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theory CRing(* Title: The algebraic hierarchy of rings Id: $Id: CRing.thy,v 1.19 2005/07/01 12:01:13 berghofe Exp $ Author: Clemens Ballarin, started 9 December 1996 Copyright: Clemens Ballarin *) header {* Abelian Groups *} theory CRing imports FiniteProduct uses ("ringsimp.ML") begin record 'a ring = "'a monoid" + zero :: 'a ("\<zero>\<index>") add :: "['a, 'a] => 'a" (infixl "⊕\<index>" 65) text {* Derived operations. *} constdefs (structure R) a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80) "a_inv R == m_inv (| carrier = carrier R, mult = add R, one = zero R |)" minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65) "[| x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus> y == x ⊕ (\<ominus> y)" locale abelian_monoid = struct G + assumes a_comm_monoid: "comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)" text {* The following definition is redundant but simple to use. *} 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: includes struct R 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: prems) lemma abelian_groupI: includes struct R 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: prems) 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 comm_group.axioms a_comm_group) lemmas monoid_record_simps = partial_object.simps monoid.simps lemma (in abelian_monoid) a_closed [intro, simp]: "[| x ∈ carrier G; y ∈ carrier G |] ==> x ⊕ y ∈ carrier G" by (rule monoid.m_closed [OF a_monoid, simplified monoid_record_simps]) lemma (in abelian_monoid) zero_closed [intro, simp]: "\<zero> ∈ carrier G" by (rule monoid.one_closed [OF a_monoid, simplified monoid_record_simps]) lemma (in abelian_group) a_inv_closed [intro, simp]: "x ∈ carrier G ==> \<ominus> x ∈ carrier G" by (simp add: a_inv_def group.inv_closed [OF a_group, simplified monoid_record_simps]) lemma (in abelian_group) minus_closed [intro, simp]: "[| x ∈ carrier G; y ∈ carrier G |] ==> x \<ominus> y ∈ carrier G" by (simp add: minus_def) lemma (in abelian_group) a_l_cancel [simp]: "[| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (x ⊕ y = x ⊕ z) = (y = z)" by (rule group.l_cancel [OF a_group, simplified monoid_record_simps]) lemma (in abelian_group) a_r_cancel [simp]: "[| x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (y ⊕ x = z ⊕ x) = (y = z)" by (rule group.r_cancel [OF a_group, simplified monoid_record_simps]) lemma (in abelian_monoid) a_assoc: "[|x ∈ carrier G; y ∈ carrier G; z ∈ carrier G|] ==> (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)" by (rule monoid.m_assoc [OF a_monoid, simplified monoid_record_simps]) lemma (in abelian_monoid) l_zero [simp]: "x ∈ carrier G ==> \<zero> ⊕ x = x" by (rule monoid.l_one [OF a_monoid, simplified monoid_record_simps]) lemma (in abelian_group) l_neg: "x ∈ carrier G ==> \<ominus> x ⊕ x = \<zero>" by (simp add: a_inv_def group.l_inv [OF a_group, simplified monoid_record_simps]) lemma (in abelian_monoid) a_comm: "[|x ∈ carrier G; y ∈ carrier G|] ==> x ⊕ y = y ⊕ x" by (rule comm_monoid.m_comm [OF a_comm_monoid, simplified monoid_record_simps]) lemma (in abelian_monoid) a_lcomm: "[|x ∈ carrier G; y ∈ carrier G; z ∈ carrier G|] ==> x ⊕ (y ⊕ z) = y ⊕ (x ⊕ z)" by (rule comm_monoid.m_lcomm [OF a_comm_monoid, simplified monoid_record_simps]) lemma (in abelian_monoid) r_zero [simp]: "x ∈ carrier G ==> x ⊕ \<zero> = x" using monoid.r_one [OF a_monoid] by simp lemma (in abelian_group) r_neg: "x ∈ carrier G ==> x ⊕ (\<ominus> x) = \<zero>" using group.r_inv [OF a_group] by (simp add: a_inv_def) lemma (in abelian_group) minus_zero [simp]: "\<ominus> \<zero> = \<zero>" by (simp add: a_inv_def group.inv_one [OF a_group, simplified monoid_record_simps]) lemma (in