src/HOL/Library/Lattice_Algebras.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 57862 8f074e6e22fc child 58881 b9556a055632 permissions -rw-r--r--
simpler proof
```     1 (* Author: Steven Obua, TU Muenchen *)
```
```     2
```
```     3 header {* Various algebraic structures combined with a lattice *}
```
```     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   assume "a \<le> c" "b \<le> c"
```
```    84   then show "- inf (- a) (- b) \<le> c"
```
```    85     by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)
```
```    86 qed
```
```    87
```
```    88 lemma neg_inf_eq_sup: "- inf a b = sup (- a) (- b)"
```
```    89   by (simp add: inf_eq_neg_sup)
```
```    90
```
```    91 lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"
```
```    92   using neg_inf_eq_sup [of b c, symmetric] by simp
```
```    93
```
```    94 lemma neg_sup_eq_inf: "- sup a b = inf (- a) (- b)"
```
```    95   by (simp add: sup_eq_neg_inf)
```
```    96
```
```    97 lemma diff_sup_eq_inf: "a - sup b c = a + inf (- b) (- c)"
```
```    98   using neg_sup_eq_inf [of b c, symmetric] by simp
```
```    99
```
```   100 lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
```
```   101 proof -
```
```   102   have "0 = - inf 0 (a - b) + inf (a - b) 0"
```
```   103     by (simp add: inf_commute)
```
```   104   then have "0 = sup 0 (b - a) + inf (a - b) 0"
```
```   105     by (simp add: inf_eq_neg_sup)
```
```   106   then have "0 = (- a + sup a b) + (inf a b + (- b))"
```
```   107     by (simp only: add_sup_distrib_left add_inf_distrib_right) simp
```
```   108   then show ?thesis
```
```   109     by (simp add: algebra_simps)
```
```   110 qed
```
```   111
```
```   112
```
```   113 subsection {* Positive Part, Negative Part, Absolute Value *}
```
```   114
```
```   115 definition nprt :: "'a \<Rightarrow> 'a"
```
```   116   where "nprt x = inf x 0"
```
```   117
```
```   118 definition pprt :: "'a \<Rightarrow> 'a"
```
```   119   where "pprt x = sup x 0"
```
```   120
```
```   121 lemma pprt_neg: "pprt (- x) = - nprt x"
```
```   122 proof -
```
```   123   have "sup (- x) 0 = sup (- x) (- 0)"
```
```   124     unfolding minus_zero ..
```
```   125   also have "\<dots> = - inf x 0"
```
```   126     unfolding neg_inf_eq_sup ..
```
```   127   finally have "sup (- x) 0 = - inf x 0" .
```
```   128   then show ?thesis
```
```   129     unfolding pprt_def nprt_def .
```
```   130 qed
```
```   131
```
```   132 lemma nprt_neg: "nprt (- x) = - pprt x"
```
```   133 proof -
```
```   134   from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
```
```   135   then have "pprt x = - nprt (- x)" by simp
```
```   136   then show ?thesis by simp
```
```   137 qed
```
```   138
```
```   139 lemma prts: "a = pprt a + nprt a"
```
```   140   by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
```
```   141
```
```   142 lemma zero_le_pprt[simp]: "0 \<le> pprt a"
```
```   143   by (simp add: pprt_def)
```
```   144
```
```   145 lemma nprt_le_zero[simp]: "nprt a \<le> 0"
```
```   146   by (simp add: nprt_def)
```
```   147
```
```   148 lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
```
```   149 proof
```
```   150   assume ?l
```
```   151   then show ?r
```
```   152     apply -
```
```   153     apply (rule add_le_imp_le_right[of _ "uminus b" _])
```
```   154     apply (simp add: add.assoc)
```
```   155     done
```
```   156 next
```
```   157   assume ?r
```
```   158   then show ?l
```
```   159     apply -
```
```   160     apply (rule add_le_imp_le_right[of _ "b" _])
```
```   161     apply simp
```
```   162     done
```
```   163 qed
```
```   164
```
```   165 lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
```
```   166 lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
```
```   167
```
```   168 lemma pprt_eq_id [simp, no_atp]: "0 \<le> x \<Longrightarrow> pprt x = x"
```
```   169   by (simp add: pprt_def sup_absorb1)
```
```   170
```
```   171 lemma nprt_eq_id [simp, no_atp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
