# HG changeset patch
# User haftmann
# Date 1265634411 -3600
# Node ID b8c8d01cc20d4bd4eb497854f488dbf17ec19900
# Parent  2c159d2cdae7787bc83fca9aba44340b747e504b
separate library theory for type classes combining lattices with various algebraic structures; more simp rules

diff -r 2c159d2cdae7 -r b8c8d01cc20d src/HOL/OrderedGroup.thy
--- a/src/HOL/OrderedGroup.thy	Mon Feb 08 14:06:48 2010 +0100
+++ b/src/HOL/OrderedGroup.thy	Mon Feb 08 14:06:51 2010 +0100
@@ -710,7 +710,7 @@
 
 subclass linordered_cancel_ab_semigroup_add ..
 
-lemma neg_less_eq_nonneg:
+lemma neg_less_eq_nonneg [simp]:
   "- a \<le> a \<longleftrightarrow> 0 \<le> a"
 proof
   assume A: "- a \<le> a" show "0 \<le> a"
@@ -728,8 +728,27 @@
     show "0 \<le> a" using A .
   qed
 qed
-  
-lemma less_eq_neg_nonpos:
+
+lemma neg_less_nonneg [simp]:
+  "- a < a \<longleftrightarrow> 0 < a"
+proof
+  assume A: "- a < a" show "0 < a"
+  proof (rule classical)
+    assume "\<not> 0 < a"
+    then have "a \<le> 0" by auto
+    with A have "- a < 0" by (rule less_le_trans)
+    then show ?thesis by auto
+  qed
+next
+  assume A: "0 < a" show "- a < a"
+  proof (rule less_trans)
+    show "- a < 0" using A by (simp add: minus_le_iff)
+  next
+    show "0 < a" using A .
+  qed
+qed
+
+lemma less_eq_neg_nonpos [simp]:
   "a \<le> - a \<longleftrightarrow> a \<le> 0"
 proof
   assume A: "a \<le> - a" show "a \<le> 0"
@@ -748,7 +767,7 @@
   qed
 qed
 
-lemma equal_neg_zero:
+lemma equal_neg_zero [simp]:
   "a = - a \<longleftrightarrow> a = 0"
 proof
   assume "a = 0" then show "a = - a" by simp
@@ -765,9 +784,81 @@
   qed
 qed
 
-lemma neg_equal_zero:
+lemma neg_equal_zero [simp]:
   "- a = a \<longleftrightarrow> a = 0"
-  unfolding equal_neg_zero [symmetric] by auto
+  by (auto dest: sym)
+
+lemma double_zero [simp]:
+  "a + a = 0 \<longleftrightarrow> a = 0"
+proof
+  assume assm: "a + a = 0"
+  then have a: "- a = a" by (rule minus_unique)
+  then show "a = 0" by (simp add: neg_equal_zero)
+qed simp
+
+lemma double_zero_sym [simp]:
+  "0 = a + a \<longleftrightarrow> a = 0"
+  by (rule, drule sym) simp_all
+
+lemma zero_less_double_add_iff_zero_less_single_add [simp]:
+  "0 < a + a \<longleftrightarrow> 0 < a"
+proof
+  assume "0 < a + a"
+  then have "0 - a < a" by (simp only: diff_less_eq)
+  then have "- a < a" by simp
+  then show "0 < a" by (simp add: neg_less_nonneg)
+next
+  assume "0 < a"
+  with this have "0 + 0 < a + a"
+    by (rule add_strict_mono)
+  then show "0 < a + a" by simp
+qed
+
+lemma zero_le_double_add_iff_zero_le_single_add [simp]:
+  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
+  by (auto simp add: le_less)
+
+lemma double_add_less_zero_iff_single_add_less_zero [simp]:
+  "a + a < 0 \<longleftrightarrow> a < 0"
+proof -
+  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
+    by (simp add: not_less)
+  then show ?thesis by simp
+qed
+
+lemma double_add_le_zero_iff_single_add_le_zero [simp]:
+  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
+proof -
+  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
+    by (simp add: not_le)
+  then show ?thesis by simp
+qed
+
+lemma le_minus_self_iff:
+  "a \<le> - a \<longleftrightarrow> a \<le> 0"
+proof -
+  from add_le_cancel_left [of "- a" "a + a" 0]
+  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" 
+    by (simp add: add_assoc [symmetric])
+  thus ?thesis by simp
+qed
+
+lemma minus_le_self_iff:
+  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
+proof -
+  from add_le_cancel_left [of "- a" 0 "a + a"]
+  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" 
+    by (simp add: add_assoc [symmetric])
+  thus ?thesis by simp
+qed
+
+lemma minus_max_eq_min:
+  "- max x y = min (-x) (-y)"
+  by (auto simp add: max_def min_def)
+
+lemma minus_min_eq_max:
+  "- min x y = max (-x) (-y)"
+  by (auto simp add: max_def min_def)
 
