haftmann@35050: (*  Title:   HOL/Groups.thy
wenzelm@29269:     Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad
obua@14738: *)
obua@14738: 
haftmann@35050: header {* Groups, also combined with orderings *}
obua@14738: 
haftmann@35050: theory Groups
haftmann@35092: imports Orderings
haftmann@35267: uses ("~~/src/Provers/Arith/abel_cancel.ML")
nipkow@15131: begin
obua@14738: 
haftmann@35301: subsection {* Fact collections *}
haftmann@35301: 
haftmann@35301: ML {*
haftmann@36343: structure Ac_Simps = Named_Thms(
haftmann@36343:   val name = "ac_simps"
haftmann@36343:   val description = "associativity and commutativity simplification rules"
haftmann@36343: )
haftmann@36343: *}
haftmann@36343: 
haftmann@36343: setup Ac_Simps.setup
haftmann@36343: 
haftmann@36348: text{* The rewrites accumulated in @{text algebra_simps} deal with the
haftmann@36348: classical algebraic structures of groups, rings and family. They simplify
haftmann@36348: terms by multiplying everything out (in case of a ring) and bringing sums and
haftmann@36348: products into a canonical form (by ordered rewriting). As a result it decides
haftmann@36348: group and ring equalities but also helps with inequalities.
haftmann@36348: 
haftmann@36348: Of course it also works for fields, but it knows nothing about multiplicative
haftmann@36348: inverses or division. This is catered for by @{text field_simps}. *}
haftmann@36348: 
haftmann@36343: ML {*
haftmann@35301: structure Algebra_Simps = Named_Thms(
haftmann@35301:   val name = "algebra_simps"
haftmann@35301:   val description = "algebra simplification rules"
haftmann@35301: )
haftmann@35301: *}
haftmann@35301: 
haftmann@35301: setup Algebra_Simps.setup
haftmann@35301: 
haftmann@36348: text{* Lemmas @{text field_simps} multiply with denominators in (in)equations
haftmann@36348: if they can be proved to be non-zero (for equations) or positive/negative
haftmann@36348: (for inequations). Can be too aggressive and is therefore separate from the
haftmann@36348: more benign @{text algebra_simps}. *}
haftmann@35301: 
haftmann@36348: ML {*
haftmann@36348: structure Field_Simps = Named_Thms(
haftmann@36348:   val name = "field_simps"
haftmann@36348:   val description = "algebra simplification rules for fields"
haftmann@36348: )
haftmann@36348: *}
haftmann@36348: 
haftmann@36348: setup Field_Simps.setup
haftmann@35301: 
haftmann@35301: 
haftmann@35301: subsection {* Abstract structures *}
haftmann@35301: 
haftmann@35301: text {*
haftmann@35301:   These locales provide basic structures for interpretation into
haftmann@35301:   bigger structures;  extensions require careful thinking, otherwise
haftmann@35301:   undesired effects may occur due to interpretation.
haftmann@35301: *}
haftmann@35301: 
haftmann@35301: locale semigroup =
haftmann@35301:   fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
haftmann@35301:   assumes assoc [ac_simps]: "a * b * c = a * (b * c)"
haftmann@35301: 
haftmann@35301: locale abel_semigroup = semigroup +
haftmann@35301:   assumes commute [ac_simps]: "a * b = b * a"
haftmann@35301: begin
haftmann@35301: 
haftmann@35301: lemma left_commute [ac_simps]:
haftmann@35301:   "b * (a * c) = a * (b * c)"
haftmann@35301: proof -
haftmann@35301:   have "(b * a) * c = (a * b) * c"
haftmann@35301:     by (simp only: commute)
haftmann@35301:   then show ?thesis
haftmann@35301:     by (simp only: assoc)
haftmann@35301: qed
haftmann@35301: 
haftmann@35301: end
haftmann@35301: 
haftmann@35720: locale monoid = semigroup +
haftmann@35723:   fixes z :: 'a ("1")
haftmann@35723:   assumes left_neutral [simp]: "1 * a = a"
haftmann@35723:   assumes right_neutral [simp]: "a * 1 = a"
haftmann@35720: 
haftmann@35720: locale comm_monoid = abel_semigroup +
haftmann@35723:   fixes z :: 'a ("1")
haftmann@35723:   assumes comm_neutral: "a * 1 = a"
haftmann@35720: 
haftmann@35720: sublocale comm_monoid < monoid proof
haftmann@35720: qed (simp_all add: commute comm_neutral)
haftmann@35720: 
haftmann@35301: 
haftmann@35267: subsection {* Generic operations *}
haftmann@35267: 
haftmann@35267: class zero = 
haftmann@35267:   fixes zero :: 'a  ("0")
haftmann@35267: 
haftmann@35267: class one =
haftmann@35267:   fixes one  :: 'a  ("1")
haftmann@35267: 
wenzelm@36176: hide_const (open) zero one
haftmann@35267: 
haftmann@35267: syntax
haftmann@35267:   "_index1"  :: index    ("\<^sub>1")
haftmann@35267: translations
haftmann@35267:   (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
haftmann@35267: 
haftmann@35267: lemma Let_0 [simp]: "Let 0 f = f 0"
haftmann@35267:   unfolding Let_def ..
haftmann@35267: 
haftmann@35267: lemma Let_1 [simp]: "Let 1 f = f 1"
haftmann@35267:   unfolding Let_def ..
