1.1 --- a/src/HOL/IMP/Abs_Int2_ivl.thy Wed Mar 06 14:10:07 2013 +0100
1.2 +++ b/src/HOL/IMP/Abs_Int2_ivl.thy Wed Mar 06 16:10:56 2013 +0100
1.3 @@ -1,297 +1,341 @@
1.4 (* Author: Tobias Nipkow *)
1.5
1.6 theory Abs_Int2_ivl
1.7 -imports Abs_Int2
1.8 +imports "~~/src/HOL/Library/Quotient_List"
1.9 + "~~/src/HOL/Library/Extended"
1.10 + Abs_Int2
1.11 begin
1.12
1.13 subsection "Interval Analysis"
1.14
1.15 -datatype lb = Minf | Lb int
1.16 -datatype ub = Pinf | Ub int
1.17 +type_synonym eint = "int extended"
1.18 +type_synonym eint2 = "eint * eint"
1.19
1.20 -datatype ivl = Ivl lb ub
1.21 +definition \<gamma>_rep :: "eint2 \<Rightarrow> int set" where
1.22 +"\<gamma>_rep p = (let (l,h) = p in {i. l \<le> Fin i \<and> Fin i \<le> h})"
1.23
1.24 -definition "\<gamma>_ivl i = (case i of
1.25 - Ivl (Lb l) (Ub h) \<Rightarrow> {l..h} |
1.26 - Ivl (Lb l) Pinf \<Rightarrow> {l..} |
1.27 - Ivl Minf (Ub h) \<Rightarrow> {..h} |
1.28 - Ivl Minf Pinf \<Rightarrow> UNIV)"
1.29 +definition eq_ivl :: "eint2 \<Rightarrow> eint2 \<Rightarrow> bool" where
1.30 +"eq_ivl p1 p2 = (\<gamma>_rep p1 = \<gamma>_rep p2)"
1.31
1.32 -abbreviation Ivl_Lb_Ub :: "int \<Rightarrow> int \<Rightarrow> ivl" ("{_\<dots>_}") where
1.33 -"{lo\<dots>hi} == Ivl (Lb lo) (Ub hi)"
1.34 -abbreviation Ivl_Lb_Pinf :: "int \<Rightarrow> ivl" ("{_\<dots>}") where
1.35 -"{lo\<dots>} == Ivl (Lb lo) Pinf"
1.36 -abbreviation Ivl_Minf_Ub :: "int \<Rightarrow> ivl" ("{\<dots>_}") where
1.37 -"{\<dots>hi} == Ivl Minf (Ub hi)"
1.38 -abbreviation Ivl_Minf_Pinf :: "ivl" ("{\<dots>}") where
1.39 -"{\<dots>} == Ivl Minf Pinf"
1.40 +lemma refl_eq_ivl[simp]: "eq_ivl p p"
1.41 +by(auto simp: eq_ivl_def)
1.42
1.43 -lemmas lub_splits = lb.splits ub.splits
1.44 -
1.45 -definition "num_ivl n = {n\<dots>n}"
1.46 +quotient_type ivl = eint2 / eq_ivl
1.47 +by(rule equivpI)(auto simp: reflp_def symp_def transp_def eq_ivl_def)
1.48
1.49 -fun in_ivl :: "int \<Rightarrow> ivl \<Rightarrow> bool" where
1.50 -"in_ivl k (Ivl (Lb l) (Ub h)) \<longleftrightarrow> l \<le> k \<and> k \<le> h" |
1.51 -"in_ivl k (Ivl (Lb l) Pinf) \<longleftrightarrow> l \<le> k" |
1.52 -"in_ivl k (Ivl Minf (Ub h)) \<longleftrightarrow> k \<le> h" |
1.53 -"in_ivl k (Ivl Minf Pinf) \<longleftrightarrow> True"
1.54 +lift_definition \<gamma>_ivl :: "ivl \<Rightarrow> int set" is \<gamma>_rep
1.55 +by(simp add: eq_ivl_def)
1.56
1.57 -
1.58 -instantiation lb :: linorder
1.59 -begin
1.60 +abbreviation ivl_abbr :: "eint \<Rightarrow> eint \<Rightarrow> ivl" ("[_\<dots>_]") where
1.61 +"[l\<dots>h] == abs_ivl(l,h)"
1.62
1.63 -definition less_eq_lb where
1.64 -"l1 \<le> l2 = (case l1 of Minf \<Rightarrow> True | Lb i1 \<Rightarrow> (case l2 of Minf \<Rightarrow> False | Lb i2 \<Rightarrow> i1 \<le> i2))"
1.65 +lift_definition num_ivl :: "int \<Rightarrow> ivl" is "\<lambda>i. (Fin i, Fin i)"
1.66 +by(auto simp: eq_ivl_def)
1.67
1.68 -definition less_lb :: "lb \<Rightarrow> lb \<Rightarrow> bool" where
1.69 -"((l1::lb) < l2) = (l1 \<le> l2 & ~ l1 \<ge> l2)"
1.70 +fun in_ivl_rep :: "int \<Rightarrow> eint2 \<Rightarrow> bool" where
1.71 +"in_ivl_rep k (l,h) \<longleftrightarrow> l \<le> Fin k \<and> Fin k \<le> h"
1.72
1.73 -instance
1.74 -proof
1.75 - case goal1 show ?case by(rule less_lb_def)
1.76 -next
1.77 - case goal2 show ?case by(auto simp: less_eq_lb_def split:lub_splits)
1.78 -next
