src/HOL/Library/Lattice_Algebras.thy
 author wenzelm Wed Mar 08 10:50:59 2017 +0100 (2017-03-08) changeset 65151 a7394aa4d21c parent 61546 53bb4172c7f7 child 68406 6beb45f6cf67 permissions -rw-r--r--
tuned proofs;
```     1 (*  Author:     Steven Obua, TU Muenchen *)
```
```     2
```
```     3 section \<open>Various algebraic structures combined with a lattice\<close>
```
```     4
```
```     5 theory Lattice_Algebras
```
```     6   imports Complex_Main
```
```     7 begin
```
```     8
```
```     9 class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf
```
```    10 begin
```
```    11
```
```    12 lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + c)"
```
```    13   apply (rule antisym)
```
```    14    apply (simp_all add: le_infI)
```
```    15   apply (rule add_le_imp_le_left [of "uminus a"])
```
```    16   apply (simp only: add.assoc [symmetric], simp add: diff_le_eq add.commute)
```
```    17   done
```
```    18
```
```    19 lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)"
```
```    20 proof -
```
```    21   have "c + inf a b = inf (c + a) (c + b)"
```
```    22     by (simp add: add_inf_distrib_left)
```
```    23   then show ?thesis
```
```    24     by (simp add: add.commute)
```
```    25 qed
```
```    26
```
```    27 end
```
```    28
```
```    29 class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup
```
```    30 begin
```
```    31
```
```    32 lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a + c)"
```
```    33   apply (rule antisym)
```
```    34    apply (rule add_le_imp_le_left [of "uminus a"])
```
```    35    apply (simp only: add.assoc [symmetric], simp)
```
```    36    apply (simp add: le_diff_eq add.commute)
```
```    37   apply (rule le_supI)
```
```    38    apply (rule add_le_imp_le_left [of "a"], simp only: add.assoc[symmetric], simp)+
```
```    39   done
```
```    40
```
```    41 lemma add_sup_distrib_right: "sup a b + c = sup (a + c) (b + c)"
```
```    42 proof -
```
```    43   have "c + sup a b = sup (c+a) (c+b)"
```
```    44     by (simp add: add_sup_distrib_left)
```
```    45   then show ?thesis
```
```    46     by (simp add: add.commute)
```
```    47 qed
```
```    48
```
```    49 end
```
```    50
```
```    51 class lattice_ab_group_add = ordered_ab_group_add + lattice
```
```    52 begin
```
```    53
```
```    54 subclass semilattice_inf_ab_group_add ..
```
```    55 subclass semilattice_sup_ab_group_add ..
```
```    56
```
```    57 lemmas add_sup_inf_distribs =
```
```    58   add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
```
```    59
```
```    60 lemma inf_eq_neg_sup: "inf a b = - sup (- a) (- b)"
```
```    61 proof (rule inf_unique)
```
```    62   fix a b c :: 'a
```
```    63   show "- sup (- a) (- b) \<le> a"
```
```    64     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
```
```    65       (simp, simp add: add_sup_distrib_left)
```
```    66   show "- sup (-a) (-b) \<le> b"
```
```    67     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
```
```    68       (simp, simp add: add_sup_distrib_left)
```
```    69   assume "a \<le> b" "a \<le> c"
```
```    70   then show "a \<le> - sup (-b) (-c)"
```
```    71     by (subst neg_le_iff_le [symmetric]) (simp add: le_supI)
```
```    72 qed
```
```    73
```
```    74 lemma sup_eq_neg_inf: "sup a b = - inf (- a) (- b)"
```
```    75 proof (rule sup_unique)
```
```    76   fix a b c :: 'a
```
```    77   show "a \<le> - inf (- a) (- b)"
```
```    78     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
```
```    79       (simp, simp add: add_inf_distrib_left)
```
```    80   show "b \<le> - inf (- a) (- b)"
```
```    81     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
```
```    82       (simp, simp add: add_inf_distrib_left)
```
```    83   show "- inf (- a) (- b) \<le> c" if "a \<le> c" "b \<le> c"
```
```    84     using that by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)
```
```    85 qed
```
```    86
```
```    87 lemma neg_inf_eq_sup: "- inf a b = sup (- a) (- b)"
```
```    88   by (simp add: inf_eq_neg_sup)
```
```    89
```
```    90 lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"
```
```    91   using neg_inf_eq_sup [of b c, symmetric] by simp
```
```    92
```
```    93 lemma neg_sup_eq_inf: "- sup a b = inf (- a) (- b)"
```
```    94   by (simp add: sup_eq_neg_inf)
```
```    95
```
```    96 lemma diff_sup_eq_inf: "a - sup b c = a + inf (- b) (- c)"
```
```    97   using neg_sup_eq_inf [of b c, symmetric] by simp
```
```    98
```
```    99 lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
```
```   100 proof -
```
```   101   have "0 = - inf 0 (a - b) + inf (a - b) 0"
```
```   102     by (simp add: inf_commute)
```
```   103   then have "0 = sup 0 (b - a) + inf (a - b) 0"
```
