(* Title: HOL/Algebra/Ring.thy Author: Clemens Ballarin, started 9 December 1996 Copyright: Clemens Ballarin *) theory Ring imports FiniteProduct begin section {* 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.simps text {* 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) auto context abelian_monoid begin lemmas a_closed = add.m_closed lemmas zero_closed = add.one_closed lemmas a_assoc = add.m_assoc lemmas l_zero = add.l_one lemmas r_zero = add.r_one lemmas minus_unique = add.inv_unique end sublocale 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 begin lemmas a_comm = add.m_comm lemmas a_lcomm = add.m_lcomm lemmas a_ac = a_assoc a_comm a_lcomm lemmas finsum_empty = add.finprod_empty lemmas finsum_insert = add.finprod_insert lemmas finsum_zero = add.finprod_one lemmas finsum_closed = add.finprod_closed lemmas finsum_Un_Int = add.finprod_Un_Int lemmas finsum_Un_disjoint = add.finprod_Un_disjoint lemmas finsum_addf = add.finprod_multf lemmas finsum_cong' = add.finprod_cong' lemmas finsum_0 = add.finprod_0 lemmas finsum_Suc = add.finprod_Suc lemmas finsum_Suc2 = add.finprod_Suc2 lemmas finsum_add = add.finprod_mult lemmas finsum_infinite = add.finprod_infinite lemmas finsum_cong = add.finprod_cong text {*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 auto done *) lemmas finsum_singleton = add.finprod_singleton end sublocale 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 begin lemmas a_inv_closed = add.inv_closed lemma 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_cancel lemmas a_r_cancel = add.r_cancel lemmas l_neg = add.l_inv [simp del] lemmas r_neg = add.r_inv [simp del] lemmas minus_zero = add.inv_one lemmas minus_minus = add.inv_inv lemmas a_inv_inj = add.inv_inj lemmas minus_equality = add.inv_equality end sublocale 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" .. qed subsection {* Rings: Basic Definitions *} locale semiring = abelian_monoid 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" and l_null[simp]: "x ∈ carrier R ==> \<zero> ⊗ x = \<zero>" and r_null[simp]: "x ∈ carrier R ==> x ⊗ \<zero> = \<zero>" locale ring = abelian_group R + monoid R for R (structure) + assumes "[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> (x ⊕ y) ⊗ z = x ⊗ z ⊕ y ⊗ z" and "[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> z ⊗ (x ⊕ y) = z ⊗ x ⊕ z ⊗ y" locale cring = ring + comm_monoid R locale "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 begin lemma 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) end lemmas ring_record_simps = monoid_record_simps ring.simps lemma 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) qed lemma (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) qed context ring begin text {* The following proofs are from Jacobson, Basic Algebra I, pp.~88--89. *} sublocale semiring proof - note [simp] = ring_axioms[unfolded ring_def ring_axioms_def] show "semiring R" proof (unfold_locales) fix x assume R: "x ∈ carrier R" then have "\<zero> ⊗ x ⊕ \<zero> ⊗ x = (\<zero> ⊕ \<zero>) ⊗ x" by (simp 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 "\<zero> ⊗ x = \<zero>" by (simp del: r_zero) from R have "x ⊗ \<zero> ⊕ x ⊗ \<zero> = x ⊗ (\<zero> ⊕ \<zero>)" by (simp 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 "x ⊗ \<zero> = \<zero>" by (simp del: r_zero) qed auto qed lemma 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) qed lemma 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 ) qed end lemma (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" attribute_setup algebra = {* Scan.lift ((Args.add >> K true || Args.del >> K false) --| Args.colon || Scan.succeed true) -- Scan.lift Args.name -- Scan.repeat Args.term >> (fn ((b, n), ts) => if b then Ringsimp.add_struct (n, ts) else Ringsimp.del_struct (n, ts)) *} "theorems controlling algebra method" method_setup algebra = {* Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac) *} "normalisation of algebraic structure" lemmas (in semiring) semiring_simprules [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] = a_closed zero_closed m_closed one_closed a_assoc l_zero a_comm m_assoc l_one l_distr r_zero a_lcomm r_distr l_null r_null 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_minus lemmas (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_minus lemma (in semiring) nat_pow_zero: "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>" by (induct n) simp_all context semiring begin lemma 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 fast qed fast lemma one_zeroI: assumes carrzero: "carrier R = {\<zero>}" shows "\<one> = \<zero>" proof - from one_closed and carrzero show "\<one> = \<zero>" by simp qed lemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)" apply rule apply (erule one_zeroI) apply (erule one_zeroD) done lemma carrier_one_not_zero: "(carrier R ≠ {\<zero>}) = (\<one> ≠ \<zero>)" by (simp add: carrier_one_zero) end text {* 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 algebra qed lemma 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 algebra qed subsubsection {* Sums over Finite Sets *} lemma (in semiring) 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 simp next case (insert x F) then show ?case by (simp add: Pi_def l_distr) qed lemma (in semiring) 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 simp next case (insert x F) then show ?case by (simp add: Pi_def r_distr) qed subsection {* Integral Domains *} context "domain" begin lemma zero_not_one [simp]: "\<zero> ~= \<one>" by (rule not_sym) simp lemma 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 auto qed lemma 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 simp qed lemma 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 algebra qed end subsection {* 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 force next 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 qed qed (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 fast qed subsection {* 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) qed lemma (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 simp qed lemma (in ring_hom_cring) hom_finsum [simp]: "f ∈ A -> carrier R ==> h (finsum R f A) = finsum S (h o f) A" by (induct A rule: infinite_finite_induct, auto simp: Pi_def) lemma (in ring_hom_cring) hom_finprod: "f ∈ A -> carrier R ==> h (finprod R f A) = finprod S (h o f) A" by (induct A rule: infinite_finite_induct, auto simp: Pi_def) declare ring_hom_cring.hom_finprod [simp] lemma id_ring_hom [simp]: "id ∈ ring_hom R R" by (auto intro!: ring_hom_memI) end