abelian_group) minus_minus [simp]: "x ∈ carrier G ==> \<ominus> (\<ominus> x) = x" using group.inv_inv [OF a_group, simplified monoid_record_simps] by (simp add: a_inv_def) lemma (in abelian_group) a_inv_inj: "inj_on (a_inv G) (carrier G)" using group.inv_inj [OF a_group, simplified monoid_record_simps] by (simp add: a_inv_def) lemma (in abelian_group) minus_add: "[| x ∈ carrier G; y ∈ carrier G |] ==> \<ominus> (x ⊕ y) = \<ominus> x ⊕ \<ominus> y" using comm_group.inv_mult [OF a_comm_group] by (simp add: a_inv_def) lemmas (in abelian_monoid) a_ac = a_assoc a_comm a_lcomm subsection {* Sums over Finite Sets *} text {* This definition makes it easy to lift lemmas from @{term finprod}. *} constdefs finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" "finsum G f A == finprod (| carrier = carrier G, mult = add G, one = zero G |) f A" 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" == "finsum \<struct>\<index> (%i. b) A" -- {* Beware of argument permutation! *} (* lemmas (in abelian_monoid) finsum_empty [simp] = comm_monoid.finprod_empty [OF a_comm_monoid, simplified] is dangeous, because attributes (like simplified) are applied upon opening the locale, simplified refers to the simpset at that time!!! lemmas (in abelian_monoid) finsum_empty [simp] = abelian_monoid.finprod_empty [OF a_abelian_monoid, folded finsum_def, simplified monoid_record_simps] makes the locale slow, because proofs are repeated for every "lemma (in abelian_monoid)" command. When lemma is used time in UnivPoly.thy from beginning to UP_cring goes down from 110 secs to 60 secs. *) lemma (in abelian_monoid) finsum_empty [simp]: "finsum G f {} = \<zero>" by (rule comm_monoid.finprod_empty [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_insert [simp]: "[| finite F; a ∉ F; f ∈ F -> carrier G; f a ∈ carrier G |] ==> finsum G f (insert a F) = f a ⊕ finsum G f F" by (rule comm_monoid.finprod_insert [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_zero [simp]: "finite A ==> (\<Oplus>i∈A. \<zero>) = \<zero>" by (rule comm_monoid.finprod_one [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_closed [simp]: fixes A assumes fin: "finite A" and f: "f ∈ A -> carrier G" shows "finsum G f A ∈ carrier G" by (rule comm_monoid.finprod_closed [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_Un_Int: "[| finite A; finite B; g ∈ A -> carrier G; g ∈ B -> carrier G |] ==> finsum G g (A Un B) ⊕ finsum G g (A Int B) = finsum G g A ⊕ finsum G g B" by (rule comm_monoid.finprod_Un_Int [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_Un_disjoint: "[| finite A; finite B; A Int B = {}; g ∈ A -> carrier G; g ∈ B -> carrier G |] ==> finsum G g (A Un B) = finsum G g A ⊕ finsum G g B" by (rule comm_monoid.finprod_Un_disjoint [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_addf: "[| finite A; f ∈ A -> carrier G; g ∈ A -> carrier G |] ==> finsum G (%x. f x ⊕ g x) A = (finsum G f A ⊕ finsum G g A)" by (rule comm_monoid.finprod_multf [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_cong': "[| A = B; g : B -> carrier G; !!i. i : B ==> f i = g i |] ==> finsum G f A = finsum G g B" by (rule comm_monoid.finprod_cong' [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) auto lemma (in abelian_monoid) finsum_0 [simp]: "f : {0::nat} -> carrier G ==> finsum G f {..0} = f 0" by (rule comm_monoid.finprod_0 [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_Suc [simp]: "f : {..Suc n} -> carrier G ==> finsum G f {..Suc n} = (f (Suc n) ⊕ finsum G f {..n})" by (rule comm_monoid.finprod_Suc [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_Suc2: "f : {..Suc n} -> carrier G ==> finsum G f {..Suc n} = (finsum G (%i. f (Suc i)) {..n} ⊕ f 0)" by (rule comm_monoid.finprod_Suc2 [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_add [simp]: "[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==> finsum G (%i. f i ⊕ g i) {..n::nat} = finsum G f {..n} ⊕ finsum G g {..n}" by (rule comm_monoid.finprod_mult [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) lemma (in abelian_monoid) finsum_cong: "[| A = B; f : B -> carrier G; !!i. i : B =simp=> f i = g i |] ==> finsum G f A = finsum G g B" by (rule comm_monoid.finprod_cong [OF a_comm_monoid, folded finsum_def, simplified monoid_record_simps]) (auto simp add: simp_implies_def) 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. *} section {* The Algebraic Hierarchy of Rings *} subsection {* Basic Definitions *} locale ring = abelian_group R + monoid R + 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 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 {* Basic Facts of Rings *} lemma ringI: includes struct R 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 prems) lemma (in ring) is_abelian_group: "abelian_group R" by (auto intro!: abelian_groupI a_assoc a_comm l_neg) lemma (in ring) is_monoid: "monoid R" by (auto intro!: monoidI m_assoc) lemma cringI: includes struct R 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 (rule cring.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 qed (auto intro: cring.intro abelian_group.axioms comm_monoid.axioms ring_axioms.intro prems) lemma (in cring) is_comm_monoid: "comm_monoid R" by (auto intro!: comm_monoidI m_assoc m_comm) subsection {* 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 text {* The following proofs are from Jacobson, Basic Algebra I, pp.~88--89 *} lemma (in ring) 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) qed lemma (in ring) 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) qed lemma (in ring) 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 l_null) 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 (in ring) 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 r_null) 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 lemma (in ring) minus_eq: "[| x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus> y = x ⊕ \<ominus> y" by (simp only: minus_def) lemmas (in ring) ring_simprules = 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) cring_simprules = 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 use "ringsimp.ML" method_setup algebra = {* Method.ctxt_args cring_normalise *} {* computes distributive normal form in locale context cring *} lemma (in cring) nat_pow_zero: "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>" by (induct n) simp_all text {* Two examples for use of method algebra *} lemma includes ring R + cring S shows "[| a ∈ carrier R; b ∈ carrier R; c ∈ carrier S; d ∈ carrier S |] ==> a ⊕ \<ominus> (a ⊕ \<ominus> b) = b & c ⊗S d = d ⊗S c" by algebra lemma includes cring shows "[| a ∈ carrier R; b ∈ carrier R |] ==> a \<ominus> (a \<ominus> b) = b" by algebra subsection {* Sums over Finite Sets *} lemma (in cring) 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: Finites) 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 cring) 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: Finites) case empty then show ?case by simp next case (insert x F) then show ?case by (simp add: Pi_def r_distr) qed subsection {* Facts of Integral Domains *} lemma (in "domain") zero_not_one [simp]: "\<zero> ~= \<one>" by (rule not_sym) simp lemma (in "domain") 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 (in "domain") 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 (in "domain") 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 subsection {* Morphisms *} constdefs (structure R S) ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set" "ring_hom