haftmann@35267: 
haftmann@35267: setup {*
haftmann@35267:   Reorient_Proc.add
haftmann@35267:     (fn Const(@{const_name Groups.zero}, _) => true
haftmann@35267:       | Const(@{const_name Groups.one}, _) => true
haftmann@35267:       | _ => false)
haftmann@35267: *}
haftmann@35267: 
haftmann@35267: simproc_setup reorient_zero ("0 = x") = Reorient_Proc.proc
haftmann@35267: simproc_setup reorient_one ("1 = x") = Reorient_Proc.proc
haftmann@35267: 
haftmann@35267: typed_print_translation {*
haftmann@35267: let
haftmann@35267:   fun tr' c = (c, fn show_sorts => fn T => fn ts =>
haftmann@35267:     if (not o null) ts orelse T = dummyT
haftmann@35267:       orelse not (! show_types) andalso can Term.dest_Type T
haftmann@35267:     then raise Match
haftmann@35267:     else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
haftmann@35267: in map tr' [@{const_syntax Groups.one}, @{const_syntax Groups.zero}] end;
haftmann@35267: *} -- {* show types that are presumably too general *}
haftmann@35267: 
haftmann@35267: class plus =
haftmann@35267:   fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "+" 65)
haftmann@35267: 
haftmann@35267: class minus =
haftmann@35267:   fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "-" 65)
haftmann@35267: 
haftmann@35267: class uminus =
haftmann@35267:   fixes uminus :: "'a \<Rightarrow> 'a"  ("- _" [81] 80)
haftmann@35267: 
haftmann@35267: class times =
haftmann@35267:   fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "*" 70)
haftmann@35267: 
haftmann@35267: use "~~/src/Provers/Arith/abel_cancel.ML"
haftmann@35267: 
haftmann@35092: 
nipkow@23085: subsection {* Semigroups and Monoids *}
obua@14738: 
haftmann@22390: class semigroup_add = plus +
haftmann@36348:   assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)"
haftmann@34973: 
haftmann@35723: sublocale semigroup_add < add!: semigroup plus proof
haftmann@34973: qed (fact add_assoc)
haftmann@22390: 
haftmann@22390: class ab_semigroup_add = semigroup_add +
haftmann@36348:   assumes add_commute [algebra_simps, field_simps]: "a + b = b + a"
haftmann@34973: 
haftmann@35723: sublocale ab_semigroup_add < add!: abel_semigroup plus proof
haftmann@34973: qed (fact add_commute)
haftmann@34973: 
haftmann@34973: context ab_semigroup_add
haftmann@25062: begin
obua@14738: 
haftmann@36348: lemmas add_left_commute [algebra_simps, field_simps] = add.left_commute
haftmann@25062: 
haftmann@25062: theorems add_ac = add_assoc add_commute add_left_commute
haftmann@25062: 
haftmann@25062: end
obua@14738: 
obua@14738: theorems add_ac = add_assoc add_commute add_left_commute
obua@14738: 
haftmann@22390: class semigroup_mult = times +
haftmann@36348:   assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)"
haftmann@34973: 
haftmann@35723: sublocale semigroup_mult < mult!: semigroup times proof
haftmann@34973: qed (fact mult_assoc)
obua@14738: 
haftmann@22390: class ab_semigroup_mult = semigroup_mult +
haftmann@36348:   assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a"
haftmann@34973: 
haftmann@35723: sublocale ab_semigroup_mult < mult!: abel_semigroup times proof
haftmann@34973: qed (fact mult_commute)
haftmann@34973: 
haftmann@34973: context ab_semigroup_mult
haftmann@23181: begin
obua@14738: 
haftmann@36348: lemmas mult_left_commute [algebra_simps, field_simps] = mult.left_commute
haftmann@25062: 
haftmann@25062: theorems mult_ac = mult_assoc mult_commute mult_left_commute
haftmann@23181: 
haftmann@23181: end
obua@14738: 
obua@14738: theorems mult_ac = mult_assoc mult_commute mult_left_commute
obua@14738: 
nipkow@23085: class monoid_add = zero + semigroup_add +
haftmann@35720:   assumes add_0_left: "0 + a = a"
haftmann@35720:     and add_0_right: "a + 0 = a"
haftmann@35720: 
haftmann@35723: sublocale monoid_add < add!: monoid plus 0 proof
haftmann@35720: qed (fact add_0_left add_0_right)+
nipkow@23085: 
haftmann@26071: lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"
nipkow@29667: by (rule eq_commute)
haftmann@26071: 
haftmann@22390: class comm_monoid_add = zero + ab_semigroup_add +
haftmann@25062:   assumes add_0: "0 + a = a"
nipkow@23085: 
haftmann@35723: sublocale comm_monoid_add < add!: comm_monoid plus 0 proof
haftmann@35720: qed (insert add_0, simp add: ac_simps)
haftmann@25062: 
haftmann@35720: subclass (in comm_monoid_add) monoid_add proof
haftmann@35723: qed (fact add.left_neutral add.right_neutral)+
obua@14738: 
haftmann@22390: class monoid_mult = one + semigroup_mult +
haftmann@35720:   assumes mult_1_left: "1 * a  = a"
haftmann@35720:     and mult_1_right: "a * 1 = a"
haftmann@35720: 
haftmann@35723: sublocale monoid_mult < mult!: monoid times 1 proof
haftmann@35720: qed (fact mult_1_left mult_1_right)+
obua@14738: 
haftmann@26071: lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"
nipkow@29667: by (rule eq_commute)
haftmann@26071: 
haftmann@22390: class comm_monoid_mult = one + ab_semigroup_mult +
haftmann@25062:   assumes mult_1: "1 * a = a"
obua@14738: 
haftmann@35723: sublocale comm_monoid_mult < mult!: comm_monoid times 1 proof
haftmann@35720: qed (insert mult_1, simp add: ac_simps)
haftmann@25062: 
haftmann@35720: subclass (in comm_monoid_mult) monoid_mult proof
haftmann@35723: qed (fact mult.left_neutral mult.right_neutral)+
obua@14738: 
haftmann@22390: class cancel_semigroup_add = semigroup_add +
haftmann@25062:   assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
haftmann@25062:   assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
huffman@27474: begin
huffman@27474: 
huffman@27474: lemma add_left_cancel [simp]:
huffman@27474:   "a + b = a + c \<longleftrightarrow> b = c"
nipkow@29667: by (blast dest: add_left_imp_eq)
huffman@27474: 
huffman@27474: lemma add_right_cancel [simp]:
huffman@27474:   "b + a = c + a \<longleftrightarrow> b = c"
nipkow@29667: by (blast dest: add_right_imp_eq)
huffman@27474: 
huffman@27474: end
obua@14738: 
haftmann@22390: class cancel_ab_semigroup_add = ab_semigroup_add +
haftmann@25062:   assumes add_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
haftmann@25267: begin
obua@14738: 
haftmann@25267: subclass cancel_semigroup_add
haftmann@28823: proof
haftmann@22390:   fix a b c :: 'a
haftmann@22390:   assume "a + b = a + c" 
haftmann@22390:   then show "b = c" by (rule add_imp_eq)
haftmann@22390: next
obua@14738:   fix a b c :: 'a
obua@14738:   assume "b + a = c + a"
haftmann@22390:   then have "a + b = a + c" by (simp only: add_commute)
haftmann@22390:   then show "b = c" by (rule add_imp_eq)
obua@14738: qed
obua@14738: 
haftmann@25267: end
haftmann@25267: 
huffman@29904: class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add
huffman@29904: 
huffman@29904: 
nipkow@23085: subsection {* Groups *}
nipkow@23085: 
haftmann@25762: class group_add = minus + uminus + monoid_add +
haftmann@25062:   assumes left_minus [simp]: "- a + a = 0"
haftmann@25062:   assumes diff_minus: "a - b = a + (- b)"
haftmann@25062: begin
nipkow@23085: 
huffman@34147: lemma minus_unique:
huffman@34147:   assumes "a + b = 0" shows "- a = b"
huffman@34147: proof -
huffman@34147:   have "- a = - a + (a + b)" using assms by simp
huffman@34147:   also have "\<dots> = b" by (simp add: add_assoc [symmetric])
huffman@34147:   finally show ?thesis .
