src/HOL/Library/Lattice_Algebras.thy
author wenzelm
Thu, 09 Jul 2015 00:40:57 +0200
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child 61546 53bb4172c7f7
permissions -rw-r--r--
tuned proofs;
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(* Author: Steven Obua, TU Muenchen *)
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section \<open>Various algebraic structures combined with a lattice\<close>
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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: "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 add: diff_le_eq add.commute)
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  done
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lemma add_inf_distrib_right: "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)"
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    by (simp add: add_inf_distrib_left)
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  then show ?thesis
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    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: "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 (simp add: le_diff_eq add.commute)
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  apply (rule le_supI)
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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_sup_distrib_right: "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)"
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    by (simp add: add_sup_distrib_left)
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  then show ?thesis
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    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 =
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  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 c :: '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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  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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  assume "a \<le> b" "a \<le> c"
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  then show "a \<le> - sup (-b) (-c)"
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    by (subst neg_le_iff_le [symmetric]) (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 c :: '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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  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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  assume "a \<le> c" "b \<le> c"
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  then show "- inf (- a) (- b) \<le> c"
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    by (subst neg_le_iff_le [symmetric]) (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 diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"
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  using neg_inf_eq_sup [of b c, symmetric] by simp
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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 diff_sup_eq_inf: "a - sup b c = a + inf (- b) (- c)"
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  using neg_sup_eq_inf [of b c, symmetric] by simp
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lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
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proof -
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  have "0 = - inf 0 (a - b) + inf (a - b) 0"
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    by (simp add: inf_commute)
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  then have "0 = sup 0 (b - a) + inf (a - b) 0"
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    by (simp add: inf_eq_neg_sup)
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  then have "0 = (- a + sup a b) + (inf a b + (- b))"
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    by (simp only: add_sup_distrib_left add_inf_distrib_right) simp
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  then show ?thesis
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    by (simp add: algebra_simps)
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qed
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subsection \<open>Positive Part, Negative Part, Absolute Value\<close>
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definition nprt :: "'a \<Rightarrow> 'a"
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  where "nprt x = inf x 0"
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definition pprt :: "'a \<Rightarrow> 'a"
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  where "pprt x = sup x 0"
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lemma pprt_neg: "pprt (- x) = - nprt x"
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proof -
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  have "sup (- x) 0 = sup (- x) (- 0)"
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    unfolding minus_zero ..
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  also have "\<dots> = - inf x 0"
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    unfolding neg_inf_eq_sup ..
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  finally have "sup (- x) 0 = - inf x 0" .
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  then show ?thesis
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    unfolding pprt_def nprt_def .
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qed
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lemma nprt_neg: "nprt (- x) = - pprt x"
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proof -
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  from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
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   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"
53240
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wenzelm
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diff changeset
   140
  by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
35040
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"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   143
  by (simp add: pprt_def)
35040
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"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   146
  by (simp add: nprt_def)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   147
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   148
lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   149
  (is "?l = ?r")
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   150
proof
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   151
  assume ?l
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   152
  then show ?r
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   153
    apply -
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   154
    apply (rule add_le_imp_le_right[of _ "uminus b" _])
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56228
diff changeset
   155
    apply (simp add: add.assoc)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   156
    done
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   157
next
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   158
  assume ?r
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   159
  then show ?l
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   160
    apply -
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   161
    apply (rule add_le_imp_le_right[of _ "b" _])
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   162
    apply simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   163
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   164
qed
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_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
   167
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
   168
35828
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blanchet
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diff changeset
   169
lemma pprt_eq_id [simp, no_atp]: "0 \<le> x \<Longrightarrow> pprt x = x"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   170
  by (simp add: pprt_def sup_absorb1)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   171
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   172
lemma nprt_eq_id [simp, no_atp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   173
  by (simp add: nprt_def inf_absorb1)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   174
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   175
lemma pprt_eq_0 [simp, no_atp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   176
  by (simp add: pprt_def sup_absorb2)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   177
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   178
lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   179
