wenzelm@29269
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(* Title: HOL/Algebra/abstract/Ring2.thy
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wenzelm@29269
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Author: Clemens Ballarin
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wenzelm@29269
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wenzelm@29269
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The algebraic hierarchy of rings as axiomatic classes.
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ballarin@20318
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*)
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haftmann@27542
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header {* The algebraic hierarchy of rings as type classes *}
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theory Ring2
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haftmann@27542
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imports Main
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wenzelm@21423
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begin
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ballarin@20318
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subsection {* Ring axioms *}
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haftmann@31001
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class ring = zero + one + plus + minus + uminus + times + inverse + power + dvd +
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haftmann@27542
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assumes a_assoc: "(a + b) + c = a + (b + c)"
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and l_zero: "0 + a = a"
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and l_neg: "(-a) + a = 0"
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and a_comm: "a + b = b + a"
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assumes m_assoc: "(a * b) * c = a * (b * c)"
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and l_one: "1 * a = a"
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assumes l_distr: "(a + b) * c = a * c + b * c"
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assumes m_comm: "a * b = b * a"
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assumes minus_def: "a - b = a + (-b)"
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and inverse_def: "inverse a = (if a dvd 1 then THE x. a*x = 1 else 0)"
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and divide_def: "a / b = a * inverse b"
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begin
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definition assoc :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "assoc" 50) where
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assoc_def: "a assoc b \<longleftrightarrow> a dvd b & b dvd a"
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definition irred :: "'a \<Rightarrow> bool" where
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irred_def: "irred a \<longleftrightarrow> a ~= 0 & ~ a dvd 1
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& (ALL d. d dvd a --> d dvd 1 | a dvd d)"
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definition prime :: "'a \<Rightarrow> bool" where
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prime_def: "prime p \<longleftrightarrow> p ~= 0 & ~ p dvd 1
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& (ALL a b. p dvd (a*b) --> p dvd a | p dvd b)"
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end
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subsection {* Integral domains *}
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class "domain" = ring +
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assumes one_not_zero: "1 ~= 0"
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and integral: "a * b = 0 ==> a = 0 | b = 0"
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subsection {* Factorial domains *}
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class factorial = "domain" +
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(*
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Proper definition using divisor chain condition currently not supported.
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factorial_divisor: "wf {(a, b). a dvd b & ~ (b dvd a)}"
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*)
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(*assumes factorial_divisor: "True"*)
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assumes factorial_prime: "irred a ==> prime a"
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subsection {* Euclidean domains *}
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(*
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axclass
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euclidean < "domain"
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euclidean_ax: "b ~= 0 ==> Ex (% (q, r, e_size::('a::ringS)=>nat).
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a = b * q + r & e_size r < e_size b)"
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Nothing has been proved about Euclidean domains, yet.
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Design question:
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Fix quo, rem and e_size as constants that are axiomatised with
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euclidean_ax?
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- advantage: more pragmatic and easier to use
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- disadvantage: for every type, one definition of quo and rem will
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be fixed, users may want to use differing ones;
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also, it seems not possible to prove that fields are euclidean
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domains, because that would require generic (type-independent)
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definitions of quo and rem.
