src/HOLCF/Lift.thy
 changeset 12026 0b1d80ada4ab parent 2640 ee4dfce170a0 child 12338 de0f4a63baa5
```     1.1 --- a/src/HOLCF/Lift.thy	Sat Nov 03 01:36:19 2001 +0100
1.2 +++ b/src/HOLCF/Lift.thy	Sat Nov 03 01:38:11 2001 +0100
1.3 @@ -1,16 +1,339 @@
1.4  (*  Title:      HOLCF/Lift.thy
1.5      ID:         \$Id\$
1.6 -    Author:     Oscar Slotosch
1.7 -    Copyright   1997 Technische Universitaet Muenchen
1.8 +    Author:     Olaf Mueller
1.10  *)
1.11
1.12 -Lift = Lift3 +
1.13 +header {* Lifting types of class term to flat pcpo's *}
1.14 +
1.15 +theory Lift = Cprod3:
1.16 +
1.17 +defaultsort "term"
1.18 +
1.19 +
1.20 +typedef 'a lift = "UNIV :: 'a option set" ..
1.21 +
1.22 +constdefs
1.23 +  Undef :: "'a lift"
1.24 +  "Undef == Abs_lift None"
1.25 +  Def :: "'a => 'a lift"
1.26 +  "Def x == Abs_lift (Some x)"
1.27 +
1.28 +instance lift :: ("term") sq_ord ..
1.29 +
1.31 +  less_lift_def: "x << y == (x=y | x=Undef)"
1.32 +
1.33 +instance lift :: ("term") po
1.34 +proof
1.35 +  fix x y z :: "'a lift"
1.36 +  show "x << x" by (unfold less_lift_def) blast
1.37 +  { assume "x << y" and "y << x" thus "x = y" by (unfold less_lift_def) blast }
1.38 +  { assume "x << y" and "y << z" thus "x << z" by (unfold less_lift_def) blast }
1.39 +qed
1.40 +
1.41 +lemma inst_lift_po: "(op <<) = (\<lambda>x y. x = y | x = Undef)"
1.42 +  -- {* For compatibility with old HOLCF-Version. *}
1.43 +  by (simp only: less_lift_def [symmetric])
1.44 +
1.45 +
1.46 +subsection {* Type lift is pointed *}
1.47 +
1.48 +lemma minimal_lift [iff]: "Undef << x"
1.49 +  by (simp add: inst_lift_po)
1.50 +
1.51 +lemma UU_lift_def: "(SOME u. \<forall>y. u \<sqsubseteq> y) = Undef"
1.52 +  apply (rule minimal2UU [symmetric])
1.53 +  apply (rule minimal_lift)
1.54 +  done
1.55 +
1.56 +lemma least_lift: "EX x::'a lift. ALL y. x << y"
1.57 +  apply (rule_tac x = Undef in exI)
1.58 +  apply (rule minimal_lift [THEN allI])
1.59 +  done
1.60 +
1.61 +
1.62 +subsection {* Type lift is a cpo *}
1.63 +
1.64 +text {*
1.65 +  The following lemmas have already been proved in @{text Pcpo.ML} and
1.66 +  @{text Fix.ML}, but there class @{text pcpo} is assumed, although
1.67 +  only @{text po} is necessary and a least element. Therefore they are
1.68 +  redone here for the @{text po} lift with least element @{text
1.69 +  Undef}.
