src/HOL/Library/Lattice_Algebras.thy
author haftmann
Mon, 08 Feb 2010 15:25:00 +0100
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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(* Author: Steven Obua, TU Muenchen *)
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header {* Various algebraic structures combined with a lattice *}
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theory Lattice_Algebras
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imports Complex_Main
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begin
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class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf
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begin
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lemma add_inf_distrib_left:
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  "a + inf b c = inf (a + b) (a + c)"
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apply (rule antisym)
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apply (simp_all add: le_infI)
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apply (rule add_le_imp_le_left [of "uminus a"])
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apply (simp only: add_assoc [symmetric], simp)
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apply rule
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apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
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done
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lemma add_inf_distrib_right:
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  "inf a b + c = inf (a + c) (b + c)"
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proof -
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  have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
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  thus ?thesis by (simp add: add_commute)
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qed
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end
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class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup
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begin
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lemma add_sup_distrib_left:
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  "a + sup b c = sup (a + b) (a + c)" 
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apply (rule antisym)
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apply (rule add_le_imp_le_left [of "uminus a"])
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apply (simp only: add_assoc[symmetric], simp)
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apply rule
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apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
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apply (rule le_supI)
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apply (simp_all)
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done
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lemma add_sup_distrib_right:
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  "sup a b + c = sup (a+c) (b+c)"
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proof -
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  have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
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  thus ?thesis by (simp add: add_commute)
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qed
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end
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class lattice_ab_group_add = ordered_ab_group_add + lattice
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begin
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subclass semilattice_inf_ab_group_add ..
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subclass semilattice_sup_ab_group_add ..
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lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
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lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
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proof (rule inf_unique)
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  fix a b :: 'a
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  show "- sup (-a) (-b) \<le> a"
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    by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
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      (simp, simp add: add_sup_distrib_left)
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next
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  fix a b :: 'a
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  show "- sup (-a) (-b) \<le> b"
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    by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
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      (simp, simp add: add_sup_distrib_left)
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next
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  fix a b c :: 'a
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  assume "a \<le> b" "a \<le> c"
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  then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
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    (simp add: le_supI)
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qed
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lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
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proof (rule sup_unique)
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  fix a b :: 'a
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  show "a \<le> - inf (-a) (-b)"
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    by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
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      (simp, simp add: add_inf_distrib_left)
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next
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  fix a b :: 'a
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  show "b \<le> - inf (-a) (-b)"
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    by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
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      (simp, simp add: add_inf_distrib_left)
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next
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  fix a b c :: 'a
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  assume "a \<le> c" "b \<le> c"
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  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
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    (simp add: le_infI)
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qed
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lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
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by (simp add: inf_eq_neg_sup)
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lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
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by (simp add: sup_eq_neg_inf)
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lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
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   105
proof -
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   106
  have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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   107
  hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
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   108
  hence "0 = (-a + sup a b) + (inf a b + (-b))"
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    by (simp add: add_sup_distrib_left add_inf_distrib_right)
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       (simp add: algebra_simps)
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   111
  thus ?thesis by (simp add: algebra_simps)
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diff changeset
   112
qed
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   113
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subsection {* Positive Part, Negative Part, Absolute Value *}
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definition
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  nprt :: "'a \<Rightarrow> 'a" where
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   118
  "nprt x = inf x 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   119
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   120
definition
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   121
  pprt :: "'a \<Rightarrow> 'a" where
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   122
  "pprt x = sup x 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   123
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   124
lemma pprt_neg: "pprt (- x) = - nprt x"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   125
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   126
  have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero ..
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   127
  also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup ..
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   128
  finally have "sup (- x) 0 = - inf x 0" .
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   129
  then show ?thesis unfolding pprt_def nprt_def .