  by (simp add: nprt_def inf_absorb2)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   180
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   181
lemma sup_0_imp_0:
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   182
  assumes "sup a (- a) = 0"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   183
  shows "a = 0"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   184
proof -
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   185
  have p: "0 \<le> a" if "sup a (- a) = 0" for a :: 'a
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   186
  proof -
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   187
    from that have "sup a (- a) + a = a"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   188
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   189
    then have "sup (a + a) 0 = a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   190
      by (simp add: add_sup_distrib_right)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   191
    then have "sup (a + a) 0 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   192
      by simp
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   193
    then show ?thesis
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   194
      by (blast intro: order_trans inf_sup_ord)
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   195
  qed
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   196
  from assms have **: "sup (-a) (-(-a)) = 0"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   197
    by (simp add: sup_commute)
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   198
  from p[OF assms] p[OF **] show "a = 0"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   199
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   200
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   201
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   202
lemma inf_0_imp_0: "inf a (- a) = 0 \<Longrightarrow> a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   203
  apply (simp add: inf_eq_neg_sup)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   204
  apply (simp add: sup_commute)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   205
  apply (erule sup_0_imp_0)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   206
  done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   207
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   208
lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   209
  apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   210
  apply (erule inf_0_imp_0)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   211
  apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   212
  done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   213
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   214
lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   215
  apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   216
  apply (erule sup_0_imp_0)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   217
  apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   218
  done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   219
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   220
lemma zero_le_double_add_iff_zero_le_single_add [simp]: "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   221
  (is "?lhs \<longleftrightarrow> ?rhs")
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   222
proof
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   223
  show ?rhs if ?lhs
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   224
  proof -
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   225
    from that have a: "inf (a + a) 0 = 0"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   226
      by (simp add: inf_commute inf_absorb1)
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   227
    have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a"  (is "?l=_")
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   228
      by (simp add: add_sup_inf_distribs inf_aci)
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   229
    then have "?l = 0 + inf a 0"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   230
      by (simp add: a, simp add: inf_commute)
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   231
    then have "inf a 0 = 0"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   232
      by (simp only: add_right_cancel)
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   233
    then show ?thesis
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   234
      unfolding le_iff_inf by (simp add: inf_commute)
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   235
  qed
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   236
  show ?lhs if ?rhs
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   237
    by (simp add: add_mono[OF that that, simplified])
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   238
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   239
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   240
lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   241
  (is "?lhs \<longleftrightarrow> ?rhs")
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   242
proof
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   243
  show ?rhs if ?lhs
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   244
  proof -
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   245
    from that have "a + a + - a = - a"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   246
      by simp
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   247
    then have "a + (a + - a) = - a"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   248
      by (simp only: add.assoc)
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   249
    then have a: "- a = a"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   250
      by simp
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   251
    show ?thesis
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   252
      apply (rule antisym)
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   253
      apply (unfold neg_le_iff_le [symmetric, of a])
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   254
      unfolding a
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   255
      apply simp
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   256
      unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   257
      unfolding that
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   258
      unfolding le_less
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   259
      apply simp_all
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   260
      done
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   261
  qed
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   262
  show ?lhs if ?rhs
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   263
    using that by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   264
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   265
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   266
lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   267
proof (cases "a = 0")
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   268
  case True
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   269
  then show ?thesis by auto
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   270
next
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   271
  case False
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   272
  then show ?thesis
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   273
    unfolding less_le
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   274
    apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   275
    apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   276
    apply clarify
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   277
    apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   278
    apply assumption
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   279
    apply (rule notI)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   280
    unfolding double_zero [symmetric, of a]
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   281
    apply blast
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   282
    done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   283
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   284
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   285
lemma double_add_le_zero_iff_single_add_le_zero [simp]: "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   286
proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   287
  have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)"
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   288
    by (subst le_minus_iff) simp
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   289