1.70 +*}
1.71 +
1.72 +lemma notUndef_I: "[| x<<y; x ~= Undef |] ==> y ~= Undef"
1.73 +  -- {* Tailoring @{text notUU_I} of @{text Pcpo.ML} to @{text Undef} *}
1.74 +  by (blast intro: antisym_less)
1.75 +
1.76 +lemma chain_mono2_po: "[| EX j.~Y(j)=Undef; chain(Y::nat=>('a)lift) |]
1.77 +         ==> EX j. ALL i. j<i-->~Y(i)=Undef"
1.78 +  -- {* Tailoring @{text chain_mono2} of @{text Pcpo.ML} to @{text Undef} *}
1.79 +  apply safe
1.80 +  apply (rule exI)
1.81 +  apply (intro strip)
1.82 +  apply (rule notUndef_I)
1.83 +   apply (erule (1) chain_mono)
1.84 +  apply assumption
1.85 +  done
1.86 +
1.87 +lemma flat_imp_chfin_poo: "(ALL Y. chain(Y::nat=>('a)lift)-->(EX n. max_in_chain n Y))"
1.88 +  -- {* Tailoring @{text flat_imp_chfin} of @{text Fix.ML} to @{text lift} *}
1.89 +  apply (unfold max_in_chain_def)
1.90 +  apply (intro strip)
1.91 +  apply (rule_tac P = "ALL i. Y (i) = Undef" in case_split)
1.92 +   apply (rule_tac x = 0 in exI)
1.93 +   apply (intro strip)
1.94 +   apply (rule trans)
1.95 +    apply (erule spec)
1.96 +   apply (rule sym)
1.97 +   apply (erule spec)
1.98 +  apply (subgoal_tac "ALL x y. x << (y:: ('a) lift) --> x=Undef | x=y")
1.99 +   prefer 2 apply (simp add: inst_lift_po)
1.100 +  apply (rule chain_mono2_po [THEN exE])
1.101 +    apply fast
1.102 +   apply assumption
1.103 +  apply (rule_tac x = "Suc x" in exI)
1.104 +  apply (intro strip)
1.105 +  apply (rule disjE)
1.106 +    prefer 3 apply assumption
1.107 +   apply (rule mp)
1.108 +    apply (drule spec)
1.109 +    apply (erule spec)
1.110 +   apply (erule le_imp_less_or_eq [THEN disjE])
1.111 +    apply (erule chain_mono)
1.112 +    apply auto
1.113 +  done
1.114 +
1.115 +theorem cpo_lift: "chain (Y::nat => 'a lift) ==> EX x. range Y <<| x"
1.116 +  apply (cut_tac flat_imp_chfin_poo)
1.117 +  apply (blast intro: lub_finch1)
1.118 +  done
1.119 +
1.120 +instance lift :: ("term") pcpo
1.121 +  apply intro_classes
1.122 +   apply (erule cpo_lift)
1.123 +  apply (rule least_lift)
1.124 +  done
1.125 +
1.126 +lemma inst_lift_pcpo: "UU = Undef"
1.127 +  -- {* For compatibility with old HOLCF-Version. *}
1.128 +  by (simp add: UU_def UU_lift_def)
1.129 +
1.130 +
1.131 +subsection {* Lift as a datatype *}
1.132 +
1.133 +lemma lift_distinct1: "UU ~= Def x"
1.134 +  by (simp add: Undef_def Def_def Abs_lift_inject lift_def inst_lift_pcpo)
1.135 +
1.136 +lemma lift_distinct2: "Def x ~= UU"
1.137 +  by (simp add: Undef_def Def_def Abs_lift_inject lift_def inst_lift_pcpo)
1.138 +
1.139 +lemma Def_inject: "(Def x = Def x') = (x = x')"
1.140 +  by (simp add: Def_def Abs_lift_inject lift_def)
1.141 +
1.142 +lemma lift_induct: "P UU ==> (!!x. P (Def x)) ==> P y"
1.143 +  apply (induct y)
1.144 +  apply (induct_tac y)
1.145 +   apply (simp_all add: Undef_def Def_def inst_lift_pcpo)
1.146 +  done
1.147 +