```
```   172   by (simp add: nprt_def inf_absorb1)
```
```   173
```
```   174 lemma pprt_eq_0 [simp, no_atp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
```
```   175   by (simp add: pprt_def sup_absorb2)
```
```   176
```
```   177 lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
```
```   178   by (simp add: nprt_def inf_absorb2)
```
```   179
```
```   180 lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
```
```   181 proof -
```
```   182   {
```
```   183     fix a :: 'a
```
```   184     assume hyp: "sup a (- a) = 0"
```
```   185     then have "sup a (- a) + a = a"
```
```   186       by simp
```
```   187     then have "sup (a + a) 0 = a"
```
```   188       by (simp add: add_sup_distrib_right)
```
```   189     then have "sup (a + a) 0 \<le> a"
```
```   190       by simp
```
```   191     then have "0 \<le> a"
```
```   192       by (blast intro: order_trans inf_sup_ord)
```
```   193   }
```
```   194   note p = this
```
```   195   assume hyp:"sup a (-a) = 0"
```
```   196   then have hyp2:"sup (-a) (-(-a)) = 0"
```
```   197     by (simp add: sup_commute)
```
```   198   from p[OF hyp] p[OF hyp2] show "a = 0"
```
```   199     by simp
```
```   200 qed
```
```   201
```
```   202 lemma inf_0_imp_0: "inf a (- a) = 0 \<Longrightarrow> a = 0"
```
```   203   apply (simp add: inf_eq_neg_sup)
```
```   204   apply (simp add: sup_commute)
```
```   205   apply (erule sup_0_imp_0)
```
```   206   done
```
```   207
```
```   208 lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
```
```   209   apply rule
```
```   210   apply (erule inf_0_imp_0)
```
```   211   apply simp
```
```   212   done
```
```   213
```
```   214 lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
```
```   215   apply rule
```
```   216   apply (erule sup_0_imp_0)
```
```   217   apply simp
```
```   218   done
```
```   219
```
```   220 lemma zero_le_double_add_iff_zero_le_single_add [simp]:
```
```   221   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
```
```   222 proof
```
```   223   assume "0 \<le> a + a"
```
```   224   then have a: "inf (a + a) 0 = 0"
```
```   225     by (simp add: inf_commute inf_absorb1)
```
```   226   have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a"  (is "?l=_")
```
```   227     by (simp add: add_sup_inf_distribs inf_aci)
```
```   228   then have "?l = 0 + inf a 0"
```
```   229     by (simp add: a, simp add: inf_commute)
```
```   230   then have "inf a 0 = 0"
```
```   231     by (simp only: add_right_cancel)
```
```   232   then show "0 \<le> a"
```
```   233     unfolding le_iff_inf by (simp add: inf_commute)
```
```   234 next
```
```   235   assume a: "0 \<le> a"
```
```   236   show "0 \<le> a + a"
```
```   237     by (simp add: add_mono[OF a a, simplified])
```
```   238 qed
```
```   239
```
```   240 lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
```
```   241 proof
```
```   242   assume assm: "a + a = 0"
```
```   243   then have "a + a + - a = - a"
```
```   244     by simp
```
```   245   then have "a + (a + - a) = - a"
```
```   246     by (simp only: add.assoc)
```
```   247   then have a: "- a = a"
```
```   248     by simp
```
```   249   show "a = 0"
```
```   250     apply (rule antisym)
```
```   251     apply (unfold neg_le_iff_le [symmetric, of a])
```
```   252     unfolding a
```
```   253     apply simp
```
```   254     unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
```
```   255     unfolding assm
```
```   256     unfolding le_less
```
```   257     apply simp_all
```
```   258     done
```
```   259 next
```
```   260   assume "a = 0"
```
```   261   then show "a + a = 0"
```
```   262     by simp
```
```   263 qed
```
```   264
```
```   265 lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
```
```   266 proof (cases "a = 0")
```
```   267   case True
```
```   268   then show ?thesis by auto
```
```   269 next
```
```   270   case False
```
```   271   then show ?thesis
```
```   272     unfolding less_le
```
```   273     apply simp
```