R S == {h. h ∈ carrier R -> carrier S & (ALL x y. x ∈ carrier R & y ∈ carrier R --> h (x ⊗ y) = h x ⊗S h y & h (x ⊕ y) = h x ⊕S h y) & h \<one> = \<one>S}" lemma ring_hom_memI: includes struct R + struct S 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 prems 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: includes struct R + struct S 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: includes struct R + struct S 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: includes struct R + struct S shows "h ∈ ring_hom R S ==> h \<one> = \<one>S" by (simp add: ring_hom_def) locale ring_hom_cring = cring R + cring S + var 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]: "[| finite A; f ∈ A -> carrier R |] ==> h (finsum R f A) = finsum S (h o f) A" proof (induct set: Finites) case empty then show ?case by simp next case insert then show ?case by (simp add: Pi_def) qed lemma (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: Finites) case empty then show ?case by simp next case insert then show ?case by (simp add: Pi_def) qed declare ring_hom_cring.hom_finprod [simp] lemma id_ring_hom [simp]: "id ∈ ring_hom R R" by (auto intro!: ring_hom_memI) end
lemma abelian_monoidI:
[| !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R y ∈ carrier R; \<zero>R ∈ carrier R; !!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊕R y ⊕R z = x ⊕R (y ⊕R z); !!x. x ∈ carrier R ==> \<zero>R ⊕R x = x; !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R y = y ⊕R x |] ==> abelian_monoid R
lemma abelian_groupI:
[| !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R y ∈ carrier R; \<zero>R ∈ carrier R; !!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊕R y ⊕R z = x ⊕R (y ⊕R z); !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R y = y ⊕R x; !!x. x ∈ carrier R ==> \<zero>R ⊕R x = x; !!x. x ∈ carrier R ==> ∃y∈carrier R. y ⊕R x = \<zero>R |] ==> abelian_group R
lemma a_monoid:
abelian_monoid G ==> monoid (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |)
lemma a_group:
abelian_group G ==> group (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |)
lemmas monoid_record_simps:
carrier (| carrier = carrier, ... = more |) = carrier
partial_object.more (| carrier = carrier, ... = more |) = more
(| carrier = carrier, ... = more |)(| carrier := carrier' |) = (| carrier = carrier', ... = more |)
(| carrier = carrier, ... = more |)(| partial_object.more := more' |) = (| carrier = carrier, ... = more' |)
op ⊗(| carrier = carrier, mult = mult, one = one, ... = more |) = mult
\<one>(| carrier = carrier, mult = mult, one = one, ... = more |) = one
monoid.more (| carrier = carrier, mult = mult, one = one, ... = more |) = more
(| carrier = carrier, mult = mult, one = one, ... = more |)(| mult := mult' |) = (| carrier = carrier, mult = mult', one = one, ... = more |)
(| carrier = carrier, mult = mult, one = one, ... = more |)(| one := one' |) = (| carrier = carrier, mult = mult, one = one', ... = more |)
(| carrier = carrier, mult = mult, one = one, ... = more |) (| monoid.more := more' |) = (| carrier = carrier, mult = mult, one = one, ... = more' |)
lemmas monoid_record_simps:
carrier (| carrier = carrier, ... = more |) = carrier
partial_object.more (| carrier = carrier, ... = more |) = more
(| carrier = carrier, ... = more |)(| carrier := carrier' |) = (| carrier = carrier', ... = more |)
(| carrier = carrier, ... = more |)(| partial_object.more := more' |) = (| carrier = carrier, ... = more' |)
op ⊗(| carrier = carrier, mult = mult, one = one, ... = more |) = mult
\<one>(| carrier = carrier, mult = mult, one = one, ... = more |) = one
monoid.more (| carrier = carrier, mult = mult, one = one, ... = more |) = more