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*)
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subsection {* Fields *}
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class field = ring +
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assumes field_one_not_zero: "1 ~= 0"
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(* Avoid a common superclass as the first thing we will
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prove about fields is that they are domains. *)
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and field_ax: "a ~= 0 ==> a dvd 1"
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section {* Basic facts *}
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subsection {* Normaliser for rings *}
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(* derived rewrite rules *)
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lemma a_lcomm: "(a::'a::ring)+(b+c) = b+(a+c)"
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apply (rule a_comm [THEN trans])
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apply (rule a_assoc [THEN trans])
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apply (rule a_comm [THEN arg_cong])
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done
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lemma r_zero: "(a::'a::ring) + 0 = a"
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apply (rule a_comm [THEN trans])
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apply (rule l_zero)
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done
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lemma r_neg: "(a::'a::ring) + (-a) = 0"
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apply (rule a_comm [THEN trans])
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apply (rule l_neg)
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done
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lemma r_neg2: "(a::'a::ring) + (-a + b) = b"
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apply (rule a_assoc [symmetric, THEN trans])
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apply (simp add: r_neg l_zero)
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done
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lemma r_neg1: "-(a::'a::ring) + (a + b) = b"
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apply (rule a_assoc [symmetric, THEN trans])
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apply (simp add: l_neg l_zero)
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done
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(* auxiliary *)
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lemma a_lcancel: "!! a::'a::ring. a + b = a + c ==> b = c"
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apply (rule box_equals)
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prefer 2
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apply (rule l_zero)
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prefer 2
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apply (rule l_zero)
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apply (rule_tac a1 = a in l_neg [THEN subst])
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apply (simp add: a_assoc)
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done
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lemma minus_add: "-((a::'a::ring) + b) = (-a) + (-b)"
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apply (rule_tac a = "a + b" in a_lcancel)
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wenzelm@21423
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apply (simp add: r_neg l_neg l_zero a_assoc a_comm a_lcomm)
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done
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lemma minus_minus: "-(-(a::'a::ring)) = a"
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apply (rule a_lcancel)
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wenzelm@21423
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apply (rule r_neg [THEN trans])
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wenzelm@21423
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apply (rule l_neg [symmetric])
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done
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wenzelm@21423
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lemma minus0: "- 0 = (0::'a::ring)"
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apply (rule a_lcancel)
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apply (rule r_neg [THEN trans])
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apply (rule l_zero [symmetric])
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done
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(* derived rules for multiplication *)
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lemma m_lcomm: "(a::'a::ring)*(b*c) = b*(a*c)"
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apply (rule m_comm [THEN trans])
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apply (rule m_assoc [THEN trans])
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apply (rule m_comm [THEN arg_cong])
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done
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lemma r_one: "(a::'a::ring) * 1 = a"
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wenzelm@21423
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apply (rule m_comm [THEN trans])
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apply (rule l_one)
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done
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lemma r_distr: "(a::'a::ring) * (b + c) = a * b + a * c"
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wenzelm@21423
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apply (rule m_comm [THEN trans])
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wenzelm@21423
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apply (rule l_distr [THEN trans])
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apply (simp add: m_comm)
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done
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(* the following proof is from Jacobson, Basic Algebra I, pp. 88-89 *)
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lemma l_null: "0 * (a::'a::ring) = 0"
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apply (rule a_lcancel)
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apply (rule l_distr [symmetric, THEN trans])
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apply (simp add: r_zero)
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done
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lemma r_null: "(a::'a::ring) * 0 = 0"
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apply (rule m_comm [THEN trans])
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apply (rule l_null)
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done
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wenzelm@21423
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wenzelm@21423
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lemma l_minus: "(-(a::'a::ring)) * b = - (a * b)"
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wenzelm@21423
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apply (rule a_lcancel)
