wenzelm@14770: (*  Title:   HOL/OrderedGroup.thy
wenzelm@29269:     Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad
obua@14738: *)
obua@14738: 
obua@14738: header {* Ordered Groups *}
obua@14738: 
nipkow@15131: theory OrderedGroup
haftmann@22452: imports Lattices
wenzelm@19798: uses "~~/src/Provers/Arith/abel_cancel.ML"
nipkow@15131: begin
obua@14738: 
obua@14738: text {*
obua@14738:   The theory of partially ordered groups is taken from the books:
obua@14738:   \begin{itemize}
obua@14738:   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
obua@14738:   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
obua@14738:   \end{itemize}
obua@14738:   Most of the used notions can also be looked up in 
obua@14738:   \begin{itemize}
wenzelm@14770:   \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
obua@14738:   \item \emph{Algebra I} by van der Waerden, Springer.
obua@14738:   \end{itemize}
obua@14738: *}
obua@14738: 
nipkow@29667: ML{*
nipkow@29667: structure AlgebraSimps =
nipkow@29667:   NamedThmsFun(val name = "algebra_simps"
nipkow@29667:                val description = "algebra simplification rules");
nipkow@29667: *}
nipkow@29667: 
nipkow@29667: setup AlgebraSimps.setup
nipkow@29667: 
nipkow@29667: text{* The rewrites accumulated in @{text algebra_simps} deal with the
nipkow@29667: classical algebraic structures of groups, rings and family. They simplify
nipkow@29667: terms by multiplying everything out (in case of a ring) and bringing sums and
nipkow@29667: products into a canonical form (by ordered rewriting). As a result it decides
nipkow@29667: group and ring equalities but also helps with inequalities.
nipkow@29667: 
nipkow@29667: Of course it also works for fields, but it knows nothing about multiplicative
nipkow@29667: inverses or division. This is catered for by @{text field_simps}. *}
nipkow@29667: 
nipkow@23085: subsection {* Semigroups and Monoids *}
obua@14738: 
haftmann@22390: class semigroup_add = plus +
nipkow@29667:   assumes add_assoc[algebra_simps]: "(a + b) + c = a + (b + c)"
haftmann@22390: 
haftmann@22390: class ab_semigroup_add = semigroup_add +
nipkow@29667:   assumes add_commute[algebra_simps]: "a + b = b + a"
haftmann@25062: begin
obua@14738: 
nipkow@29667: lemma add_left_commute[algebra_simps]: "a + (b + c) = b + (a + c)"
nipkow@29667: by (rule mk_left_commute [of "plus", OF add_assoc add_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 +
nipkow@29667:   assumes mult_assoc[algebra_simps]: "(a * b) * c = a * (b * c)"
obua@14738: 
haftmann@22390: class ab_semigroup_mult = semigroup_mult +
nipkow@29667:   assumes mult_commute[algebra_simps]: "a * b = b * a"
haftmann@23181: begin
obua@14738: 
nipkow@29667: lemma mult_left_commute[algebra_simps]: "a * (b * c) = b * (a * c)"
nipkow@29667: by (rule mk_left_commute [of "times", OF mult_assoc mult_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: 
haftmann@26015: class ab_semigroup_idem_mult = ab_semigroup_mult +
nipkow@29667:   assumes mult_idem[simp]: "x * x = x"
haftmann@26015: begin
haftmann@26015: 
nipkow@29667: lemma mult_left_idem[simp]: "x * (x * y) = x * y"
haftmann@26015:   unfolding mult_assoc [symmetric, of x] mult_idem ..