huffman@34147: qed
huffman@34147: 
huffman@34147: lemmas equals_zero_I = minus_unique (* legacy name *)
obua@14738: 
haftmann@25062: lemma minus_zero [simp]: "- 0 = 0"
obua@14738: proof -
huffman@34147:   have "0 + 0 = 0" by (rule add_0_right)
huffman@34147:   thus "- 0 = 0" by (rule minus_unique)
obua@14738: qed
obua@14738: 
haftmann@25062: lemma minus_minus [simp]: "- (- a) = a"
nipkow@23085: proof -
huffman@34147:   have "- a + a = 0" by (rule left_minus)
huffman@34147:   thus "- (- a) = a" by (rule minus_unique)
nipkow@23085: qed
obua@14738: 
haftmann@25062: lemma right_minus [simp]: "a + - a = 0"
obua@14738: proof -
haftmann@25062:   have "a + - a = - (- a) + - a" by simp
haftmann@25062:   also have "\<dots> = 0" by (rule left_minus)
obua@14738:   finally show ?thesis .
obua@14738: qed
obua@14738: 
huffman@34147: lemma minus_add_cancel: "- a + (a + b) = b"
huffman@34147: by (simp add: add_assoc [symmetric])
huffman@34147: 
huffman@34147: lemma add_minus_cancel: "a + (- a + b) = b"
huffman@34147: by (simp add: add_assoc [symmetric])
huffman@34147: 
huffman@34147: lemma minus_add: "- (a + b) = - b + - a"
huffman@34147: proof -
huffman@34147:   have "(a + b) + (- b + - a) = 0"
huffman@34147:     by (simp add: add_assoc add_minus_cancel)
huffman@34147:   thus "- (a + b) = - b + - a"
huffman@34147:     by (rule minus_unique)
huffman@34147: qed
huffman@34147: 
haftmann@25062: lemma right_minus_eq: "a - b = 0 \<longleftrightarrow> a = b"
obua@14738: proof
nipkow@23085:   assume "a - b = 0"
nipkow@23085:   have "a = (a - b) + b" by (simp add:diff_minus add_assoc)
nipkow@23085:   also have "\<dots> = b" using `a - b = 0` by simp
nipkow@23085:   finally show "a = b" .
obua@14738: next
nipkow@23085:   assume "a = b" thus "a - b = 0" by (simp add: diff_minus)
obua@14738: qed
obua@14738: 
haftmann@25062: lemma diff_self [simp]: "a - a = 0"
nipkow@29667: by (simp add: diff_minus)
obua@14738: 
haftmann@25062: lemma diff_0 [simp]: "0 - a = - a"
nipkow@29667: by (simp add: diff_minus)
obua@14738: 
haftmann@25062: lemma diff_0_right [simp]: "a - 0 = a" 
nipkow@29667: by (simp add: diff_minus)
obua@14738: 
haftmann@25062: lemma diff_minus_eq_add [simp]: "a - - b = a + b"
nipkow@29667: by (simp add: diff_minus)
obua@14738: 
haftmann@25062: lemma neg_equal_iff_equal [simp]:
haftmann@25062:   "- a = - b \<longleftrightarrow> a = b" 
obua@14738: proof 
obua@14738:   assume "- a = - b"
nipkow@29667:   hence "- (- a) = - (- b)" by simp
haftmann@25062:   thus "a = b" by simp
obua@14738: next
haftmann@25062:   assume "a = b"
haftmann@25062:   thus "- a = - b" by simp
obua@14738: qed
obua@14738: 
haftmann@25062: lemma neg_equal_0_iff_equal [simp]:
haftmann@25062:   "- a = 0 \<longleftrightarrow> a = 0"
nipkow@29667: by (subst neg_equal_iff_equal [symmetric], simp)
obua@14738: 
haftmann@25062: lemma neg_0_equal_iff_equal [simp]:
haftmann@25062:   "0 = - a \<longleftrightarrow> 0 = a"
nipkow@29667: by (subst neg_equal_iff_equal [symmetric], simp)
obua@14738: 
obua@14738: text{*The next two equations can make the simplifier loop!*}
obua@14738: 
haftmann@25062: lemma equation_minus_iff:
haftmann@25062:   "a = - b \<longleftrightarrow> b = - a"
obua@14738: proof -
haftmann@25062:   have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
haftmann@25062:   thus ?thesis by (simp add: eq_commute)
haftmann@25062: qed
haftmann@25062: 
haftmann@25062: lemma minus_equation_iff:
haftmann@25062:   "- a = b \<longleftrightarrow> - b = a"
haftmann@25062: proof -
haftmann@25062:   have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
obua@14738:   thus ?thesis by (simp add: eq_commute)
obua@14738: qed
obua@14738: 
huffman@28130: lemma diff_add_cancel: "a - b + b = a"
nipkow@29667: by (simp add: diff_minus add_assoc)
huffman@28130: 
huffman@28130: lemma add_diff_cancel: "a + b - b = a"
nipkow@29667: by (simp add: diff_minus add_assoc)
nipkow@29667: 
haftmann@36348: declare diff_minus[symmetric, algebra_simps, field_simps]
huffman@28130: 
huffman@29914: lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914: proof
huffman@29914:   assume "a = - b" then show "a + b = 0" by simp
huffman@29914: next
huffman@29914:   assume "a + b = 0"
huffman@29914:   moreover have "a + (b + - b) = (a + b) + - b"
huffman@29914:     by (simp only: add_assoc)
huffman@29914:   ultimately show "a = - b" by simp
huffman@29914: qed
huffman@29914: 
haftmann@25062: end
haftmann@25062: 
haftmann@25762: class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062:   assumes ab_left_minus: "- a + a = 0"
haftmann@25062:   assumes ab_diff_minus: "a - b = a + (- b)"
haftmann@25267: begin
haftmann@25062: 
haftmann@25267: subclass group_add
haftmann@28823:   proof qed (simp_all add: ab_left_minus ab_diff_minus)
haftmann@25062: 
huffman@29904: subclass cancel_comm_monoid_add
haftmann@28823: proof
haftmann@25062:   fix a b c :: 'a
haftmann@25062:   assume "a + b = a + c"
haftmann@25062:   then have "- a + a + b = - a + a + c"
haftmann@25062:     unfolding add_assoc by simp
haftmann@25062:   then show "b = c" by simp
haftmann@25062: qed
haftmann@25062: 
haftmann@36348: lemma uminus_add_conv_diff[algebra_simps, field_simps]:
haftmann@25062:   "- a + b = b - a"
nipkow@29667: by (simp add:diff_minus add_commute)
haftmann@25062: 
haftmann@25062: lemma minus_add_distrib [simp]:
haftmann@25062:   "- (a + b) = - a + - b"
huffman@34146: by (rule minus_unique) (simp add: add_ac)
haftmann@25062: 
haftmann@25062: lemma minus_diff_eq [simp]:
haftmann@25062:   "- (a - b) = b - a"
nipkow@29667: by (simp add: diff_minus add_commute)
haftmann@25077: 
haftmann@36348: lemma add_diff_eq[algebra_simps, field_simps]: "a + (b - c) = (a + b) - c"
nipkow@29667: by (simp add: diff_minus add_ac)
haftmann@25077: 
haftmann@36348: lemma diff_add_eq[algebra_simps, field_simps]: "(a - b) + c = (a + c) - b"
nipkow@29667: by (simp add: diff_minus add_ac)
haftmann@25077: 
haftmann@36348: lemma diff_eq_eq[algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b"
nipkow@29667: by (auto simp add: diff_minus add_assoc)
haftmann@25077: 
haftmann@36348: lemma eq_diff_eq[algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c"
nipkow@29667: by (auto simp add: diff_minus add_assoc)
haftmann@25077: 
haftmann@36348: lemma diff_diff_eq[algebra_simps, field_simps]: "(a - b) - c = a - (b + c)"
nipkow@29667: by (simp add: diff_minus add_ac)
haftmann@25077: 
haftmann@36348: lemma diff_diff_eq2[algebra_simps, field_simps]: "a - (b - c) = (a + c) - b"
nipkow@29667: by (simp add: diff_minus add_ac)
haftmann@25077: 