```   104     by (simp add: inf_eq_neg_sup)
```
```   105   then have "0 = (- a + sup a b) + (inf a b + (- b))"
```
```   106     by (simp only: add_sup_distrib_left add_inf_distrib_right) simp
```
```   107   then show ?thesis
```
```   108     by (simp add: algebra_simps)
```
```   109 qed
```
```   110
```
```   111
```
```   112 subsection \<open>Positive Part, Negative Part, Absolute Value\<close>
```
```   113
```
```   114 definition nprt :: "'a \<Rightarrow> 'a"
```
```   115   where "nprt x = inf x 0"
```
```   116
```
```   117 definition pprt :: "'a \<Rightarrow> 'a"
```
```   118   where "pprt x = sup x 0"
```
```   119
```
```   120 lemma pprt_neg: "pprt (- x) = - nprt x"
```
```   121 proof -
```
```   122   have "sup (- x) 0 = sup (- x) (- 0)"
```
```   123     by (simp only: minus_zero)
```
```   124   also have "\<dots> = - inf x 0"
```
```   125     by (simp only: neg_inf_eq_sup)
```
```   126   finally have "sup (- x) 0 = - inf x 0" .
```
```   127   then show ?thesis
```
```   128     by (simp only: pprt_def nprt_def)
```
```   129 qed
```
```   130
```
```   131 lemma nprt_neg: "nprt (- x) = - pprt x"
```
```   132 proof -
```
```   133   from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
```
```   134   then have "pprt x = - nprt (- x)" by simp
```
```   135   then show ?thesis by simp
```
```   136 qed
```
```   137
```
```   138 lemma prts: "a = pprt a + nprt a"
```
```   139   by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
```
```   140
```
```   141 lemma zero_le_pprt[simp]: "0 \<le> pprt a"
```
```   142   by (simp add: pprt_def)
```
```   143
```
```   144 lemma nprt_le_zero[simp]: "nprt a \<le> 0"
```
```   145   by (simp add: nprt_def)
```
```   146
```
```   147 lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0"
```
```   148   (is "?lhs = ?rhs")
```
```   149 proof
```
```   150   assume ?lhs
```
```   151   show ?rhs
```
```   152     by (rule add_le_imp_le_right[of _ "uminus b" _]) (simp add: add.assoc \<open>?lhs\<close>)
```
```   153 next
```
```   154   assume ?rhs
```
```   155   show ?lhs
```
```   156     by (rule add_le_imp_le_right[of _ "b" _]) (simp add: \<open>?rhs\<close>)
```
```   157 qed
```
```   158
```
```   159 lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
```
```   160 lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
```
```   161
```
```   162 lemma pprt_eq_id [simp, no_atp]: "0 \<le> x \<Longrightarrow> pprt x = x"
```
```   163   by (simp add: pprt_def sup_absorb1)
```
```   164
```
```   165 lemma nprt_eq_id [simp, no_atp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
```
```   166   by (simp add: nprt_def inf_absorb1)
```
```   167
```
```   168 lemma pprt_eq_0 [simp, no_atp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
```
```   169   by (simp add: pprt_def sup_absorb2)
```
```   170
```
```   171 lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
```
```   172   by (simp add: nprt_def inf_absorb2)
```
```   173
```
```   174 lemma sup_0_imp_0:
```
```   175   assumes "sup a (- a) = 0"
```
```   176   shows "a = 0"
```
```   177 proof -
```
```   178   have pos: "0 \<le> a" if "sup a (- a) = 0" for a :: 'a
```
```   179   proof -
```
```   180     from that have "sup a (- a) + a = a"
```
```   181       by simp
```
```   182     then have "sup (a + a) 0 = a"
```
```   183       by (simp add: add_sup_distrib_right)
```
```   184     then have "sup (a + a) 0 \<le> a"
```
```   185       by simp
```
```   186     then show ?thesis
```
```   187       by (blast intro: order_trans inf_sup_ord)
```
```   188   qed
```
```   189   from assms have **: "sup (-a) (-(-a)) = 0"
```
```   190     by (simp add: sup_commute)
```
```   191   from pos[OF assms] pos[OF **] show "a = 0"
```
```   192     by simp
```
```   193 qed
```
```   194
```
```   195 lemma inf_0_imp_0: "inf a (- a) = 0 \<Longrightarrow> a = 0"
```
```   196   apply (simp add: inf_eq_neg_sup)
```
```   197   apply (simp add: sup_commute)
```
```   198   apply (erule sup_0_imp_0)
```
```   199   done
```
```   200
```
```   201 lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
```
```   202   apply (rule iffI)
```
```   203    apply (erule inf_0_imp_0)
```
```   204   apply simp
```
```   205   done
```
```   206
```
```   207 lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
```
```   208   apply (rule iffI)
```
```   209    apply (erule sup_0_imp_0)
```
```   210   apply simp
```
```   211   done
```
```   212
```
```   213 lemma zero_le_double_add_iff_zero_le_single_add [simp]: "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
```
```   214   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   215 proof
```
```   216   show ?rhs if ?lhs
```
```   217   proof -
```
```   218     from that have a: "inf (a + a) 0 = 0"
```