1.79 - case goal3 thus ?case by(auto simp: less_eq_lb_def split:lub_splits)
1.80 -next
1.81 - case goal4 thus ?case by(auto simp: less_eq_lb_def split:lub_splits)
1.82 -next
1.83 - case goal5 thus ?case by(auto simp: less_eq_lb_def split:lub_splits)
1.84 -qed
1.85 +lift_definition in_ivl :: "int \<Rightarrow> ivl \<Rightarrow> bool" is in_ivl_rep
1.86 +by(auto simp: eq_ivl_def \<gamma>_rep_def)
1.87 +
1.88 +definition is_empty_rep :: "eint2 \<Rightarrow> bool" where
1.89 +"is_empty_rep p = (let (l,h) = p in l>h | l=Pinf & h=Pinf | l=Minf & h=Minf)"
1.90
1.91 -end
1.92 -
1.93 -instantiation ub :: linorder
1.94 -begin
1.95 +lemma \<gamma>_rep_cases: "\<gamma>_rep p = (case p of (Fin i,Fin j) => {i..j} | (Fin i,Pinf) => {i..} |
1.96 + (Minf,Fin i) \<Rightarrow> {..i} | (Minf,Pinf) \<Rightarrow> UNIV | _ \<Rightarrow> {})"
1.97 +by(auto simp add: \<gamma>_rep_def split: prod.splits extended.splits)
1.98
1.99 -definition less_eq_ub where
1.100 -"u1 \<le> u2 = (case u2 of Pinf \<Rightarrow> True | Ub i2 \<Rightarrow> (case u1 of Pinf \<Rightarrow> False | Ub i1 \<Rightarrow> i1 \<le> i2))"
1.101 -
1.102 -definition less_ub :: "ub \<Rightarrow> ub \<Rightarrow> bool" where
1.103 -"((u1::ub) < u2) = (u1 \<le> u2 & ~ u1 \<ge> u2)"
1.104 +lift_definition is_empty_ivl :: "ivl \<Rightarrow> bool" is is_empty_rep
1.105 +apply(auto simp: eq_ivl_def \<gamma>_rep_cases is_empty_rep_def)
1.106 +apply(auto simp: not_less less_eq_extended_cases split: extended.splits)
1.107 +done
1.108
1.109 -instance
1.110 -proof
1.111 - case goal1 show ?case by(rule less_ub_def)
1.112 -next
1.113 - case goal2 show ?case by(auto simp: less_eq_ub_def split:lub_splits)
1.114 -next
1.115 - case goal3 thus ?case by(auto simp: less_eq_ub_def split:lub_splits)
1.116 -next
1.117 - case goal4 thus ?case by(auto simp: less_eq_ub_def split:lub_splits)
1.118 -next
1.119 - case goal5 thus ?case by(auto simp: less_eq_ub_def split:lub_splits)
1.120 -qed
1.121 +lemma eq_ivl_iff: "eq_ivl p1 p2 = (is_empty_rep p1 & is_empty_rep p2 | p1 = p2)"
1.122 +by(auto simp: eq_ivl_def is_empty_rep_def \<gamma>_rep_cases Icc_eq_Icc split: prod.splits extended.splits)
1.123
1.124 -end
1.125 +definition empty_rep :: eint2 where "empty_rep = (Pinf,Minf)"
1.126
1.127 -lemmas le_lub_defs = less_eq_lb_def less_eq_ub_def
1.128 +lift_definition empty_ivl :: ivl is empty_rep
1.129 +by simp
1.130
1.131 -lemma le_lub_simps[simp]:
1.132 - "Minf \<le> l" "Lb i \<le> Lb j = (i \<le> j)" "~ Lb i \<le> Minf"
1.133 - "h \<le> Pinf" "Ub i \<le> Ub j = (i \<le> j)" "~ Pinf \<le> Ub j"
1.134 -by(auto simp: le_lub_defs split: lub_splits)
1.135 -
1.136 -definition empty where "empty = {1\<dots>0}"
1.137 +lemma is_empty_empty_rep[simp]: "is_empty_rep empty_rep"
1.138 +by(auto simp add: is_empty_rep_def empty_rep_def)
1.139
1.140 -fun is_empty where
1.141 -"is_empty {l\<dots>h} = (h<l)" |
1.142 -"is_empty _ = False"
1.143 +lemma is_empty_rep_iff: "is_empty_rep p = (\<gamma>_rep p = {})"
1.144 +by(auto simp add: \<gamma>_rep_cases is_empty_rep_def split: prod.splits extended.splits)
1.145
1.146 -lemma [simp]: "is_empty(Ivl l h) =
1.147 - (case l of Lb l \<Rightarrow> (case h of Ub h \<Rightarrow> h<l | Pinf \<Rightarrow> False) | Minf \<Rightarrow> False)"
1.148 -by(auto split: lub_splits)
1.149 -
1.150 -lemma [simp]: "is_empty i \<Longrightarrow> \<gamma>_ivl i = {}"
1.151 -by(auto simp add: \<gamma>_ivl_def split: ivl.split lub_splits)