haftmann@26015: 
haftmann@26015: end
haftmann@26015: 
nipkow@23085: class monoid_add = zero + semigroup_add +
haftmann@25062:   assumes add_0_left [simp]: "0 + a = a"
haftmann@25062:     and add_0_right [simp]: "a + 0 = a"
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"
haftmann@25062: begin
nipkow@23085: 
haftmann@25062: subclass monoid_add
haftmann@28823:   proof qed (insert add_0, simp_all add: add_commute)
haftmann@25062: 
haftmann@25062: end
obua@14738: 
haftmann@22390: class monoid_mult = one + semigroup_mult +
haftmann@25062:   assumes mult_1_left [simp]: "1 * a  = a"
haftmann@25062:   assumes mult_1_right [simp]: "a * 1 = a"
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"
haftmann@25062: begin
obua@14738: 
haftmann@25062: subclass monoid_mult
haftmann@28823:   proof qed (insert mult_1, simp_all add: mult_commute)
haftmann@25062: 
haftmann@25062: end
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: 
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: 
haftmann@25062: lemma minus_add_cancel: "- a + (a + b) = b"
nipkow@29667: by (simp add: add_assoc[symmetric])
obua@14738: 
haftmann@25062: lemma minus_zero [simp]: "- 0 = 0"
obua@14738: proof -
haftmann@25062:   have "- 0 = - 0 + (0 + 0)" by (simp only: add_0_right)
haftmann@25062:   also have "\<dots> = 0" by (rule minus_add_cancel)
obua@14738:   finally show ?thesis .
obua@14738: qed
obua@14738: 
haftmann@25062: lemma minus_minus [simp]: "- (- a) = a"
nipkow@23085: proof -
haftmann@25062:   have "- (- a) = - (- a) + (- a + a)" by simp
haftmann@25062:   also have "\<dots> = a" by (rule minus_add_cancel)
nipkow@23085:   finally show ?thesis .
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: 
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 equals_zero_I:
nipkow@29667:   assumes "a + b = 0" shows "- a = b"
nipkow@23085: proof -
haftmann@25062:   have "- a = - a + (a + b)" using assms by simp
haftmann@25062:   also have "\<dots> = b" by (simp add: add_assoc[symmetric])
nipkow@23085:   finally show ?thesis .
nipkow@23085: 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: 
nipkow@29667: declare diff_minus[symmetric, algebra_simps]
huffman@28130: 
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: 
haftmann@25267: subclass cancel_ab_semigroup_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: 
nipkow@29667: lemma uminus_add_conv_diff[algebra_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"
nipkow@29667: by (rule equals_zero_I) (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: 
nipkow@29667: lemma add_diff_eq[algebra_simps]: "a + (b - c) = (a + b) - c"
nipkow@29667: by (simp add: diff_minus add_ac)
haftmann@25077: 
nipkow@29667: lemma diff_add_eq[algebra_simps]: "(a - b) + c = (a + c) - b"
nipkow@29667: by (simp add: diff_minus add_ac)
haftmann@25077: 
nipkow@29667: lemma diff_eq_eq[algebra_simps]: "a - b = c \<longleftrightarrow> a = c + b"
nipkow@29667: by (auto simp add: diff_minus add_assoc)
haftmann@25077: 
nipkow@29667: lemma eq_diff_eq[algebra_simps]: "a = c - b \<longleftrightarrow> a + b = c"
nipkow@29667: by (auto simp add: diff_minus add_assoc)
haftmann@25077: 
nipkow@29667: lemma diff_diff_eq[algebra_simps]: "(a - b) - c = a - (b + c)"
nipkow@29667: by (simp add: diff_minus add_ac)
haftmann@25077: 
nipkow@29667: lemma diff_diff_eq2[algebra_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: 
haftmann@25062: end
obua@14738: 
obua@14738: subsection {* (Partially) Ordered Groups *} 
obua@14738: 
haftmann@22390: class pordered_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@25062: class pordered_cancel_ab_semigroup_add =
haftmann@25062:   pordered_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@25062: class pordered_ab_semigroup_add_imp_le =
haftmann@25062:   pordered_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@25303: class pordered_comm_monoid_add =
haftmann@25303:   pordered_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: 
haftmann@25303: 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: 
haftmann@25303: end
haftmann@25303: 
haftmann@25062: class pordered_ab_group_add =
haftmann@25062:   ab_group_add + pordered_ab_semigroup_add
haftmann@25062: begin
haftmann@25062: 
huffman@27516: subclass pordered_cancel_ab_semigroup_add ..