```   219       by (simp add: inf_commute inf_absorb1)
```
```   220     have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a"  (is "?l = _")
```
```   221       by (simp add: add_sup_inf_distribs inf_aci)
```
```   222     then have "?l = 0 + inf a 0"
```
```   223       by (simp add: a, simp add: inf_commute)
```
```   224     then have "inf a 0 = 0"
```
```   225       by (simp only: add_right_cancel)
```
```   226     then show ?thesis
```
```   227       unfolding le_iff_inf by (simp add: inf_commute)
```
```   228   qed
```
```   229   show ?lhs if ?rhs
```
```   230     by (simp add: add_mono[OF that that, simplified])
```
```   231 qed
```
```   232
```
```   233 lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
```
```   234   using add_nonneg_eq_0_iff eq_iff by auto
```
```   235
```
```   236 lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
```
```   237   by (meson le_less_trans less_add_same_cancel2 less_le_not_le
```
```   238       zero_le_double_add_iff_zero_le_single_add)
```
```   239
```
```   240 lemma double_add_le_zero_iff_single_add_le_zero [simp]: "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   241 proof -
```
```   242   have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)"
```
```   243     by (subst le_minus_iff) simp
```
```   244   moreover have "\<dots> \<longleftrightarrow> a \<le> 0"
```
```   245     by (simp only: minus_add_distrib zero_le_double_add_iff_zero_le_single_add) simp
```
```   246   ultimately show ?thesis
```
```   247     by blast
```
```   248 qed
```
```   249
```
```   250 lemma double_add_less_zero_iff_single_less_zero [simp]: "a + a < 0 \<longleftrightarrow> a < 0"
```
```   251 proof -
```
```   252   have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)"
```
```   253     by (subst less_minus_iff) simp
```
```   254   moreover have "\<dots> \<longleftrightarrow> a < 0"
```
```   255     by (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp
```
```   256   ultimately show ?thesis
```
```   257     by blast
```
```   258 qed
```
```   259
```
```   260 declare neg_inf_eq_sup [simp]
```
```   261   and neg_sup_eq_inf [simp]
```
```   262   and diff_inf_eq_sup [simp]
```
```   263   and diff_sup_eq_inf [simp]
```
```   264
```
```   265 lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
```
```   266 proof -
```
```   267   from add_le_cancel_left [of "uminus a" "plus a a" zero]
```
```   268   have "a \<le> - a \<longleftrightarrow> a + a \<le> 0"
```
```   269     by (simp add: add.assoc[symmetric])
```
```   270   then show ?thesis
```
```   271     by simp
```
```   272 qed
```
```   273
```
```   274 lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
```
```   275 proof -
```
```   276   have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"
```
```   277     using add_le_cancel_left [of "uminus a" zero "plus a a"]
```
```   278     by (simp add: add.assoc[symmetric])
```
```   279   then show ?thesis
```
```   280     by simp
```
```   281 qed
```
```   282
```
```   283 lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
```
```   284   unfolding le_iff_inf by (simp add: nprt_def inf_commute)
```
```   285
```
```   286 lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
```
```   287   unfolding le_iff_sup by (simp add: pprt_def sup_commute)
```
```   288
```
```   289 lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
```
```   290   unfolding le_iff_sup by (simp add: pprt_def sup_commute)
```
```   291
```
```   292 lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
```
```   293   unfolding le_iff_inf by (simp add: nprt_def inf_commute)
```
```   294
```
```   295 lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
```
```   296   unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
```
```   297
```
```   298 lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
```
```   299   unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
```
```   300
```
```   301 end
```
```   302
```
```   303 lemmas add_sup_inf_distribs =
```
```   304   add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
```
```   305
```
```   306
```
```   307 class lattice_ab_group_add_abs = lattice_ab_group_add + abs +
```
```   308   assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
```
```   309 begin
```
```   310
```
```   311 lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
```
```   312 proof -
```
```   313   have "0 \<le> \<bar>a\<bar>"
```
```   314   proof -
```
```   315     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
```
```   316       by (auto simp add: abs_lattice)
```
```   317     show ?thesis
```
```   318       by (rule add_mono [OF a b, simplified])
```
```   319   qed
```
```   320   then have "0 \<le> sup a (- a)"
```
```   321     unfolding abs_lattice .