1.152 +declare is_empty_rep_iff[THEN iffD1, simp]
1.153
1.154
1.155 instantiation ivl :: semilattice
1.156 begin
1.157
1.158 -fun le_aux where
1.159 -"le_aux (Ivl l1 h1) (Ivl l2 h2) = (l2 \<le> l1 & h1 \<le> h2)"
1.160 +definition le_rep :: "eint2 \<Rightarrow> eint2 \<Rightarrow> bool" where
1.161 +"le_rep p1 p2 = (let (l1,h1) = p1; (l2,h2) = p2 in
1.162 + if is_empty_rep(l1,h1) then True else
1.163 + if is_empty_rep(l2,h2) then False else l1 \<ge> l2 & h1 \<le> h2)"
1.164 +
1.165 +lemma le_iff_subset: "le_rep p1 p2 \<longleftrightarrow> \<gamma>_rep p1 \<subseteq> \<gamma>_rep p2"
1.166 +apply rule
1.167 +apply(auto simp: is_empty_rep_def le_rep_def \<gamma>_rep_def split: if_splits prod.splits)[1]
1.168 +apply(auto simp: is_empty_rep_def \<gamma>_rep_cases le_rep_def)
1.169 +apply(auto simp: not_less split: extended.splits)
1.170 +done
1.171 +
1.172 +lift_definition less_eq_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> bool" is le_rep
1.173 +by(auto simp: eq_ivl_def le_iff_subset)
1.174
1.175 -definition le_ivl where
1.176 -"i1 \<sqsubseteq> i2 =
1.177 - (if is_empty i1 then True else
1.178 - if is_empty i2 then False else le_aux i1 i2)"
1.179 +definition less_ivl where "i1 < i2 = (i1 \<le> i2 \<and> \<not> i2 \<le> (i1::ivl))"
1.180 +
1.181 +definition join_rep :: "eint2 \<Rightarrow> eint2 \<Rightarrow> eint2" where
1.182 +"join_rep p1 p2 = (if is_empty_rep p1 then p2 else if is_empty_rep p2 then p1
1.183 + else let (l1,h1) = p1; (l2,h2) = p2 in (min l1 l2, max h1 h2))"
1.184
1.185 -definition "i1 \<squnion> i2 =
1.186 - (if is_empty i1 then i2 else if is_empty i2 then i1
1.187 - else case (i1,i2) of (Ivl l1 h1, Ivl l2 h2) \<Rightarrow> Ivl (min l1 l2) (max h1 h2))"
1.188 +lift_definition join_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> ivl" is join_rep
1.189 +by(auto simp: eq_ivl_iff join_rep_def)
1.190
1.191 -definition "\<top> = {\<dots>}"
1.192 +lift_definition top_ivl :: ivl is "(Minf,Pinf)"
1.193 +by(auto simp: eq_ivl_def)
1.194 +
1.195 +lemma is_empty_min_max:
1.196 + "\<not> is_empty_rep (l1,h1) \<Longrightarrow> \<not> is_empty_rep (l2, h2) \<Longrightarrow> \<not> is_empty_rep (min l1 l2, max h1 h2)"
1.197 +by(auto simp add: is_empty_rep_def max_def min_def split: if_splits)
1.198
1.199 instance
1.200 proof
1.201 - case goal1 thus ?case
1.202 - by(cases x, simp add: le_ivl_def)
1.203 + case goal1 show ?case by (rule less_ivl_def)
1.204 +next
1.205 + case goal2 show ?case by transfer (simp add: le_rep_def split: prod.splits)
1.206 next
1.207 - case goal2 thus ?case
1.208 - by(cases x, cases y, cases z, auto simp: le_ivl_def split: if_splits)
1.209 + case goal3 thus ?case by transfer (auto simp: le_rep_def split: if_splits)
1.210 next
1.211 - case goal3 thus ?case
1.212 - by(cases x, cases y, simp add: le_ivl_def join_ivl_def le_lub_defs min_def max_def split: lub_splits)
1.213 + case goal4 thus ?case by transfer (auto simp: le_rep_def eq_ivl_iff split: if_splits)
1.214 next
1.215 - case goal4 thus ?case
1.216 - by(cases x, cases y, simp add: le_ivl_def join_ivl_def le_lub_defs min_def max_def split: lub_splits)
1.217 + case goal6 thus ?case by transfer (auto simp add: le_rep_def join_rep_def is_empty_min_max)
1.218 +next
1.219 + case goal7 thus ?case by transfer (auto simp add: le_rep_def join_rep_def is_empty_min_max)
1.220 next
1.221 - case goal5 thus ?case