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   130
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   131
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   132
lemma nprt_neg: "nprt (- x) = - pprt x"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   133
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   134
  from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   135
  then have "pprt x = - nprt (- x)" by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   136
  then show ?thesis by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   137
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   138
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   139
lemma prts: "a = pprt a + nprt a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   140
by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   141
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   142
lemma zero_le_pprt[simp]: "0 \<le> pprt a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   143
by (simp add: pprt_def)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   144
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   145
lemma nprt_le_zero[simp]: "nprt a \<le> 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   146
by (simp add: nprt_def)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   147
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   148
lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   149
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   150
  have a: "?l \<longrightarrow> ?r"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   151
    apply (auto)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   152
    apply (rule add_le_imp_le_right[of _ "uminus b" _])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   153
    apply (simp add: add_assoc)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   154
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   155
  have b: "?r \<longrightarrow> ?l"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   156
    apply (auto)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   157
    apply (rule add_le_imp_le_right[of _ "b" _])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   158
    apply (simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   159
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   160
  from a b show ?thesis by blast
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   161
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   162
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   163
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   164
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   165
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   166
lemma pprt_eq_id [simp, noatp]: "0 \<le> x \<Longrightarrow> pprt x = x"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   167
  by (simp add: pprt_def sup_aci sup_absorb1)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   168
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   169
lemma nprt_eq_id [simp, noatp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   170
  by (simp add: nprt_def inf_aci inf_absorb1)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   171
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   172
lemma pprt_eq_0 [simp, noatp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   173
  by (simp add: pprt_def sup_aci sup_absorb2)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   174
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   175
lemma nprt_eq_0 [simp, noatp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   176
  by (simp add: nprt_def inf_aci inf_absorb2)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   177
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   178
lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   179
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   180
  {
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   181
    fix a::'a
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   182
    assume hyp: "sup a (-a) = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   183
    hence "sup a (-a) + a = a" by (simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   184
    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right) 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   185
    hence "sup (a+a) 0 <= a" by (simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   186
    hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   187
  }
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   188
  note p = this
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   189
  assume hyp:"sup a (-a) = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   190
  hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   191
  from p[OF hyp] p[OF hyp2] show "a = 0" by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   192
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   193
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   194
lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   195
apply (simp add: inf_eq_neg_sup)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   196
apply (simp add: sup_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   197
apply (erule sup_0_imp_0)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   198
done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   199
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   200
lemma inf_0_eq_0 [simp, noatp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   201
by (rule, erule inf_0_imp_0) simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   202
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   203
lemma sup_0_eq_0 [simp, noatp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   204
by (rule, erule sup_0_imp_0) simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   205
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   206
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   207
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   208
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   209
  assume "0 <= a + a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   210
  hence a:"inf (a+a) 0 = 0" by (simp add: inf_commute inf_absorb1)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   211
  have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_")
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   212
    by (simp add: add_sup_inf_distribs inf_aci)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   213
  hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   214
  hence "inf a 0 = 0" by (simp only: add_right_cancel)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   215
  then show "0 <= a" unfolding le_iff_inf by (simp add: inf_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   216
next
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   217
  assume a: "0 <= a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   218
  show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   219
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   220
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   221
lemma double_zero [simp]:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   222
  "a + a = 0 \<longleftrightarrow> a = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   223
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   224
  assume assm: "a + a = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   225
  then have "a + a + - a = - a" by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   226
  then have "a + (a + - a) = - a" by (simp only: add_assoc)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   227
  then have a: "- a = a" by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   228
  show "a = 0" apply (rule antisym)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   229
  apply (unfold neg_le_iff_le [symmetric, of a])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   230
  unfolding a apply simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   231
  unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   232
  unfolding assm unfolding le_less apply simp_all done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   233
next
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   234
  assume "a = 0" then show "a + a = 0" by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   235
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   236
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   237
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   238
  "0 < a + a \<longleftrightarrow> 0 < a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   239
proof (cases "a = 0")
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   240
  case True then show ?thesis by auto
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   241
next
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   242
  case False then show ?thesis (*FIXME tune proof*)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   243
  unfolding less_le apply simp apply rule
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   244
  apply clarify
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   245
  apply rule
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   246
  apply assumption
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   247
  apply (rule notI)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   248
  unfolding double_zero [symmetric, of a] apply simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   249
  done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   250
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   251
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   252
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   253
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   254
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   255
  have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   256
  moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by (simp add: zero_le_double_add_iff_zero_le_single_add)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   257
  ultimately show ?thesis by blast
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   258
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   259
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   260
lemma double_add_less_zero_iff_single_less_zero [simp]:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   261
  "a + a < 0 \<longleftrightarrow> a < 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   262
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   263
  have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   264
  moreover have "\<dots> \<longleftrightarrow> a < 0" by (simp add: zero_less_double_add_iff_zero_less_single_add)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   265
  ultimately show ?thesis by blast
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   266
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   267
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   268
declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   269
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   270
lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   271
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   272
  from add_le_cancel_left [of "uminus a" "plus a a" zero]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   273
  have "(a <= -a) = (a+a <= 0)" 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   274
    by (simp add: add_assoc[symmetric])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   275
  thus ?thesis by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   276
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   277
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   278
lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   279
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   280
  from add_le_cancel_left [of "uminus a" zero "plus a a"]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   281
  have "(-a <= a) = (0 <= a+a)" 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   282
    by (simp add: add_assoc[symmetric])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   283
  thus ?thesis by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   284
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   285
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   286
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   287
unfolding le_iff_inf by (simp add: nprt_def inf_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   288
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   289
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   290
unfolding le_iff_sup by (simp add: pprt_def sup_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   291
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   292
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   293
unfolding le_iff_sup by (simp add: pprt_def sup_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   294
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   295
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   296
unfolding le_iff_inf by (simp add: nprt_def inf_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   297
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   298
lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   299
unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   300
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   301
lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   302
unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   303
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   304
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   305
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   306
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   307
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   308
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   309
class lattice_ab_group_add_abs = lattice_ab_group_add + abs +
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   310
  assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   311
begin
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   312
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   313
lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   314
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   315
  have "0 \<le> \<bar>a\<bar>"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   316
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   317
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   318
    show ?thesis by (rule add_mono [OF a b, simplified])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   319
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   320
  then have "0 \<le> sup a (- a)" unfolding abs_lattice .