1.148 +rep_datatype lift
1.149 +  distinct lift_distinct1 lift_distinct2
1.150 +  inject Def_inject
1.151 +  induction lift_induct
1.152 +
1.153 +lemma Def_not_UU: "Def a ~= UU"
1.154 +  by simp
1.155 +
1.156 +
1.157 +subsection {* Further operations *}
1.158 +
1.159 +consts
1.160 + flift1      :: "('a => 'b::pcpo) => ('a lift -> 'b)"
1.161 + flift2      :: "('a => 'b)       => ('a lift -> 'b lift)"
1.162 + liftpair    ::"'a::term lift * 'b::term lift => ('a * 'b) lift"
1.163
1.164 -instance lift :: (term)flat (ax_flat_lift)
1.165 +defs
1.166 + flift1_def:
1.167 +  "flift1 f == (LAM x. (case x of
1.168 +                   UU => UU
1.169 +                 | Def y => (f y)))"
1.170 + flift2_def:
1.171 +  "flift2 f == (LAM x. (case x of
1.172 +                   UU => UU
1.173 +                 | Def y => Def (f y)))"
1.174 + liftpair_def:
1.175 +  "liftpair x  == (case (cfst\$x) of
1.176 +                  UU  => UU
1.177 +                | Def x1 => (case (csnd\$x) of
1.178 +                               UU => UU
1.179 +                             | Def x2 => Def (x1,x2)))"
1.180 +
1.181 +
1.182 +declare inst_lift_pcpo [symmetric, simp]
1.183 +
1.184 +
1.185 +lemma less_lift: "(x::'a lift) << y = (x=y | x=UU)"
1.186 +  by (simp add: inst_lift_po)
1.187 +
1.188 +
1.189 +text {* @{text UU} and @{text Def} *}
1.190 +
1.191 +lemma Lift_exhaust: "x = UU | (EX y. x = Def y)"
1.192 +  by (induct x) simp_all
1.193 +
1.194 +lemma Lift_cases: "[| x = UU ==> P; ? a. x = Def a ==> P |] ==> P"
1.195 +  by (insert Lift_exhaust) blast
1.196 +
1.197 +lemma not_Undef_is_Def: "(x ~= UU) = (EX y. x = Def y)"
1.198 +  by (cases x) simp_all
1.199 +
1.200 +text {*
1.201 +  For @{term "x ~= UU"} in assumptions @{text def_tac} replaces @{text
1.202 +  x} by @{text "Def a"} in conclusion. *}
1.203 +
1.204 +ML {*
1.205 +  local val not_Undef_is_Def = thm "not_Undef_is_Def"
1.206 +  in val def_tac = SIMPSET' (fn ss =>
1.207 +    etac (not_Undef_is_Def RS iffD1 RS exE) THEN' asm_simp_tac ss)
1.208 +  end;
1.209 +*}
1.210 +
1.211 +lemma Undef_eq_UU: "Undef = UU"
1.212 +  by (rule inst_lift_pcpo [symmetric])
1.213 +
1.214 +lemma DefE: "Def x = UU ==> R"
1.215 +  by simp
1.216 +
1.217 +lemma DefE2: "[| x = Def s; x = UU |] ==> R"
1.218 +  by simp
1.219 +
1.220 +lemma Def_inject_less_eq: "Def x << Def y = (x = y)"
1.221 +  by (simp add: less_lift_def)
1.222 +
1.223 +lemma Def_less_is_eq [simp]: "Def x << y = (Def x = y)"
1.224 +  by (simp add: less_lift)
1.225 +
1.226 +
1.227 +subsection {* Lift is flat *}
1.228 +
1.229 +instance lift :: ("term") flat
1.230 +proof
1.231 +  show "ALL x y::'a lift. x << y --> x = UU | x = y"
1.232 +    by (simp add: less_lift)
1.233 +qed
1.234 +
1.235 +defaultsort pcpo
1.236 +
1.237 +
1.238 +text {*
1.239 +  \medskip Two specific lemmas for the combination of LCF and HOL
1.240 +  terms.