1.222 - by(cases x, cases y, cases z, auto simp add: le_ivl_def join_ivl_def le_lub_defs min_def max_def split: lub_splits if_splits)
1.223 + case goal8 thus ?case by transfer (auto simp add: le_rep_def join_rep_def)
1.224 next
1.225 - case goal6 thus ?case
1.226 - by(cases x, simp add: Top_ivl_def le_ivl_def le_lub_defs split: lub_splits)
1.227 + case goal5 show ?case by transfer (simp add: le_rep_def is_empty_rep_def)
1.228 qed
1.229
1.230 end
1.231
1.232 +text{* Implement (naive) executable equality: *}
1.233 +instantiation ivl :: equal
1.234 +begin
1.235 +
1.236 +definition equal_ivl where
1.237 +"equal_ivl i1 (i2::ivl) = (i1\<le>i2 \<and> i2 \<le> i1)"
1.238 +
1.239 +instance
1.240 +proof
1.241 + case goal1 show ?case by(simp add: equal_ivl_def eq_iff)
1.242 +qed
1.243 +
1.244 +end
1.245 +
1.246 +lemma [simp]: fixes x :: "'a::linorder extended" shows "(\<not> x < Pinf) = (x = Pinf)"
1.247 +by(simp add: not_less)
1.248 +lemma [simp]: fixes x :: "'a::linorder extended" shows "(\<not> Minf < x) = (x = Minf)"
1.249 +by(simp add: not_less)
1.250
1.251 instantiation ivl :: lattice
1.252 begin
1.253
1.254 -definition "i1 \<sqinter> i2 = (if is_empty i1 \<or> is_empty i2 then empty else
1.255 - case (i1,i2) of (Ivl l1 h1, Ivl l2 h2) \<Rightarrow> Ivl (max l1 l2) (min h1 h2))"
1.256 +definition meet_rep :: "eint2 \<Rightarrow> eint2 \<Rightarrow> eint2" where
1.257 +"meet_rep p1 p2 = (let (l1,h1) = p1; (l2,h2) = p2 in (max l1 l2, min h1 h2))"
1.258
1.259 -definition "\<bottom> = empty"
1.260 +lemma \<gamma>_meet_rep: "\<gamma>_rep(meet_rep p1 p2) = \<gamma>_rep p1 \<inter> \<gamma>_rep p2"
1.261 +by(auto simp:meet_rep_def \<gamma>_rep_cases split: prod.splits extended.splits)
1.262 +
1.263 +lift_definition meet_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> ivl" is meet_rep
1.264 +by(auto simp: \<gamma>_meet_rep eq_ivl_def)
1.265 +
1.266 +definition "\<bottom> = empty_ivl"
1.267
1.268 instance
1.269 proof
1.270 case goal2 thus ?case
1.271 - by (simp add:meet_ivl_def empty_def le_ivl_def le_lub_defs max_def min_def split: ivl.splits lub_splits)
1.272 + unfolding meet_rep_def by transfer (auto simp: le_iff_subset \<gamma>_meet_rep)
1.273 next
1.274 case goal3 thus ?case
1.275 - by (simp add: empty_def meet_ivl_def le_ivl_def le_lub_defs max_def min_def split: ivl.splits lub_splits)
1.276 + unfolding meet_rep_def by transfer (auto simp: le_iff_subset \<gamma>_meet_rep)
1.277 next
1.278 case goal4 thus ?case
1.279 - by (cases x, cases y, cases z, auto simp add: le_ivl_def meet_ivl_def empty_def le_lub_defs max_def min_def split: lub_splits if_splits)
1.280 + unfolding meet_rep_def by transfer (auto simp: le_iff_subset \<gamma>_meet_rep)
1.281 next
1.282 - case goal1 show ?case by(cases x, simp add: bot_ivl_def empty_def le_ivl_def)
1.283 + case goal1 show ?case unfolding bot_ivl_def by transfer (auto simp: le_iff_subset)
1.284 qed
1.285
1.286 end
1.287
1.288
1.289 -instantiation lb :: plus
1.290 +lemma eq_ivl_empty: "eq_ivl p empty_rep = is_empty_rep p"
1.291 +by (metis eq_ivl_iff is_empty_empty_rep)
1.292 +
1.293 +lemma le_ivl_nice: "[l1\<dots>h1] \<le> [l2\<dots>h2] \<longleftrightarrow>
1.294 + (if [l1\<dots>h1] = \<bottom> then True else
1.295 + if [l2\<dots>h2] = \<bottom> then False else l1 \<ge> l2 & h1 \<le> h2)"
1.296 +unfolding bot_ivl_def by transfer (simp add: le_rep_def eq_ivl_empty)
1.297 +