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wenzelm@21423
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apply (rule r_neg [symmetric, THEN [2] trans])
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wenzelm@21423
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apply (rule l_distr [symmetric, THEN trans])
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wenzelm@21423
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apply (simp add: l_null r_neg)
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wenzelm@21423
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done
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lemma r_minus: "(a::'a::ring) * (-b) = - (a * b)"
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apply (rule a_lcancel)
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wenzelm@21423
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apply (rule r_neg [symmetric, THEN [2] trans])
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wenzelm@21423
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apply (rule r_distr [symmetric, THEN trans])
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wenzelm@21423
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apply (simp add: r_null r_neg)
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done
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wenzelm@21423
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wenzelm@21423
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(*** Term order for commutative rings ***)
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ML {*
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wenzelm@21423
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fun ring_ord (Const (a, _)) =
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wenzelm@21423
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find_index (fn a' => a = a')
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haftmann@22997
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[@{const_name HOL.zero}, @{const_name HOL.plus}, @{const_name HOL.uminus},
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@{const_name HOL.minus}, @{const_name HOL.one}, @{const_name HOL.times}]
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| ring_ord _ = ~1;
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wenzelm@21423
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wenzelm@29269
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fun termless_ring (a, b) = (TermOrd.term_lpo ring_ord (a, b) = LESS);
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wenzelm@21423
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wenzelm@21423
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val ring_ss = HOL_basic_ss settermless termless_ring addsimps
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wenzelm@21423
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[thm "a_assoc", thm "l_zero", thm "l_neg", thm "a_comm", thm "m_assoc",
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thm "l_one", thm "l_distr", thm "m_comm", thm "minus_def",
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wenzelm@21423
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thm "r_zero", thm "r_neg", thm "r_neg2", thm "r_neg1", thm "minus_add",
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wenzelm@21423
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thm "minus_minus", thm "minus0", thm "a_lcomm", thm "m_lcomm", (*thm "r_one",*)
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wenzelm@21423
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thm "r_distr", thm "l_null", thm "r_null", thm "l_minus", thm "r_minus"];
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*} (* Note: r_one is not necessary in ring_ss *)
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method_setup ring =
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wenzelm@30549
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{* Scan.succeed (K (SIMPLE_METHOD' (full_simp_tac ring_ss))) *}
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ballarin@20318
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{* computes distributive normal form in rings *}
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subsection {* Rings and the summation operator *}
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(* Basic facts --- move to HOL!!! *)
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(* needed because natsum_cong (below) disables atMost_0 *)
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lemma natsum_0 [simp]: "setsum f {..(0::nat)} = (f 0::'a::comm_monoid_add)"
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by simp
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(*
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ballarin@20318
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lemma natsum_Suc [simp]:
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"setsum f {..Suc n} = (f (Suc n) + setsum f {..n}::'a::comm_monoid_add)"
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by (simp add: atMost_Suc)
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*)
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lemma natsum_Suc2:
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"setsum f {..Suc n} = (f 0::'a::comm_monoid_add) + (setsum (%i. f (Suc i)) {..n})"
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proof (induct n)
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ballarin@20318
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case 0 show ?case by simp
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next
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haftmann@22384
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case Suc thus ?case by (simp add: add_assoc)
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qed
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ballarin@20318
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ballarin@20318
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lemma natsum_cong [cong]:
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ballarin@20318
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"!!k. [| j = k; !!i::nat. i <= k ==> f i = (g i::'a::comm_monoid_add) |] ==>
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ballarin@20318
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setsum f {..j} = setsum g {..k}"
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ballarin@20318
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by (induct j) auto
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ballarin@20318
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ballarin@20318
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lemma natsum_zero [simp]: "setsum (%i. 0) {..n::nat} = (0::'a::comm_monoid_add)"
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ballarin@20318
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by (induct n) simp_all
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ballarin@20318
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ballarin@20318
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lemma natsum_add [simp]:
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ballarin@20318
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"!!f::nat=>'a::comm_monoid_add.