haftmann@25062: 
haftmann@25062: subclass pordered_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: 
huffman@27516: subclass pordered_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: 
nipkow@29667: lemma diff_less_eq[algebra_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: 
nipkow@29667: lemma less_diff_eq[algebra_simps]: "a < c - b \<longleftrightarrow> a + b < c"
haftmann@25077: apply (subst less_iff_diff_less_0 [of "plus 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: 
nipkow@29667: lemma diff_le_eq[algebra_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: 
nipkow@29667: lemma le_diff_eq[algebra_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: 
nipkow@29667: text{*Legacy - use @{text algebra_simps} *}
nipkow@29667: lemmas group_simps = algebra_simps
haftmann@25230: 
haftmann@25077: end
haftmann@25077: 
nipkow@29667: text{*Legacy - use @{text algebra_simps} *}
nipkow@29667: lemmas group_simps = algebra_simps
haftmann@25230: 
haftmann@25062: class ordered_ab_semigroup_add =
haftmann@25062:   linorder + pordered_ab_semigroup_add
haftmann@25062: 
haftmann@25062: class ordered_cancel_ab_semigroup_add =
haftmann@25062:   linorder + pordered_cancel_ab_semigroup_add
haftmann@25267: begin
haftmann@25062: 
huffman@27516: subclass ordered_ab_semigroup_add ..
haftmann@25062: 
haftmann@25267: subclass pordered_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@25230: class ordered_ab_group_add =
haftmann@25230:   linorder + pordered_ab_group_add
haftmann@25267: begin
haftmann@25230: 
huffman@27516: subclass ordered_cancel_ab_semigroup_add ..
haftmann@25230: 
haftmann@25303: lemma neg_less_eq_nonneg:
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@25303:   
haftmann@25303: lemma less_eq_neg_nonpos:
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@25303: lemma equal_neg_zero:
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@25303: lemma neg_equal_zero:
haftmann@25303:   "- a = a \<longleftrightarrow> a = 0"
haftmann@25303:   unfolding equal_neg_zero [symmetric] by auto
haftmann@25303: 
haftmann@25267: end
haftmann@25267: 
haftmann@25077: -- {* FIXME localize the following *}
obua@14738: 
paulson@15234: lemma add_increasing:
paulson@15234:   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
paulson@15234:   shows  "[|0\<le>a; b\<le>c|] ==> b \<le> a + c"
obua@14738: by (insert add_mono [of 0 a b c], simp)
obua@14738: 
nipkow@15539: lemma add_increasing2:
nipkow@15539:   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
nipkow@15539:   shows  "[|0\<le>c; b\<le>a|] ==> b \<le> a + c"
nipkow@15539: by (simp add:add_increasing add_commute[of a])
nipkow@15539: 
paulson@15234: lemma add_strict_increasing:
paulson@15234:   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
paulson@15234:   shows "[|0<a; b\<le>c|] ==> b < a + c"
paulson@15234: by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234: 
paulson@15234: lemma add_strict_increasing2:
paulson@15234:   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
paulson@15234:   shows "[|0\<le>a; b<c|] ==> b < a + c"
paulson@15234: by (insert add_le_less_mono [of 0 a b c], simp)
paulson@15234: 
obua@14738: 
haftmann@25303: class pordered_ab_group_add_abs = pordered_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)
nipkow@29667:    (auto intro!: abs_ge_self abs_leI order_trans [of "uminus (abs a)" zero "abs a"])
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@25307:     with abs_ge_self [of "uminus 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: 
haftmann@25303: lemma abs_0_eq [simp, noatp]: "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"
nipkow@29667: by (insert abs_le_D1 [of "uminus 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>"
nipkow@29667:   apply (simp add: algebra_simps)
nipkow@29667:   apply (subgoal_tac "abs a = abs (plus b (minus a b))")
haftmann@25303:   apply (erule ssubst)
haftmann@25303:   apply (rule abs_triangle_ineq)
nipkow@29667:   apply (rule arg_cong[of _ _ abs])
nipkow@29667:   apply (simp add: algebra_simps)
avigad@16775: done