```
```   322   then have "sup (sup a (- a)) 0 = sup a (- a)"
```
```   323     by (rule sup_absorb1)
```
```   324   then show ?thesis
```
```   325     by (simp add: add_sup_inf_distribs ac_simps pprt_def nprt_def abs_lattice)
```
```   326 qed
```
```   327
```
```   328 subclass ordered_ab_group_add_abs
```
```   329 proof
```
```   330   have abs_ge_zero [simp]: "0 \<le> \<bar>a\<bar>" for a
```
```   331   proof -
```
```   332     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
```
```   333       by (auto simp add: abs_lattice)
```
```   334     show "0 \<le> \<bar>a\<bar>"
```
```   335       by (rule add_mono [OF a b, simplified])
```
```   336   qed
```
```   337   have abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" for a b
```
```   338     by (simp add: abs_lattice le_supI)
```
```   339   fix a b
```
```   340   show "0 \<le> \<bar>a\<bar>"
```
```   341     by simp
```
```   342   show "a \<le> \<bar>a\<bar>"
```
```   343     by (auto simp add: abs_lattice)
```
```   344   show "\<bar>-a\<bar> = \<bar>a\<bar>"
```
```   345     by (simp add: abs_lattice sup_commute)
```
```   346   show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" if "a \<le> b"
```
```   347     using that by (rule abs_leI)
```
```   348   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
```
```   349   proof -
```
```   350     have g: "\<bar>a\<bar> + \<bar>b\<bar> = sup (a + b) (sup (- a - b) (sup (- a + b) (a + (- b))))"
```
```   351       (is "_ = sup ?m ?n")
```
```   352       by (simp add: abs_lattice add_sup_inf_distribs ac_simps)
```
```   353     have a: "a + b \<le> sup ?m ?n"
```
```   354       by simp
```
```   355     have b: "- a - b \<le> ?n"
```
```   356       by simp
```
```   357     have c: "?n \<le> sup ?m ?n"
```
```   358       by simp
```
```   359     from b c have d: "- a - b \<le> sup ?m ?n"
```
```   360       by (rule order_trans)
```
```   361     have e: "- a - b = - (a + b)"
```
```   362       by simp
```
```   363     from a d e have "\<bar>a + b\<bar> \<le> sup ?m ?n"
```
```   364       apply -
```
```   365       apply (drule abs_leI)
```
```   366        apply (simp_all only: algebra_simps minus_add)
```
```   367       apply (metis add_uminus_conv_diff d sup_commute uminus_add_conv_diff)
```
```   368       done
```
```   369     with g[symmetric] show ?thesis by simp
```
```   370   qed
```
```   371 qed
```
```   372
```
```   373 end
```
```   374
```
```   375 lemma sup_eq_if:
```
```   376   fixes a :: "'a::{lattice_ab_group_add,linorder}"
```
```   377   shows "sup a (- a) = (if a < 0 then - a else a)"
```
```   378   using add_le_cancel_right [of a a "- a", symmetric, simplified]
```
```   379     and add_le_cancel_right [of "-a" a a, symmetric, simplified]
```
```   380   by (auto simp: sup_max max.absorb1 max.absorb2)
```
```   381
```
```   382 lemma abs_if_lattice:
```
```   383   fixes a :: "'a::{lattice_ab_group_add_abs,linorder}"
```
```   384   shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
```
```   385   by auto
```
```   386
```
```   387 lemma estimate_by_abs:
```
```   388   fixes a b c :: "'a::lattice_ab_group_add_abs"
```
```   389   assumes "a + b \<le> c"
```
```   390   shows "a \<le> c + \<bar>b\<bar>"
```
```   391 proof -
```
```   392   from assms have "a \<le> c + (- b)"
```
```   393     by (simp add: algebra_simps)
```
```   394   have "- b \<le> \<bar>b\<bar>"
```
```   395     by (rule abs_ge_minus_self)
```
```   396   then have "c + (- b) \<le> c + \<bar>b\<bar>"
```
```   397     by (rule add_left_mono)
```
```   398   with \<open>a \<le> c + (- b)\<close> show ?thesis
```
```   399     by (rule order_trans)
```
```   400 qed
```
```   401
```
```   402 class lattice_ring = ordered_ring + lattice_ab_group_add_abs
```
```   403 begin
```
```   404
```
```   405 subclass semilattice_inf_ab_group_add ..