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ballarin@20318
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setsum (%i. f i + g i) {..n::nat} = setsum f {..n} + setsum g {..n}"
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ballarin@20318
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by (induct n) (simp_all add: add_ac)
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ballarin@20318
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ballarin@20318
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(* Facts specific to rings *)
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haftmann@27542
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subclass (in ring) comm_monoid_add
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proof
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ballarin@20318
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fix x y z
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haftmann@27542
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show "x + y = y + x" by (rule a_comm)
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haftmann@27542
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show "(x + y) + z = x + (y + z)" by (rule a_assoc)
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haftmann@27542
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show "0 + x = x" by (rule l_zero)
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ballarin@20318
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qed
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ballarin@20318
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ballarin@20318
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ML {*
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local
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ballarin@20318
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val lhss =
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ballarin@20318
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["t + u::'a::ring",
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ballarin@20318
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"t - u::'a::ring",
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ballarin@20318
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"t * u::'a::ring",
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ballarin@20318
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"- t::'a::ring"];
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ballarin@20318
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fun proc ss t =
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ballarin@20318
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let val rew = Goal.prove (Simplifier.the_context ss) [] []
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ballarin@20318
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(HOLogic.mk_Trueprop
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ballarin@20318
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(HOLogic.mk_eq (t, Var (("x", Term.maxidx_of_term t + 1), fastype_of t))))
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ballarin@20318
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(fn _ => simp_tac (Simplifier.inherit_context ss ring_ss) 1)
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ballarin@20318
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|> mk_meta_eq;
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ballarin@20318
|
283 |
val (t', u) = Logic.dest_equals (Thm.prop_of rew);
|
ballarin@20318
|
284 |
in if t' aconv u
|
ballarin@20318
|
285 |
then NONE
|
ballarin@20318
|
286 |
else SOME rew
|
ballarin@20318
|
287 |
end;
|
ballarin@20318
|
288 |
in
|
ballarin@20318
|
289 |
val ring_simproc = Simplifier.simproc (the_context ()) "ring" lhss (K proc);
|
ballarin@20318
|
290 |
end;
|
ballarin@20318
|
291 |
*}
|
ballarin@20318
|
292 |
|
wenzelm@26480
|
293 |
ML {* Addsimprocs [ring_simproc] *}
|
ballarin@20318
|
294 |
|
ballarin@20318
|
295 |
lemma natsum_ldistr:
|
ballarin@20318
|
296 |
"!!a::'a::ring. setsum f {..n::nat} * a = setsum (%i. f i * a) {..n}"
|
ballarin@20318
|
297 |
by (induct n) simp_all
|
ballarin@20318
|
298 |
|
ballarin@20318
|
299 |
lemma natsum_rdistr:
|
ballarin@20318
|
300 |
"!!a::'a::ring. a * setsum f {..n::nat} = setsum (%i. a * f i) {..n}"
|
ballarin@20318
|
301 |
by (induct n) simp_all
|
ballarin@20318
|
302 |
|
ballarin@20318
|
303 |
subsection {* Integral Domains *}
|
ballarin@20318
|
304 |
|
ballarin@20318
|
305 |
declare one_not_zero [simp]
|
ballarin@20318
|
306 |
|
ballarin@20318
|
307 |
lemma zero_not_one [simp]:
|
ballarin@20318
|
308 |
"0 ~= (1::'a::domain)"
|
ballarin@20318
|
309 |
by (rule not_sym) simp
|
ballarin@20318
|
310 |
|
ballarin@20318
|
311 |
lemma integral_iff: (* not by default a simp rule! *)
|
ballarin@20318
|
312 |
"(a * b = (0::'a::domain)) = (a = 0 | b = 0)"
|
ballarin@20318
|
313 |
proof
|
ballarin@20318
|
314 |
assume "a * b = 0" then show "a = 0 | b = 0" by (simp add: integral)
|
ballarin@20318
|
315 |
next
|
ballarin@20318
|
316 |
assume "a = 0 | b = 0" then show "a * b = 0" by auto
|
ballarin@20318
|
317 |
qed
|
ballarin@20318
|
318 |
|
ballarin@20318
|
319 |
(*
|
ballarin@20318
|
320 |
lemma "(a::'a::ring) - (a - b) = b" apply simp
|
ballarin@20318
|
321 |
simproc seems to fail on this example (fixed with new term order)
|
ballarin@20318
|
322 |
*)
|
ballarin@20318
|
323 |
(*
|
ballarin@20318
|
324 |
lemma bug: "(b::'a::ring) - (b - a) = a" by simp
|
ballarin@20318
|
325 |
simproc for rings cannot prove "(a::'a::ring) - (a - b) = b"
|
ballarin@20318
|
326 |
*)
|
ballarin@20318
|
327 |
lemma m_lcancel:
|
ballarin@20318
|
328 |
assumes prem: "(a::'a::domain) ~= 0" shows conc: "(a * b = a * c) = (b = c)"
|
ballarin@20318
|
329 |
proof
|
ballarin@20318
|
330 |
assume eq: "a * b = a * c"
|
ballarin@20318
|
331 |
then have "a * (b - c) = 0" by simp
|
ballarin@20318
|
332 |
then have "a = 0 | (b - c) = 0" by (simp only: integral_iff)
|
ballarin@20318
|
333 |
with prem have "b - c = 0" by auto
|
ballarin@20318
|
334 |
then have "b = b - (b - c)" by simp
|
ballarin@20318
|
335 |
also have "b - (b - c) = c" by simp
|
ballarin@20318
|
336 |
finally show "b = c" .
