src/HOL/Library/Lattice_Algebras.thy
 changeset 53240 07593a0a27f4 parent 46986 8198cbff1771 child 54230 b1d955791529
```     1.1 --- a/src/HOL/Library/Lattice_Algebras.thy	Tue Aug 27 23:21:12 2013 +0200
1.2 +++ b/src/HOL/Library/Lattice_Algebras.thy	Tue Aug 27 23:54:23 2013 +0200
1.3 @@ -9,20 +9,19 @@
1.5  begin
1.6
1.8 -  "a + inf b c = inf (a + b) (a + c)"
1.9 -apply (rule antisym)
1.11 -apply (rule add_le_imp_le_left [of "uminus a"])
1.12 -apply (simp only: add_assoc [symmetric], simp)
1.13 -apply rule
1.15 -done
1.16 +lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + c)"
1.17 +  apply (rule antisym)
1.18 +  apply (simp_all add: le_infI)
1.19 +  apply (rule add_le_imp_le_left [of "uminus a"])
1.20 +  apply (simp only: add_assoc [symmetric], simp)
1.21 +  apply rule
1.23 +  done
1.24
1.26 -  "inf a b + c = inf (a + c) (b + c)"
1.27 +lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)"
1.28  proof -
1.29 -  have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
1.30 +  have "c + inf a b = inf (c+a) (c+b)"
1.33  qed
1.34
1.35 @@ -31,19 +30,17 @@
1.37  begin
1.38
1.40 -  "a + sup b c = sup (a + b) (a + c)"
1.41 -apply (rule antisym)
1.42 -apply (rule add_le_imp_le_left [of "uminus a"])
1.43 -apply (simp only: add_assoc[symmetric], simp)
1.44 -apply rule
1.46 -apply (rule le_supI)
1.47 -apply (simp_all)
1.48 -done
1.49 +lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a + c)"
1.50 +  apply (rule antisym)
1.51 +  apply (rule add_le_imp_le_left [of "uminus a"])
1.52 +  apply (simp only: add_assoc[symmetric], simp)
1.53 +  apply rule
1.55 +  apply (rule le_supI)
1.56 +  apply (simp_all)
1.57 +  done
1.58
1.60 -  "sup a b + c = sup (a+c) (b+c)"
1.61 +lemma add_sup_distrib_right: "sup a b + c = sup (a+c) (b+c)"
1.62  proof -
1.63    have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
1.65 @@ -57,69 +54,61 @@
1.68
1.72
1.73  lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
1.74  proof (rule inf_unique)
1.75 -  fix a b :: 'a
1.76 +  fix a b c :: 'a
1.77    show "- sup (-a) (-b) \<le> a"
1.78      by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
1.80 -next
1.81 -  fix a b :: 'a
1.82    show "- sup (-a) (-b) \<le> b"
1.83      by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
1.85 -next
1.86 -  fix a b c :: 'a
1.87    assume "a \<le> b" "a \<le> c"
1.88 -  then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
1.90 +  then show "a \<le> - sup (-b) (-c)"
1.91 +    by (subst neg_le_iff_le [symmetric]) (simp add: le_supI)
1.92  qed
1.93 -
1.94 +
1.95  lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
1.96  proof (rule sup_unique)
1.97 -  fix a b :: 'a
1.98 +  fix a b c :: 'a
1.99    show "a \<le> - inf (-a) (-b)"
1.100      by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
1.102 -next
1.103 -  fix a b :: 'a
1.104    show "b \<le> - inf (-a) (-b)"
1.105      by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
1.107 -next
1.108 -  fix a b c :: 'a
1.109    assume "a \<le> c" "b \<le> c"
1.110 -  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
1.112 +  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)
1.113  qed
1.114
1.115  lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
1.117 +  by (simp add: inf_eq_neg_sup)
1.118
1.119  lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
1.121 +  by (simp add: sup_eq_neg_inf)
1.122
1.123  lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
1.124  proof -
1.125 -  have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
1.126 -  hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
1.127 +  have "0 = - inf 0 (a-b) + inf (a-b) 0"
1.128 +    by (simp add: inf_commute)
1.129 +  hence "0 = sup 0 (b-a) + inf (a-b) 0"
1.130 +    by (simp add: inf_eq_neg_sup)
1.131    hence "0 = (-a + sup a b) + (inf a b + (-b))"
1.135    thus ?thesis by (simp add: algebra_simps)
1.136  qed
1.137
1.138 +
1.139  subsection {* Positive Part, Negative Part, Absolute Value *}
1.140
1.141 -definition
1.142 -  nprt :: "'a \<Rightarrow> 'a" where
1.143 -  "nprt x = inf x 0"
1.144 +definition nprt :: "'a \<Rightarrow> 'a"
1.145 +  where "nprt x = inf x 0"
1.146
1.147 -definition
1.148 -  pprt :: "'a \<Rightarrow> 'a" where
1.149 -  "pprt x = sup x 0"
1.150 +definition pprt :: "'a \<Rightarrow> 'a"
1.151 +  where "pprt x = sup x 0"
1.152
1.153  lemma pprt_neg: "pprt (- x) = - nprt x"
1.154  proof -