avigad@16775: 
haftmann@25303: lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@25303:   apply (subst abs_le_iff)
haftmann@25303:   apply auto
haftmann@25303:   apply (rule abs_triangle_ineq2)
haftmann@25303:   apply (subst abs_minus_commute)
haftmann@25303:   apply (rule abs_triangle_ineq2)
avigad@16775: done
avigad@16775: 
haftmann@25303: lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303: proof -
nipkow@29667:   have "abs(a - b) = abs(a + - b)" by (subst diff_minus, rule refl)
nipkow@29667:   also have "... <= abs a + abs (- b)" 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: 
haftmann@22452: 
obua@14738: subsection {* Lattice Ordered (Abelian) Groups *}
obua@14738: 
haftmann@25303: class lordered_ab_group_add_meet = pordered_ab_group_add + lower_semilattice
haftmann@25090: begin
obua@14738: 
haftmann@25090: lemma add_inf_distrib_left:
haftmann@25090:   "a + inf b c = inf (a + b) (a + c)"
haftmann@25090: apply (rule antisym)
haftmann@22422: apply (simp_all add: le_infI)
haftmann@25090: apply (rule add_le_imp_le_left [of "uminus a"])
haftmann@25090: apply (simp only: add_assoc [symmetric], simp)
nipkow@21312: apply rule
nipkow@21312: apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
obua@14738: done
obua@14738: 
haftmann@25090: lemma add_inf_distrib_right:
haftmann@25090:   "inf a b + c = inf (a + c) (b + c)"
haftmann@25090: proof -
haftmann@25090:   have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
haftmann@25090:   thus ?thesis by (simp add: add_commute)
haftmann@25090: qed
haftmann@25090: 
haftmann@25090: end
haftmann@25090: 
haftmann@25303: class lordered_ab_group_add_join = pordered_ab_group_add + upper_semilattice
haftmann@25090: begin
haftmann@25090: 
haftmann@25090: lemma add_sup_distrib_left:
haftmann@25090:   "a + sup b c = sup (a + b) (a + c)" 
haftmann@25090: apply (rule antisym)
haftmann@25090: apply (rule add_le_imp_le_left [of "uminus a"])
obua@14738: apply (simp only: add_assoc[symmetric], simp)
nipkow@21312: apply rule
nipkow@21312: apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
haftmann@22422: apply (rule le_supI)
nipkow@21312: apply (simp_all)
obua@14738: done
obua@14738: 
haftmann@25090: lemma add_sup_distrib_right:
haftmann@25090:   "sup a b + c = sup (a+c) (b+c)"
obua@14738: proof -
haftmann@22452:   have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
obua@14738:   thus ?thesis by (simp add: add_commute)
obua@14738: qed
obua@14738: 
haftmann@25090: end
haftmann@25090: 
haftmann@25303: class lordered_ab_group_add = pordered_ab_group_add + lattice
haftmann@25090: begin
haftmann@25090: 
huffman@27516: subclass lordered_ab_group_add_meet ..
huffman@27516: subclass lordered_ab_group_add_join ..
haftmann@25090: 
haftmann@22422: lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
obua@14738: 
haftmann@25090: lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
haftmann@22452: proof (rule inf_unique)
haftmann@22452:   fix a b :: 'a
haftmann@25090:   show "- sup (-a) (-b) \<le> a"
haftmann@25090:     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
haftmann@25090:       (simp, simp add: add_sup_distrib_left)
haftmann@22452: next
haftmann@22452:   fix a b :: 'a
haftmann@25090:   show "- sup (-a) (-b) \<le> b"
haftmann@25090:     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
haftmann@25090:       (simp, simp add: add_sup_distrib_left)
haftmann@22452: next
haftmann@22452:   fix a b c :: 'a
haftmann@22452:   assume "a \<le> b" "a \<le> c"
haftmann@22452:   then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
haftmann@22452:     (simp add: le_supI)
haftmann@22452: qed
haftmann@22452:   
haftmann@25090: lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
haftmann@22452: proof (rule sup_unique)
haftmann@22452:   fix a b :: 'a
haftmann@25090:   show "a \<le> - inf (-a) (-b)"
haftmann@25090:     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
haftmann@25090:       (simp, simp add: add_inf_distrib_left)
haftmann@22452: next
haftmann@22452:   fix a b :: 'a
haftmann@25090:   show "b \<le> - inf (-a) (-b)"