haftmann@25077: lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
nipkow@29667: by (simp add: algebra_simps)
haftmann@25077: 
huffman@35216: (* FIXME: duplicates right_minus_eq from class group_add *)
huffman@35216: (* but only this one is declared as a simp rule. *)
blanchet@35828: lemma diff_eq_0_iff_eq [simp, no_atp]: "a - b = 0 \<longleftrightarrow> a = b"
huffman@30629: by (simp add: algebra_simps)
huffman@30629: 
haftmann@25062: end
obua@14738: 
obua@14738: subsection {* (Partially) Ordered Groups *} 
obua@14738: 
haftmann@35301: text {*
haftmann@35301:   The theory of partially ordered groups is taken from the books:
haftmann@35301:   \begin{itemize}
haftmann@35301:   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301:   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301:   \end{itemize}
haftmann@35301:   Most of the used notions can also be looked up in 
haftmann@35301:   \begin{itemize}
haftmann@35301:   \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35301:   \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301:   \end{itemize}
haftmann@35301: *}
haftmann@35301: 
haftmann@35028: class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062:   assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062: begin
haftmann@24380: 
haftmann@25062: lemma add_right_mono:
haftmann@25062:   "a \<le> b \<Longrightarrow> a + c \<le> b + c"
nipkow@29667: by (simp add: add_commute [of _ c] add_left_mono)
obua@14738: 
obua@14738: text {* non-strict, in both arguments *}
obua@14738: lemma add_mono:
haftmann@25062:   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738:   apply (erule add_right_mono [THEN order_trans])
obua@14738:   apply (simp add: add_commute add_left_mono)
obua@14738:   done
obua@14738: 
haftmann@25062: end
haftmann@25062: 
haftmann@35028: class ordered_cancel_ab_semigroup_add =
haftmann@35028:   ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062: begin
haftmann@25062: 
obua@14738: lemma add_strict_left_mono:
haftmann@25062:   "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667: by (auto simp add: less_le add_left_mono)
obua@14738: 
obua@14738: lemma add_strict_right_mono:
haftmann@25062:   "a < b \<Longrightarrow> a + c < b + c"
nipkow@29667: by (simp add: add_commute [of _ c] add_strict_left_mono)
obua@14738: 
obua@14738: text{*Strict monotonicity in both arguments*}
haftmann@25062: lemma add_strict_mono:
haftmann@25062:   "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062: apply (erule add_strict_right_mono [THEN less_trans])
obua@14738: apply (erule add_strict_left_mono)
obua@14738: done
obua@14738: 
obua@14738: lemma add_less_le_mono:
haftmann@25062:   "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062: apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062: apply (erule add_left_mono)
obua@14738: done
obua@14738: 
obua@14738: lemma add_le_less_mono:
haftmann@25062:   "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062: apply (erule add_right_mono [THEN le_less_trans])
obua@14738: apply (erule add_strict_left_mono) 
obua@14738: done
obua@14738: 
haftmann@25062: end
haftmann@25062: 
haftmann@35028: class ordered_ab_semigroup_add_imp_le =
haftmann@35028:   ordered_cancel_ab_semigroup_add +
haftmann@25062:   assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062: begin
haftmann@25062: 
obua@14738: lemma add_less_imp_less_left:
nipkow@29667:   assumes less: "c + a < c + b" shows "a < b"
obua@14738: proof -
obua@14738:   from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738:   have "a <= b" 
obua@14738:     apply (insert le)
obua@14738:     apply (drule add_le_imp_le_left)
obua@14738:     by (insert le, drule add_le_imp_le_left, assumption)
obua@14738:   moreover have "a \<noteq> b"
obua@14738:   proof (rule ccontr)
obua@14738:     assume "~(a \<noteq> b)"
obua@14738:     then have "a = b" by simp
obua@14738:     then have "c + a = c + b" by simp
obua@14738:     with less show "False"by simp
obua@14738:   qed
obua@14738:   ultimately show "a < b" by (simp add: order_le_less)
obua@14738: qed
obua@14738: 
obua@14738: lemma add_less_imp_less_right:
haftmann@25062:   "a + c < b + c \<Longrightarrow> a < b"
obua@14738: apply (rule add_less_imp_less_left [of c])
obua@14738: apply (simp add: add_commute)  
obua@14738: done
obua@14738: 
obua@14738: lemma add_less_cancel_left [simp]:
haftmann@25062:   "c + a < c + b \<longleftrightarrow> a < b"
nipkow@29667: by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738: 
obua@14738: lemma add_less_cancel_right [simp]:
haftmann@25062:   "a + c < b + c \<longleftrightarrow> a < b"
nipkow@29667: by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738: 
obua@14738: lemma add_le_cancel_left [simp]:
haftmann@25062:   "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
nipkow@29667: by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738: 
obua@14738: lemma add_le_cancel_right [simp]:
haftmann@25062:   "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
nipkow@29667: by (simp add: add_commute [of a c] add_commute [of b c])
obua@14738: 
obua@14738: lemma add_le_imp_le_right:
haftmann@25062:   "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667: by simp
haftmann@25062: 
haftmann@25077: lemma max_add_distrib_left:
haftmann@25077:   "max x y + z = max (x + z) (y + z)"
haftmann@25077:   unfolding max_def by auto
haftmann@25077: 
haftmann@25077: lemma min_add_distrib_left:
haftmann@25077:   "min x y + z = min (x + z) (y + z)"
haftmann@25077:   unfolding min_def by auto
haftmann@25077: 
haftmann@25062: end
haftmann@25062: 
haftmann@25303: subsection {* Support for reasoning about signs *}
haftmann@25303: 
haftmann@35028: class ordered_comm_monoid_add =
haftmann@35028:   ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303: begin
haftmann@25303: 
haftmann@25303: lemma add_pos_nonneg:
nipkow@29667:   assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303: proof -
haftmann@25303:   have "0 + 0 < a + b" 
haftmann@25303:     using assms by (rule add_less_le_mono)
haftmann@25303:   then show ?thesis by simp
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: lemma add_pos_pos:
nipkow@29667:   assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667: by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303: 
haftmann@25303: lemma add_nonneg_pos:
nipkow@29667:   assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303: proof -
haftmann@25303:   have "0 + 0 < a + b" 
haftmann@25303:     using assms by (rule add_le_less_mono)
haftmann@25303:   then show ?thesis by simp
