# Theory Ring

Up to index of Isabelle/HOL/Free-Groups

theory Ring
imports FiniteProduct
`(*  Title:      HOL/Algebra/Ring.thy    Author:     Clemens Ballarin, started 9 December 1996    Copyright:  Clemens Ballarin*)theory Ringimports FiniteProductbeginsection {* The Algebraic Hierarchy of Rings *}subsection {* Abelian Groups *}record 'a ring = "'a monoid" +  zero :: 'a ("\<zero>\<index>")  add :: "['a, 'a] => 'a" (infixl "⊕\<index>" 65)text {* Derived operations. *}definition  a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80)  where "a_inv R = m_inv (| carrier = carrier R, mult = add R, one = zero R |)"definition  a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65)  where "[| x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus>⇘R⇙ y = x ⊕⇘R⇙ (\<ominus>⇘R⇙ y)"locale abelian_monoid =  fixes G (structure)  assumes a_comm_monoid:     "comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)"definition  finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where  "finsum G = finprod (| carrier = carrier G, mult = add G, one = zero G |)"syntax  "_finsum" :: "index => idt => 'a set => 'b => 'b"      ("(3\<Oplus>__:_. _)" [1000, 0, 51, 10] 10)syntax (xsymbols)  "_finsum" :: "index => idt => 'a set => 'b => 'b"      ("(3\<Oplus>__∈_. _)" [1000, 0, 51, 10] 10)syntax (HTML output)  "_finsum" :: "index => idt => 'a set => 'b => 'b"      ("(3\<Oplus>__∈_. _)" [1000, 0, 51, 10] 10)translations  "\<Oplus>\<index>i:A. b" == "CONST finsum \<struct>\<index> (%i. b) A"  -- {* Beware of argument permutation! *}locale abelian_group = abelian_monoid +  assumes a_comm_group:     "comm_group (| carrier = carrier G, mult = add G, one = zero G |)"subsection {* Basic Properties *}lemma abelian_monoidI:  fixes R (structure)  assumes a_closed:      "!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕ y ∈ carrier R"    and zero_closed: "\<zero> ∈ carrier R"    and a_assoc:      "!!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==>      (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)"    and l_zero: "!!x. x ∈ carrier R ==> \<zero> ⊕ x = x"    and a_comm:      "!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕ y = y ⊕ x"  shows "abelian_monoid R"  by (auto intro!: abelian_monoid.intro comm_monoidI intro: assms)lemma abelian_groupI:  fixes R (structure)  assumes a_closed:      "!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕ y ∈ carrier R"    and zero_closed: "zero R ∈ carrier R"    and a_assoc:      "!!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==>      (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)"    and a_comm:      "!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕ y = y ⊕ x"    and l_zero: "!!x. x ∈ carrier R ==> \<zero> ⊕ x = x"    and l_inv_ex: "!!x. x ∈ carrier R ==> EX y : carrier R. y ⊕ x = \<zero>"  shows "abelian_group R"  by (auto intro!: abelian_group.intro abelian_monoidI      abelian_group_axioms.intro comm_monoidI comm_groupI    intro: assms)lemma (in abelian_monoid) a_monoid:  "monoid (| carrier = carrier G, mult = add G, one = zero G |)"by (rule comm_monoid.axioms, rule a_comm_monoid) lemma (in abelian_group) a_group:  "group (| carrier = carrier G, mult = add G, one = zero G |)"  by (simp add: group_def a_monoid)    (simp add: comm_group.axioms group.axioms a_comm_group)lemmas monoid_record_simps = partial_object.simps monoid.simpstext {* Transfer facts from multiplicative structures via