1.298 +lemma join_ivl_nice: "[l1\<dots>h1] \<squnion> [l2\<dots>h2] =
1.299 + (if [l1\<dots>h1] = \<bottom> then [l2\<dots>h2] else
1.300 + if [l2\<dots>h2] = \<bottom> then [l1\<dots>h1] else [min l1 l2\<dots>max h1 h2])"
1.301 +unfolding bot_ivl_def by transfer (simp add: join_rep_def eq_ivl_empty)
1.302 +
1.303 +lemma meet_nice: "[l1\<dots>h1] \<sqinter> [l2\<dots>h2] = [max l1 l2\<dots>min h1 h2]"
1.304 +by transfer (simp add: meet_rep_def)
1.305 +
1.306 +
1.307 +instantiation ivl :: plus
1.308 begin
1.309
1.310 -fun plus_lb where
1.311 -"Lb x + Lb y = Lb(x+y)" |
1.312 -"_ + _ = Minf"
1.313 +definition plus_rep :: "eint2 \<Rightarrow> eint2 \<Rightarrow> eint2" where
1.314 +"plus_rep p1 p2 =
1.315 + (if is_empty_rep p1 \<or> is_empty_rep p2 then empty_rep else
1.316 + let (l1,h1) = p1; (l2,h2) = p2 in (l1+l2, h1+h2))"
1.317 +
1.318 +lift_definition plus_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> ivl" is plus_rep
1.319 +by(auto simp: plus_rep_def eq_ivl_iff)
1.320
1.321 instance ..
1.322 end
1.323
1.324 -instantiation ub :: plus
1.325 -begin
1.326 -
1.327 -fun plus_ub where
1.328 -"Ub x + Ub y = Ub(x+y)" |
1.329 -"_ + _ = Pinf"
1.330 -
1.331 -instance ..
1.332 -end
1.333 -
1.334 -instantiation ivl :: plus
1.335 -begin
1.336 +lemma plus_ivl_nice: "[l1\<dots>h1] + [l2\<dots>h2] =
1.337 + (if [l1\<dots>h1] = \<bottom> \<or> [l2\<dots>h2] = \<bottom> then \<bottom> else [l1+l2 \<dots> h1+h2])"
1.338 +unfolding bot_ivl_def by transfer (auto simp: plus_rep_def eq_ivl_empty)
1.339
1.340 -definition "i1+i2 = (if is_empty i1 | is_empty i2 then empty else
1.341 - case (i1,i2) of (Ivl l1 h1, Ivl l2 h2) \<Rightarrow> Ivl (l1+l2) (h1+h2))"
1.342 -
1.343 -instance ..
1.344 -end
1.345 +lemma uminus_eq_Minf[simp]: "-x = Minf \<longleftrightarrow> x = Pinf"
1.346 +by(cases x) auto
1.347 +lemma uminus_eq_Pinf[simp]: "-x = Pinf \<longleftrightarrow> x = Minf"
1.348 +by(cases x) auto
1.349
1.350 -fun uminus_ub :: "ub \<Rightarrow> lb" where
1.351 -"uminus_ub(Ub( x)) = Lb(-x)" |
1.352 -"uminus_ub Pinf = Minf"
1.353 -
1.354 -fun uminus_lb :: "lb \<Rightarrow> ub" where
1.355 -"uminus_lb(Lb( x)) = Ub(-x)" |
1.356 -"uminus_lb Minf = Pinf"
1.357 +lemma uminus_le_Fin_iff: "- x \<le> Fin(-y) \<longleftrightarrow> Fin y \<le> (x::'a::ordered_ab_group_add extended)"
1.358 +by(cases x) auto
1.359 +lemma Fin_uminus_le_iff: "Fin(-y) \<le> -x \<longleftrightarrow> x \<le> ((Fin y)::'a::ordered_ab_group_add extended)"
1.360 +by(cases x) auto
1.361
1.362 instantiation ivl :: uminus
1.363 begin
1.364
1.365 -fun uminus_ivl where
1.366 -"-(Ivl l h) = Ivl (uminus_ub h) (uminus_lb l)"
1.367 -
1.368 -instance ..
1.369 -end
1.370 +definition uminus_rep :: "eint2 \<Rightarrow> eint2" where
1.371 +"uminus_rep p = (let (l,h) = p in (-h, -l))"
1.372
1.373 -instantiation ivl :: minus
1.374 -begin
1.375 +lemma \<gamma>_uminus_rep: "i : \<gamma>_rep p \<Longrightarrow> -i \<in> \<gamma>_rep(uminus_rep p)"
1.376 +by(auto simp: uminus_rep_def \<gamma>_rep_def image_def uminus_le_Fin_iff Fin_uminus_le_iff
1.377 + split: prod.split)
1.378
1.379 -definition minus_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> ivl" where
1.380 -"(i1::ivl) - i2 = i1 + -i2"
1.381 +lift_definition uminus_ivl :: "ivl \<Rightarrow> ivl" is uminus_rep
1.382 +by (auto simp: uminus_rep_def eq_ivl_def \<gamma>_rep_cases)
1.383 + (auto simp: Icc_eq_Icc split: extended.splits)
1.384
1.385 instance ..