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: lemma add_nonneg_nonneg:
nipkow@29667:   assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303: proof -
haftmann@25303:   have "0 + 0 \<le> a + b" 
haftmann@25303:     using assms by (rule add_mono)
haftmann@25303:   then show ?thesis by simp
haftmann@25303: qed
haftmann@25303: 
huffman@30691: lemma add_neg_nonpos:
nipkow@29667:   assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303: proof -
haftmann@25303:   have "a + b < 0 + 0"
haftmann@25303:     using assms by (rule add_less_le_mono)
haftmann@25303:   then show ?thesis by simp
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: lemma add_neg_neg: 
nipkow@29667:   assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667: by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303: 
haftmann@25303: lemma add_nonpos_neg:
nipkow@29667:   assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303: proof -
haftmann@25303:   have "a + b < 0 + 0"
haftmann@25303:     using assms by (rule add_le_less_mono)
haftmann@25303:   then show ?thesis by simp
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: lemma add_nonpos_nonpos:
nipkow@29667:   assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303: proof -
haftmann@25303:   have "a + b \<le> 0 + 0"
haftmann@25303:     using assms by (rule add_mono)
haftmann@25303:   then show ?thesis by simp
haftmann@25303: qed
haftmann@25303: 
huffman@30691: lemmas add_sign_intros =
huffman@30691:   add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691:   add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691: 
huffman@29886: lemma add_nonneg_eq_0_iff:
huffman@29886:   assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886:   shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886: proof (intro iffI conjI)
huffman@29886:   have "x = x + 0" by simp
huffman@29886:   also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886:   also assume "x + y = 0"
huffman@29886:   also have "0 \<le> x" using x .
huffman@29886:   finally show "x = 0" .
huffman@29886: next
huffman@29886:   have "y = 0 + y" by simp
huffman@29886:   also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886:   also assume "x + y = 0"
huffman@29886:   also have "0 \<le> y" using y .
huffman@29886:   finally show "y = 0" .
huffman@29886: next
huffman@29886:   assume "x = 0 \<and> y = 0"
huffman@29886:   then show "x + y = 0" by simp
huffman@29886: qed
huffman@29886: 
haftmann@25303: end
haftmann@25303: 
haftmann@35028: class ordered_ab_group_add =
haftmann@35028:   ab_group_add + ordered_ab_semigroup_add
haftmann@25062: begin
haftmann@25062: 
haftmann@35028: subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062: 
haftmann@35028: subclass ordered_ab_semigroup_add_imp_le
haftmann@28823: proof
haftmann@25062:   fix a b c :: 'a
haftmann@25062:   assume "c + a \<le> c + b"
haftmann@25062:   hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@25062:   hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
haftmann@25062:   thus "a \<le> b" by simp
haftmann@25062: qed
haftmann@25062: 
haftmann@35028: subclass ordered_comm_monoid_add ..
haftmann@25303: 
haftmann@25077: lemma max_diff_distrib_left:
haftmann@25077:   shows "max x y - z = max (x - z) (y - z)"
nipkow@29667: by (simp add: diff_minus, rule max_add_distrib_left) 
haftmann@25077: 
haftmann@25077: lemma min_diff_distrib_left:
haftmann@25077:   shows "min x y - z = min (x - z) (y - z)"
nipkow@29667: by (simp add: diff_minus, rule min_add_distrib_left) 
haftmann@25077: 
haftmann@25077: lemma le_imp_neg_le:
nipkow@29667:   assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077: proof -
nipkow@29667:   have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
nipkow@29667:   hence "0 \<le> -a+b" by simp
nipkow@29667:   hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
nipkow@29667:   thus ?thesis by (simp add: add_assoc)
haftmann@25077: qed
haftmann@25077: 
haftmann@25077: lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077: proof 
haftmann@25077:   assume "- b \<le> - a"
nipkow@29667:   hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077:   thus "a\<le>b" by simp
haftmann@25077: next
haftmann@25077:   assume "a\<le>b"
haftmann@25077:   thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077: qed
haftmann@25077: 
haftmann@25077: lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667: by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077: 
haftmann@25077: lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667: by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077: 
haftmann@25077: lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667: by (force simp add: less_le) 
haftmann@25077: 
haftmann@25077: lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667: by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077: 
haftmann@25077: lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667: by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077: 
haftmann@25077: text{*The next several equations can make the simplifier loop!*}
haftmann@25077: 
haftmann@25077: lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077: proof -
haftmann@25077:   have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077:   thus ?thesis by simp
haftmann@25077: qed
haftmann@25077: 
haftmann@25077: lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077: proof -
haftmann@25077:   have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077:   thus ?thesis by simp
haftmann@25077: qed
haftmann@25077: 
haftmann@25077: lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077: proof -
haftmann@25077:   have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077:   have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077:     apply (auto simp only: le_less)
haftmann@25077:     apply (drule mm)
haftmann@25077:     apply (simp_all)
haftmann@25077:     apply (drule mm[simplified], assumption)
haftmann@25077:     done
haftmann@25077:   then show ?thesis by simp
haftmann@25077: qed
haftmann@25077: 
haftmann@25077: lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667: by (auto simp add: le_less minus_less_iff)
haftmann@25077: 
haftmann@25077: lemma less_iff_diff_less_0: "a < b \<longleftrightarrow> a - b < 0"
haftmann@25077: proof -
haftmann@25077:   have  "(a < b) = (a + (- b) < b + (-b))"  
haftmann@25077:     by (simp only: add_less_cancel_right)
haftmann@25077:   also have "... =  (a - b < 0)" by (simp add: diff_minus)
haftmann@25077:   finally show ?thesis .