interpretation. *}sublocale abelian_monoid <  add!: monoid "(| carrier = carrier G, mult = add G, one = zero G |)"  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"  by (rule a_monoid) autocontext abelian_monoid beginlemmas a_closed = add.m_closed lemmas zero_closed = add.one_closedlemmas a_assoc = add.m_assoclemmas l_zero = add.l_onelemmas r_zero = add.r_onelemmas minus_unique = add.inv_uniqueendsublocale abelian_monoid <  add!: comm_monoid "(| carrier = carrier G, mult = add G, one = zero G |)"  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"    and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = finsum G"  by (rule a_comm_monoid) (auto simp: finsum_def)context abelian_monoid beginlemmas a_comm = add.m_commlemmas a_lcomm = add.m_lcommlemmas a_ac = a_assoc a_comm a_lcommlemmas finsum_empty = add.finprod_emptylemmas finsum_insert = add.finprod_insertlemmas finsum_zero = add.finprod_onelemmas finsum_closed = add.finprod_closedlemmas finsum_Un_Int = add.finprod_Un_Intlemmas finsum_Un_disjoint = add.finprod_Un_disjointlemmas finsum_addf = add.finprod_multflemmas finsum_cong' = add.finprod_cong'lemmas finsum_0 = add.finprod_0lemmas finsum_Suc = add.finprod_Suclemmas finsum_Suc2 = add.finprod_Suc2lemmas finsum_add = add.finprod_multlemmas finsum_cong = add.finprod_congtext {*Usually, if this rule causes a failed congruence proof error,   the reason is that the premise @{text "g ∈ B -> carrier G"} cannot be shown.   Adding @{thm [source] Pi_def} to the simpset is often useful. *}lemmas finsum_reindex = add.finprod_reindex(* The following would be wrong.  Needed is the equivalent of (^) for addition,  or indeed the canonical embedding from Nat into the monoid.lemma finsum_const:  assumes fin [simp]: "finite A"      and a [simp]: "a : carrier G"    shows "finsum G (%x. a) A = a (^) card A"  using fin apply induct  apply force  apply (subst finsum_insert)  apply auto  apply (force simp add: Pi_def)  apply (subst m_comm)  apply autodone*)lemmas finsum_singleton = add.finprod_singletonendsublocale abelian_group <  add!: group "(| carrier = carrier G, mult = add G, one = zero G |)"  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"    and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"  by (rule a_group) (auto simp: m_inv_def a_inv_def)context abelian_group beginlemmas a_inv_closed = add.inv_closedlemma minus_closed [intro, simp]:  "[| x ∈ carrier G; y ∈ carrier G |] ==> x \<ominus> y ∈ carrier G"  by (simp add: a_minus_def)lemmas a_l_cancel = add.l_cancellemmas a_r_cancel = add.r_cancellemmas l_neg = add.l_inv [simp del]lemmas r_neg = add.r_inv [simp del]lemmas minus_zero = add.inv_onelemmas minus_minus = add.inv_invlemmas a_inv_inj = add.inv_injlemmas minus_equality = add.inv_equalityendsublocale abelian_group <  add!: comm_group "(| carrier = carrier G, mult = add G, one = zero G |)"  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"    and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"    and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = finsum G"  by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def)lemmas (in abelian_group) minus_add = add.inv_mult text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *}lemma comm_group_abelian_groupI:  fixes G (structure)  assumes cg: "comm_group (|carrier = carrier G, mult = add G, one = zero G|)),"  shows "abelian_group G"proof -  interpret comm_group "(|carrier = carrier G, mult = add G, one = zero G|)),"    by (rule cg)  show "abelian_group G" ..qedsubsection {* Rings: Basic Definitions *}locale ring = abelian_group R + monoid R for R (structure) +  assumes l_distr: "[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]      ==> (x ⊕ y) ⊗ z = x ⊗ z ⊕ y ⊗ z"    and r_distr: "[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]      ==> z ⊗ (x ⊕ y) = z ⊗ x ⊕ z ⊗ y"locale cring = ring + comm_monoid Rlocale "domain" = cring +  assumes one_not_zero [simp]: "\<one> ~= \<zero>"    and integral: "[| a ⊗ b = \<zero>; a ∈ carrier R; b ∈ carrier R |] ==>                  a = \<zero> | b = \<zero>"locale field = "domain" +  assumes field_Units: "Units R = carrier R - {\<zero>}"subsection {* Rings *}lemma ringI:  fixes R (structure)  assumes abelian_group: "abelian_group R"    and monoid: "monoid R"    and l_distr: "!!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]      ==> (x ⊕ y) ⊗ z = x ⊗ z ⊕ y ⊗ z"    and r_distr: "!!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]      ==> z ⊗ (x ⊕ y) = z ⊗ x ⊕ z ⊗ y"  shows "ring R"  by (auto intro: ring.intro    abelian_group.axioms ring_axioms.intro assms)context ring beginlemma is_abelian_group: "abelian_group R" ..lemma is_monoid: "monoid R"  by (auto intro!: monoidI m_assoc)lemma is_ring: "ring R"  by (rule ring_axioms)endlemmas ring_record_simps = monoid_record_simps ring.simpslemma cringI:  fixes R (structure)  assumes abelian_group: "abelian_group R"    and comm_monoid: "comm_monoid R"    and l_distr: "!!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]      ==> (x ⊕ y) ⊗ z = x ⊗ z ⊕ y ⊗ z"  shows "cring R"proof (intro cring.intro ring.intro)  show "ring_axioms R"    -- {* Right-distributivity follows from left-distributivity and          commutativity. *}  proof (rule ring_axioms.intro)    fix x y z    assume R: "x ∈ carrier R" "y ∈ carrier R" "z ∈ carrier R"    note [simp] = comm_monoid.axioms [OF comm_monoid]      abelian_group.axioms [OF abelian_group]      abelian_monoid.a_closed            from R have "z ⊗ (x ⊕ y) = (x ⊕ y) ⊗ z"      by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])    also from R have "... = x ⊗ z ⊕ y ⊗ z" by (simp add: l_distr)    also from R have "... = z ⊗ x ⊕ z ⊗ y"      by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])    finally show "z ⊗ (x ⊕ y) = z ⊗ x ⊕ z ⊗ y" .  qed (rule l_distr)qed (auto intro: cring.intro  abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms)(*lemma (in cring) is_comm_monoid:  "comm_monoid R"  by (auto intro!: comm_monoidI m_assoc m_comm)*)lemma (in cring) is_cring:  "cring R" by (rule cring_axioms)subsubsection {* Normaliser for Rings *}lemma (in abelian_group) r_neg2:  "[| x ∈ carrier G; y ∈ carrier G |] ==> x ⊕ (\<ominus> x ⊕ y) = y"proof -  assume G: "x ∈ carrier G" "y ∈ carrier G"  then have "(x ⊕ \<ominus> x) ⊕ y = y"    by (simp only: r_neg l_zero)  with G show ?thesis    by (simp add: a_ac)qedlemma (in abelian_group) r_neg1:  "[| x ∈ carrier G; y ∈ carrier G |] ==> \<ominus> x ⊕ (x ⊕ y) = y"proof -  assume G: "x ∈ carrier G" "y ∈ carrier G"  then have "(\<ominus> x ⊕ x) ⊕ y = y"     by (simp only: l_neg l_zero)  with G show ?thesis by (simp add: a_ac)qedcontext ring begintext {*   The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.