```   274     apply rule
```
```   275     apply clarify
```
```   276     apply rule
```
```   277     apply assumption
```
```   278     apply (rule notI)
```
```   279     unfolding double_zero [symmetric, of a]
```
```   280     apply blast
```
```   281     done
```
```   282 qed
```
```   283
```
```   284 lemma double_add_le_zero_iff_single_add_le_zero [simp]:
```
```   285   "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   286 proof -
```
```   287   have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)"
```
```   288     by (subst le_minus_iff, simp)
```
```   289   moreover have "\<dots> \<longleftrightarrow> a \<le> 0"
```
```   290     by (simp only: minus_add_distrib zero_le_double_add_iff_zero_le_single_add) simp
```
```   291   ultimately show ?thesis
```
```   292     by blast
```
```   293 qed
```
```   294
```
```   295 lemma double_add_less_zero_iff_single_less_zero [simp]:
```
```   296   "a + a < 0 \<longleftrightarrow> a < 0"
```
```   297 proof -
```
```   298   have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)"
```
```   299     by (subst less_minus_iff) simp
```
```   300   moreover have "\<dots> \<longleftrightarrow> a < 0"
```
```   301     by (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp
```
```   302   ultimately show ?thesis
```
```   303     by blast
```
```   304 qed
```
```   305
```
```   306 declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp] diff_inf_eq_sup [simp] diff_sup_eq_inf [simp]
```
```   307
```
```   308 lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
```
```   309 proof -
```
```   310   from add_le_cancel_left [of "uminus a" "plus a a" zero]
```
```   311   have "a \<le> - a \<longleftrightarrow> a + a \<le> 0"
```
```   312     by (simp add: add.assoc[symmetric])
```
```   313   then show ?thesis
```
```   314     by simp
```
```   315 qed
```
```   316
```
```   317 lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
```
```   318 proof -
```
```   319   from add_le_cancel_left [of "uminus a" zero "plus a a"]
```
```   320   have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"
```
```   321     by (simp add: add.assoc[symmetric])
```
```   322   then show ?thesis
```
```   323     by simp
```
```   324 qed
```
```   325
```
```   326 lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
```
```   327   unfolding le_iff_inf by (simp add: nprt_def inf_commute)
```
```   328
```
```   329 lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
```
```   330   unfolding le_iff_sup by (simp add: pprt_def sup_commute)
```
```   331
```
```   332 lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
```
```   333   unfolding le_iff_sup by (simp add: pprt_def sup_commute)
```
```   334
```
```   335 lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
```
```   336   unfolding le_iff_inf by (simp add: nprt_def inf_commute)
```
```   337
```
```   338 lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
```
```   339   unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
```
```   340
```
```   341 lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
```
```   342   unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
```
```   343
```
```   344 end
```
```   345
```
```   346 lemmas add_sup_inf_distribs =
```
```   347   add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
```
```   348
```
```   349
```
```   350 class lattice_ab_group_add_abs = lattice_ab_group_add + abs +
```
```   351   assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
```
```   352 begin
```
```   353
```
```   354 lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
```
```   355 proof -
```
```   356   have "0 \<le> \<bar>a\<bar>"
```
```   357   proof -
```
```   358     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
```
```   359       by (auto simp add: abs_lattice)
```
```   360     show ?thesis
```
```   361       by (rule add_mono [OF a b, simplified])
```
```   362   qed
```
```   363   then have "0 \<le> sup a (- a)"
```
```   364     unfolding abs_lattice .