1.386 end
1.387
1.388 -lemma minus_ivl_cases: "i1 - i2 = (if is_empty i1 | is_empty i2 then empty else
1.389 - case (i1,i2) of (Ivl l1 h1, Ivl l2 h2) \<Rightarrow> Ivl (l1 + uminus_ub h2) (h1 + uminus_lb l2))"
1.390 -by(auto simp: plus_ivl_def minus_ivl_def split: ivl.split lub_splits)
1.391 +lemma uminus_nice: "-[l\<dots>h] = [-h\<dots>-l]"
1.392 +by transfer (simp add: uminus_rep_def)
1.393 +
1.394 +instantiation ivl :: minus
1.395 +begin
1.396 +definition minus_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> ivl" where "(iv1::ivl) - iv2 = iv1 + -iv2"
1.397 +instance ..
1.398 +end
1.399 +
1.400
1.401 -lemma gamma_minus_ivl:
1.402 - "n1 : \<gamma>_ivl i1 \<Longrightarrow> n2 : \<gamma>_ivl i2 \<Longrightarrow> n1-n2 : \<gamma>_ivl(i1 - i2)"
1.403 -by(auto simp add: minus_ivl_def plus_ivl_def \<gamma>_ivl_def split: ivl.splits lub_splits)
1.404 +definition filter_plus_ivl :: "ivl \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl*ivl" where
1.405 +"filter_plus_ivl iv iv1 iv2 = (iv1 \<sqinter> (iv - iv2), iv2 \<sqinter> (iv - iv1))"
1.406 +
1.407 +definition filter_less_rep :: "bool \<Rightarrow> eint2 \<Rightarrow> eint2 \<Rightarrow> eint2 * eint2" where
1.408 +"filter_less_rep res p1 p2 =
1.409 + (if is_empty_rep p1 \<or> is_empty_rep p2 then (empty_rep,empty_rep) else
1.410 + let (l1,h1) = p1; (l2,h2) = p2 in
1.411 + if res
1.412 + then ((l1, min h1 (h2 + Fin -1)), (max (l1 + Fin 1) l2, h2))
1.413 + else ((max l1 l2, h1), (l2, min h1 h2)))"
1.414
1.415 -definition "filter_plus_ivl i i1 i2 = ((*if is_empty i then empty else*)
1.416 - i1 \<sqinter> (i - i2), i2 \<sqinter> (i - i1))"
1.417 +lift_definition filter_less_ivl :: "bool \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl * ivl" is filter_less_rep
1.418 +by(auto simp: prod_pred_def filter_less_rep_def eq_ivl_iff)
1.419 +declare filter_less_ivl.abs_eq[code] (* bug in lifting *)
1.420 +
1.421 +lemma filter_less_ivl_nice: "filter_less_ivl b [l1\<dots>h1] [l2\<dots>h2] =
1.422 + (if [l1\<dots>h1] = \<bottom> \<or> [l2\<dots>h2] = \<bottom> then (\<bottom>,\<bottom>) else
1.423 + if b
1.424 + then ([l1 \<dots> min h1 (h2 + Fin -1)], [max (l1 + Fin 1) l2 \<dots> h2])
1.425 + else ([max l1 l2 \<dots> h1], [l2 \<dots> min h1 h2]))"
1.426 +unfolding prod_rel_eq[symmetric] bot_ivl_def
1.427 +by transfer (auto simp: filter_less_rep_def eq_ivl_empty)
1.428
1.429 -fun filter_less_ivl :: "bool \<Rightarrow> ivl \<Rightarrow> ivl \<Rightarrow> ivl * ivl" where
1.430 -"filter_less_ivl res (Ivl l1 h1) (Ivl l2 h2) =
1.431 - (if is_empty(Ivl l1 h1) \<or> is_empty(Ivl l2 h2) then (empty, empty) else
1.432 - if res
1.433 - then (Ivl l1 (min h1 (h2 + Ub -1)), Ivl (max (l1 + Lb 1) l2) h2)
1.434 - else (Ivl (max l1 l2) h1, Ivl l2 (min h1 h2)))"
1.435 +lemma add_mono_le_Fin:
1.436 + "\<lbrakk>x1 \<le> Fin y1; x2 \<le> Fin y2\<rbrakk> \<Longrightarrow> x1 + x2 \<le> Fin (y1 + (y2::'a::ordered_ab_group_add))"
1.437 +by(drule (1) add_mono) simp
1.438 +
1.439 +lemma add_mono_Fin_le:
1.440 + "\<lbrakk>Fin y1 \<le> x1; Fin y2 \<le> x2\<rbrakk> \<Longrightarrow> Fin(y1 + y2::'a::ordered_ab_group_add) \<le> x1 + x2"
1.441 +by(drule (1) add_mono) simp
1.442 +
1.443 +lemma plus_rep_plus:
1.444 + "\<lbrakk> i1 \<in> \<gamma>_rep (l1,h1); i2 \<in> \<gamma>_rep (l2, h2)\<rbrakk> \<Longrightarrow> i1 + i2 \<in> \<gamma>_rep (l1 + l2, h1 + h2)"
1.445 +by(simp add: \<gamma>_rep_def add_mono_Fin_le add_mono_le_Fin)
1.446
1.447 interpretation Val_abs
