src/HOL/Library/Lattice_Algebras.thy
author haftmann
Fri, 04 Jul 2014 20:18:47 +0200
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permissions -rw-r--r--
reduced name variants for assoc and commute on plus and mult
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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: "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 {* Positive Part, Negative Part, Absolute Value *}
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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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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"
53240
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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
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")
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   149
proof
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   150
  assume ?l
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   151
  then show ?r
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   152
    apply -
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   153
    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
   154
    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
   155
    done
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   156
next
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   157
  assume ?r
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   158
  then show ?l
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   159
    apply -
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   160
    apply (rule add_le_imp_le_right[of _ "b" _])
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   161
    apply simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   162
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   163
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   164
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   165
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
   166
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
   167
35828
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blanchet
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diff changeset
   168
lemma pprt_eq_id [simp, no_atp]: "0 \<le> x \<Longrightarrow> pprt x = x"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   169
  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
   170
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   171
lemma nprt_eq_id [simp, no_atp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   172
  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
   173
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   174
lemma pprt_eq_0 [simp, no_atp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   175
  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
   176
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   177
lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   178
  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
   179
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   180
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
   181
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   182
  {
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   183
    fix a :: 'a
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   184
    assume hyp: "sup a (- a) = 0"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   185
    then have "sup a (- a) + a = a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   186
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   187
    then have "sup (a + a) 0 = a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   188
      by (simp add: add_sup_distrib_right)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   189
    then have "sup (a + a) 0 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   190
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   191
    then have "0 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   192
      by (blast intro: order_trans inf_sup_ord)
35040
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
  note p = this
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   195
  assume hyp:"sup a (-a) = 0"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   196
  then have hyp2:"sup (-a) (-(-a)) = 0"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   197
    by (simp add: sup_commute)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   198
  from p[OF hyp] p[OF hyp2] show "a = 0"
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
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   220
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
   221
  "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
   222
proof
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   223
  assume "0 \<le> a + a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   224
  then have a: "inf (a + a) 0 = 0"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   225
    by (simp add: inf_commute inf_absorb1)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   226
  have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a"  (is "?l=_")
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   227
    by (simp add: add_sup_inf_distribs inf_aci)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   228
  then have "?l = 0 + inf a 0"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   229
    by (simp add: a, simp add: inf_commute)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   230
  then have "inf a 0 = 0"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   231
    by (simp only: add_right_cancel)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   232
  then show "0 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   233
    unfolding le_iff_inf by (simp add: inf_commute)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   234
next
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   235
  assume a: "0 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   236
  show "0 \<le> a + a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   237
    by (simp add: add_mono[OF a a, 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"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   241
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   242
  assume assm: "a + a = 0"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   243
  then have "a + a + - a = - a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   244
    by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   245
  then have "a + (a + - a) = - a"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56228
diff changeset
   246
    by (simp only: add.assoc)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   247
  then have a: "- a = a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   248
    by simp
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   249
  show "a = 0"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   250
    apply (rule antisym)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   251
    apply (unfold neg_le_iff_le [symmetric, of a])
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   252
    unfolding a
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   253
    apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   254
    unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   255
    unfolding assm
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   256
    unfolding le_less
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   257
    apply simp_all
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   258
    done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   259
next
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   260
  assume "a = 0"
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   261
  then show "a + a = 0"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   262
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   263
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   264
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   265
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
   266
proof (cases "a = 0")
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   267
  case True
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   268
  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
   269
next
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   270
  case False
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   271
  then show ?thesis
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   272
    unfolding less_le
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   273
    apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   274
    apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   275
    apply clarify
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   276
    apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   277
    apply assumption
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   278
    apply (rule notI)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   279
    unfolding double_zero [symmetric, of a]
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   280
    apply blast
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   281
    done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   282
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   283
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   284
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   285
  "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)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
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
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   295
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
   296
  "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
   297
proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   298
  have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   299
    by (subst less_minus_iff) simp
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   300
  moreover have "\<dots> \<longleftrightarrow> a < 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   301
    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
   302
  ultimately show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   303
    by blast
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   304
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   305
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   306
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
   307
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   308
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
   309
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   310
  from add_le_cancel_left [of "uminus a" "plus a a" zero]
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   311
  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
   312
    by (simp add: add.assoc[symmetric])
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   313
  then show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   314
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   315
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   316
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   317
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
   318
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   319
  from add_le_cancel_left [of "uminus a" zero "plus a a"]
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   320
  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56228
diff changeset
   321
    by (simp add: add.assoc[symmetric])
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   322
  then show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   323
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   324
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   325
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   326
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   327
  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
   328
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   329
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   330
  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
   331
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   332
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   333
  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
   334
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   335
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   336
  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
   337
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   338
lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   339
  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
   340
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   341
lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   342
  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
   343
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   344
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   345
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   346
lemmas add_sup_inf_distribs =
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   347
  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
   348
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   349
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   350
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
   351
  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
   352
begin
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   353
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   354
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
   355
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   356
  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
   357
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   358
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   359
      by (auto simp add: abs_lattice)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   360
    show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   361
      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
   362
  qed
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   363
  then have "0 \<le> sup a (- a)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   364
    unfolding abs_lattice .