(| carrier = carrier, mult = mult, one = one, ... = more |)(| mult := mult' |) = (| carrier = carrier, mult = mult', one = one, ... = more |)
(| carrier = carrier, mult = mult, one = one, ... = more |)(| one := one' |) = (| carrier = carrier, mult = mult, one = one', ... = more |)
(| carrier = carrier, mult = mult, one = one, ... = more |) (| monoid.more := more' |) = (| carrier = carrier, mult = mult, one = one, ... = more' |)
lemma a_closed:
[| abelian_monoid G; x ∈ carrier G; y ∈ carrier G |] ==> x ⊕G y ∈ carrier G
lemma zero_closed:
abelian_monoid G ==> \<zero>G ∈ carrier G
lemma a_inv_closed:
[| abelian_group G; x ∈ carrier G |] ==> \<ominus>G x ∈ carrier G
lemma minus_closed:
[| abelian_group G; x ∈ carrier G; y ∈ carrier G |] ==> x \<ominus>G y ∈ carrier G
lemma a_l_cancel:
[| abelian_group G; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (x ⊕G y = x ⊕G z) = (y = z)
lemma a_r_cancel:
[| abelian_group G; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (y ⊕G x = z ⊕G x) = (y = z)
lemma a_assoc:
[| abelian_monoid G; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> x ⊕G y ⊕G z = x ⊕G (y ⊕G z)
lemma l_zero:
[| abelian_monoid G; x ∈ carrier G |] ==> \<zero>G ⊕G x = x
lemma l_neg:
[| abelian_group G; x ∈ carrier G |] ==> \<ominus>G x ⊕G x = \<zero>G
lemma a_comm:
[| abelian_monoid G; x ∈ carrier G; y ∈ carrier G |] ==> x ⊕G y = y ⊕G x
lemma a_lcomm:
[| abelian_monoid G; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> x ⊕G (y ⊕G z) = y ⊕G (x ⊕G z)
lemma r_zero:
[| abelian_monoid G; x ∈ carrier G |] ==> x ⊕G \<zero>G = x
lemma r_neg:
[| abelian_group G; x ∈ carrier G |] ==> x ⊕G \<ominus>G x = \<zero>G
lemma minus_zero:
abelian_group G ==> \<ominus>G \<zero>G = \<zero>G
lemma minus_minus:
[| abelian_group G; x ∈ carrier G |] ==> \<ominus>G (\<ominus>G x) = x
lemma a_inv_inj:
abelian_group G ==> inj_on (a_inv G) (carrier G)
lemma minus_add:
[| abelian_group G; x ∈ carrier G; y ∈ carrier G |] ==> \<ominus>G (x ⊕G y) = \<ominus>G x ⊕G \<ominus>G y
lemmas a_ac:
[| abelian_monoid G; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> x ⊕G y ⊕G z = x ⊕G (y ⊕G z)
[| abelian_monoid G; x ∈ carrier G; y ∈ carrier G |] ==> x ⊕G y = y ⊕G x
[| abelian_monoid G; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> x ⊕G (y ⊕G z) = y ⊕G (x ⊕G z)
lemma finsum_empty:
abelian_monoid G ==> finsum G f {} = \<zero>G
lemma finsum_insert:
[| abelian_monoid G; finite F; a ∉ F; f ∈ F -> carrier G; f a ∈ carrier G |] ==> finsum G f (insert a F) = f a ⊕G finsum G f F
lemma finsum_zero:
[| abelian_monoid G; finite A |] ==> (\<Oplus>Gi∈A. \<zero>G) = \<zero>G
lemma finsum_closed:
[| abelian_monoid G; finite A; f ∈ A -> carrier G |] ==> finsum G f A ∈ carrier G
lemma finsum_Un_Int:
[| abelian_monoid G; finite A; finite B; g ∈ A -> carrier G; g ∈ B -> carrier G |] ==> finsum G g (A ∪ B) ⊕G finsum G g (A ∩ B) = finsum G g A ⊕G finsum G g B
lemma finsum_Un_disjoint:
[| abelian_monoid G; finite A; finite B; A ∩ B = {}; g ∈ A -> carrier G; g ∈ B -> carrier G |] ==> finsum G g (A ∪ B) = finsum G g A ⊕G finsum G g B
lemma finsum_addf:
[| abelian_monoid G; finite A; f ∈ A -> carrier G; g ∈ A -> carrier G |] ==> (\<Oplus>Gx∈A. f x ⊕G g x) = finsum G f A ⊕G finsum G g A
lemma finsum_cong':
[| abelian_monoid G; A = B; g ∈ B -> carrier G; !!i. i ∈ B ==> f i = g i |] ==> finsum G f A = finsum G g B
lemma finsum_0:
[| abelian_monoid G; f ∈ {0} -> carrier G |] ==> finsum G f {..0} = f 0
lemma finsum_Suc:
[| abelian_monoid G; f ∈ {..Suc n} -> carrier G |] ==> finsum G f {..Suc n} = f (Suc n) ⊕G finsum G f {..n}
lemma finsum_Suc2:
[| abelian_monoid G; f ∈ {..Suc n} -> carrier G |] ==> finsum G f {..Suc n} = (\<Oplus>Gi∈{..n}. f (Suc i)) ⊕G f 0
lemma finsum_add:
[| abelian_monoid G; f ∈ {..n} -> carrier G; g ∈ {..n} -> carrier G |] ==> (\<Oplus>Gi∈{..n}. f i ⊕G g i) = finsum G f {..n} ⊕G finsum G g {..n}
lemma finsum_cong:
[| abelian_monoid G; A = B; f ∈ B -> carrier G; !!i. i ∈ B =simp=> f i = g i |] ==> finsum G f A = finsum G g B
lemma ringI:
[| abelian_group R; monoid R; !!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> (x ⊕R y) ⊗R z = x ⊗R z ⊕R y ⊗R z; !!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> z ⊗R (x ⊕R y) = z ⊗R x ⊕R z ⊗R y |] ==> ring R
lemma is_abelian_group:
ring R ==> abelian_group R
lemma is_monoid:
ring R ==> monoid R
lemma cringI:
[| abelian_group R; comm_monoid R; !!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> (x ⊕R y) ⊗R z = x ⊗R z ⊕R y ⊗R z |] ==> cring R
lemma is_comm_monoid:
cring R ==> comm_monoid R
lemma r_neg2:
[| abelian_group G; x ∈ carrier G; y ∈ carrier G |] ==> x ⊕G (\<ominus>G x ⊕G y) = y
lemma r_neg1:
[| abelian_group G; x ∈ carrier G; y ∈ carrier G |] ==> \<ominus>G x ⊕G (x ⊕G y) = y
lemma l_null:
[| ring R; x ∈ carrier R |] ==> \<zero>R ⊗R x = \<zero>R
lemma r_null:
[| ring R; x ∈ carrier R |] ==> x ⊗R \<zero>R = \<zero>R
lemma l_minus:
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> \<ominus>R x ⊗R y = \<ominus>R (x ⊗R y)
lemma r_minus:
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊗R \<ominus>R y = \<ominus>R (x ⊗R y)
lemma minus_eq:
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus>R y = x ⊕R \<ominus>R y
lemmas ring_simprules:
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R y ∈ carrier R
ring R ==> \<zero>R ∈ carrier R
[| ring R; x ∈ carrier R |] ==> \<ominus>R x ∈ carrier R
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus>R y ∈ carrier R
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊗R y ∈ carrier R
ring R ==> \<one>R ∈ carrier R
[| ring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊕R y ⊕R z = x ⊕R (y ⊕R z)
[| ring R; x ∈ carrier R |] ==> \<zero>R ⊕R x = x
[| ring R; x ∈ carrier R |] ==> \<ominus>R x ⊕R x = \<zero>R
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R y = y ⊕R x
[| ring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊗R y ⊗R z = x ⊗R (y ⊗R z)
[| ring R; x ∈ carrier R |] ==> \<one>R ⊗R x = x
[| ring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> (x ⊕R y) ⊗R z = x ⊗R z ⊕R y ⊗R z
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus>R y = x ⊕R \<ominus>R y
[| ring R; x ∈ carrier R |] ==> x ⊕R \<zero>R = x
[| ring R; x ∈ carrier R |] ==> x ⊕R \<ominus>R x = \<zero>R
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R (\<ominus>R x ⊕R y) = y
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> \<ominus>R x ⊕R (x ⊕R y) = y
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> \<ominus>R (x ⊕R y) = \<ominus>R x ⊕R \<ominus>R y
[| ring R; x ∈ carrier R |] ==> \<ominus>R (\<ominus>R x) = x
ring R ==> \<ominus>R \<zero>R = \<zero>R
[| ring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊕R (y ⊕R z) = y ⊕R (x ⊕R z)
[| ring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> z ⊗R (x ⊕R y) = z ⊗R x ⊕R z ⊗R y
[| ring R; x ∈ carrier R |] ==> \<zero>R ⊗R x = \<zero>R
[| ring R; x ∈ carrier R |] ==> x ⊗R \<zero>R = \<zero>R
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> \<ominus>R x ⊗R y = \<ominus>R (x ⊗R y)
[| ring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊗R \<ominus>R y = \<ominus>R (x ⊗R y)
lemmas cring_simprules:
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R y ∈ carrier R
cring R ==> \<zero>R ∈ carrier R
[| cring R; x ∈ carrier R |] ==> \<ominus>R x ∈ carrier R
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus>R y ∈ carrier R
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊗R y ∈ carrier R
cring R ==> \<one>R ∈ carrier R