*}lemma l_null [simp]:  "x ∈ carrier R ==> \<zero> ⊗ x = \<zero>"proof -  assume R: "x ∈ carrier R"  then have "\<zero> ⊗ x ⊕ \<zero> ⊗ x = (\<zero> ⊕ \<zero>) ⊗ x"    by (simp add: l_distr del: l_zero r_zero)  also from R have "... = \<zero> ⊗ x ⊕ \<zero>" by simp  finally have "\<zero> ⊗ x ⊕ \<zero> ⊗ x = \<zero> ⊗ x ⊕ \<zero>" .  with R show ?thesis by (simp del: r_zero)qedlemma r_null [simp]:  "x ∈ carrier R ==> x ⊗ \<zero> = \<zero>"proof -  assume R: "x ∈ carrier R"  then have "x ⊗ \<zero> ⊕ x ⊗ \<zero> = x ⊗ (\<zero> ⊕ \<zero>)"    by (simp add: r_distr del: l_zero r_zero)  also from R have "... = x ⊗ \<zero> ⊕ \<zero>" by simp  finally have "x ⊗ \<zero> ⊕ x ⊗ \<zero> = x ⊗ \<zero> ⊕ \<zero>" .  with R show ?thesis by (simp del: r_zero)qedlemma l_minus:  "[| x ∈ carrier R; y ∈ carrier R |] ==> \<ominus> x ⊗ y = \<ominus> (x ⊗ y)"proof -  assume R: "x ∈ carrier R" "y ∈ carrier R"  then have "(\<ominus> x) ⊗ y ⊕ x ⊗ y = (\<ominus> x ⊕ x) ⊗ y" by (simp add: l_distr)  also from R have "... = \<zero>" by (simp add: l_neg)  finally have "(\<ominus> x) ⊗ y ⊕ x ⊗ y = \<zero>" .  with R have "(\<ominus> x) ⊗ y ⊕ x ⊗ y ⊕ \<ominus> (x ⊗ y) = \<zero> ⊕ \<ominus> (x ⊗ y)" by simp  with R show ?thesis by (simp add: a_assoc r_neg)qedlemma r_minus:  "[| x ∈ carrier R; y ∈ carrier R |] ==> x ⊗ \<ominus> y = \<ominus> (x ⊗ y)"proof -  assume R: "x ∈ carrier R" "y ∈ carrier R"  then have "x ⊗ (\<ominus> y) ⊕ x ⊗ y = x ⊗ (\<ominus> y ⊕ y)" by (simp add: r_distr)  also from R have "... = \<zero>" by (simp add: l_neg)  finally have "x ⊗ (\<ominus> y) ⊕ x ⊗ y = \<zero>" .  with R have "x ⊗ (\<ominus> y) ⊕ x ⊗ y ⊕ \<ominus> (x ⊗ y) = \<zero> ⊕ \<ominus> (x ⊗ y)" by simp  with R show ?thesis by (simp add: a_assoc r_neg )qedendlemma (in abelian_group) minus_eq:  "[| x ∈ carrier G; y ∈ carrier G |] ==> x \<ominus> y = x ⊕ \<ominus> y"  by (simp only: a_minus_def)text {* Setup algebra method:  compute distributive normal form in locale contexts *}ML_file "ringsimp.ML"setup Algebra.attrib_setupmethod_setup algebra = {*  Scan.succeed (SIMPLE_METHOD' o Algebra.algebra_tac)*} "normalisation of algebraic structure"lemmas (in ring) ring_simprules  [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero  a_lcomm r_distr l_null r_null l_minus r_minuslemmas (in cring)  [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =  _lemmas (in cring) cring_simprules  [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero  a_lcomm m_lcomm r_distr l_null r_null l_minus r_minuslemma (in cring) nat_pow_zero:  "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"  by (induct n) simp_allcontext ring beginlemma one_zeroD:  assumes onezero: "\<one> = \<zero>"  shows "carrier R = {\<zero>}"proof (rule, rule)  fix x  assume xcarr: "x ∈ carrier R"  from xcarr have "x = x ⊗ \<one>" by simp  with onezero have "x = x ⊗ \<zero>" by simp  with xcarr have "x = \<zero>" by simp  then show "x ∈ {\<zero>}" by fastqed fastlemma one_zeroI:  assumes carrzero: "carrier R = {\<zero>}"  shows "\<one> = \<zero>"proof -  from one_closed and carrzero      show "\<one> = \<zero>" by simpqedlemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)"  apply rule   apply (erule one_zeroI)  apply (erule one_zeroD)  donelemma carrier_one_not_zero: "(carrier R ≠ {\<zero>}) = (\<one> ≠ \<zero>)"  by (simp add: carrier_one_zero)endtext {* Two examples for use of method algebra *}lemma  fixes R (structure) and S (structure)  assumes "ring R" "cring S"  assumes RS: "a ∈ carrier R" "b ∈ carrier R" "c ∈ carrier S" "d ∈ carrier S"  shows "a ⊕ \<ominus> (a ⊕ \<ominus> b) = b & c ⊗⇘S⇙ d = d ⊗⇘S⇙ c"proof -  interpret ring R by fact  interpret cring S by fact  from RS show ?thesis by algebraqedlemma  fixes R (structure)  assumes "ring R"  assumes R: "a ∈ carrier R" "b ∈ carrier R"  shows "a \<ominus> (a \<ominus> b) = b"proof -  interpret ring R by fact  from R show ?thesis by algebraqedsubsubsection {* Sums over Finite Sets *}lemma (in ring) finsum_ldistr:  "[| finite A; a ∈ carrier R; f ∈ A -> carrier R |] ==>   finsum R f A ⊗ a = finsum R (%i. f i ⊗ a) A"proof (induct set: finite)  case empty then show ?case by simpnext  case (insert x F) then show ?case by (simp add: Pi_def l_distr)qedlemma (in ring) finsum_rdistr:  "[| finite A; a ∈ carrier R; f ∈ A -> carrier R |] ==>   a ⊗ finsum R f A = finsum R (%i. a ⊗ f i) A"proof (induct set: finite)  case empty then show ?case by simpnext  case (insert x F) then show ?case by (simp add: Pi_def r_distr)qedsubsection {* Integral Domains *}context "domain" beginlemma zero_not_one [simp]:  "\<zero> ~= \<one>"  by (rule not_sym) simplemma integral_iff: (* not by default a simp rule! *)  "[| a ∈ carrier R; b ∈ carrier R |] ==> (a ⊗ b = \<zero>) = (a = \<zero> | b = \<zero>)"proof  assume "a ∈ carrier R" "b ∈ carrier R" "a ⊗ b = \<zero>"  then show "a = \<zero> | b = \<zero>" by (simp add: integral)next  assume "a ∈ carrier R" "b ∈ carrier R" "a = \<zero> | b = \<zero>"  then show "a ⊗ b = \<zero>" by autoqedlemma m_lcancel:  assumes prem: "a ~= \<zero>"    and R: "a ∈ carrier R" "b ∈ carrier R" "c ∈ carrier R"  shows "(a ⊗ b = a ⊗ c) = (b = c)"proof  assume eq: "a ⊗ b = a ⊗ c"  with R have "a ⊗ (b \<ominus> c) = \<zero>" by algebra  with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff)  with prem and R have "b \<ominus> c = \<zero>" by auto   with R have "b = b \<ominus> (b \<ominus> c)" by algebra   also from R have "b \<ominus> (b \<ominus> c) = c" by algebra  finally show "b = c" .next  assume "b = c" then show "a ⊗ b = a ⊗ c" by simpqedlemma m_rcancel:  assumes prem: "a ~= \<zero>"    and R: "a ∈ carrier R" "b ∈ carrier R" "c ∈ carrier R"  shows conc: "(b ⊗ a = c ⊗ a) = (b = c)"proof -  from prem and R have "(a ⊗ b = a ⊗ c) = (b = c)" by (rule m_lcancel)  with R show ?thesis by algebraqedendsubsection {* Fields *}text {* Field would not need to be derived from domain, the properties  for domain follow from the assumptions of field *}lemma (in cring) cring_fieldI:  assumes field_Units: "Units R = carrier R - {\<zero>}"  shows "field R"proof  from field_Units have "\<zero> ∉ Units R" by fast  moreover have "\<one> ∈ Units R" by fast  ultimately show "\<one> ≠ \<zero>" by forcenext  fix a b  assume acarr: "a ∈ carrier R"    and bcarr: "b ∈ carrier R"    and ab: "a ⊗ b = \<zero>"  show "a = \<zero> ∨ b = \<zero>"  proof (cases "a = \<zero>", simp)    assume "a ≠ \<zero>"    with field_Units and acarr have aUnit: "a ∈ Units R" by fast    from bcarr have "b = \<one> ⊗ b" by algebra    also from aUnit acarr have "... = (inv a ⊗ a) ⊗ b" by simp    also from acarr bcarr aUnit[THEN Units_inv_closed]    have "... = (inv a) ⊗ (a ⊗ b)" by algebra    also from ab and acarr bcarr aUnit have "... = (inv a) ⊗ \<zero>" by simp    also from aUnit[THEN Units_inv_closed] have "... = \<zero>" by algebra    finally have "b = \<zero>" .    