|
ballarin@20318
|
337 |
next
|
ballarin@20318
|
338 |
assume "b = c" then show "a * b = a * c" by simp
|
ballarin@20318
|
339 |
qed
|
ballarin@20318
|
340 |
|
ballarin@20318
|
341 |
lemma m_rcancel:
|
ballarin@20318
|
342 |
"(a::'a::domain) ~= 0 ==> (b * a = c * a) = (b = c)"
|
ballarin@20318
|
343 |
by (simp add: m_lcancel)
|
ballarin@20318
|
344 |
|
haftmann@27542
|
345 |
declare power_Suc [simp]
|
haftmann@21416
|
346 |
|
haftmann@21416
|
347 |
lemma power_one [simp]:
|
haftmann@21416
|
348 |
"1 ^ n = (1::'a::ring)" by (induct n) simp_all
|
haftmann@21416
|
349 |
|
haftmann@21416
|
350 |
lemma power_zero [simp]:
|
haftmann@21416
|
351 |
"n \<noteq> 0 \<Longrightarrow> 0 ^ n = (0::'a::ring)" by (induct n) simp_all
|
haftmann@21416
|
352 |
|
haftmann@21416
|
353 |
lemma power_mult [simp]:
|
haftmann@21416
|
354 |
"(a::'a::ring) ^ m * a ^ n = a ^ (m + n)"
|
haftmann@21416
|
355 |
by (induct m) simp_all
|
haftmann@21416
|
356 |
|
haftmann@21416
|
357 |
|
haftmann@21416
|
358 |
section "Divisibility"
|
haftmann@21416
|
359 |
|
haftmann@21416
|
360 |
lemma dvd_zero_right [simp]:
|
haftmann@21416
|
361 |
"(a::'a::ring) dvd 0"
|
haftmann@21416
|
362 |
proof
|
haftmann@21416
|
363 |
show "0 = a * 0" by simp
|
haftmann@21416
|
364 |
qed
|
haftmann@21416
|
365 |
|
haftmann@21416
|
366 |
lemma dvd_zero_left:
|
haftmann@21416
|
367 |
"0 dvd (a::'a::ring) \<Longrightarrow> a = 0" unfolding dvd_def by simp
|
haftmann@21416
|
368 |
|
haftmann@21416
|
369 |
lemma dvd_refl_ring [simp]:
|
haftmann@21416
|
370 |
"(a::'a::ring) dvd a"
|
haftmann@21416
|
371 |
proof
|
haftmann@21416
|
372 |
show "a = a * 1" by simp
|
haftmann@21416
|
373 |
qed
|
haftmann@21416
|
374 |
|
haftmann@21416
|
375 |
lemma dvd_trans_ring:
|
haftmann@21416
|
376 |
fixes a b c :: "'a::ring"
|
haftmann@21416
|
377 |
assumes a_dvd_b: "a dvd b"
|
haftmann@21416
|
378 |
and b_dvd_c: "b dvd c"
|
haftmann@21416
|
379 |
shows "a dvd c"
|
haftmann@21416
|
380 |
proof -
|
haftmann@21416
|
381 |
from a_dvd_b obtain l where "b = a * l" using dvd_def by blast
|
haftmann@21416
|
382 |
moreover from b_dvd_c obtain j where "c = b * j" using dvd_def by blast
|
haftmann@21416
|
383 |
ultimately have "c = a * (l * j)" by simp
|
haftmann@21416
|
384 |
then have "\<exists>k. c = a * k" ..
