove that some predicate p holds for all x : M it suffices to verify p x for x ∈ s, verify p 0, and verify that p x and p y imply p (x + y).

                                theorem Submonoid.dense_induction {M : Type u_1} [MulOneClass M] {p : MProp} (x : M) {s : Set M} (hs : Submonoid.closure s = ) (mem : xs, p x) (one : p 1) (mul : ∀ (x y : M), p xp yp (x * y)) :
                                p x

                                If s is a dense set in a monoid M, Submonoid.closure s = ⊤, then in order to prove that some predicate p holds for all x : M it suffices to verify p x for x ∈ s, verify p 1, and verify that p x and p y imply p (x * y).

                                theorem AddSubmonoid.gi.proof_1 (M : Type u_1) [AddZeroClass M] :
                                ∀ (x : AddSubmonoid M), x (AddSubmonoid.closure x)
                                theorem AddSubmonoid.gi.proof_2 (M : Type u_1) [AddZeroClass M] :
                                ∀ (x : Set M) (x_1 : (AddSubmonoid.closure x) x), (fun (s : Set M) (x : (AddSubmonoid.closure s) s) => AddSubmonoid.closure s) x x_1 = (fun (s : Set M) (x : (AddSubmonoid.closure s) s) => AddSubmonoid.closure s) x x_1
                                def AddSubmonoid.gi (M : Type u_1) [AddZeroClass M] :
                                GaloisInsertion AddSubmonoid.closure SetLike.coe

                                closure forms a Galois insertion with the coercion to set.

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                                  def Submonoid.gi (M : Type u_1) [MulOneClass M] :
                                  GaloisInsertion Submonoid.closure SetLike.coe

                                  closure forms a Galois insertion with the coercion to set.

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                                    @[simp]

                                    Additive closure of an additive submonoid S equals S

                                    @[simp]
                                    theorem Submonoid.closure_eq {M : Type u_1} [MulOneClass M] (S : Submonoid M) :

                                    Closure of a submonoid S equals S.

                                    @[simp]
                                    theorem Submonoid.sup_eq_closure {M : Type u_1} [MulOneClass M] (N : Submonoid M) (N' : Submonoid M) :
                                    N N' = Submonoid.closure (N N')
                                    theorem AddSubmonoid.closure_iUnion {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} (s : ιSet M) :
                                    AddSubmonoid.closure (⋃ (i : ι), s i) = ⨆ (i : ι), AddSubmonoid.closure (s i)
                                    theorem Submonoid.closure_iUnion {M : Type u_1} [MulOneClass M] {ι : Sort u_4} (s : ιSet M) :
                                    Submonoid.closure (⋃ (i : ι), s i) = ⨆ (i : ι), Submonoid.closure (s i)
                                    theorem AddSubmonoid.mem_iSup {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} (p : ιAddSubmonoid M) {m : M} :
                                    m ⨆ (i : ι), p i ∀ (N : AddSubmonoid M), (∀ (i : ι), p i N)m N
                                    theorem Submonoid.mem_iSup {M : Type u_1} [MulOneClass M] {ι : Sort u_4} (p : ιSubmonoid M) {m : M} :
                                    m ⨆ (i : ι), p i ∀ (N : Submonoid M), (∀ (i : ι), p i N)m N
                                    theorem AddSubmonoid.iSup_eq_closure {M : Type u_1} [AddZeroClass M] {ι : Sort u_4} (p : ιAddSubmonoid M) :
                                    ⨆ (i : ι), p i = AddSubmonoid.closure (⋃ (i : ι), (p i))
                                    theorem Submonoid.iSup_eq_closure {M : Type u_1} [MulOneClass M] {ι : Sort u_4} (p : ιSubmonoid M) :
                                    ⨆ (i : ι), p i = Submonoid.closure (⋃ (i : ι), (p i))
                                    theorem AddSubmonoid.disjoint_def {M : Type u_1} [AddZeroClass M] {p₁ : AddSubmonoid M} {p₂ : AddSubmonoid M} :
                                    Disjoint p₁ p₂ ∀ {x : M}, x p₁x p₂x = 0
                                    theorem Submonoid.disjoint_def {M : Type u_1} [MulOneClass M] {p₁ : Submonoid M} {p₂ : Submonoid M} :
                                    Disjoint p₁ p₂ ∀ {x : M}, x p₁x p₂x = 1
                                    theorem AddSubmonoid.disjoint_def' {M : Type u_1} [AddZeroClass M] {p₁ : AddSubmonoid M} {p₂ : AddSubmonoid M} :
                                    Disjoint p₁ p₂ ∀ {x y : M}, x p₁y p₂x = yx = 0
                                    theorem Submonoid.disjoint_def' {M : Type u_1} [MulOneClass M] {p₁ : Submonoid M} {p₂ : Submonoid M} :
                                    Disjoint p₁ p₂ ∀ {x y : M}, x p₁y p₂x = yx = 1
                                    def AddMonoidHom.eqLocusM {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (g : M →+ N) :