```
```   365   then have "sup (sup a (- a)) 0 = sup a (- a)"
```
```   366     by (rule sup_absorb1)
```
```   367   then show ?thesis
```
```   368     by (simp add: add_sup_inf_distribs ac_simps pprt_def nprt_def abs_lattice)
```
```   369 qed
```
```   370
```
```   371 subclass ordered_ab_group_add_abs
```
```   372 proof
```
```   373   have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
```
```   374   proof -
```
```   375     fix a b
```
```   376     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
```
```   377       by (auto simp add: abs_lattice)
```
```   378     show "0 \<le> \<bar>a\<bar>"
```
```   379       by (rule add_mono [OF a b, simplified])
```
```   380   qed
```
```   381   have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
```
```   382     by (simp add: abs_lattice le_supI)
```
```   383   fix a b
```
```   384   show "0 \<le> \<bar>a\<bar>"
```
```   385     by simp
```
```   386   show "a \<le> \<bar>a\<bar>"
```
```   387     by (auto simp add: abs_lattice)
```
```   388   show "\<bar>-a\<bar> = \<bar>a\<bar>"
```
```   389     by (simp add: abs_lattice sup_commute)
```
```   390   {
```
```   391     assume "a \<le> b"
```
```   392     then show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
```
```   393       by (rule abs_leI)
```
```   394   }
```
```   395   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
```
```   396   proof -
```
```   397     have g: "\<bar>a\<bar> + \<bar>b\<bar> = sup (a + b) (sup (- a - b) (sup (- a + b) (a + (- b))))"
```
```   398       (is "_=sup ?m ?n")
```
```   399       by (simp add: abs_lattice add_sup_inf_distribs ac_simps)
```
```   400     have a: "a + b \<le> sup ?m ?n"
```
```   401       by simp
```
```   402     have b: "- a - b \<le> ?n"
```
```   403       by simp
```
```   404     have c: "?n \<le> sup ?m ?n"
```
```   405       by simp
```
```   406     from b c have d: "- a - b \<le> sup ?m ?n"
```
```   407       by (rule order_trans)
```
```   408     have e: "- a - b = - (a + b)"
```
```   409       by simp
```
```   410     from a d e have "\<bar>a + b\<bar> \<le> sup ?m ?n"
```
```   411       apply -
```
```   412       apply (drule abs_leI)
```
```   413       apply (simp_all only: algebra_simps minus_add)
```
```   414       apply (metis add_uminus_conv_diff d sup_commute uminus_add_conv_diff)
```
```   415       done
```
```   416     with g[symmetric] show ?thesis by simp
```
```   417   qed
```
```   418 qed
```
```   419
```
```   420 end
```
```   421
```
```   422 lemma sup_eq_if:
```
```   423   fixes a :: "'a::{lattice_ab_group_add, linorder}"
```
```   424   shows "sup a (- a) = (if a < 0 then - a else a)"
```
```   425 proof -
```
```   426   note add_le_cancel_right [of a a "- a", symmetric, simplified]
```
```   427   moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
```
```   428   then show ?thesis by (auto simp: sup_max max.absorb1 max.absorb2)
```
```   429 qed
```
```   430
```
```   431 lemma abs_if_lattice:
```
```   432   fixes a :: "'a::{lattice_ab_group_add_abs, linorder}"
```
```   433   shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
```
```   434   by auto
```
```   435
```
```   436 lemma estimate_by_abs:
```
```   437   fixes a b c :: "'a::lattice_ab_group_add_abs"
```
```   438   shows "a + b \<le> c \<Longrightarrow> a \<le> c + \<bar>b\<bar>"
```
```   439 proof -
```
```   440   assume "a + b \<le> c"
```
```   441   then have "a \<le> c + (- b)"
```
```   442     by (simp add: algebra_simps)
```
```   443   have "- b \<le> \<bar>b\<bar>"
```
```   444     by (rule abs_ge_minus_self)
```
```   445   then have "c + (- b) \<le> c + \<bar>b\<bar>"
```
```   446     by (rule add_left_mono)
```
```   447   with `a \<le> c + (- b)` show ?thesis
```
```   448     by (rule order_trans)
```
```   449 qed
```
```   450
```
```   451 class lattice_ring = ordered_ring + lattice_ab_group_add_abs
```
```   452 begin
```
```   453
```
```   454 subclass semilattice_inf_ab_group_add ..