  moreover have "\<dots> \<longleftrightarrow> a \<le> 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   290
    by (simp only: minus_add_distrib zero_le_double_add_iff_zero_le_single_add) simp
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   291
  ultimately show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   292
    by blast
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   293
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   294
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   295
lemma double_add_less_zero_iff_single_less_zero [simp]: "a + a < 0 \<longleftrightarrow> a < 0"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   296
proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   297
  have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   298
    by (subst less_minus_iff) simp
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   299
  moreover have "\<dots> \<longleftrightarrow> a < 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   300
    by (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   301
  ultimately show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   302
    by blast
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   303
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   304
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   305
declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp] diff_inf_eq_sup [simp] diff_sup_eq_inf [simp]
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   306
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   307
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
   308
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   309
  from add_le_cancel_left [of "uminus a" "plus a a" zero]
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   310
  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56228
diff changeset
   311
    by (simp add: add.assoc[symmetric])
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   312
  then show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   313
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   314
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   315
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   316
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
   317
proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   318
  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   319
    using add_le_cancel_left [of "uminus a" zero "plus a a"]
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56228
diff changeset
   320
    by (simp add: add.assoc[symmetric])
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   321
  then show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   322
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   323
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   324
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   325
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   326
  unfolding le_iff_inf by (simp add: nprt_def inf_commute)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   327
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   328
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   329
  unfolding le_iff_sup by (simp add: pprt_def sup_commute)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   330
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   331
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   332
  unfolding le_iff_sup by (simp add: pprt_def sup_commute)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   333
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   334
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   335
  unfolding le_iff_inf by (simp add: nprt_def inf_commute)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   336
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   337
lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   338
  unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   339
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   340
lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   341
  unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   342
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   343
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   344
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   345
lemmas add_sup_inf_distribs =
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   346
  add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   347
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   348
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   349
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
   350
  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
   351
begin
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   352
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   353
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
   354
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   355
  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
   356
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   357
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   358
      by (auto simp add: abs_lattice)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   359
    show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   360
      by (rule add_mono [OF a b, simplified])
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   361
  qed
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   362
  then have "0 \<le> sup a (- a)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   363
    unfolding abs_lattice .
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   364
  then have "sup (sup a (- a)) 0 = sup a (- a)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   365
    by (rule sup_absorb1)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   366
  then show ?thesis
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   367
    by (simp add: add_sup_inf_distribs ac_simps pprt_def nprt_def abs_lattice)
35040
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
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
   371
proof
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   372
  have abs_ge_zero [simp]: "0 \<le> \<bar>a\<bar>" for a
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   373
  proof -
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   374
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   375
      by (auto simp add: abs_lattice)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   376
    show "0 \<le> \<bar>a\<bar>"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   377
      by (rule add_mono [OF a b, simplified])
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   378
  qed
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   379
  have abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" for a b
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   380
    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
   381
  fix a b
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   382
  show "0 \<le> \<bar>a\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   383
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   384
  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
   385
    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
   386
  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
   387
    by (simp add: abs_lattice sup_commute)
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   388
  show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" if "a \<le> b"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   389
    using that by (rule abs_leI)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   390
  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
   391
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   392
    have g: "\<bar>a\<bar> + \<bar>b\<bar> = sup (a + b) (sup (- a - b) (sup (- a + b) (a + (- b))))"
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   393
      (is "_ = sup ?m ?n")
57862
8f074e6e22fc tuned proofs;
wenzelm
parents: 57512
diff changeset
   394
      by (simp add: abs_lattice add_sup_inf_distribs ac_simps)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   395
    have a: "a + b \<le> sup ?m ?n"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   396
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   397
    have b: "- a - b \<le> ?n"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   398
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   399
    have c: "?n \<le> sup ?m ?n"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   400
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   401
    from b c have d: "- a - b \<le> sup ?m ?n"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   402
