wneuper/isa: src/HOL/IMPP/Natural.thy@a6dc1769fdda

     1 (*  Title:      HOL/IMPP/Natural.thy

     2     ID:         $Id$

     3     Author:     David von Oheimb (based on a theory by Tobias Nipkow et al), TUM

     4     Copyright   1999 TUM

     5 *)

     7 header {* Natural semantics of commands *}

     9 theory Natural

    10 imports Com

    11 begin

    13 (** Execution of commands **)

    15 consts

    16   newlocs :: locals

    17   setlocs :: "state => locals => state"

    18   getlocs :: "state => locals"

    19   update  :: "state => vname => val => state"     ("_/[_/::=/_]" [900,0,0] 900)

    21 abbreviation

    22   loc :: "state => locals"  ("_<_>" [75,0] 75) where

    23   "s<X> == getlocs s X"

    25 inductive

    26   evalc :: "[com,state,    state] => bool"  ("<_,_>/ -c-> _" [0,0,  51] 51)

    27   where

    28     Skip:    "<SKIP,s> -c-> s"

    30   | Assign:  "<X :== a,s> -c-> s[X::=a s]"

    32   | Local:   "<c, s0[Loc Y::= a s0]> -c-> s1 ==>

    33               <LOCAL Y := a IN c, s0> -c-> s1[Loc Y::=s0<Y>]"

    35   | Semi:    "[| <c0,s0> -c-> s1; <c1,s1> -c-> s2 |] ==>

    36               <c0;; c1, s0> -c-> s2"

    38   | IfTrue:  "[| b s; <c0,s> -c-> s1 |] ==>

    39               <IF b THEN c0 ELSE c1, s> -c-> s1"

    41   | IfFalse: "[| ~b s; <c1,s> -c-> s1 |] ==>

    42               <IF b THEN c0 ELSE c1, s> -c-> s1"

    44   | WhileFalse: "~b s ==> <WHILE b DO c,s> -c-> s"

    46   | WhileTrue:  "[| b s0;  <c,s0> -c-> s1;  <WHILE b DO c, s1> -c-> s2 |] ==>

    47                  <WHILE b DO c, s0> -c-> s2"

    49   | Body:       "<the (body pn), s0> -c-> s1 ==>

    50                  <BODY pn, s0> -c-> s1"

    52   | Call:       "<BODY pn, (setlocs s0 newlocs)[Loc Arg::=a s0]> -c-> s1 ==>

    53                  <X:=CALL pn(a), s0> -c-> (setlocs s1 (getlocs s0))

    54                                           [X::=s1<Res>]"

    56 inductive

    57   evaln :: "[com,state,nat,state] => bool"  ("<_,_>/ -_-> _" [0,0,0,51] 51)

    58   where

    59     Skip:    "<SKIP,s> -n-> s"

    61   | Assign:  "<X :== a,s> -n-> s[X::=a s]"

    63   | Local:   "<c, s0[Loc Y::= a s0]> -n-> s1 ==>

    64               <LOCAL Y := a IN c, s0> -n-> s1[Loc Y::=s0<Y>]"

    66   | Semi:    "[| <c0,s0> -n-> s1; <c1,s1> -n-> s2 |] ==>

    67               <c0;; c1, s0> -n-> s2"

    69   | IfTrue:  "[| b s; <c0,s> -n-> s1 |] ==>

    70               <IF b THEN c0 ELSE c1, s> -n-> s1"

    72   | IfFalse: "[| ~b s; <c1,s> -n-> s1 |] ==>

    73               <IF b THEN c0 ELSE c1, s> -n-> s1"

    75   | WhileFalse: "~b s ==> <WHILE b DO c,s> -n-> s"

    77   | WhileTrue:  "[| b s0;  <c,s0> -n-> s1;  <WHILE b DO c, s1> -n-> s2 |] ==>

    78                  <WHILE b DO c, s0> -n-> s2"

    80   | Body:       "<the (body pn), s0> -    n-> s1 ==>

    81                  <BODY pn, s0> -Suc n-> s1"

    83   | Call:       "<BODY pn, (setlocs s0 newlocs)[Loc Arg::=a s0]> -n-> s1 ==>

    84                  <X:=CALL pn(a), s0> -n-> (setlocs s1 (getlocs s0))

    85                                           [X::=s1<Res>]"