haftmann@25077: qed
haftmann@25077: 
haftmann@36348: lemma diff_less_eq[algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077: apply (subst less_iff_diff_less_0 [of a])
haftmann@25077: apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@25077: apply (simp add: diff_minus add_ac)
haftmann@25077: done
haftmann@25077: 
haftmann@36348: lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302: apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077: apply (subst less_iff_diff_less_0 [of a])
haftmann@25077: apply (simp add: diff_minus add_ac)
haftmann@25077: done
haftmann@25077: 
haftmann@36348: lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
nipkow@29667: by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
haftmann@25077: 
haftmann@36348: lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
nipkow@29667: by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
haftmann@25077: 
haftmann@25077: lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"
nipkow@29667: by (simp add: algebra_simps)
haftmann@25077: 
haftmann@25077: end
haftmann@25077: 
haftmann@35028: class linordered_ab_semigroup_add =
haftmann@35028:   linorder + ordered_ab_semigroup_add
haftmann@25062: 
haftmann@35028: class linordered_cancel_ab_semigroup_add =
haftmann@35028:   linorder + ordered_cancel_ab_semigroup_add
haftmann@25267: begin
haftmann@25062: 
haftmann@35028: subclass linordered_ab_semigroup_add ..
haftmann@25062: 
haftmann@35028: subclass ordered_ab_semigroup_add_imp_le
haftmann@28823: proof
haftmann@25062:   fix a b c :: 'a
haftmann@25062:   assume le: "c + a <= c + b"  
haftmann@25062:   show "a <= b"
haftmann@25062:   proof (rule ccontr)
haftmann@25062:     assume w: "~ a \<le> b"
haftmann@25062:     hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062:     hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062:     have "a = b" 
haftmann@25062:       apply (insert le)
haftmann@25062:       apply (insert le2)
haftmann@25062:       apply (drule antisym, simp_all)
haftmann@25062:       done
haftmann@25062:     with w show False 
haftmann@25062:       by (simp add: linorder_not_le [symmetric])
haftmann@25062:   qed
haftmann@25062: qed
haftmann@25062: 
haftmann@25267: end
haftmann@25267: 
haftmann@35028: class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267: begin
haftmann@25230: 
haftmann@35028: subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230: 
haftmann@35036: lemma neg_less_eq_nonneg [simp]:
haftmann@25303:   "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@25303: proof
haftmann@25303:   assume A: "- a \<le> a" show "0 \<le> a"
haftmann@25303:   proof (rule classical)
haftmann@25303:     assume "\<not> 0 \<le> a"
haftmann@25303:     then have "a < 0" by auto
haftmann@25303:     with A have "- a < 0" by (rule le_less_trans)
haftmann@25303:     then show ?thesis by auto
haftmann@25303:   qed
haftmann@25303: next
haftmann@25303:   assume A: "0 \<le> a" show "- a \<le> a"
haftmann@25303:   proof (rule order_trans)
haftmann@25303:     show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@25303:   next
haftmann@25303:     show "0 \<le> a" using A .
haftmann@25303:   qed
haftmann@25303: qed
haftmann@35036: 
haftmann@35036: lemma neg_less_nonneg [simp]:
haftmann@35036:   "- a < a \<longleftrightarrow> 0 < a"
haftmann@35036: proof
haftmann@35036:   assume A: "- a < a" show "0 < a"
haftmann@35036:   proof (rule classical)
haftmann@35036:     assume "\<not> 0 < a"
haftmann@35036:     then have "a \<le> 0" by auto
haftmann@35036:     with A have "- a < 0" by (rule less_le_trans)
haftmann@35036:     then show ?thesis by auto
haftmann@35036:   qed
haftmann@35036: next
haftmann@35036:   assume A: "0 < a" show "- a < a"
haftmann@35036:   proof (rule less_trans)
haftmann@35036:     show "- a < 0" using A by (simp add: minus_le_iff)
haftmann@35036:   next
haftmann@35036:     show "0 < a" using A .
haftmann@35036:   qed
haftmann@35036: qed
haftmann@35036: 
haftmann@35036: lemma less_eq_neg_nonpos [simp]:
haftmann@25303:   "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@25303: proof
haftmann@25303:   assume A: "a \<le> - a" show "a \<le> 0"
haftmann@25303:   proof (rule classical)
haftmann@25303:     assume "\<not> a \<le> 0"
haftmann@25303:     then have "0 < a" by auto
haftmann@25303:     then have "0 < - a" using A by (rule less_le_trans)
haftmann@25303:     then show ?thesis by auto
haftmann@25303:   qed
haftmann@25303: next
haftmann@25303:   assume A: "a \<le> 0" show "a \<le> - a"
haftmann@25303:   proof (rule order_trans)
haftmann@25303:     show "0 \<le> - a" using A by (simp add: minus_le_iff)
haftmann@25303:   next
haftmann@25303:     show "a \<le> 0" using A .