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   321
  then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   322
  then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   323
    by (simp add: add_sup_inf_distribs sup_aci
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   324
      pprt_def nprt_def diff_minus abs_lattice)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   325
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   326
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   327
subclass ordered_ab_group_add_abs
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   328
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   329
  have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   330
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   331
    fix a b
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   332
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   333
    show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   334
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   335
  have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   336
    by (simp add: abs_lattice le_supI)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   337
  fix a b
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   338
  show "0 \<le> \<bar>a\<bar>" by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   339
  show "a \<le> \<bar>a\<bar>"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   340
    by (auto simp add: abs_lattice)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   341
  show "\<bar>-a\<bar> = \<bar>a\<bar>"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   342
    by (simp add: abs_lattice sup_commute)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   343
  show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (fact abs_leI)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   344
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   345
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   346
    have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   347
      by (simp add: abs_lattice add_sup_inf_distribs sup_aci diff_minus)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   348
    have a:"a+b <= sup ?m ?n" by (simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   349
    have b:"-a-b <= ?n" by (simp) 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   350
    have c:"?n <= sup ?m ?n" by (simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   351
    from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   352
    have e:"-a-b = -(a+b)" by (simp add: diff_minus)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   353
    from a d e have "abs(a+b) <= sup ?m ?n" 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   354
      by (drule_tac abs_leI, auto)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   355
    with g[symmetric] show ?thesis by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   356
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   357
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   358
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   359
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   360
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   361
lemma sup_eq_if:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   362
  fixes a :: "'a\<Colon>{lattice_ab_group_add, linorder}"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   363
  shows "sup a (- a) = (if a < 0 then - a else a)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   364
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   365
  note add_le_cancel_right [of a a "- a", symmetric, simplified]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   366
  moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   367
  then show ?thesis by (auto simp: sup_max min_max.sup_absorb1 min_max.sup_absorb2)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   368
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   369
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   370
lemma abs_if_lattice:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   371
  fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   372
  shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   373
by auto
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   374
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   375
lemma estimate_by_abs:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   376
  "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b" 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   377
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   378
  assume "a+b <= c"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   379
  hence 2: "a <= c+(-b)" by (simp add: algebra_simps)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   380
  have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   381
  show ?thesis by (rule le_add_right_mono[OF 2 3])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   382
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   383
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   384
class lattice_ring = ordered_ring + lattice_ab_group_add_abs
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   385
begin
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   386
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   387
subclass semilattice_inf_ab_group_add ..
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   388
subclass semilattice_sup_ab_group_add ..