1.155 @@ -137,27 +126,29 @@
1.156  qed
1.157
1.158  lemma prts: "a = pprt a + nprt a"
1.161
1.162  lemma zero_le_pprt[simp]: "0 \<le> pprt a"
1.164 +  by (simp add: pprt_def)
1.165
1.166  lemma nprt_le_zero[simp]: "nprt a \<le> 0"
1.168 +  by (simp add: nprt_def)
1.169
1.170  lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
1.171 -proof -
1.172 -  have a: "?l \<longrightarrow> ?r"
1.173 -    apply (auto)
1.174 +proof
1.175 +  assume ?l
1.176 +  then show ?r
1.177 +    apply -
1.178      apply (rule add_le_imp_le_right[of _ "uminus b" _])
1.180      done
1.181 -  have b: "?r \<longrightarrow> ?l"
1.182 -    apply (auto)
1.183 +next
1.184 +  assume ?r
1.185 +  then show ?l
1.186 +    apply -
1.187      apply (rule add_le_imp_le_right[of _ "b" _])
1.188 -    apply (simp)
1.189 +    apply simp
1.190      done
1.191 -  from a b show ?thesis by blast
1.192  qed
1.193
1.194  lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
1.195 @@ -181,7 +172,7 @@
1.196      fix a::'a
1.197      assume hyp: "sup a (-a) = 0"
1.198      hence "sup a (-a) + a = a" by (simp)
1.199 -    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)
1.200 +    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)
1.201      hence "sup (a+a) 0 <= a" by (simp)
1.202      hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
1.203    }
1.204 @@ -192,16 +183,22 @@
1.205  qed
1.206
1.207  lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
1.210 -apply (erule sup_0_imp_0)
1.211 -done
1.212 +  apply (simp add: inf_eq_neg_sup)
1.213 +  apply (simp add: sup_commute)
1.214 +  apply (erule sup_0_imp_0)
1.215 +  done
1.216
1.217  lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
1.218 -by (rule, erule inf_0_imp_0) simp
1.219 +  apply rule
1.220 +  apply (erule inf_0_imp_0)
1.221 +  apply simp
1.222 +  done
1.223
1.224  lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
1.225 -by (rule, erule sup_0_imp_0) simp
1.226 +  apply rule
1.227 +  apply (erule sup_0_imp_0)
1.228 +  apply simp
1.229 +  done
1.230
1.232    "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
1.233 @@ -218,39 +215,48 @@
1.234    show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
1.235  qed
1.236
1.237 -lemma double_zero [simp]:
1.238 -  "a + a = 0 \<longleftrightarrow> a = 0"
1.239 +lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
1.240  proof
1.241    assume assm: "a + a = 0"
1.242    then have "a + a + - a = - a" by simp
1.243    then have "a + (a + - a) = - a" by (simp only: add_assoc)
1.244    then have a: "- a = a" by simp
1.245 -  show "a = 0" apply (rule antisym)
1.246 -  apply (unfold neg_le_iff_le [symmetric, of a])
1.247 -  unfolding a apply simp
1.249 -  unfolding assm unfolding le_less apply simp_all done
1.250 +  show "a = 0"
1.251 +    apply (rule antisym)
1.252 +    apply (unfold neg_le_iff_le [symmetric, of a])
1.253 +    unfolding a
1.254 +    apply simp
1.256 +    unfolding assm
1.257 +    unfolding le_less
1.258 +    apply simp_all
1.259 +    done
1.260  next
1.261 -  assume "a = 0" then show "a + a = 0" by simp
1.262 +  assume "a = 0"
1.263 +  then show "a + a = 0" by simp
1.264  qed
1.265
1.267 -  "0 < a + a \<longleftrightarrow> 0 < a"
1.268 +lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
1.269  proof (cases "a = 0")
1.270 -  case True then show ?thesis by auto
1.271 +  case True
1.272 +  then show ?thesis by auto
1.273  next
1.274 -  case False then show ?thesis (*FIXME tune proof*)
1.275 -  unfolding less_le apply simp apply rule
1.276 -  apply clarify
1.277 -  apply rule
1.278 -  apply assumption
1.279 -  apply (rule notI)
1.280 -  unfolding double_zero [symmetric, of a] apply simp
1.281 -  done
1.282 +  case False
1.283 +  then show ?thesis
1.284 +    unfolding less_le
1.285 +    apply simp
1.286 +    apply rule
1.287 +    apply clarify
1.288 +    apply rule
1.289 +    apply assumption
1.290 +    apply (rule notI)
1.291 +    unfolding double_zero [symmetric, of a]
1.292 +    apply simp
1.293 +    done
1.294  qed
1.295
1.297 -  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
1.298 +  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
1.299  proof -
1.300    have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
1.301    moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by simp
1.302 @@ -270,7 +276,7 @@
1.303  lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
1.304  proof -