      by (rule order_trans)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   403
    have e: "- a - b = - (a + b)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   404
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   405
    from a d e have "\<bar>a + b\<bar> \<le> sup ?m ?n"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   406
      apply -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   407
      apply (drule abs_leI)
57862
8f074e6e22fc tuned proofs;
wenzelm
parents: 57512
diff changeset
   408
      apply (simp_all only: algebra_simps minus_add)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   409
      apply (metis add_uminus_conv_diff d sup_commute uminus_add_conv_diff)
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   410
      done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   411
    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
   412
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   413
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   414
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   415
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   416
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   417
lemma sup_eq_if:
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   418
  fixes a :: "'a::{lattice_ab_group_add,linorder}"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   419
  shows "sup a (- a) = (if a < 0 then - a else a)"
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   420
  using add_le_cancel_right [of a a "- a", symmetric, simplified]
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   421
    and add_le_cancel_right [of "-a" a a, symmetric, simplified]
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   422
  by (auto simp: sup_max max.absorb1 max.absorb2)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   423
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   424
lemma abs_if_lattice:
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   425
  fixes a :: "'a::{lattice_ab_group_add_abs,linorder}"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   426
  shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   427
  by auto
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   428
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   429
lemma estimate_by_abs:
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   430
  fixes a b c :: "'a::lattice_ab_group_add_abs"
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   431
  assumes "a + b \<le> c"
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   432
  shows "a \<le> c + \<bar>b\<bar>"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   433
proof -
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   434
  from assms have "a \<le> c + (- b)"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   435
    by (simp add: algebra_simps)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   436
  have "- b \<le> \<bar>b\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   437
    by (rule abs_ge_minus_self)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   438
  then have "c + (- b) \<le> c + \<bar>b\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   439
    by (rule add_left_mono)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   440
  with \<open>a \<le> c + (- b)\<close> show ?thesis
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   441
    by (rule order_trans)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   442
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   443
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   444
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
   445
begin
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   446
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   447
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
   448
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
   449
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   450
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   451
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   452
lemma abs_le_mult:
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   453
  fixes a b :: "'a::lattice_ring"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   454
  shows "\<bar>a * b\<bar> \<le> \<bar>a\<bar> * \<bar>b\<bar>"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   455
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   456
  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
   457
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   458
  have a: "\<bar>a\<bar> * \<bar>b\<bar> = ?x"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   459
    by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   460
  have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow>
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   461
            u * v = pprt a * pprt b + pprt a * nprt b +
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   462
                    nprt a * pprt b + nprt a * nprt b" for u v :: 'a
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   463
    apply (subst prts[of u], subst prts[of v])
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   464
    apply (simp add: algebra_simps)
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   465
    done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   466
  note b = this[OF refl[of a] refl[of b]]
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   467
  have xy: "- ?x \<le> ?y"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   468
    apply simp
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   469
    apply (metis (full_types) add_increasing add_uminus_conv_diff
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   470
      lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   471
      mult_nonpos_nonpos nprt_le_zero zero_le_pprt)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   472
    done
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   473
  have yx: "?y \<le> ?x"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   474
    apply simp
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   475
    apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   476
      lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   477
      mult_nonpos_nonneg nprt_le_zero zero_le_pprt)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   478
    done
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   479
  have i1: "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   480
    by (simp only: a b yx)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   481
  have i2: "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   482
    by (simp only: a b xy)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   483
  show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   484
    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
   485
    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
   486
    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
   487
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   488
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   489
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   490
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
   491
proof
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   492
  fix a b :: "'a::lattice_ring"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   493
  assume a: "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   494
  show "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   495
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   496
    have s: "(0 \<le> a * b) \<or> (a * b \<le> 0)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   497
      apply auto
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   498
      apply (rule_tac split_mult_pos_le)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   499
      apply (rule_tac contrapos_np[of "a * b \<le> 0"])
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   500
      apply simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   501
      apply (rule_tac split_mult_neg_le)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   502
      using a
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   503
      apply blast
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   504
      done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   505
    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
   506
      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
   507
    show ?thesis
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   508
    proof (cases "0 \<le> a * b")
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   509
      case True
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   510
      then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   511