```
```   406 subclass semilattice_sup_ab_group_add ..
```
```   407
```
```   408 end
```
```   409
```
```   410 lemma abs_le_mult:
```
```   411   fixes a b :: "'a::lattice_ring"
```
```   412   shows "\<bar>a * b\<bar> \<le> \<bar>a\<bar> * \<bar>b\<bar>"
```
```   413 proof -
```
```   414   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
```
```   415   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
```
```   416   have a: "\<bar>a\<bar> * \<bar>b\<bar> = ?x"
```
```   417     by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
```
```   418   have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow>
```
```   419             u * v = pprt a * pprt b + pprt a * nprt b +
```
```   420                     nprt a * pprt b + nprt a * nprt b" for u v :: 'a
```
```   421     apply (subst prts[of u], subst prts[of v])
```
```   422     apply (simp add: algebra_simps)
```
```   423     done
```
```   424   note b = this[OF refl[of a] refl[of b]]
```
```   425   have xy: "- ?x \<le> ?y"
```
```   426     apply simp
```
```   427     apply (metis (full_types) add_increasing add_uminus_conv_diff
```
```   428       lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg
```
```   429       mult_nonpos_nonpos nprt_le_zero zero_le_pprt)
```
```   430     done
```
```   431   have yx: "?y \<le> ?x"
```
```   432     apply simp
```
```   433     apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff
```
```   434       lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos
```
```   435       mult_nonpos_nonneg nprt_le_zero zero_le_pprt)
```
```   436     done
```
```   437   have i1: "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>"
```
```   438     by (simp only: a b yx)
```
```   439   have i2: "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b"
```
```   440     by (simp only: a b xy)
```
```   441   show ?thesis
```
```   442     apply (rule abs_leI)
```
```   443     apply (simp add: i1)
```
```   444     apply (simp add: i2[simplified minus_le_iff])
```
```   445     done
```
```   446 qed
```
```   447
```
```   448 instance lattice_ring \<subseteq> ordered_ring_abs
```
```   449 proof
```
```   450   fix a b :: "'a::lattice_ring"
```
```   451   assume a: "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
```
```   452   show "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
```
```   453   proof -
```
```   454     have s: "(0 \<le> a * b) \<or> (a * b \<le> 0)"
```
```   455       apply auto
```
```   456       apply (rule_tac split_mult_pos_le)
```
```   457       apply (rule_tac contrapos_np[of "a * b \<le> 0"])
```
```   458       apply simp
```
```   459       apply (rule_tac split_mult_neg_le)
```
```   460       using a
```
```   461       apply blast
```
```   462       done
```
```   463     have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
```
```   464       by (simp add: prts[symmetric])
```
```   465     show ?thesis
```
```   466     proof (cases "0 \<le> a * b")
```
```   467       case True
```
```   468       then show ?thesis
```
```   469         apply (simp_all add: mulprts abs_prts)
```
```   470         using a
```
```   471         apply (auto simp add:
```
```   472           algebra_simps
```
```   473           iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
```
```   474           iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
```
```   475         apply(drule (1) mult_nonneg_nonpos[of a b], simp)
```
```   476         apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
```
```   477         done
```
```   478     next
```
```   479       case False
```
```   480       with s have "a * b \<le> 0"
```
```   481         by simp
```
```   482       then show ?thesis
```
```   483         apply (simp_all add: mulprts abs_prts)
```
```   484         apply (insert a)
```
```   485         apply (auto simp add: algebra_simps)
```
```   486         apply(drule (1) mult_nonneg_nonneg[of a b],simp)