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   365
  then have "sup (sup a (- a)) 0 = sup a (- a)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   366
    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
   367
  then show ?thesis
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   368
    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
   369
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   370
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   371
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
   372
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   373
  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
   374
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   375
    fix a b
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   376
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   377
      by (auto simp add: abs_lattice)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   378
    show "0 \<le> \<bar>a\<bar>"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   379
      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
   380
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   381
  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
   382
    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
   383
  fix a b
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   384
  show "0 \<le> \<bar>a\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   385
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   386
  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
   387
    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
   388
  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
   389
    by (simp add: abs_lattice sup_commute)
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   390
  {
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   391
    assume "a \<le> b"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   392
    then show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   393
      by (rule abs_leI)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   394
  }
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   395
  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
   396
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   397
    have g: "\<bar>a\<bar> + \<bar>b\<bar> = sup (a + b) (sup (- a - b) (sup (- a + b) (a + (- b))))"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   398
      (is "_=sup ?m ?n")
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   399
      by (simp add: abs_lattice add_sup_inf_distribs sup_aci ac_simps)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   400
    have a: "a + b \<le> sup ?m ?n"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   401
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   402
    have b: "- a - b \<le> ?n"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   403
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   404
    have c: "?n \<le> sup ?m ?n"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   405
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   406
    from b c have d: "- a - b \<le> sup ?m ?n"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   407
      by (rule order_trans)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   408
    have e: "- a - b = - (a + b)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   409
      by simp
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   410
    from a d e have "\<bar>a + b\<bar> \<le> sup ?m ?n"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   411
      apply -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   412
      apply (drule abs_leI)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   413
      apply (simp_all only: algebra_simps ac_simps minus_add)
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   414
      apply (metis add_uminus_conv_diff d sup_commute uminus_add_conv_diff)
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   415
      done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   416
    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
   417
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   418
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   419
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   420
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   421
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   422
lemma sup_eq_if:
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   423
  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
   424
  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
   425
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   426
  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
   427
  moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54230
diff changeset
   428
  then show ?thesis 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
   429
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   430
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   431
lemma abs_if_lattice:
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   432
  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
   433
  shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   434
  by auto
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   435
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   436
lemma estimate_by_abs:
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   437
  fixes a b c :: "'a::lattice_ab_group_add_abs"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   438
  shows "a + b \<le> c \<Longrightarrow> 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
   439
proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   440
  assume "a + b \<le> c"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   441
  then have "a \<le> c + (- b)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   442
    by (simp add: algebra_simps)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   443
  have "- b \<le> \<bar>b\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   444
    by (rule abs_ge_minus_self)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   445
  then have "c + (- b) \<le> c + \<bar>b\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   446
    by (rule add_left_mono)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   447
  with `a \<le> c + (- b)` show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   448
    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
   449
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   450
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   451
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
   452
begin
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   453
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   454
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
   455
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
   456
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   457
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   458
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   459
lemma abs_le_mult:
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   460
  fixes a b :: "'a::lattice_ring"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   461
  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
   462
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   463
  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
   464
  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
   465
  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
   466
    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
   467
  {
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   468
    fix u v :: 'a
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   469
    have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow>
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   470
              u * v = pprt a * pprt b + pprt a * nprt b +
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   471
                      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
   472
      apply (subst prts[of u], subst prts[of v])
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   473
      apply (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
   474
      done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   475
  }
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   476
  note b = this[OF refl[of a] refl[of b]]
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   477
  have xy: "- ?x \<le> ?y"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   478
    apply simp
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   479
    apply (metis (full_types) add_increasing add_uminus_conv_diff
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   480
      lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   481
      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
   482
    done
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   483
  have yx: "?y \<le> ?x"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   484
    apply simp
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   485
    apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   486
      lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   487
      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
   488
    done
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   489
  have i1: "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   490
    by (simp only: a b yx)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   491
  have i2: "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   492
    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
   493
  show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   494
    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
   495
    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
   496
    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
   497
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   498
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   499
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   500