                                    The additive submonoid of elements x : M such that f x = g x

                                    Equations
                                    • AddMonoidHom.eqLocusM f g = { toAddSubsemigroup := { carrier := {x : M | f x = g x}, add_mem' := }, zero_mem' := }
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                                      theorem AddMonoidHom.eqLocusM.proof_2 {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (g : M →+ N) :
                                      0 { carrier := {x : M | f x = g x}, add_mem' := }.carrier
                                      theorem AddMonoidHom.eqLocusM.proof_1 {M : Type u_2} {N : Type u_1} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (g : M →+ N) :
                                      ∀ {a b : M}, f a = g af b = g bf (a + b) = g (a + b)
                                      def MonoidHom.eqLocusM {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] (f : M →* N) (g : M →* N) :

                                      The submonoid of elements x : M such that f x = g x

                                      Equations
                                      • MonoidHom.eqLocusM f g = { toSubsemigroup := { carrier := {x : M | f x = g x}, mul_mem' := }, one_mem' := }
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                                        @[simp]
                                        theorem AddMonoidHom.eqLocusM_same {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) :
                                        @[simp]
                                        theorem MonoidHom.eqLocusM_same {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] (f : M →* N) :
                                        theorem AddMonoidHom.eqOn_closureM {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] {f : M →+ N} {g : M →+ N} {s : Set M} (h : Set.EqOn (f) (g) s) :

                                        If two monoid homomorphisms are equal on a set, then they are equal on its submonoid closure.

                                        theorem MonoidHom.eqOn_closureM {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] {f : M →* N} {g : M →* N} {s : Set M} (h : Set.EqOn (f) (g) s) :
                                        Set.EqOn f g (Submonoid.closure s)

                                        If two monoid homomorphisms are equal on a set, then they are equal on its submonoid closure.

                                        theorem AddMonoidHom.eq_of_eqOn_topM {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] {f : M →+ N} {g : M →+ N} (h : Set.EqOn f g ) :
                                        f = g
                                        theorem MonoidHom.eq_of_eqOn_topM {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] {f : M →* N} {g : M →* N} (h : Set.EqOn f g ) :
                                        f = g
                                        theorem AddMonoidHom.eq_of_eqOn_denseM {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] {s : Set M} (hs : AddSubmonoid.closure s = ) {f : M →+ N} {g : M →+ N} (h : Set.EqOn (f) (g) s) :
                                        f = g
                                        theorem MonoidHom.eq_of_eqOn_denseM {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] {s : Set M} (hs : Submonoid.closure s = ) {f : M →* N} {g : M →* N} (h : Set.EqOn (f) (g) s) :
                                        f = g

                                        The additive submonoid consisting of the additive units of an additive monoid

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                                          theorem IsAddUnit.addSubmonoid.proof_1 (M : Type u_1) [AddMonoid M] {a : M} {b : M} (ha : a setOf IsAddUnit) (hb : b setOf IsAddUnit) :
                                          a + b setOf IsAddUnit
                                          def IsUnit.submonoid (M : Type u_4) [Monoid M] :

                                          The submonoid consisting of the units of a monoid

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                                            def AddMonoidHom.ofClosureMEqTopLeft {M : Type u_4} {N : Type u_5} [AddMonoid M] [AddMonoid N] {s : Set M} (f : MN) (hs : AddSubmonoid.closure s = ) (h1 : f 0 = 0) (hmul : xs, ∀ (y : M), f (x + y) = f x + f y) :
                                            M →+ N

                                            Let s be a subset of an additive monoid M such that the closure of s is the whole monoid. Then AddMonoidHom.ofClosureEqTopLeft defines an additive monoid homomorphism from M asking for a proof of f (x + y) = f x + f y only for x ∈ s.