|
haftmann@21416
|
385 |
then show ?thesis using dvd_def by blast
|
haftmann@21416
|
386 |
qed
|
haftmann@21416
|
387 |
|
wenzelm@21423
|
388 |
|
wenzelm@21423
|
389 |
lemma unit_mult:
|
wenzelm@21423
|
390 |
"!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1"
|
wenzelm@21423
|
391 |
apply (unfold dvd_def)
|
wenzelm@21423
|
392 |
apply clarify
|
wenzelm@21423
|
393 |
apply (rule_tac x = "k * ka" in exI)
|
wenzelm@21423
|
394 |
apply simp
|
wenzelm@21423
|
395 |
done
|
wenzelm@21423
|
396 |
|
wenzelm@21423
|
397 |
lemma unit_power: "!!a::'a::ring. a dvd 1 ==> a^n dvd 1"
|
wenzelm@21423
|
398 |
apply (induct_tac n)
|
wenzelm@21423
|
399 |
apply simp
|
wenzelm@21423
|
400 |
apply (simp add: unit_mult)
|
wenzelm@21423
|
401 |
done
|
wenzelm@21423
|
402 |
|
wenzelm@21423
|
403 |
lemma dvd_add_right [simp]:
|
wenzelm@21423
|
404 |
"!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd b + c"
|
wenzelm@21423
|
405 |
apply (unfold dvd_def)
|
wenzelm@21423
|
406 |
apply clarify
|
wenzelm@21423
|
407 |
apply (rule_tac x = "k + ka" in exI)
|
wenzelm@21423
|
408 |
apply (simp add: r_distr)
|
wenzelm@21423
|
409 |
done
|
wenzelm@21423
|
410 |
|
wenzelm@21423
|
411 |
lemma dvd_uminus_right [simp]:
|
wenzelm@21423
|
412 |
"!! a::'a::ring. a dvd b ==> a dvd -b"
|
wenzelm@21423
|
413 |
apply (unfold dvd_def)
|
wenzelm@21423
|
414 |
apply clarify
|
wenzelm@21423
|
415 |
apply (rule_tac x = "-k" in exI)
|
wenzelm@21423
|
416 |
apply (simp add: r_minus)
|
wenzelm@21423
|
417 |
done
|
wenzelm@21423
|
418 |
|
wenzelm@21423
|
419 |
lemma dvd_l_mult_right [simp]:
|
wenzelm@21423
|
420 |
"!! a::'a::ring. a dvd b ==> a dvd c*b"
|
wenzelm@21423
|
421 |
apply (unfold dvd_def)
|
wenzelm@21423
|
422 |
apply clarify
|
wenzelm@21423
|
423 |
apply (rule_tac x = "c * k" in exI)
|
wenzelm@21423
|
424 |
apply simp
|
wenzelm@21423
|
425 |
done
|
wenzelm@21423
|
426 |
|
wenzelm@21423
|
427 |
lemma dvd_r_mult_right [simp]:
|
wenzelm@21423
|
428 |
"!! a::'a::ring. a dvd b ==> a dvd b*c"
|
wenzelm@21423
|
429 |
apply (unfold dvd_def)
|
wenzelm@21423
|
430 |
apply clarify
|
wenzelm@21423
|
431 |
apply (rule_tac x = "k * c" in exI)
|
wenzelm@21423
|
432 |
apply simp
|
wenzelm@21423
|
433 |
done
|
wenzelm@21423
|
434 |
|
wenzelm@21423
|
435 |
|
wenzelm@21423
|
436 |
(* Inverse of multiplication *)
|
wenzelm@21423
|
437 |
|
wenzelm@21423
|
438 |
section "inverse"
|
wenzelm@21423
|
439 |
|
wenzelm@21423
|
440 |
lemma inverse_unique: "!! a::'a::ring. [| a * x = 1; a * y = 1 |] ==> x = y"
|
wenzelm@21423
|
441 |