haftmann@25090:     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
haftmann@25090:       (simp, simp add: add_inf_distrib_left)
haftmann@22452: next
haftmann@22452:   fix a b c :: 'a
haftmann@22452:   assume "a \<le> c" "b \<le> c"
haftmann@22452:   then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
haftmann@22452:     (simp add: le_infI)
haftmann@22452: qed
obua@14738: 
haftmann@25230: lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
nipkow@29667: by (simp add: inf_eq_neg_sup)
haftmann@25230: 
haftmann@25230: lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
nipkow@29667: by (simp add: sup_eq_neg_inf)
haftmann@25230: 
haftmann@25090: lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
obua@14738: proof -
haftmann@22422:   have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
haftmann@22422:   hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
haftmann@22422:   hence "0 = (-a + sup a b) + (inf a b + (-b))"
nipkow@29667:     by (simp add: add_sup_distrib_left add_inf_distrib_right)
nipkow@29667:        (simp add: algebra_simps)
nipkow@29667:   thus ?thesis by (simp add: algebra_simps)
obua@14738: qed
obua@14738: 
obua@14738: subsection {* Positive Part, Negative Part, Absolute Value *}
obua@14738: 
haftmann@22422: definition
haftmann@25090:   nprt :: "'a \<Rightarrow> 'a" where
haftmann@22422:   "nprt x = inf x 0"
haftmann@22422: 
haftmann@22422: definition
haftmann@25090:   pprt :: "'a \<Rightarrow> 'a" where
haftmann@22422:   "pprt x = sup x 0"
obua@14738: 
haftmann@25230: lemma pprt_neg: "pprt (- x) = - nprt x"
haftmann@25230: proof -
haftmann@25230:   have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero ..
haftmann@25230:   also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup ..
haftmann@25230:   finally have "sup (- x) 0 = - inf x 0" .
haftmann@25230:   then show ?thesis unfolding pprt_def nprt_def .
haftmann@25230: qed
haftmann@25230: 
haftmann@25230: lemma nprt_neg: "nprt (- x) = - pprt x"
haftmann@25230: proof -
haftmann@25230:   from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
haftmann@25230:   then have "pprt x = - nprt (- x)" by simp
haftmann@25230:   then show ?thesis by simp
haftmann@25230: qed
haftmann@25230: 
obua@14738: lemma prts: "a = pprt a + nprt a"
nipkow@29667: by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
obua@14738: 
obua@14738: lemma zero_le_pprt[simp]: "0 \<le> pprt a"
nipkow@29667: by (simp add: pprt_def)
obua@14738: 
obua@14738: lemma nprt_le_zero[simp]: "nprt a \<le> 0"
nipkow@29667: by (simp add: nprt_def)
obua@14738: 
haftmann@25090: lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
obua@14738: proof -
obua@14738:   have a: "?l \<longrightarrow> ?r"
obua@14738:     apply (auto)
haftmann@25090:     apply (rule add_le_imp_le_right[of _ "uminus b" _])
obua@14738:     apply (simp add: add_assoc)
obua@14738:     done
obua@14738:   have b: "?r \<longrightarrow> ?l"
obua@14738:     apply (auto)
obua@14738:     apply (rule add_le_imp_le_right[of _ "b" _])
obua@14738:     apply (simp)
obua@14738:     done
obua@14738:   from a b show ?thesis by blast
obua@14738: qed
obua@14738: 
obua@15580: lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
obua@15580: lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
obua@15580: 
haftmann@25090: lemma pprt_eq_id [simp, noatp]: "0 \<le> x \<Longrightarrow> pprt x = x"
nipkow@29667: by (simp add: pprt_def le_iff_sup sup_ACI)
obua@15580: 
haftmann@25090: lemma nprt_eq_id [simp, noatp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
nipkow@29667: by (simp add: nprt_def le_iff_inf inf_ACI)
obua@15580: 
haftmann@25090: lemma pprt_eq_0 [simp, noatp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
nipkow@29667: by (simp add: pprt_def le_iff_sup sup_ACI)
obua@15580: 
haftmann@25090: lemma nprt_eq_0 [simp, noatp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
nipkow@29667: by (simp add: nprt_def le_iff_inf inf_ACI)
obua@15580: 
haftmann@25090: lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
obua@14738: proof -