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                                              theorem AddMonoidHom.ofClosureMEqTopLeft.proof_1 {M : Type u_2} {N : Type u_1} [AddMonoid M] [AddMonoid N] {s : Set M} (f : MN) (hs : AddSubmonoid.closure s = ) (h1 : f 0 = 0) (hmul : xs, ∀ (y : M), f (x + y) = f x + f y) (x : M) (y : M) :
                                              f (x + y) = f x + f y
                                              def MonoidHom.ofClosureMEqTopLeft {M : Type u_4} {N : Type u_5} [Monoid M] [Monoid N] {s : Set M} (f : MN) (hs : Submonoid.closure s = ) (h1 : f 1 = 1) (hmul : xs, ∀ (y : M), f (x * y) = f x * f y) :
                                              M →* N

                                              Let s be a subset of a monoid M such that the closure of s is the whole monoid. Then MonoidHom.ofClosureEqTopLeft defines a monoid homomorphism from M asking for a proof of f (x * y) = f x * f y only for x ∈ s.

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                                                @[simp]
                                                theorem AddMonoidHom.coe_ofClosureMEqTopLeft {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddMonoid N] {s : Set M} (f : MN) (hs : AddSubmonoid.closure s = ) (h1 : f 0 = 0) (hmul : xs, ∀ (y : M), f (x + y) = f x + f y) :
                                                @[simp]
                                                theorem MonoidHom.coe_ofClosureMEqTopLeft {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {s : Set M} (f : MN) (hs : Submonoid.closure s = ) (h1 : f 1 = 1) (hmul : xs, ∀ (y : M), f (x * y) = f x * f y) :
                                                (MonoidHom.ofClosureMEqTopLeft f hs h1 hmul) = f
                                                def AddMonoidHom.ofClosureMEqTopRight {M : Type u_4} {N : Type u_5} [AddMonoid M] [AddMonoid N] {s : Set M} (f : MN) (hs : AddSubmonoid.closure s = ) (h1 : f 0 = 0) (hmul : ∀ (x y : M), y sf (x + y) = f x + f y) :
                                                M →+ N

                                                Let s be a subset of an additive monoid M such that the closure of s is the whole monoid. Then AddMonoidHom.ofClosureEqTopRight defines an additive monoid homomorphism from M asking for a proof of f (x + y) = f x + f y only for y ∈ s.

                                                Equations
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                                                  theorem AddMonoidHom.ofClosureMEqTopRight.proof_1 {M : Type u_2} {N : Type u_1} [AddMonoid M] [AddMonoid N] {s : Set M} (f : MN) (hs : AddSubmonoid.closure s = ) (h1 : f 0 = 0) (hmul : ∀ (x y : M), y sf (x + y) = f x + f y) (x : M) (y : M) :
                                                  { toFun := f, map_zero' := h1 }.toFun (x + y) = { toFun := f, map_zero' := h1 }.toFun x + { toFun := f, map_zero' := h1 }.toFun y
                                                  def MonoidHom.ofClosureMEqTopRight {M : Type u_4} {N : Type u_5} [Monoid M] [Monoid N] {s : Set M} (f : MN) (hs : Submonoid.closure s = ) (h1 : f 1 = 1) (hmul : ∀ (x y : M), y sf (x * y) = f x * f y) :
                                                  M →* N

                                                  Let s be a subset of a monoid M such that the closure of s is the whole monoid. Then MonoidHom.ofClosureEqTopRight defines a monoid homomorphism from M asking for a proof of f (x * y) = f x * f y only for y ∈ s.

                                                  Equations
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                                                    @[simp]
                                                    theorem AddMonoidHom.coe_ofClosureMEqTopRight {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddMonoid N] {s : Set M} (f : MN) (hs : AddSubmonoid.closure s = ) (h1 : f 0 = 0) (hmul : ∀ (x y : M), y sf (x + y) = f x + f y) :
                                                    @[simp]
                                                    theorem MonoidHom.coe_ofClosureMEqTopRight {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] {s : Set M} (f : MN) (hs : Submonoid.closure s = ) (h1 : f 1 = 1) (hmul : ∀ (x y : M), y sf (x * y) = f x * f y) :
                                                    (MonoidHom.ofClosureMEqTopRight f hs h1 hmul) = f