1.241 +*}
1.242 +
1.243 +lemma cont_Rep_CFun_app: "[|cont g; cont f|] ==> cont(%x. ((f x)\$(g x)) s)"
1.244 +  apply (rule cont2cont_CF1L)
1.245 +  apply (tactic "resolve_tac cont_lemmas1 1")+
1.246 +   apply auto
1.247 +  done
1.248 +
1.249 +lemma cont_Rep_CFun_app_app: "[|cont g; cont f|] ==> cont(%x. ((f x)\$(g x)) s t)"
1.250 +  apply (rule cont2cont_CF1L)
1.251 +  apply (erule cont_Rep_CFun_app)
1.252 +  apply assumption
1.253 +  done
1.254
1.255 -default pcpo
1.256 +text {* Continuity of if-then-else. *}
1.257 +
1.258 +lemma cont_if: "[| cont f1; cont f2 |] ==> cont (%x. if b then f1 x else f2 x)"
1.259 +  by (cases b) simp_all
1.260 +
1.261 +
1.262 +subsection {* Continuity Proofs for flift1, flift2, if *}
1.263 +
1.264 +text {* Need the instance of @{text flat}. *}
1.265 +
1.266 +lemma cont_flift1_arg: "cont (lift_case UU f)"
1.267 +  -- {* @{text flift1} is continuous in its argument itself. *}
1.268 +  apply (rule flatdom_strict2cont)
1.269 +  apply simp
1.270 +  done
1.271 +
1.272 +lemma cont_flift1_not_arg: "!!f. [| !! a. cont (%y. (f y) a) |] ==>
1.273 +           cont (%y. lift_case UU (f y))"
1.274 +  -- {* @{text flift1} is continuous in a variable that occurs only
1.275 +    in the @{text Def} branch. *}
1.276 +  apply (rule cont2cont_CF1L_rev)
1.277 +  apply (intro strip)
1.278 +  apply (case_tac y)
1.279 +   apply simp
1.280 +  apply simp
1.281 +  done
1.282 +
1.283 +lemma cont_flift1_arg_and_not_arg: "!!f. [| !! a. cont (%y. (f y) a); cont g|] ==>
1.284 +    cont (%y. lift_case UU (f y) (g y))"
1.285 +  -- {* @{text flift1} is continuous in a variable that occurs either
1.286 +    in the @{text Def} branch or in the argument. *}
1.287 +  apply (rule_tac tt = g in cont2cont_app)
1.288 +    apply (rule cont_flift1_not_arg)
1.289 +    apply auto
1.290 +  apply (rule cont_flift1_arg)
1.291 +  done
1.292 +
1.293 +lemma cont_flift2_arg: "cont (lift_case UU (%y. Def (f y)))"
1.294 +  -- {* @{text flift2} is continuous in its argument itself. *}
1.295 +  apply (rule flatdom_strict2cont)
1.296 +  apply simp
1.297 +  done
1.298 +
1.299 +text {*
1.300 +  \medskip Extension of cont_tac and installation of simplifier.
1.301 +*}
1.302 +
1.303 +lemma cont2cont_CF1L_rev2: "(!!y. cont (%x. c1 x y)) ==> cont c1"
1.304 +  apply (rule cont2cont_CF1L_rev)
1.305 +  apply simp
1.306 +  done
1.307 +
1.308 +lemmas cont_lemmas_ext [simp] =
1.309 +  cont_flift1_arg cont_flift2_arg
1.310 +  cont_flift1_arg_and_not_arg cont2cont_CF1L_rev2
1.311 +  cont_Rep_CFun_app cont_Rep_CFun_app_app cont_if
1.312 +
1.313 +ML_setup {*
1.314 +val cont_lemmas2 = cont_lemmas1 @ thms "cont_lemmas_ext";
1.315 +
1.316 +fun cont_tac  i = resolve_tac cont_lemmas2 i;
1.317 +fun cont_tacR i = REPEAT (cont_tac i);
1.318 +
1.319 +local val flift1_def = thm "flift1_def" and flift2_def = thm "flift2_def"
1.320 +in fun cont_tacRs i =
1.321 +  simp_tac (simpset() addsimps [flift1_def, flift2_def]) i THEN
1.322 +  REPEAT (cont_tac i)
1.323 +end;
1.324 +
1.326 +  (mk_solver "cont_tac" (K (DEPTH_SOLVE_1 o cont_tac)));
1.327 +*}
1.328 +
1.329 +
1.330 +subsection {* flift1, flift2 *}
1.331 +
1.332 +lemma flift1_Def [simp]: "flift1 f\$(Def x) = (f x)"
1.333 +  by (simp add: flift1_def)
1.334 +
1.335 +lemma flift2_Def [simp]: "flift2 f\$(Def x) = Def (f x)"
1.336 +  by (simp add: flift2_def)
1.337 +
1.338 +lemma flift1_UU [simp]: "flift1 f\$UU = UU"
1.339 +  by (simp add: flift1_def)
1.340 +
1.341 +lemma flift2_UU [simp]: "flift2 f\$UU = UU"
1.342 +  by (simp add: flift2_def)
1.343 +
1.344 +lemma flift2_nUU [simp]: "x~=UU ==> (flift2 f)\$x~=UU"
1.345 +  by (tactic "def_tac 1")
1.346
1.347  end
1.348 -
1.349 -
1.350 -
```