haftmann@25303:   qed
haftmann@25303: qed
haftmann@25303: 
haftmann@35036: lemma equal_neg_zero [simp]:
haftmann@25303:   "a = - a \<longleftrightarrow> a = 0"
haftmann@25303: proof
haftmann@25303:   assume "a = 0" then show "a = - a" by simp
haftmann@25303: next
haftmann@25303:   assume A: "a = - a" show "a = 0"
haftmann@25303:   proof (cases "0 \<le> a")
haftmann@25303:     case True with A have "0 \<le> - a" by auto
haftmann@25303:     with le_minus_iff have "a \<le> 0" by simp
haftmann@25303:     with True show ?thesis by (auto intro: order_trans)
haftmann@25303:   next
haftmann@25303:     case False then have B: "a \<le> 0" by auto
haftmann@25303:     with A have "- a \<le> 0" by auto
haftmann@25303:     with B show ?thesis by (auto intro: order_trans)
haftmann@25303:   qed
haftmann@25303: qed
haftmann@25303: 
haftmann@35036: lemma neg_equal_zero [simp]:
haftmann@25303:   "- a = a \<longleftrightarrow> a = 0"
haftmann@35036:   by (auto dest: sym)
haftmann@35036: 
haftmann@35036: lemma double_zero [simp]:
haftmann@35036:   "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036: proof
haftmann@35036:   assume assm: "a + a = 0"
haftmann@35036:   then have a: "- a = a" by (rule minus_unique)
huffman@35216:   then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036: qed simp
haftmann@35036: 
haftmann@35036: lemma double_zero_sym [simp]:
haftmann@35036:   "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036:   by (rule, drule sym) simp_all
haftmann@35036: 
haftmann@35036: lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036:   "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036: proof
haftmann@35036:   assume "0 < a + a"
haftmann@35036:   then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036:   then have "- a < a" by simp
huffman@35216:   then show "0 < a" by (simp only: neg_less_nonneg)
haftmann@35036: next
haftmann@35036:   assume "0 < a"
haftmann@35036:   with this have "0 + 0 < a + a"
haftmann@35036:     by (rule add_strict_mono)
haftmann@35036:   then show "0 < a + a" by simp
haftmann@35036: qed
haftmann@35036: 
haftmann@35036: lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036:   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036:   by (auto simp add: le_less)
haftmann@35036: 
haftmann@35036: lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036:   "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036: proof -
haftmann@35036:   have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036:     by (simp add: not_less)
haftmann@35036:   then show ?thesis by simp
haftmann@35036: qed
haftmann@35036: 
haftmann@35036: lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036:   "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036: proof -
haftmann@35036:   have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036:     by (simp add: not_le)
haftmann@35036:   then show ?thesis by simp
haftmann@35036: qed
haftmann@35036: 
haftmann@35036: lemma le_minus_self_iff:
haftmann@35036:   "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@35036: proof -
haftmann@35036:   from add_le_cancel_left [of "- a" "a + a" 0]
haftmann@35036:   have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" 
haftmann@35036:     by (simp add: add_assoc [symmetric])
haftmann@35036:   thus ?thesis by simp
haftmann@35036: qed
haftmann@35036: 
haftmann@35036: lemma minus_le_self_iff:
haftmann@35036:   "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@35036: proof -
haftmann@35036:   from add_le_cancel_left [of "- a" 0 "a + a"]
haftmann@35036:   have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" 
haftmann@35036:     by (simp add: add_assoc [symmetric])
haftmann@35036:   thus ?thesis by simp
haftmann@35036: qed
haftmann@35036: 
haftmann@35036: lemma minus_max_eq_min:
haftmann@35036:   "- max x y = min (-x) (-y)"
haftmann@35036:   by (auto simp add: max_def min_def)
haftmann@35036: 
haftmann@35036: lemma minus_min_eq_max:
haftmann@35036:   "- min x y = max (-x) (-y)"
haftmann@35036:   by (auto simp add: max_def min_def)
haftmann@25303: 
haftmann@25267: end
haftmann@25267: 
haftmann@36302: context ordered_comm_monoid_add
haftmann@36302: begin
obua@14738: 
paulson@15234: lemma add_increasing:
haftmann@36302:   "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@36302:   by (insert add_mono [of 0 a b c], simp)
obua@14738: 
nipkow@15539: lemma add_increasing2:
haftmann@36302:   "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@36302:   by (simp add: add_increasing add_commute [of a])
nipkow@15539: 
paulson@15234: lemma add_strict_increasing:
haftmann@36302:   "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@36302:   by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234: 
paulson@15234: lemma add_strict_increasing2:
haftmann@36302:   "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36302:   by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@36302: 
haftmann@36302: end
paulson@15234: 
haftmann@35092: class abs =
haftmann@35092:   fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092: begin
haftmann@35092: 
haftmann@35092: notation (xsymbols)
haftmann@35092:   abs  ("\<bar>_\<bar>")
haftmann@35092: 
haftmann@35092: notation (HTML output)
haftmann@35092:   abs  ("\<bar>_\<bar>")
haftmann@35092: 
haftmann@35092: end
haftmann@35092: 
haftmann@35092: class sgn =
haftmann@35092:   fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092: 
haftmann@35092: class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092:   assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092: 
haftmann@35092: class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092:   assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092: begin
haftmann@35092: 
haftmann@35092: lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092:   by (simp add:sgn_if)
haftmann@35092: 
haftmann@35092: end
obua@14738: 
haftmann@35028: class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303:   assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303:     and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303:     and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303:     and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303:     and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303: begin
haftmann@25303: 
haftmann@25307: lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307:   unfolding neg_le_0_iff_le by simp
haftmann@25307: 
haftmann@25307: lemma abs_of_nonneg [simp]:
nipkow@29667:   assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307: proof (rule antisym)
haftmann@25307:   from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307:   from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307:   then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307: qed (rule abs_ge_self)
haftmann@25307: 
haftmann@25307: lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667: by (rule antisym)
haftmann@36302:    (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307: 
haftmann@25307: lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307: proof -
haftmann@25307:   have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307:   proof (rule antisym)
haftmann@25307:     assume zero: "\<bar>a\<bar> = 0"
haftmann@25307:     with abs_ge_self show "a \<le> 0" by auto
haftmann@25307:     from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302:     with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307:     with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307:   qed
haftmann@25307:   then show ?thesis by auto
haftmann@25307: qed
haftmann@25307: 
haftmann@25303: lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667: by simp
avigad@16775: 
blanchet@35828: lemma abs_0_eq [simp, no_atp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303: proof -
haftmann@25303:   have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303:   thus ?thesis by simp
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303: proof
haftmann@25303:   assume "\<bar>a\<bar> \<le> 0"
haftmann@25303:   then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303:   thus "a = 0" by simp
haftmann@25303: next
haftmann@25303:   assume "a = 0"
haftmann@25303:   thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667: by (simp add: less_le)
haftmann@25303: 
haftmann@25303: lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303: proof -
haftmann@25303:   have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303:   show ?thesis by (simp add: a)
haftmann@25303: qed
avigad@16775: 
haftmann@25303: lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303: proof -
haftmann@25303:   have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303:   then show ?thesis by simp
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: lemma abs_minus_commute: 
haftmann@25303:   "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303: proof -
haftmann@25303:   have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303:   also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303:   finally show ?thesis .
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667: by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775: 
haftmann@25303: lemma abs_of_nonpos [simp]:
nipkow@29667:   assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303: proof -
haftmann@25303:   let ?b = "- a"
haftmann@25303:   have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303:   unfolding abs_minus_cancel [of "?b"]
haftmann@25303:   unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303:   unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303:   then show ?thesis using assms by auto
haftmann@25303: qed
haftmann@25303:   
haftmann@25303: lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667: by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303: 
haftmann@25303: lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667: by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303: 
haftmann@25303: lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302: by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303: 
haftmann@25303: lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667: by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303: 
haftmann@25303: lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302: proof -
haftmann@36302:   have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@36302:     by (simp add: algebra_simps add_diff_cancel)
haftmann@36302:   then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302:     by (simp add: abs_triangle_ineq)
haftmann@36302:   then show ?thesis
haftmann@36302:     by (simp add: algebra_simps)
haftmann@36302: qed
haftmann@36302: 
haftmann@36302: lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302:   by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775: 
haftmann@25303: lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302:   by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775: 
haftmann@25303: lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303: proof -
haftmann@36302:   have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (subst diff_minus, rule refl)
haftmann@36302:   also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667:   finally show ?thesis by simp
haftmann@25303: qed
avigad@16775: 
haftmann@25303: lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303: proof -
haftmann@25303:   have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
haftmann@25303:   also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303:   finally show ?thesis .