        apply (simp_all add: mulprts abs_prts)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   512
        using a
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   513
        apply (auto simp add:
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   514
          algebra_simps
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   515
          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
   516
          iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   517
        apply(drule (1) mult_nonneg_nonpos[of a b], simp)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   518
        apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   519
        done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   520
    next
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   521
      case False
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   522
      with s have "a * b \<le> 0"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   523
        by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   524
      then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   525
        apply (simp_all add: mulprts abs_prts)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   526
        apply (insert a)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   527
        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
   528
        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
   529
        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
   530
        done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   531
    qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   532
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   533
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   534
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   535
lemma mult_le_prts:
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   536
  fixes a b :: "'a::lattice_ring"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   537
  assumes "a1 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   538
    and "a \<le> a2"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   539
    and "b1 \<le> b"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   540
    and "b \<le> b2"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   541
  shows "a * b \<le>
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   542
    pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   543
proof -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   544
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   545
    by (subst prts[symmetric])+ simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   546
  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
   547
    by (simp add: algebra_simps)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   548
  moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   549
    by (simp_all add: assms mult_mono)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   550
  moreover have "pprt a * nprt b \<le> pprt a1 * nprt b2"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   551
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   552
    have "pprt a * nprt b \<le> pprt a * nprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   553
      by (simp add: mult_left_mono assms)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   554
    moreover have "pprt a * nprt b2 \<le> pprt a1 * nprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   555
      by (simp add: mult_right_mono_neg assms)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   556
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   557
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   558
  qed
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   559
  moreover have "nprt a * pprt b \<le> nprt a2 * pprt b1"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   560
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   561
    have "nprt a * pprt b \<le> nprt a2 * pprt b"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   562
      by (simp add: mult_right_mono assms)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   563
    moreover have "nprt a2 * pprt b \<le> nprt a2 * pprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   564
      by (simp add: mult_left_mono_neg assms)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   565
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   566
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   567
  qed
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   568
  moreover have "nprt a * nprt b \<le> nprt a1 * nprt b1"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   569
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   570
    have "nprt a * nprt b \<le> nprt a * nprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   571
      by (simp add: mult_left_mono_neg assms)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   572
    moreover have "nprt a * nprt b1 \<le> nprt a1 * nprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   573
      by (simp add: mult_right_mono_neg assms)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   574
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   575
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   576
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   577
  ultimately show ?thesis
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   578
    by - (rule add_mono | simp)+
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   579
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   580
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   581
lemma mult_ge_prts:
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   582
  fixes a b :: "'a::lattice_ring"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   583
  assumes "a1 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   584
    and "a \<le> a2"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   585
    and "b1 \<le> b"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   586
    and "b \<le> b2"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   587
  shows "a * b \<ge>
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   588
    nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   589
proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   590
  from assms have a1: "- a2 \<le> -a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   591
    by auto
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   592
  from assms have a2: "- a \<le> -a1"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   593
    by auto
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   594
  from mult_le_prts[of "- a2" "- a" "- a1" "b1" b "b2",
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   595
    OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
60698
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   596
  have le: "- (a * b) \<le>
29e8bdc41f90 tuned proofs;
wenzelm
parents: 60500
diff changeset
   597
    - nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   598
    - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   599
    by simp
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   600
  then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   601
      - pprt a1 * pprt b1 + - pprt a2 * nprt b1) \<le> a * b"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   602
    by (simp only: minus_le_iff)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   603
  then show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   604
    by (simp add: algebra_simps)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   605
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   606
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   607
instance int :: lattice_ring
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   608
proof
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   609
  fix k :: int
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   610
  show "\<bar>k\<bar> = sup k (- k)"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   611
    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
   612
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   613
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   614
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
   615
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   616
  fix a :: real
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   617
  show "\<bar>a\<bar> = sup a (- a)"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   618
    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
   619
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   620
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   621
end
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   622