apply (rule_tac a = "(a*y) * x" and b = "y * (a*x)" in box_equals)
|
wenzelm@21423
|
442 |
apply (simp (no_asm))
|
wenzelm@21423
|
443 |
apply auto
|
wenzelm@21423
|
444 |
done
|
wenzelm@21423
|
445 |
|
wenzelm@21423
|
446 |
lemma r_inverse_ring: "!! a::'a::ring. a dvd 1 ==> a * inverse a = 1"
|
wenzelm@21423
|
447 |
apply (unfold inverse_def dvd_def)
|
wenzelm@26342
|
448 |
apply (tactic {* asm_full_simp_tac (@{simpset} delsimprocs [ring_simproc]) 1 *})
|
wenzelm@21423
|
449 |
apply clarify
|
wenzelm@21423
|
450 |
apply (rule theI)
|
wenzelm@21423
|
451 |
apply assumption
|
wenzelm@21423
|
452 |
apply (rule inverse_unique)
|
wenzelm@21423
|
453 |
apply assumption
|
wenzelm@21423
|
454 |
apply assumption
|
wenzelm@21423
|
455 |
done
|
wenzelm@21423
|
456 |
|
wenzelm@21423
|
457 |
lemma l_inverse_ring: "!! a::'a::ring. a dvd 1 ==> inverse a * a = 1"
|
wenzelm@21423
|
458 |
by (simp add: r_inverse_ring)
|
wenzelm@21423
|
459 |
|
wenzelm@21423
|
460 |
|
wenzelm@21423
|
461 |
(* Fields *)
|
wenzelm@21423
|
462 |
|
wenzelm@21423
|
463 |
section "Fields"
|
wenzelm@21423
|
464 |
|
wenzelm@21423
|
465 |
lemma field_unit [simp]: "!! a::'a::field. (a dvd 1) = (a ~= 0)"
|
wenzelm@21423
|
466 |
by (auto dest: field_ax dvd_zero_left simp add: field_one_not_zero)
|
wenzelm@21423
|
467 |
|
wenzelm@21423
|
468 |
lemma r_inverse [simp]: "!! a::'a::field. a ~= 0 ==> a * inverse a = 1"
|
wenzelm@21423
|
469 |
by (simp add: r_inverse_ring)
|
wenzelm@21423
|
470 |
|
wenzelm@21423
|
471 |
lemma l_inverse [simp]: "!! a::'a::field. a ~= 0 ==> inverse a * a= 1"
|
wenzelm@21423
|
472 |
by (simp add: l_inverse_ring)
|
wenzelm@21423
|
473 |
|
wenzelm@21423
|
474 |
|
wenzelm@21423
|
475 |
(* fields are integral domains *)
|
wenzelm@21423
|
476 |
|
wenzelm@21423
|
477 |
lemma field_integral: "!! a::'a::field. a * b = 0 ==> a = 0 | b = 0"
|
wenzelm@23894
|
478 |
apply (tactic "step_tac @{claset} 1")
|
wenzelm@21423
|
479 |
apply (rule_tac a = " (a*b) * inverse b" in box_equals)
|
wenzelm@21423
|
480 |
apply (rule_tac [3] refl)
|
wenzelm@21423
|
481 |
prefer 2
|
wenzelm@21423
|
482 |
apply (simp (no_asm))
|
wenzelm@21423
|
483 |
apply auto
|
wenzelm@21423
|
484 |
done
|
wenzelm@21423
|
485 |
|
wenzelm@21423
|
486 |
|
wenzelm@21423
|
487 |
(* fields are factorial domains *)
|
wenzelm@21423
|
488 |
|
wenzelm@21423
|
489 |
lemma field_fact_prime: "!! a::'a::field. irred a ==> prime a"
|
wenzelm@21423
|
490 |
unfolding prime_def irred_def by (blast intro: field_ax)
|
haftmann@21416
|
491 |
|
ballarin@20318
|
492 |
end
|