obua@14738:   {
obua@14738:     fix a::'a
haftmann@22422:     assume hyp: "sup a (-a) = 0"
haftmann@22422:     hence "sup a (-a) + a = a" by (simp)
haftmann@22422:     hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right) 
haftmann@22422:     hence "sup (a+a) 0 <= a" by (simp)
haftmann@22422:     hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
obua@14738:   }
obua@14738:   note p = this
haftmann@22422:   assume hyp:"sup a (-a) = 0"
haftmann@22422:   hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute)
obua@14738:   from p[OF hyp] p[OF hyp2] show "a = 0" by simp
obua@14738: qed
obua@14738: 
haftmann@25090: lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
haftmann@22422: apply (simp add: inf_eq_neg_sup)
haftmann@22422: apply (simp add: sup_commute)
haftmann@22422: apply (erule sup_0_imp_0)
paulson@15481: done
obua@14738: 
haftmann@25090: lemma inf_0_eq_0 [simp, noatp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
nipkow@29667: by (rule, erule inf_0_imp_0) simp
obua@14738: 
haftmann@25090: lemma sup_0_eq_0 [simp, noatp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
nipkow@29667: by (rule, erule sup_0_imp_0) simp
obua@14738: 
haftmann@25090: lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@25090:   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
obua@14738: proof
obua@14738:   assume "0 <= a + a"
haftmann@22422:   hence a:"inf (a+a) 0 = 0" by (simp add: le_iff_inf inf_commute)
haftmann@25090:   have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_")
haftmann@25090:     by (simp add: add_sup_inf_distribs inf_ACI)
haftmann@22422:   hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
haftmann@22422:   hence "inf a 0 = 0" by (simp only: add_right_cancel)
haftmann@22422:   then show "0 <= a" by (simp add: le_iff_inf inf_commute)    
obua@14738: next  
obua@14738:   assume a: "0 <= a"
obua@14738:   show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
obua@14738: qed
obua@14738: 
haftmann@25090: lemma double_zero: "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@25090: proof
haftmann@25090:   assume assm: "a + a = 0"
haftmann@25090:   then have "a + a + - a = - a" by simp
haftmann@25090:   then have "a + (a + - a) = - a" by (simp only: add_assoc)
haftmann@25090:   then have a: "- a = a" by simp (*FIXME tune proof*)
haftmann@25102:   show "a = 0" apply (rule antisym)
haftmann@25090:   apply (unfold neg_le_iff_le [symmetric, of a])
haftmann@25090:   unfolding a apply simp
haftmann@25090:   unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
haftmann@25090:   unfolding assm unfolding le_less apply simp_all done
haftmann@25090: next
haftmann@25090:   assume "a = 0" then show "a + a = 0" by simp
haftmann@25090: qed
haftmann@25090: 
haftmann@25090: lemma zero_less_double_add_iff_zero_less_single_add:
haftmann@25090:   "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@25090: proof (cases "a = 0")
haftmann@25090:   case True then show ?thesis by auto
haftmann@25090: next
haftmann@25090:   case False then show ?thesis (*FIXME tune proof*)
haftmann@25090:   unfolding less_le apply simp apply rule
haftmann@25090:   apply clarify
haftmann@25090:   apply rule
haftmann@25090:   apply assumption
haftmann@25090:   apply (rule notI)
haftmann@25090:   unfolding double_zero [symmetric, of a] apply simp
haftmann@25090:   done
haftmann@25090: qed
haftmann@25090: 
haftmann@25090: lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@25090:   "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
obua@14738: proof -
haftmann@25090:   have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
haftmann@25090:   moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by (simp add: zero_le_double_add_iff_zero_le_single_add)
obua@14738:   ultimately show ?thesis by blast
obua@14738: qed
obua@14738: 
haftmann@25090: lemma double_add_less_zero_iff_single_less_zero [simp]:
haftmann@25090:   "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@25090: proof -
haftmann@25090:   have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)