1.305    from add_le_cancel_left [of "uminus a" "plus a a" zero]
1.306 -  have "(a <= -a) = (a+a <= 0)"
1.307 +  have "(a <= -a) = (a+a <= 0)"
1.309    thus ?thesis by simp
1.310  qed
1.311 @@ -278,28 +284,28 @@
1.312  lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
1.313  proof -
1.314    from add_le_cancel_left [of "uminus a" zero "plus a a"]
1.315 -  have "(-a <= a) = (0 <= a+a)"
1.316 +  have "(-a <= a) = (0 <= a+a)"
1.318    thus ?thesis by simp
1.319  qed
1.320
1.321  lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
1.322 -unfolding le_iff_inf by (simp add: nprt_def inf_commute)
1.323 +  unfolding le_iff_inf by (simp add: nprt_def inf_commute)
1.324
1.325  lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
1.326 -unfolding le_iff_sup by (simp add: pprt_def sup_commute)
1.327 +  unfolding le_iff_sup by (simp add: pprt_def sup_commute)
1.328
1.329  lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
1.330 -unfolding le_iff_sup by (simp add: pprt_def sup_commute)
1.331 +  unfolding le_iff_sup by (simp add: pprt_def sup_commute)
1.332
1.333  lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
1.334 -unfolding le_iff_inf by (simp add: nprt_def inf_commute)
1.335 +  unfolding le_iff_inf by (simp add: nprt_def inf_commute)
1.336
1.337  lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
1.338 -unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
1.339 +  unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
1.340
1.341  lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
1.342 -unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
1.343 +  unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
1.344
1.345  end
1.346
1.347 @@ -320,8 +326,7 @@
1.348    then have "0 \<le> sup a (- a)" unfolding abs_lattice .
1.349    then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
1.350    then show ?thesis
1.352 -      pprt_def nprt_def diff_minus abs_lattice)
1.354  qed
1.355
1.357 @@ -329,8 +334,10 @@
1.358    have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
1.359    proof -
1.360      fix a b
1.361 -    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
1.362 -    show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
1.363 +    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
1.364 +      by (auto simp add: abs_lattice)
1.365 +    show "0 \<le> \<bar>a\<bar>"
1.366 +      by (rule add_mono [OF a b, simplified])
1.367    qed
1.368    have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
1.369      by (simp add: abs_lattice le_supI)
1.370 @@ -340,18 +347,25 @@
1.371      by (auto simp add: abs_lattice)
1.372    show "\<bar>-a\<bar> = \<bar>a\<bar>"
1.373      by (simp add: abs_lattice sup_commute)
1.374 -  show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (fact abs_leI)
1.375 +  {
1.376 +    assume "a \<le> b"
1.377 +    then show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
1.378 +      by (rule abs_leI)
1.379 +  }
1.380    show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
1.381    proof -
1.382      have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
1.384 -    have a:"a+b <= sup ?m ?n" by (simp)
1.385 -    have b:"-a-b <= ?n" by (simp)
1.386 -    have c:"?n <= sup ?m ?n" by (simp)
1.387 -    from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
1.388 +    have a: "a + b <= sup ?m ?n" by simp
1.389 +    have b: "- a - b <= ?n" by simp
1.390 +    have c: "?n <= sup ?m ?n" by simp
1.391 +    from b c have d: "-a-b <= sup ?m ?n" by (rule order_trans)
1.392      have e:"-a-b = -(a+b)" by (simp add: diff_minus)
1.393 -    from a d e have "abs(a+b) <= sup ?m ?n"
1.394 -      by (drule_tac abs_leI, auto)
1.395 +    from a d e have "abs(a+b) <= sup ?m ?n"
1.396 +      apply -
1.397 +      apply (drule abs_leI)
1.398 +      apply auto
1.399 +      done
1.400      with g[symmetric] show ?thesis by simp
1.401    qed
1.402  qed
1.403 @@ -370,10 +384,10 @@
1.404  lemma abs_if_lattice:
1.405    fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}"
1.406    shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
1.407 -by auto
1.408 +  by auto
1.409
1.410  lemma estimate_by_abs:
1.411 -  "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
1.412 +  "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
1.413  proof -
1.414    assume "a+b <= c"
1.415    then have "a <= c+(-b)" by (simp add: algebra_simps)