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
   501
proof
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   502
  fix a b :: "'a::lattice_ring"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   503
  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
   504
  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
   505
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   506
    have s: "(0 \<le> a * b) \<or> (a * b \<le> 0)"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   507
      apply auto
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   508
      apply (rule_tac split_mult_pos_le)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   509
      apply (rule_tac contrapos_np[of "a * b \<le> 0"])
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   510
      apply simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   511
      apply (rule_tac split_mult_neg_le)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   512
      using a
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   513
      apply blast
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   514
      done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   515
    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
   516
      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
   517
    show ?thesis
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   518
    proof (cases "0 \<le> a * b")
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   519
      case True
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   520
      then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   521
        apply (simp_all add: mulprts abs_prts)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   522
        using a
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   523
        apply (auto simp add:
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   524
          algebra_simps
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   525
          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
   526
          iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   527
        apply(drule (1) mult_nonneg_nonpos[of a b], simp)
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   528
        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
   529
        done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   530
    next
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   531
      case False
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   532
      with s have "a * b \<le> 0"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   533
        by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   534
      then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   535
        apply (simp_all add: mulprts abs_prts)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   536
        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
   537
        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
   538
        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
   539
        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
   540
        done
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
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   543
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   544
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   545
lemma mult_le_prts:
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   546
  fixes a b :: "'a::lattice_ring"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   547
  assumes "a1 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   548
    and "a \<le> a2"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   549
    and "b1 \<le> b"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   550
    and "b \<le> b2"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   551
  shows "a * b \<le>
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   552
    pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   553
proof -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   554
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   555
    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
   556
    apply simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   557
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   558
  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
   559
    by (simp add: algebra_simps)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   560
  moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   561
    by (simp_all add: assms mult_mono)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   562
  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
   563
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   564
    have "pprt a * nprt b \<le> pprt a * nprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   565
      by (simp add: mult_left_mono assms)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   566
    moreover have "pprt a * nprt b2 \<le> pprt a1 * nprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   567
      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
   568
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   569
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   570
  qed
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   571
  moreover have "nprt a * pprt b \<le> nprt a2 * pprt b1"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   572
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   573
    have "nprt a * pprt b \<le> nprt a2 * pprt b"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   574
      by (simp add: mult_right_mono assms)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   575
    moreover have "nprt a2 * pprt b \<le> nprt a2 * pprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   576
      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
   577
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   578
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   579
  qed
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   580
  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
   581
  proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   582
    have "nprt a * nprt b \<le> nprt a * nprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   583
      by (simp add: mult_left_mono_neg assms)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   584
    moreover have "nprt a * nprt b1 \<le> nprt a1 * nprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   585
      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
   586
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   587
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   588
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   589
  ultimately show ?thesis
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   590
    apply -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   591
    apply (rule add_mono | simp)+
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   592
    done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   593
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   594
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   595
lemma mult_ge_prts:
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   596
  fixes a b :: "'a::lattice_ring"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   597
  assumes "a1 \<le> a"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   598
    and "a \<le> a2"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   599
    and "b1 \<le> b"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   600
    and "b \<le> b2"
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   601
  shows "a * b \<ge>
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   602
    nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   603
proof -
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   604
  from assms have a1: "- a2 \<le> -a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   605
    by auto
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   606
  from assms have a2: "- a \<le> -a1"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   607
    by auto
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   608
  from mult_le_prts[of "- a2" "- a" "- a1" "b1" b "b2",
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   609
    OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   610
  have le: "- (a * b) \<le> - nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   611
    - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   612
    by simp
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   613
  then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   614
      - 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
   615
    by (simp only: minus_le_iff)
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   616
  then show ?thesis
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   617
    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
   618
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   619
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   620
instance int :: lattice_ring
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   621
proof
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   622
  fix k :: int
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   623
  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
   624
    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
   625
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   626
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   627
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
   628
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   629
  fix a :: real
56228
0f6dc7512023 tuned proofs;
wenzelm
parents: 54863
diff changeset
   630
  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
   631
    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
   632
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   633
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   634
end
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   635