haftmann@25090:   moreover have "\<dots> \<longleftrightarrow> a < 0" by (simp add: zero_less_double_add_iff_zero_less_single_add)
haftmann@25090:   ultimately show ?thesis by blast
obua@14738: qed
obua@14738: 
haftmann@25230: declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp]
haftmann@25230: 
haftmann@25230: lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@25230: proof -
haftmann@25230:   from add_le_cancel_left [of "uminus a" "plus a a" zero]
haftmann@25230:   have "(a <= -a) = (a+a <= 0)" 
haftmann@25230:     by (simp add: add_assoc[symmetric])
haftmann@25230:   thus ?thesis by simp
haftmann@25230: qed
haftmann@25230: 
haftmann@25230: lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@25230: proof -
haftmann@25230:   from add_le_cancel_left [of "uminus a" zero "plus a a"]
haftmann@25230:   have "(-a <= a) = (0 <= a+a)" 
haftmann@25230:     by (simp add: add_assoc[symmetric])
haftmann@25230:   thus ?thesis by simp
haftmann@25230: qed
haftmann@25230: 
haftmann@25230: lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
nipkow@29667: by (simp add: le_iff_inf nprt_def inf_commute)
haftmann@25230: 
haftmann@25230: lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
nipkow@29667: by (simp add: le_iff_sup pprt_def sup_commute)
haftmann@25230: 
haftmann@25230: lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
nipkow@29667: by (simp add: le_iff_sup pprt_def sup_commute)
haftmann@25230: 
haftmann@25230: lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
nipkow@29667: by (simp add: le_iff_inf nprt_def inf_commute)
haftmann@25230: 
haftmann@25230: lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
nipkow@29667: by (simp add: le_iff_sup pprt_def sup_ACI sup_assoc [symmetric, of a])
haftmann@25230: 
haftmann@25230: lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
nipkow@29667: by (simp add: le_iff_inf nprt_def inf_ACI inf_assoc [symmetric, of a])
haftmann@25230: 
haftmann@25090: end
haftmann@25090: 
haftmann@25090: lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
haftmann@25090: 
haftmann@25230: 
haftmann@25303: class lordered_ab_group_add_abs = lordered_ab_group_add + abs +
haftmann@25230:   assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
haftmann@25230: begin
haftmann@25230: 
haftmann@25230: lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
haftmann@25230: proof -
haftmann@25230:   have "0 \<le> \<bar>a\<bar>"
haftmann@25230:   proof -
haftmann@25230:     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
haftmann@25230:     show ?thesis by (rule add_mono [OF a b, simplified])
haftmann@25230:   qed
haftmann@25230:   then have "0 \<le> sup a (- a)" unfolding abs_lattice .
haftmann@25230:   then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
haftmann@25230:   then show ?thesis
haftmann@25230:     by (simp add: add_sup_inf_distribs sup_ACI
haftmann@25230:       pprt_def nprt_def diff_minus abs_lattice)
haftmann@25230: qed
haftmann@25230: 
haftmann@25230: subclass pordered_ab_group_add_abs
haftmann@25230: proof -
haftmann@25230:   have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
haftmann@25230:   proof -
haftmann@25230:     fix a b
haftmann@25230:     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
haftmann@25230:     show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
haftmann@25230:   qed
haftmann@25230:   have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25230:     by (simp add: abs_lattice le_supI)
haftmann@25230:   show ?thesis
haftmann@28823:   proof
haftmann@25230:     fix a
haftmann@25230:     show "0 \<le> \<bar>a\<bar>" by simp
haftmann@25230:   next
haftmann@25230:     fix a
nipkow@29667:     show "a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
haftmann@25230:   next
haftmann@25230:     fix a
nipkow@29667:     show "\<bar>-a\<bar> = \<bar>a\<bar>" by (simp add: abs_lattice sup_commute)
haftmann@25230:   next
haftmann@25230:     fix a b
haftmann@25230:     show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (erule abs_leI)
haftmann@25230:   next
haftmann@25230:     fix a b