```
```   455 subclass semilattice_sup_ab_group_add ..
```
```   456
```
```   457 end
```
```   458
```
```   459 lemma abs_le_mult:
```
```   460   fixes a b :: "'a::lattice_ring"
```
```   461   shows "\<bar>a * b\<bar> \<le> \<bar>a\<bar> * \<bar>b\<bar>"
```
```   462 proof -
```
```   463   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
```
```   464   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
```
```   465   have a: "\<bar>a\<bar> * \<bar>b\<bar> = ?x"
```
```   466     by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
```
```   467   {
```
```   468     fix u v :: 'a
```
```   469     have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow>
```
```   470               u * v = pprt a * pprt b + pprt a * nprt b +
```
```   471                       nprt a * pprt b + nprt a * nprt b"
```
```   472       apply (subst prts[of u], subst prts[of v])
```
```   473       apply (simp add: algebra_simps)
```
```   474       done
```
```   475   }
```
```   476   note b = this[OF refl[of a] refl[of b]]
```
```   477   have xy: "- ?x \<le> ?y"
```
```   478     apply simp
```
```   479     apply (metis (full_types) add_increasing add_uminus_conv_diff
```
```   480       lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg
```
```   481       mult_nonpos_nonpos nprt_le_zero zero_le_pprt)
```
```   482     done
```
```   483   have yx: "?y \<le> ?x"
```
```   484     apply simp
```
```   485     apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff
```
```   486       lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos
```
```   487       mult_nonpos_nonneg nprt_le_zero zero_le_pprt)
```
```   488     done
```
```   489   have i1: "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>"
```
```   490     by (simp only: a b yx)
```
```   491   have i2: "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b"
```
```   492     by (simp only: a b xy)
```
```   493   show ?thesis
```
```   494     apply (rule abs_leI)
```
```   495     apply (simp add: i1)
```
```   496     apply (simp add: i2[simplified minus_le_iff])
```
```   497     done
```
```   498 qed
```
```   499
```
```   500 instance lattice_ring \<subseteq> ordered_ring_abs
```
```   501 proof
```
```   502   fix a b :: "'a::lattice_ring"
```
```   503   assume a: "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
```
```   504   show "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
```
```   505   proof -
```
```   506     have s: "(0 \<le> a * b) \<or> (a * b \<le> 0)"
```
```   507       apply auto
```
```   508       apply (rule_tac split_mult_pos_le)
```
```   509       apply (rule_tac contrapos_np[of "a * b \<le> 0"])
```
```   510       apply simp
```
```   511       apply (rule_tac split_mult_neg_le)
```
```   512       using a
```
```   513       apply blast
```
```   514       done
```
```   515     have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
```
```   516       by (simp add: prts[symmetric])
```
```   517     show ?thesis
```
```   518     proof (cases "0 \<le> a * b")
```
```   519       case True
```
```   520       then show ?thesis
```
```   521         apply (simp_all add: mulprts abs_prts)
```
```   522         using a
```
```   523         apply (auto simp add:
```
```   524           algebra_simps
```
```   525           iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
```
```   526           iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
```
```   527         apply(drule (1) mult_nonneg_nonpos[of a b], simp)
```
```   528         apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
```
```   529         done
```
```   530     next
```
```   531       case False
```
```   532       with s have "a * b \<le> 0"
```
```   533         by simp
```
```   534       then show ?thesis
```
```   535         apply (simp_all add: mulprts abs_prts)
```
```   536         apply (insert a)
```
```   537         apply (auto simp add: algebra_simps)
```
```   538         apply(drule (1) mult_nonneg_nonneg[of a b],simp)
```