1.448 where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
1.449 proof
1.450 - case goal1 thus ?case
1.451 - by(auto simp: \<gamma>_ivl_def le_ivl_def le_lub_defs split: ivl.split lub_splits if_splits)
1.452 + case goal1 thus ?case by transfer (simp add: le_iff_subset)
1.453 next
1.454 - case goal2 show ?case by(simp add: \<gamma>_ivl_def Top_ivl_def)
1.455 + case goal2 show ?case by transfer (simp add: \<gamma>_rep_def)
1.456 next
1.457 - case goal3 thus ?case by(simp add: \<gamma>_ivl_def num_ivl_def)
1.458 + case goal3 show ?case by transfer (simp add: \<gamma>_rep_def)
1.459 next
1.460 case goal4 thus ?case
1.461 - by(auto simp add: \<gamma>_ivl_def plus_ivl_def split: ivl.split lub_splits)
1.462 + apply transfer
1.463 + apply(auto simp: \<gamma>_rep_def plus_rep_def add_mono_le_Fin add_mono_Fin_le)
1.464 + by(auto simp: empty_rep_def is_empty_rep_def)
1.465 qed
1.466
1.467 +
1.468 interpretation Val_abs1_gamma
1.469 where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
1.470 defines aval_ivl is aval'
1.471 proof
1.472 - case goal1 thus ?case
1.473 - by(auto simp add: \<gamma>_ivl_def meet_ivl_def empty_def min_def max_def split: ivl.split lub_splits)
1.474 + case goal1 show ?case
1.475 + by transfer (auto simp add:meet_rep_def \<gamma>_rep_cases split: prod.split extended.split)
1.476 next
1.477 - case goal2 show ?case by(auto simp add: bot_ivl_def \<gamma>_ivl_def empty_def)
1.478 + case goal2 show ?case unfolding bot_ivl_def by transfer simp
1.479 qed
1.480
1.481 -lemma mono_minus_ivl: fixes i1 :: ivl
1.482 -shows "i1 \<sqsubseteq> i1' \<Longrightarrow> i2 \<sqsubseteq> i2' \<Longrightarrow> i1 - i2 \<sqsubseteq> i1' - i2'"
1.483 -apply(auto simp add: minus_ivl_cases empty_def le_ivl_def le_lub_defs split: ivl.splits)
1.484 - apply(simp split: lub_splits)
1.485 - apply(simp split: lub_splits)
1.486 -apply(simp split: lub_splits)
1.487 -done
1.488 +lemma \<gamma>_plus_rep: "i1 : \<gamma>_rep p1 \<Longrightarrow> i2 : \<gamma>_rep p2 \<Longrightarrow> i1+i2 \<in> \<gamma>_rep(plus_rep p1 p2)"
1.489 +by(auto simp: plus_rep_def plus_rep_plus split: prod.splits)
1.490
1.491 +lemma non_empty_meet: "\<lbrakk>n1 \<in> \<gamma>_rep a1; n2 \<in> \<gamma>_rep a2; n1 + n2 \<in> \<gamma>_rep a \<rbrakk> \<Longrightarrow>
1.492 + \<not> is_empty_rep (meet_rep a1 (plus_rep a (uminus_rep a2)))"
1.493 +by (auto simp add: \<gamma>_meet_rep set_eq_iff is_empty_rep_iff simp del: all_not_in_conv)
1.494 + (metis \<gamma>_plus_rep \<gamma>_uminus_rep add_diff_cancel diff_minus)
1.495 +
1.496 +lemma filter_plus: "\<lbrakk>eq_ivl (meet_rep a1 (plus_rep a (uminus_rep a2))) b1 \<and>
1.497 + eq_ivl (meet_rep a2 (plus_rep a (uminus_rep a1))) b2;
1.498 + n1 \<in> \<gamma>_rep a1; n2 \<in> \<gamma>_rep a2; n1 + n2 \<in> \<gamma>_rep a\<rbrakk>
1.499 + \<Longrightarrow> n1 \<in> \<gamma>_rep b1 \<and> n2 \<in> \<gamma>_rep b2"
1.500 +by (auto simp: eq_ivl_iff \<gamma>_meet_rep non_empty_meet add_ac)
1.501 + (metis \<gamma>_plus_rep \<gamma>_uminus_rep add_diff_cancel diff_def add_commute)+
1.502
1.503 interpretation Val_abs1
1.504 where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
1.505 and test_num' = in_ivl
1.506 and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl
1.507 proof
1.508 - case goal1 thus ?case
1.509 - by (simp add: \<gamma>_ivl_def split: ivl.split lub_splits)
1.510 + case goal1 thus ?case by transfer (auto simp: \<gamma>_rep_def)
1.511 next