haftmann@25230:     show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25230:     proof -
haftmann@25230:       have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
haftmann@25230:         by (simp add: abs_lattice add_sup_inf_distribs sup_ACI diff_minus)
haftmann@25230:       have a:"a+b <= sup ?m ?n" by (simp)
haftmann@25230:       have b:"-a-b <= ?n" by (simp) 
haftmann@25230:       have c:"?n <= sup ?m ?n" by (simp)
haftmann@25230:       from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
haftmann@25230:       have e:"-a-b = -(a+b)" by (simp add: diff_minus)
nipkow@29667:       from a d e have "abs(a+b) <= sup ?m ?n" by (drule_tac abs_leI, auto)
haftmann@25230:       with g[symmetric] show ?thesis by simp
haftmann@25230:     qed
haftmann@25230:   qed auto
haftmann@25230: qed
haftmann@25230: 
haftmann@25230: end
haftmann@25230: 
haftmann@25090: lemma sup_eq_if:
haftmann@25303:   fixes a :: "'a\<Colon>{lordered_ab_group_add, linorder}"
haftmann@25090:   shows "sup a (- a) = (if a < 0 then - a else a)"
haftmann@25090: proof -
haftmann@25090:   note add_le_cancel_right [of a a "- a", symmetric, simplified]
haftmann@25090:   moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
haftmann@25090:   then show ?thesis by (auto simp: sup_max max_def)
haftmann@25090: qed
haftmann@25090: 
haftmann@25090: lemma abs_if_lattice:
haftmann@25303:   fixes a :: "'a\<Colon>{lordered_ab_group_add_abs, linorder}"
haftmann@25090:   shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
nipkow@29667: by auto
haftmann@25090: 
haftmann@25090: 
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: 
obua@14754: lemma less_eqI: "(x::'a::pordered_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: 
obua@14754: lemma le_eqI: "(x::'a::pordered_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')"
obua@14754: by (simp add: 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: 
obua@14754: lemma add_minus_cancel: "(a::'a::ab_group_add) + (-a + b) = b"
obua@14754: by (simp add: add_assoc[symmetric])
obua@14754: 
haftmann@25090: lemma le_add_right_mono: 
obua@15178:   assumes 
obua@15178:   "a <= b + (c::'a::pordered_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: lemma estimate_by_abs:
haftmann@25303:   "a + b <= (c::'a::lordered_ab_group_add_abs) \<Longrightarrow> a <= c + abs b" 
obua@15178: proof -
nipkow@23477:   assume "a+b <= c"
nipkow@29667:   hence 2: "a <= c+(-b)" by (simp add: algebra_simps)
obua@15178:   have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
obua@15178:   show ?thesis by (rule le_add_right_mono[OF 2 3])
obua@15178: qed
obua@15178: 
haftmann@25090: subsection {* Tools setup *}
haftmann@25090: 
haftmann@25077: lemma add_mono_thms_ordered_semiring [noatp]:
haftmann@25077:   fixes i j k :: "'a\<Colon>pordered_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: 
haftmann@25077: lemma add_mono_thms_ordered_field [noatp]:
haftmann@25077:   fixes i j k :: "'a\<Colon>pordered_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.*}
haftmann@24380: lemmas diff_less_0_iff_less [simp] = less_iff_diff_less_0 [symmetric]
haftmann@24380: lemmas diff_eq_0_iff_eq [simp, noatp] = eq_iff_diff_eq_0 [symmetric]
haftmann@24380: lemmas diff_le_0_iff_le [simp] = 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@22997:       [@{const_name HOL.zero}, @{const_name HOL.plus},
haftmann@22997:         @{const_name HOL.uminus}, @{const_name HOL.minus}]
haftmann@22482:   | agrp_ord _ = ~1;
haftmann@22482: 
wenzelm@29269: fun termless_agrp (a, b) = (TermOrd.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@22997:   fun solve_add_ac thy _ (_ $ (Const (@{const_name HOL.plus},_) $ _ $ _) $ _) =
haftmann@22482:         SOME ac1
haftmann@22997:     | solve_add_ac thy _ (_ $ x $ (Const (@{const_name HOL.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@28262:   val add_ac_proc = Simplifier.simproc (the_context ())
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 = 
haftmann@22482:   fst o HOLogic.dest_bin "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: 
obua@14738: end