    88 inductive_cases evalc_elim_cases:

    89   "<SKIP,s> -c-> t"  "<X:==a,s> -c-> t"  "<LOCAL Y:=a IN c,s> -c-> t"

    90   "<c1;;c2,s> -c-> t"  "<IF b THEN c1 ELSE c2,s> -c-> t"

    91   "<BODY P,s> -c-> s1"  "<X:=CALL P(a),s> -c-> s1"

    93 inductive_cases evaln_elim_cases:

    94   "<SKIP,s> -n-> t"  "<X:==a,s> -n-> t"  "<LOCAL Y:=a IN c,s> -n-> t"

    95   "<c1;;c2,s> -n-> t"  "<IF b THEN c1 ELSE c2,s> -n-> t"

    96   "<BODY P,s> -n-> s1"  "<X:=CALL P(a),s> -n-> s1"

    98 inductive_cases evalc_WHILE_case: "<WHILE b DO c,s> -c-> t"

    99 inductive_cases evaln_WHILE_case: "<WHILE b DO c,s> -n-> t"

   101 declare evalc.intros [intro]

   102 declare evaln.intros [intro]

   104 declare evalc_elim_cases [elim!]

   105 declare evaln_elim_cases [elim!]

   107 (* evaluation of com is deterministic *)

   108 lemma com_det [rule_format (no_asm)]: "<c,s> -c-> t ==> (!u. <c,s> -c-> u --> u=t)"

   109 apply (erule evalc.induct)

   110 apply (erule_tac [8] V = "<?c,s1> -c-> s2" in thin_rl)

   111 (*blast needs unify_search_bound = 40*)

   112 apply (best elim: evalc_WHILE_case)+

   113 done

   115 lemma evaln_evalc: "<c,s> -n-> t ==> <c,s> -c-> t"

   116 apply (erule evaln.induct)

   117 apply (tactic {* ALLGOALS (resolve_tac (thms "evalc.intros") THEN_ALL_NEW atac) *})

   118 done

   120 lemma Suc_le_D_lemma: "[| Suc n <= m'; (!!m. n <= m ==> P (Suc m)) |] ==> P m'"

   121 apply (frule Suc_le_D)

   122 apply blast

   123 done

   125 lemma evaln_nonstrict [rule_format]: "<c,s> -n-> t ==> !m. n<=m --> <c,s> -m-> t"

   126 apply (erule evaln.induct)

   127 apply (tactic {* ALLGOALS (EVERY'[strip_tac,TRY o etac (thm "Suc_le_D_lemma"), REPEAT o smp_tac 1]) *})

   128 apply (tactic {* ALLGOALS (resolve_tac (thms "evaln.intros") THEN_ALL_NEW atac) *})

   129 done

   131 lemma evaln_Suc: "<c,s> -n-> s' ==> <c,s> -Suc n-> s'"

   132 apply (erule evaln_nonstrict)

   133 apply auto

   134 done

   136 lemma evaln_max2: "[| <c1,s1> -n1-> t1;  <c2,s2> -n2-> t2 |] ==>

   137     ? n. <c1,s1> -n -> t1 & <c2,s2> -n -> t2"

   138 apply (cut_tac m = "n1" and n = "n2" in nat_le_linear)

   139 apply (blast dest: evaln_nonstrict)

   140 done

   142 lemma evalc_evaln: "<c,s> -c-> t ==> ? n. <c,s> -n-> t"

   143 apply (erule evalc.induct)

   144 apply (tactic {* ALLGOALS (REPEAT o etac exE) *})

   145 apply (tactic {* TRYALL (EVERY'[datac (thm "evaln_max2") 1, REPEAT o eresolve_tac [exE, conjE]]) *})

   146 apply (tactic {* ALLGOALS (rtac exI THEN' resolve_tac (thms "evaln.intros") THEN_ALL_NEW atac) *})

   147 done

   149 lemma eval_eq: "<c,s> -c-> t = (? n. <c,s> -n-> t)"

   150 apply (fast elim: evalc_evaln evaln_evalc)

   151 done

   153 end

author	wenzelm
	Wed, 25 Jun 2008 22:01:34 +0200
changeset 27362	a6dc1769fdda
parent 24178	4ff1dc2aa18d
child 39406	0dec18004e75
permissions	-rw-r--r--