1.512 - case goal2 thus ?case
1.513 - by(auto simp add: filter_plus_ivl_def)
1.514 - (metis gamma_minus_ivl add_diff_cancel add_commute)+
1.515 + case goal2 thus ?case unfolding filter_plus_ivl_def minus_ivl_def prod_rel_eq[symmetric]
1.516 + by transfer (simp add: filter_plus)
1.517 next
1.518 case goal3 thus ?case
1.519 - by(cases a1, cases a2, auto simp: \<gamma>_ivl_def min_def max_def split: if_splits lub_splits)
1.520 + unfolding prod_rel_eq[symmetric]
1.521 + apply transfer
1.522 + apply (auto simp add: filter_less_rep_def eq_ivl_iff max_def min_def split: if_splits)
1.523 + apply(auto simp: \<gamma>_rep_cases is_empty_rep_def split: extended.splits)
1.524 + done
1.525 qed
1.526
1.527 interpretation Abs_Int1
1.528 @@ -308,25 +352,41 @@
1.529
1.530 text{* Monotonicity: *}
1.531
1.532 +lemma mono_meet_rep: "le_rep p1 p2 \<Longrightarrow> le_rep q1 q2 \<Longrightarrow> le_rep (meet_rep p1 q1) (meet_rep p2 q2)"
1.533 +by(auto simp add: le_iff_subset \<gamma>_meet_rep)
1.534 +
1.535 +lemma mono_plus_rep: "le_rep p1 p2 \<Longrightarrow> le_rep q1 q2 \<Longrightarrow> le_rep (plus_rep p1 q1) (plus_rep p2 q2)"
1.536 +apply(auto simp: plus_rep_def le_iff_subset split: if_splits)
1.537 +by(auto simp: is_empty_rep_iff \<gamma>_rep_cases split: extended.splits)
1.538 +
1.539 +lemma mono_minus_rep: "le_rep p1 p2 \<Longrightarrow> le_rep (uminus_rep p1) (uminus_rep p2)"
1.540 +apply(auto simp: uminus_rep_def le_iff_subset split: if_splits prod.split)
1.541 +by(auto simp: \<gamma>_rep_cases split: extended.splits)
1.542 +
1.543 interpretation Abs_Int1_mono
1.544 where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
1.545 and test_num' = in_ivl
1.546 and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl
1.547 proof
1.548 - case goal1 thus ?case
1.549 - by(auto simp: plus_ivl_def le_ivl_def le_lub_defs empty_def split: if_splits ivl.splits lub_splits)
1.550 + case goal1 thus ?case by transfer (rule mono_plus_rep)
1.551 +next
1.552 + case goal2 thus ?case unfolding filter_plus_ivl_def minus_ivl_def less_eq_prod_def
1.553 + by transfer (auto simp: mono_meet_rep mono_plus_rep mono_minus_rep)
1.554 next
1.555 - case goal2 thus ?case
1.556 - by(auto simp: filter_plus_ivl_def le_prod_def mono_meet mono_minus_ivl)
1.557 -next
1.558 - case goal3 thus ?case
1.559 - apply(cases a1, cases b1, cases a2, cases b2, auto simp: le_prod_def)
1.560 - by(auto simp add: empty_def le_ivl_def le_lub_defs min_def max_def split: lub_splits)
1.561 + case goal3 thus ?case unfolding less_eq_prod_def
1.562 + apply transfer
1.563 + apply(auto simp:filter_less_rep_def le_iff_subset min_def max_def split: if_splits)
1.564 + by(auto simp:is_empty_rep_iff \<gamma>_rep_cases split: extended.splits)
1.565 qed
1.566
1.567
1.568 subsubsection "Tests"
1.569
1.570 +(* Hide Fin in numerals on output *)
1.571 +declare
1.572 +Fin_numeral [code_post] Fin_neg_numeral [code_post]
1.573 +zero_extended_def[symmetric, code_post] one_extended_def[symmetric, code_post]
1.574 +
1.575 value "show_acom_opt (AI_ivl test1_ivl)"
1.576
1.577 text{* Better than @{text AI_const}: *}
1.578 @@ -334,7 +394,7 @@
1.579 value "show_acom_opt (AI_ivl test4_const)"
1.580 value "show_acom_opt (AI_ivl test6_const)"
1.581
1.582 -definition "steps c i = (step_ivl(top(vars c)) ^^ i) (bot c)"
1.583 +definition "steps c i = (step_ivl(Top(vars c)) ^^ i) (bot c)"
1.584
1.585 value "show_acom_opt (AI_ivl test2_ivl)"
1.586 value "show_acom (steps test2_ivl 0)"