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   389
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   390
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   391
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   392
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))" 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   393
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   394
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   395
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   396
  have a: "(abs a) * (abs b) = ?x"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   397
    by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   398
  {
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   399
    fix u v :: 'a
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   400
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   401
              u * v = pprt a * pprt b + pprt a * nprt b + 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   402
                      nprt a * pprt b + nprt a * nprt b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   403
      apply (subst prts[of u], subst prts[of v])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   404
      apply (simp add: algebra_simps) 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   405
      done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   406
  }
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   407
  note b = this[OF refl[of a] refl[of b]]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   408
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   409
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   410
  have xy: "- ?x <= ?y"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   411
    apply (simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   412
    apply (rule_tac y="0::'a" in order_trans)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   413
    apply (rule addm2)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   414
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   415
    apply (rule addm)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   416
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   417
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   418
  have yx: "?y <= ?x"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   419
    apply (simp add:diff_def)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   420
    apply (rule_tac y=0 in order_trans)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   421
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   422
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   423
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   424
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   425
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   426
  show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   427
    apply (rule abs_leI)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   428
    apply (simp add: i1)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   429
    apply (simp add: i2[simplified minus_le_iff])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   430
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   431
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   432
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   433
instance lattice_ring \<subseteq> ordered_ring_abs
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   434
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   435
  fix a b :: "'a\<Colon> lattice_ring"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   436
  assume "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   437
  show "abs (a*b) = abs a * abs b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   438
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   439
    have s: "(0 <= a*b) | (a*b <= 0)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   440
      apply (auto)    
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   441
      apply (rule_tac split_mult_pos_le)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   442
      apply (rule_tac contrapos_np[of "a*b <= 0"])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   443
      apply (simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   444
      apply (rule_tac split_mult_neg_le)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   445
      apply (insert prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   446
      apply (blast)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   447
      done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   448
    have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   449
      by (simp add: prts[symmetric])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   450
    show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   451
    proof cases
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   452
      assume "0 <= a * b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   453
      then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   454
        apply (simp_all add: mulprts abs_prts)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   455
        apply (insert prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   456
        apply (auto simp add: 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   457
          algebra_simps 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   458
          iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   459
          iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   460
          apply(drule (1) mult_nonneg_nonpos[of a b], simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   461
          apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   462
        done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   463
    next
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   464
      assume "~(0 <= a*b)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   465
      with s have "a*b <= 0" by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   466
      then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   467
        apply (simp_all add: mulprts abs_prts)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   468
        apply (insert prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   469
        apply (auto simp add: algebra_simps)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   470
        apply(drule (1) mult_nonneg_nonneg[of a b],simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   471
        apply(drule (1) mult_nonpos_nonpos[of a b],simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   472
        done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   473
    qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   474
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   475
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   476
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   477
lemma mult_le_prts:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   478
  assumes
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   479
  "a1 <= (a::'a::lattice_ring)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   480
  "a <= a2"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   481
  "b1 <= b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   482
  "b <= b2"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   483
  shows
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   484
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   485
proof - 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   486
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   487
    apply (subst prts[symmetric])+
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   488
    apply simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   489
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   490
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   491
    by (simp add: algebra_simps)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   492
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   493
    by (simp_all add: prems mult_mono)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   494
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   495
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   496
    have "pprt a * nprt b <= pprt a * nprt b2"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   497
      by (simp add: mult_left_mono prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   498
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   499
      by (simp add: mult_right_mono_neg prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   500
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   501
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   502
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   503
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   504
  proof - 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   505
    have "nprt a * pprt b <= nprt a2 * pprt b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   506
      by (simp add: mult_right_mono prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   507
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   508
      by (simp add: mult_left_mono_neg prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   509
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   510
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   511
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   512
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   513
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   514
    have "nprt a * nprt b <= nprt a * nprt b1"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   515
      by (simp add: mult_left_mono_neg prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   516
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   517
      by (simp add: mult_right_mono_neg prems)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   518
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   519
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   520
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   521
  ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   522
    by - (rule add_mono | simp)+
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   523
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   524
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   525
lemma mult_ge_prts:
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   526
  assumes
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   527
  "a1 <= (a::'a::lattice_ring)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   528
  "a <= a2"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   529
  "b1 <= b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   530
  "b <= b2"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   531
  shows
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   532
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   533
proof - 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   534
  from prems have a1:"- a2 <= -a" by auto
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   535
  from prems have a2: "-a <= -a1" by auto
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   536
  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg] 
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   537
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   538
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   539
    by (simp only: minus_le_iff)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   540
  then show ?thesis by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   541
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   542
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   543
instance int :: lattice_ring
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   544
proof  
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   545
  fix k :: int
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   546
  show "abs k = sup k (- k)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   547
    by (auto simp add: sup_int_def)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   548
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   549
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   550
instance real :: lattice_ring
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   551
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   552
  fix a :: real
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   553
  show "abs a = sup a (- a)"
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   554
    by (auto simp add: sup_real_def)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   555
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   556
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   557
end