 end
 
@@ -941,375 +1032,6 @@
 
 end
 
-
-subsection {* Lattice Ordered (Abelian) Groups *}
-
-class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf
-begin
-
-lemma add_inf_distrib_left:
-  "a + inf b c = inf (a + b) (a + c)"
-apply (rule antisym)
-apply (simp_all add: le_infI)
-apply (rule add_le_imp_le_left [of "uminus a"])
-apply (simp only: add_assoc [symmetric], simp)
-apply rule
-apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
-done
-
-lemma add_inf_distrib_right:
-  "inf a b + c = inf (a + c) (b + c)"
-proof -
-  have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
-  thus ?thesis by (simp add: add_commute)
-qed
-
-end
-
-class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup
-begin
-
-lemma add_sup_distrib_left:
-  "a + sup b c = sup (a + b) (a + c)" 
-apply (rule antisym)
-apply (rule add_le_imp_le_left [of "uminus a"])
-apply (simp only: add_assoc[symmetric], simp)
-apply rule
-apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
-apply (rule le_supI)
-apply (simp_all)
-done
-
-lemma add_sup_distrib_right:
-  "sup a b + c = sup (a+c) (b+c)"
-proof -
-  have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
-  thus ?thesis by (simp add: add_commute)
-qed
-
-end
-
-class lattice_ab_group_add = ordered_ab_group_add + lattice
-begin
-
-subclass semilattice_inf_ab_group_add ..
-subclass semilattice_sup_ab_group_add ..
-
-lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
-
-lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
-proof (rule inf_unique)
-  fix a b :: 'a
-  show "- sup (-a) (-b) \<le> a"
-    by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
-      (simp, simp add: add_sup_distrib_left)
-next
-  fix a b :: 'a
-  show "- sup (-a) (-b) \<le> b"
-    by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
-      (simp, simp add: add_sup_distrib_left)
-next
-  fix a b c :: 'a
-  assume "a \<le> b" "a \<le> c"
-  then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
-    (simp add: le_supI)
-qed
-  
-lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
-proof (rule sup_unique)
-  fix a b :: 'a
-  show "a \<le> - inf (-a) (-b)"
-    by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
-      (simp, simp add: add_inf_distrib_left)
-next
-  fix a b :: 'a
-  show "b \<le> - inf (-a) (-b)"
-    by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
-      (simp, simp add: add_inf_distrib_left)
-next
-  fix a b c :: 'a
-  assume "a \<le> c" "b \<le> c"
-  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
-    (simp add: le_infI)
-qed
-
-lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
-by (simp add: inf_eq_neg_sup)
-
-lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
-by (simp add: sup_eq_neg_inf)
-
-lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
-proof -
-  have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
-  hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
-  hence "0 = (-a + sup a b) + (inf a b + (-b))"
-    by (simp add: add_sup_distrib_left add_inf_distrib_right)
-       (simp add: algebra_simps)
-  thus ?thesis by (simp add: algebra_simps)
-qed
-
-subsection {* Positive Part, Negative Part, Absolute Value *}
-
-definition
-  nprt :: "'a \<Rightarrow> 'a" where
-  "nprt x = inf x 0"
-
-definition
-  pprt :: "'a \<Rightarrow> 'a" where
-  "pprt x = sup x 0"
-
-lemma pprt_neg: "pprt (- x) = - nprt x"
-proof -
-  have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero ..
-  also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup ..
-  finally have "sup (- x) 0 = - inf x 0" .
-  then show ?thesis unfolding pprt_def nprt_def .
-qed
-
-lemma nprt_neg: "nprt (- x) = - pprt x"
-proof -
-  from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
-  then have "pprt x = - nprt (- x)" by simp
-  then show ?thesis by simp
-qed
-
-lemma prts: "a = pprt a + nprt a"
-by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
-
-lemma zero_le_pprt[simp]: "0 \<le> pprt a"
-by (simp add: pprt_def)
-
-lemma nprt_le_zero[simp]: "nprt a \<le> 0"
-by (simp add: nprt_def)
-
-lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
-proof -