```   539         apply(drule (1) mult_nonpos_nonpos[of a b],simp)
```
```   540         done
```
```   541     qed
```
```   542   qed
```
```   543 qed
```
```   544
```
```   545 lemma mult_le_prts:
```
```   546   fixes a b :: "'a::lattice_ring"
```
```   547   assumes "a1 \<le> a"
```
```   548     and "a \<le> a2"
```
```   549     and "b1 \<le> b"
```
```   550     and "b \<le> b2"
```
```   551   shows "a * b \<le>
```
```   552     pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
```
```   553 proof -
```
```   554   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
```
```   555     apply (subst prts[symmetric])+
```
```   556     apply simp
```
```   557     done
```
```   558   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
```
```   559     by (simp add: algebra_simps)
```
```   560   moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2"
```
```   561     by (simp_all add: assms mult_mono)
```
```   562   moreover have "pprt a * nprt b \<le> pprt a1 * nprt b2"
```
```   563   proof -
```
```   564     have "pprt a * nprt b \<le> pprt a * nprt b2"
```
```   565       by (simp add: mult_left_mono assms)
```
```   566     moreover have "pprt a * nprt b2 \<le> pprt a1 * nprt b2"
```
```   567       by (simp add: mult_right_mono_neg assms)
```
```   568     ultimately show ?thesis
```
```   569       by simp
```
```   570   qed
```
```   571   moreover have "nprt a * pprt b \<le> nprt a2 * pprt b1"
```
```   572   proof -
```
```   573     have "nprt a * pprt b \<le> nprt a2 * pprt b"
```
```   574       by (simp add: mult_right_mono assms)
```
```   575     moreover have "nprt a2 * pprt b \<le> nprt a2 * pprt b1"
```
```   576       by (simp add: mult_left_mono_neg assms)
```
```   577     ultimately show ?thesis
```
```   578       by simp
```
```   579   qed
```
```   580   moreover have "nprt a * nprt b \<le> nprt a1 * nprt b1"
```
```   581   proof -
```
```   582     have "nprt a * nprt b \<le> nprt a * nprt b1"
```
```   583       by (simp add: mult_left_mono_neg assms)
```
```   584     moreover have "nprt a * nprt b1 \<le> nprt a1 * nprt b1"
```
```   585       by (simp add: mult_right_mono_neg assms)
```
```   586     ultimately show ?thesis
```
```   587       by simp
```
```   588   qed
```
```   589   ultimately show ?thesis
```
```   590     apply -
```
```   591     apply (rule add_mono | simp)+
```
```   592     done
```
```   593 qed
```
```   594
```
```   595 lemma mult_ge_prts:
```
```   596   fixes a b :: "'a::lattice_ring"
```
```   597   assumes "a1 \<le> a"
```
```   598     and "a \<le> a2"
```
```   599     and "b1 \<le> b"
```
```   600     and "b \<le> b2"
```
```   601   shows "a * b \<ge>
```
```   602     nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
```
```   603 proof -
```
```   604   from assms have a1: "- a2 \<le> -a"
```
```   605     by auto
```
```   606   from assms have a2: "- a \<le> -a1"
```
```   607     by auto
```
```   608   from mult_le_prts[of "- a2" "- a" "- a1" "b1" b "b2",
```
```   609     OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
```
```   610   have le: "- (a * b) \<le> - nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
```
```   611     - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
```
```   612     by simp
```
```   613   then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
```
```   614       - pprt a1 * pprt b1 + - pprt a2 * nprt b1) \<le> a * b"
```
```   615     by (simp only: minus_le_iff)
```
```   616   then show ?thesis
```
```   617     by (simp add: algebra_simps)
```
```   618 qed
```
```   619
```
```   620 instance int :: lattice_ring
```
```   621 proof
```
```   622   fix k :: int
```
```   623   show "\<bar>k\<bar> = sup k (- k)"
```
```   624     by (auto simp add: sup_int_def)
```
```   625 qed
```
```   626
```
```   627 instance real :: lattice_ring
```
```   628 proof
```
```   629   fix a :: real
```
```   630   show "\<bar>a\<bar> = sup a (- a)"
```
```   631     by (auto simp add: sup_real_def)
```
```   632 qed
```
```   633
```
```   634 end
```
```   635
```