haftmann@25303: qed
avigad@16775: 
haftmann@25303: lemma abs_add_abs [simp]:
haftmann@25303:   "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303: proof (rule antisym)
haftmann@25303:   show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303: next
haftmann@25303:   have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303:   also have "\<dots> = ?R" by simp
haftmann@25303:   finally show "?L \<le> ?R" .
haftmann@25303: qed
haftmann@25303: 
haftmann@25303: end
obua@14738: 
obua@14754: text {* Needed for abelian cancellation simprocs: *}
obua@14754: 
obua@14754: lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"
obua@14754: apply (subst add_left_commute)
obua@14754: apply (subst add_left_cancel)
obua@14754: apply simp
obua@14754: done
obua@14754: 
obua@14754: lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))"
obua@14754: apply (subst add_cancel_21[of _ _ _ 0, simplified])
obua@14754: apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified])
obua@14754: done
obua@14754: 
haftmann@35028: lemma less_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
obua@14754: by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])
obua@14754: 
haftmann@35028: lemma le_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
obua@14754: apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of  y' x'])
obua@14754: apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])
obua@14754: done
obua@14754: 
obua@14754: lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"
huffman@30629: by (simp only: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])
obua@14754: 
obua@14754: lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)"
obua@14754: by (simp add: diff_minus)
obua@14754: 
haftmann@25090: lemma le_add_right_mono: 
obua@15178:   assumes 
haftmann@35028:   "a <= b + (c::'a::ordered_ab_group_add)"
obua@15178:   "c <= d"    
obua@15178:   shows "a <= b + d"
obua@15178:   apply (rule_tac order_trans[where y = "b+c"])
obua@15178:   apply (simp_all add: prems)
obua@15178:   done
obua@15178: 
obua@15178: 
haftmann@25090: subsection {* Tools setup *}
haftmann@25090: 
blanchet@35828: lemma add_mono_thms_linordered_semiring [no_atp]:
haftmann@35028:   fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
haftmann@25077:   shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077:     and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077:     and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077:     and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077: by (rule add_mono, clarify+)+
haftmann@25077: 
blanchet@35828: lemma add_mono_thms_linordered_field [no_atp]:
haftmann@35028:   fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
haftmann@25077:   shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077:     and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077:     and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077:     and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077:     and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077: by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077:   add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077: 
paulson@17085: text{*Simplification of @{term "x-y < 0"}, etc.*}
blanchet@35828: lemmas diff_less_0_iff_less [simp, no_atp] = less_iff_diff_less_0 [symmetric]
blanchet@35828: lemmas diff_le_0_iff_le [simp, no_atp] = le_iff_diff_le_0 [symmetric]
paulson@17085: 
haftmann@22482: ML {*
wenzelm@27250: structure ab_group_add_cancel = Abel_Cancel
wenzelm@27250: (
haftmann@22482: 
haftmann@22482: (* term order for abelian groups *)
haftmann@22482: 
haftmann@22482: fun agrp_ord (Const (a, _)) = find_index (fn a' => a = a')
haftmann@35267:       [@{const_name Groups.zero}, @{const_name Groups.plus},
haftmann@35267:         @{const_name Groups.uminus}, @{const_name Groups.minus}]
haftmann@22482:   | agrp_ord _ = ~1;
haftmann@22482: 
wenzelm@35408: fun termless_agrp (a, b) = (Term_Ord.term_lpo agrp_ord (a, b) = LESS);
haftmann@22482: 
haftmann@22482: local
haftmann@22482:   val ac1 = mk_meta_eq @{thm add_assoc};
haftmann@22482:   val ac2 = mk_meta_eq @{thm add_commute};
haftmann@22482:   val ac3 = mk_meta_eq @{thm add_left_commute};
haftmann@35267:   fun solve_add_ac thy _ (_ $ (Const (@{const_name Groups.plus},_) $ _ $ _) $ _) =
haftmann@22482:         SOME ac1
haftmann@35267:     | solve_add_ac thy _ (_ $ x $ (Const (@{const_name Groups.plus},_) $ y $ z)) =
haftmann@22482:         if termless_agrp (y, x) then SOME ac3 else NONE
haftmann@22482:     | solve_add_ac thy _ (_ $ x $ y) =
haftmann@22482:         if termless_agrp (y, x) then SOME ac2 else NONE
haftmann@22482:     | solve_add_ac thy _ _ = NONE
haftmann@22482: in
wenzelm@32010:   val add_ac_proc = Simplifier.simproc @{theory}
haftmann@22482:     "add_ac_proc" ["x + y::'a::ab_semigroup_add"] solve_add_ac;
haftmann@22482: end;
haftmann@22482: 
wenzelm@27250: val eq_reflection = @{thm eq_reflection};
wenzelm@27250:   
wenzelm@27250: val T = @{typ "'a::ab_group_add"};
wenzelm@27250: 
haftmann@22482: val cancel_ss = HOL_basic_ss settermless termless_agrp
haftmann@22482:   addsimprocs [add_ac_proc] addsimps
nipkow@23085:   [@{thm add_0_left}, @{thm add_0_right}, @{thm diff_def},
haftmann@22482:    @{thm minus_add_distrib}, @{thm minus_minus}, @{thm minus_zero},
haftmann@22482:    @{thm right_minus}, @{thm left_minus}, @{thm add_minus_cancel},
haftmann@22482:    @{thm minus_add_cancel}];
wenzelm@27250: 
wenzelm@27250: val sum_pats = [@{cterm "x + y::'a::ab_group_add"}, @{cterm "x - y::'a::ab_group_add"}];
haftmann@22482:   
haftmann@22548: val eqI_rules = [@{thm less_eqI}, @{thm le_eqI}, @{thm eq_eqI}];
haftmann@22482: 
haftmann@22482: val dest_eqI = 
wenzelm@35364:   fst o HOLogic.dest_bin @{const_name "op ="} HOLogic.boolT o HOLogic.dest_Trueprop o concl_of;
haftmann@22482: 
wenzelm@27250: );
haftmann@22482: *}
haftmann@22482: 
wenzelm@26480: ML {*
haftmann@22482:   Addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv];
haftmann@22482: *}
paulson@17085: 
haftmann@33364: code_modulename SML
haftmann@35050:   Groups Arith
haftmann@33364: 
haftmann@33364: code_modulename OCaml
haftmann@35050:   Groups Arith
haftmann@33364: 
haftmann@33364: code_modulename Haskell
haftmann@35050:   Groups Arith
haftmann@33364: 
obua@14738: end