-  have a: "?l \<longrightarrow> ?r"
-    apply (auto)
-    apply (rule add_le_imp_le_right[of _ "uminus b" _])
-    apply (simp add: add_assoc)
-    done
-  have b: "?r \<longrightarrow> ?l"
-    apply (auto)
-    apply (rule add_le_imp_le_right[of _ "b" _])
-    apply (simp)
-    done
-  from a b show ?thesis by blast
-qed
-
-lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
-lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
-
-lemma pprt_eq_id [simp, noatp]: "0 \<le> x \<Longrightarrow> pprt x = x"
-  by (simp add: pprt_def sup_aci sup_absorb1)
-
-lemma nprt_eq_id [simp, noatp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
-  by (simp add: nprt_def inf_aci inf_absorb1)
-
-lemma pprt_eq_0 [simp, noatp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
-  by (simp add: pprt_def sup_aci sup_absorb2)
-
-lemma nprt_eq_0 [simp, noatp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
-  by (simp add: nprt_def inf_aci inf_absorb2)
-
-lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
-proof -
-  {
-    fix a::'a
-    assume hyp: "sup a (-a) = 0"
-    hence "sup a (-a) + a = a" by (simp)
-    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right) 
-    hence "sup (a+a) 0 <= a" by (simp)
-    hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
-  }
-  note p = this
-  assume hyp:"sup a (-a) = 0"
-  hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute)
-  from p[OF hyp] p[OF hyp2] show "a = 0" by simp
-qed
-
-lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
-apply (simp add: inf_eq_neg_sup)
-apply (simp add: sup_commute)
-apply (erule sup_0_imp_0)
-done
-
-lemma inf_0_eq_0 [simp, noatp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
-by (rule, erule inf_0_imp_0) simp
-
-lemma sup_0_eq_0 [simp, noatp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
-by (rule, erule sup_0_imp_0) simp
-
-lemma zero_le_double_add_iff_zero_le_single_add [simp]:
-  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
-proof
-  assume "0 <= a + a"
-  hence a:"inf (a+a) 0 = 0" by (simp add: inf_commute inf_absorb1)
-  have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_")
-    by (simp add: add_sup_inf_distribs inf_aci)
-  hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
-  hence "inf a 0 = 0" by (simp only: add_right_cancel)
-  then show "0 <= a" unfolding le_iff_inf by (simp add: inf_commute)
-next
-  assume a: "0 <= a"
-  show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
-qed
-
-lemma double_zero: "a + a = 0 \<longleftrightarrow> a = 0"
-proof
-  assume assm: "a + a = 0"
-  then have "a + a + - a = - a" by simp
-  then have "a + (a + - a) = - a" by (simp only: add_assoc)
-  then have a: "- a = a" by simp
-  show "a = 0" apply (rule antisym)
-  apply (unfold neg_le_iff_le [symmetric, of a])
-  unfolding a apply simp
-  unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
-  unfolding assm unfolding le_less apply simp_all done
-next
-  assume "a = 0" then show "a + a = 0" by simp
-qed
-
-lemma zero_less_double_add_iff_zero_less_single_add:
-  "0 < a + a \<longleftrightarrow> 0 < a"
-proof (cases "a = 0")
-  case True then show ?thesis by auto
-next
-  case False then show ?thesis (*FIXME tune proof*)
-  unfolding less_le apply simp apply rule
-  apply clarify
-  apply rule
-  apply assumption
-  apply (rule notI)
-  unfolding double_zero [symmetric, of a] apply simp
-  done
-qed
-
-lemma double_add_le_zero_iff_single_add_le_zero [simp]:
-  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
-proof -
-  have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
-  moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by (simp add: zero_le_double_add_iff_zero_le_single_add)
-  ultimately show ?thesis by blast
-qed
-
-lemma double_add_less_zero_iff_single_less_zero [simp]:
-  "a + a < 0 \<longleftrightarrow> a < 0"
-proof -
-  have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)
-  moreover have "\<dots> \<longleftrightarrow> a < 0" by (simp add: zero_less_double_add_iff_zero_less_single_add)
-  ultimately show ?thesis by blast
-qed
-
-declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp]
-
-lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
-proof -
-  from add_le_cancel_left [of "uminus a" "plus a a" zero]
-  have "(a <= -a) = (a+a <= 0)" 
-    by (simp add: add_assoc[symmetric])
-  thus ?thesis by simp
-qed
-
-lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
-proof -
-  from add_le_cancel_left [of "uminus a" zero "plus a a"]
-  have "(-a <= a) = (0 <= a+a)" 
-    by (simp add: add_assoc[symmetric])
-  thus ?thesis by simp
-qed
-
-lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
-unfolding le_iff_inf by (simp add: nprt_def inf_commute)
-
-lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
-unfolding le_iff_sup by (simp add: pprt_def sup_commute)
-
-lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
-unfolding le_iff_sup by (simp add: pprt_def sup_commute)
-
-lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
-unfolding le_iff_inf by (simp add: nprt_def inf_commute)
-
-lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
-unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
-
-lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
-unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
-
-end
-
-lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
-
-
-class lattice_ab_group_add_abs = lattice_ab_group_add + abs +
-  assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
-begin
-
-lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
-proof -
-  have "0 \<le> \<bar>a\<bar>"
-  proof -
-    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
-    show ?thesis by (rule add_mono [OF a b, simplified])
-  qed
-  then have "0 \<le> sup a (- a)" unfolding abs_lattice .
-  then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
-  then show ?thesis
-    by (simp add: add_sup_inf_distribs sup_aci
-      pprt_def nprt_def diff_minus abs_lattice)
-qed
-
-subclass ordered_ab_group_add_abs
-proof
-  have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
-  proof -
-    fix a b
-    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
-    show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
-  qed
-  have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
-    by (simp add: abs_lattice le_supI)
-  fix a b
-  show "0 \<le> \<bar>a\<bar>" by simp
-  show "a \<le> \<bar>a\<bar>"
-    by (auto simp add: abs_lattice)
-  show "\<bar>-a\<bar> = \<bar>a\<bar>"
-    by (simp add: abs_lattice sup_commute)
-  show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (fact abs_leI)
-  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
-  proof -
-    have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
-      by (simp add: abs_lattice add_sup_inf_distribs sup_aci diff_minus)
-    have a:"a+b <= sup ?m ?n" by (simp)
-    have b:"-a-b <= ?n" by (simp) 
-    have c:"?n <= sup ?m ?n" by (simp)
-    from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
-    have e:"-a-b = -(a+b)" by (simp add: diff_minus)
-    from a d e have "abs(a+b) <= sup ?m ?n" 
-      by (drule_tac abs_leI, auto)
-    with g[symmetric] show ?thesis by simp
-  qed
-qed
-
-end
-
-lemma sup_eq_if:
-  fixes a :: "'a\<Colon>{lattice_ab_group_add, linorder}"
-  shows "sup a (- a) = (if a < 0 then - a else a)"
-proof -
-  note add_le_cancel_right [of a a "- a", symmetric, simplified]
-  moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
-  then show ?thesis by (auto simp: sup_max min_max.sup_absorb1 min_max.sup_absorb2)
-qed
-
-lemma abs_if_lattice:
-  fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}"
-  shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
-by auto
-
-
 text {* Needed for abelian cancellation simprocs: *}
 
 lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"
@@ -1346,14 +1068,6 @@
   apply (simp_all add: prems)
   done
 
-lemma estimate_by_abs:
-  "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b" 
-proof -
-  assume "a+b <= c"
-  hence 2: "a <= c+(-b)" by (simp add: algebra_simps)
-  have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
-  show ?thesis by (rule le_add_right_mono[OF 2 3])
-qed
 
 subsection {* Tools setup *}