then show "a = \<zero> ∨ b = \<zero>" by simp  qedqed (rule field_Units)text {* Another variant to show that something is a field *}lemma (in cring) cring_fieldI2:  assumes notzero: "\<zero> ≠ \<one>"  and invex: "!!a. [|a ∈ carrier R; a ≠ \<zero>|] ==> ∃b∈carrier R. a ⊗ b = \<one>"  shows "field R"  apply (rule cring_fieldI, simp add: Units_def)  apply (rule, clarsimp)  apply (simp add: notzero)proof (clarsimp)  fix x  assume xcarr: "x ∈ carrier R"    and "x ≠ \<zero>"  then have "∃y∈carrier R. x ⊗ y = \<one>" by (rule invex)  then obtain y where ycarr: "y ∈ carrier R" and xy: "x ⊗ y = \<one>" by fast  from xy xcarr ycarr have "y ⊗ x = \<one>" by (simp add: m_comm)  with ycarr and xy show "∃y∈carrier R. y ⊗ x = \<one> ∧ x ⊗ y = \<one>" by fastqedsubsection {* Morphisms *}definition  ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"  where "ring_hom R S =    {h. h ∈ carrier R -> carrier S &      (ALL x y. x ∈ carrier R & y ∈ carrier R -->        h (x ⊗⇘R⇙ y) = h x ⊗⇘S⇙ h y & h (x ⊕⇘R⇙ y) = h x ⊕⇘S⇙ h y) &      h \<one>⇘R⇙ = \<one>⇘S⇙}"lemma ring_hom_memI:  fixes R (structure) and S (structure)  assumes hom_closed: "!!x. x ∈ carrier R ==> h x ∈ carrier S"    and hom_mult: "!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==>      h (x ⊗ y) = h x ⊗⇘S⇙ h y"    and hom_add: "!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==>      h (x ⊕ y) = h x ⊕⇘S⇙ h y"    and hom_one: "h \<one> = \<one>⇘S⇙"  shows "h ∈ ring_hom R S"  by (auto simp add: ring_hom_def assms Pi_def)lemma ring_hom_closed:  "[| h ∈ ring_hom R S; x ∈ carrier R |] ==> h x ∈ carrier S"  by (auto simp add: ring_hom_def funcset_mem)lemma ring_hom_mult:  fixes R (structure) and S (structure)  shows    "[| h ∈ ring_hom R S; x ∈ carrier R; y ∈ carrier R |] ==>    h (x ⊗ y) = h x ⊗⇘S⇙ h y"    by (simp add: ring_hom_def)lemma ring_hom_add:  fixes R (structure) and S (structure)  shows    "[| h ∈ ring_hom R S; x ∈ carrier R; y ∈ carrier R |] ==>    h (x ⊕ y) = h x ⊕⇘S⇙ h y"    by (simp add: ring_hom_def)lemma ring_hom_one:  fixes R (structure) and S (structure)  shows "h ∈ ring_hom R S ==> h \<one> = \<one>⇘S⇙"  by (simp add: ring_hom_def)locale ring_hom_cring = R: cring R + S: cring S    for R (structure) and S (structure) +  fixes h  assumes homh [simp, intro]: "h ∈ ring_hom R S"  notes hom_closed [simp, intro] = ring_hom_closed [OF homh]    and hom_mult [simp] = ring_hom_mult [OF homh]    and hom_add [simp] = ring_hom_add [OF homh]    and hom_one [simp] = ring_hom_one [OF homh]lemma (in ring_hom_cring) hom_zero [simp]:  "h \<zero> = \<zero>⇘S⇙"proof -  have "h \<zero> ⊕⇘S⇙ h \<zero> = h \<zero> ⊕⇘S⇙ \<zero>⇘S⇙"    by (simp add: hom_add [symmetric] del: hom_add)  then show ?thesis by (simp del: S.r_zero)qedlemma (in ring_hom_cring) hom_a_inv [simp]:  "x ∈ carrier R ==> h (\<ominus> x) = \<ominus>⇘S⇙ h x"proof -  assume R: "x ∈ carrier R"  then have "h x ⊕⇘S⇙ h (\<ominus> x) = h x ⊕⇘S⇙ (\<ominus>⇘S⇙ h x)"    by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)  with R show ?thesis by simpqedlemma (in ring_hom_cring) hom_finsum [simp]:  "[| finite A; f ∈ A -> carrier R |] ==>  h (finsum R f A) = finsum S (h o f) A"proof (induct set: finite)  case empty then show ?case by simpnext  case insert then show ?case by (simp add: Pi_def)qedlemma (in ring_hom_cring) hom_finprod:  "[| finite A; f ∈ A -> carrier R |] ==>  h (finprod R f A) = finprod S (h o f) A"proof (induct set: finite)  case empty then show ?case by simpnext  case insert then show ?case by (simp add: Pi_def)qeddeclare ring_hom_cring.hom_finprod [simp]lemma id_ring_hom [simp]:  "id ∈ ring_hom R R"  by (auto intro!: ring_hom_memI)end`