[| cring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊕R y ⊕R z = x ⊕R (y ⊕R z)
[| cring R; x ∈ carrier R |] ==> \<zero>R ⊕R x = x
[| cring R; x ∈ carrier R |] ==> \<ominus>R x ⊕R x = \<zero>R
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R y = y ⊕R x
[| cring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊗R y ⊗R z = x ⊗R (y ⊗R z)
[| cring R; x ∈ carrier R |] ==> \<one>R ⊗R x = x
[| cring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> (x ⊕R y) ⊗R z = x ⊗R z ⊕R y ⊗R z
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊗R y = y ⊗R x
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus>R y = x ⊕R \<ominus>R y
[| cring R; x ∈ carrier R |] ==> x ⊕R \<zero>R = x
[| cring R; x ∈ carrier R |] ==> x ⊕R \<ominus>R x = \<zero>R
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊕R (\<ominus>R x ⊕R y) = y
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> \<ominus>R x ⊕R (x ⊕R y) = y
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> \<ominus>R (x ⊕R y) = \<ominus>R x ⊕R \<ominus>R y
[| cring R; x ∈ carrier R |] ==> \<ominus>R (\<ominus>R x) = x
cring R ==> \<ominus>R \<zero>R = \<zero>R
[| cring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊕R (y ⊕R z) = y ⊕R (x ⊕R z)
[| cring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> x ⊗R (y ⊗R z) = y ⊗R (x ⊗R z)
[| cring R; x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==> z ⊗R (x ⊕R y) = z ⊗R x ⊕R z ⊗R y
[| cring R; x ∈ carrier R |] ==> \<zero>R ⊗R x = \<zero>R
[| cring R; x ∈ carrier R |] ==> x ⊗R \<zero>R = \<zero>R
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> \<ominus>R x ⊗R y = \<ominus>R (x ⊗R y)
[| cring R; x ∈ carrier R; y ∈ carrier R |] ==> x ⊗R \<ominus>R y = \<ominus>R (x ⊗R y)
lemma nat_pow_zero:
[| cring R; n ≠ 0 |] ==> \<zero>R (^)R n = \<zero>R
lemma
[| ring R; cring S; a ∈ carrier R; b ∈ carrier R; c ∈ carrier S; d ∈ carrier S |] ==> a ⊕R \<ominus>R (a ⊕R \<ominus>R b) = b ∧ c ⊗S d = d ⊗S c
lemma
[| cring R; a ∈ carrier R; b ∈ carrier R |] ==> a \<ominus>R (a \<ominus>R b) = b
lemma finsum_ldistr:
[| cring R; finite A; a ∈ carrier R; f ∈ A -> carrier R |] ==> finsum R f A ⊗R a = (\<Oplus>Ri∈A. f i ⊗R a)
lemma finsum_rdistr:
[| cring R; finite A; a ∈ carrier R; f ∈ A -> carrier R |] ==> a ⊗R finsum R f A = (\<Oplus>Ri∈A. a ⊗R f i)
lemma zero_not_one:
domain R ==> \<zero>R ≠ \<one>R
lemma integral_iff:
[| domain R; a ∈ carrier R; b ∈ carrier R |] ==> (a ⊗R b = \<zero>R) = (a = \<zero>R ∨ b = \<zero>R)
lemma m_lcancel:
[| domain R; a ≠ \<zero>R; a ∈ carrier R; b ∈ carrier R; c ∈ carrier R |] ==> (a ⊗R b = a ⊗R c) = (b = c)
lemma conc:
[| domain R; a ≠ \<zero>R; a ∈ carrier R; b ∈ carrier R; c ∈ carrier R |] ==> (b ⊗R a = c ⊗R a) = (b = c)
lemma ring_hom_memI:
[| !!x. x ∈ carrier R ==> h x ∈ carrier S; !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> h (x ⊗R y) = h x ⊗S h y; !!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> h (x ⊕R y) = h x ⊕S h y; h \<one>R = \<one>S |] ==> h ∈ ring_hom R S
lemma ring_hom_closed:
[| h ∈ ring_hom R S; x ∈ carrier R |] ==> h x ∈ carrier S
lemma ring_hom_mult:
[| h ∈ ring_hom R S; x ∈ carrier R; y ∈ carrier R |] ==> h (x ⊗R y) = h x ⊗S h y
lemma ring_hom_add:
[| h ∈ ring_hom R S; x ∈ carrier R; y ∈ carrier R |] ==> h (x ⊕R y) = h x ⊕S h y
lemma ring_hom_one:
h ∈ ring_hom R S ==> h \<one>R = \<one>S
lemma hom_zero:
ring_hom_cring R S h ==> h \<zero>R = \<zero>S
lemma hom_a_inv:
[| ring_hom_cring R S h; x ∈ carrier R |] ==> h (\<ominus>R x) = \<ominus>S h x
lemma hom_finsum:
[| ring_hom_cring R S h; finite A; f ∈ A -> carrier R |] ==> h (finsum R f A) = finsum S (h o f) A
lemma hom_finprod:
[| ring_hom_cring R S h; finite A; f ∈ A -> carrier R |] ==> h (finprod R f A) = finprod S (h o f) A
lemma id_ring_hom:
id ∈ ring_hom R R