```
```   487         apply(drule (1) mult_nonpos_nonpos[of a b],simp)
```
```   488         done
```
```   489     qed
```
```   490   qed
```
```   491 qed
```
```   492
```
```   493 lemma mult_le_prts:
```
```   494   fixes a b :: "'a::lattice_ring"
```
```   495   assumes "a1 \<le> a"
```
```   496     and "a \<le> a2"
```
```   497     and "b1 \<le> b"
```
```   498     and "b \<le> b2"
```
```   499   shows "a * b \<le>
```
```   500     pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
```
```   501 proof -
```
```   502   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
```
```   503     by (subst prts[symmetric])+ simp
```
```   504   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
```
```   505     by (simp add: algebra_simps)
```
```   506   moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2"
```
```   507     by (simp_all add: assms mult_mono)
```
```   508   moreover have "pprt a * nprt b \<le> pprt a1 * nprt b2"
```
```   509   proof -
```
```   510     have "pprt a * nprt b \<le> pprt a * nprt b2"
```
```   511       by (simp add: mult_left_mono assms)
```
```   512     moreover have "pprt a * nprt b2 \<le> pprt a1 * nprt b2"
```
```   513       by (simp add: mult_right_mono_neg assms)
```
```   514     ultimately show ?thesis
```
```   515       by simp
```
```   516   qed
```
```   517   moreover have "nprt a * pprt b \<le> nprt a2 * pprt b1"
```
```   518   proof -
```
```   519     have "nprt a * pprt b \<le> nprt a2 * pprt b"
```
```   520       by (simp add: mult_right_mono assms)
```
```   521     moreover have "nprt a2 * pprt b \<le> nprt a2 * pprt b1"
```
```   522       by (simp add: mult_left_mono_neg assms)
```
```   523     ultimately show ?thesis
```
```   524       by simp
```
```   525   qed
```
```   526   moreover have "nprt a * nprt b \<le> nprt a1 * nprt b1"
```
```   527   proof -
```
```   528     have "nprt a * nprt b \<le> nprt a * nprt b1"
```
```   529       by (simp add: mult_left_mono_neg assms)
```
```   530     moreover have "nprt a * nprt b1 \<le> nprt a1 * nprt b1"
```
```   531       by (simp add: mult_right_mono_neg assms)
```
```   532     ultimately show ?thesis
```
```   533       by simp
```
```   534   qed
```
```   535   ultimately show ?thesis
```
```   536     by - (rule add_mono | simp)+
```
```   537 qed
```
```   538
```
```   539 lemma mult_ge_prts:
```
```   540   fixes a b :: "'a::lattice_ring"
```
```   541   assumes "a1 \<le> a"
```
```   542     and "a \<le> a2"
```
```   543     and "b1 \<le> b"
```
```   544     and "b \<le> b2"
```
```   545   shows "a * b \<ge>
```
```   546     nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
```
```   547 proof -
```
```   548   from assms have a1: "- a2 \<le> -a"
```
```   549     by auto
```
```   550   from assms have a2: "- a \<le> -a1"
```
```   551     by auto
```
```   552   from mult_le_prts[of "- a2" "- a" "- a1" "b1" b "b2",
```
```   553     OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
```
```   554   have le: "- (a * b) \<le>
```
```   555     - nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
```
```   556     - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
```
```   557     by simp
```
```   558   then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
```
```   559       - pprt a1 * pprt b1 + - pprt a2 * nprt b1) \<le> a * b"
```
```   560     by (simp only: minus_le_iff)
```
```   561   then show ?thesis
```
```   562     by (simp add: algebra_simps)
```
```   563 qed
```
```   564
```
```   565 instance int :: lattice_ring
```
```   566 proof
```
```   567   show "\<bar>k\<bar> = sup k (- k)" for k :: int
```
```   568     by (auto simp add: sup_int_def)
```
```   569 qed
```
```   570
```
```   571 instance real :: lattice_ring
```
```   572 proof
```
```   573   show "\<bar>a\<bar> = sup a (- a)" for a :: real
```
```   574     by (auto simp add: sup_real_def)
```
```   575 qed
```
```   576
```
```   577 end
```