1.416 @@ -390,7 +404,7 @@
1.417
1.418  end
1.419
1.420 -lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
1.421 +lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
1.422  proof -
1.423    let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
1.424    let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
1.425 @@ -398,11 +412,11 @@
1.426      by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
1.427    {
1.428      fix u v :: 'a
1.429 -    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
1.430 -              u * v = pprt a * pprt b + pprt a * nprt b +
1.431 +    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
1.432 +              u * v = pprt a * pprt b + pprt a * nprt b +
1.433                        nprt a * pprt b + nprt a * nprt b"
1.434        apply (subst prts[of u], subst prts[of v])
1.435 -      apply (simp add: algebra_simps)
1.436 +      apply (simp add: algebra_simps)
1.437        done
1.438    }
1.439    note b = this[OF refl[of a] refl[of b]]
1.440 @@ -432,7 +446,7 @@
1.441    show "abs (a*b) = abs a * abs b"
1.442    proof -
1.443      have s: "(0 <= a*b) | (a*b <= 0)"
1.444 -      apply (auto)
1.445 +      apply (auto)
1.446        apply (rule_tac split_mult_pos_le)
1.447        apply (rule_tac contrapos_np[of "a*b <= 0"])
1.448        apply (simp)
1.449 @@ -448,8 +462,8 @@
1.450        then show ?thesis
1.451          apply (simp_all add: mulprts abs_prts)
1.452          apply (insert a)
1.453 -        apply (auto simp add:
1.454 -          algebra_simps
1.455 +        apply (auto simp add:
1.456 +          algebra_simps
1.457            iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
1.458            iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
1.459            apply(drule (1) mult_nonneg_nonpos[of a b], simp)
1.460 @@ -470,15 +484,14 @@
1.461  qed
1.462
1.463  lemma mult_le_prts:
1.464 -  assumes
1.465 -  "a1 <= (a::'a::lattice_ring)"
1.466 -  "a <= a2"
1.467 -  "b1 <= b"
1.468 -  "b <= b2"
1.469 -  shows
1.470 -  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
1.471 -proof -
1.472 -  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
1.473 +  assumes "a1 <= (a::'a::lattice_ring)"
1.474 +    and "a <= a2"
1.475 +    and "b1 <= b"
1.476 +    and "b <= b2"
1.477 +  shows "a * b <=
1.478 +    pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
1.479 +proof -
1.480 +  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
1.481      apply (subst prts[symmetric])+
1.482      apply simp
1.483      done
1.484 @@ -496,7 +509,7 @@
1.485        by simp
1.486    qed
1.487    moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
1.488 -  proof -
1.489 +  proof -
1.490      have "nprt a * pprt b <= nprt a2 * pprt b"
1.491        by (simp add: mult_right_mono assms)
1.492      moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
1.493 @@ -514,29 +527,33 @@
1.494        by simp
1.495    qed
1.496    ultimately show ?thesis
1.497 -    by - (rule add_mono | simp)+
1.498 +    apply -
1.499 +    apply (rule add_mono | simp)+
1.500 +    done
1.501  qed
1.502
1.503  lemma mult_ge_prts:
1.504 -  assumes
1.505 -  "a1 <= (a::'a::lattice_ring)"
1.506 -  "a <= a2"
1.507 -  "b1 <= b"
1.508 -  "b <= b2"
1.509 -  shows
1.510 -  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
1.511 -proof -
1.512 -  from assms have a1:"- a2 <= -a" by auto
1.513 -  from assms have a2: "-a <= -a1" by auto
1.514 -  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
1.515 -  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp
1.516 +  assumes "a1 <= (a::'a::lattice_ring)"
1.517 +    and "a <= a2"
1.518 +    and "b1 <= b"
1.519 +    and "b <= b2"
1.520 +  shows "a * b >=
1.521 +    nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
1.522 +proof -
1.523 +  from assms have a1:"- a2 <= -a"
1.524 +    by auto
1.525 +  from assms have a2: "-a <= -a1"
1.526 +    by auto
1.527 +  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
1.528 +  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
1.529 +    by simp
1.530    then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
1.531      by (simp only: minus_le_iff)
1.532    then show ?thesis by simp
1.533  qed
1.534
1.535  instance int :: lattice_ring
1.536 -proof
1.537 +proof
1.538    fix k :: int
1.539    show "abs k = sup k (- k)"
1.540      by (auto simp add: sup_int_def)
```