(* Title: HOL/Induct/Com
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge
Example of Mutual Induction via Iteratived Inductive Definitions: Commands
*)
header{*Mutual Induction via Iteratived Inductive Definitions*}
theory Com imports Main begin
typedecl loc
types state = "loc => nat"
n2n2n = "nat => nat => nat"
arities loc :: type
datatype
exp = N nat
| X loc
| Op n2n2n exp exp
| valOf com exp ("VALOF _ RESULTIS _" 60)
and
com = SKIP
| ":=" loc exp (infixl 60)
| Semi com com ("_;;_" [60, 60] 60)
| Cond exp com com ("IF _ THEN _ ELSE _" 60)
| While exp com ("WHILE _ DO _" 60)
subsection {* Commands *}
text{* Execution of commands *}
consts exec :: "((exp*state) * (nat*state)) set => ((com*state)*state)set"
"@exec" :: "((exp*state) * (nat*state)) set =>
[com*state,state] => bool" ("_/ -[_]-> _" [50,0,50] 50)
translations "csig -[eval]-> s" == "(csig,s) \<in> exec eval"
syntax eval' :: "[exp*state,nat*state] =>
((exp*state) * (nat*state)) set => bool"
("_/ -|[_]-> _" [50,0,50] 50)
translations
"esig -|[eval]-> ns" => "(esig,ns) \<in> eval"
text{*Command execution. Natural numbers represent Booleans: 0=True, 1=False*}
inductive "exec eval"
intros
Skip: "(SKIP,s) -[eval]-> s"
Assign: "(e,s) -|[eval]-> (v,s') ==> (x := e, s) -[eval]-> s'(x:=v)"
Semi: "[| (c0,s) -[eval]-> s2; (c1,s2) -[eval]-> s1 |]
==> (c0 ;; c1, s) -[eval]-> s1"
IfTrue: "[| (e,s) -|[eval]-> (0,s'); (c0,s') -[eval]-> s1 |]
==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"
IfFalse: "[| (e,s) -|[eval]-> (Suc 0, s'); (c1,s') -[eval]-> s1 |]
==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"
WhileFalse: "(e,s) -|[eval]-> (Suc 0, s1)
==> (WHILE e DO c, s) -[eval]-> s1"
WhileTrue: "[| (e,s) -|[eval]-> (0,s1);
(c,s1) -[eval]-> s2; (WHILE e DO c, s2) -[eval]-> s3 |]
==> (WHILE e DO c, s) -[eval]-> s3"
declare exec.intros [intro]
inductive_cases
[elim!]: "(SKIP,s) -[eval]-> t"
and [elim!]: "(x:=a,s) -[eval]-> t"
and [elim!]: "(c1;;c2, s) -[eval]-> t"
and [elim!]: "(IF e THEN c1 ELSE c2, s) -[eval]-> t"
and exec_WHILE_case: "(WHILE b DO c,s) -[eval]-> t"
text{*Justifies using "exec" in the inductive definition of "eval"*}
lemma exec_mono: "A<=B ==> exec(A) <= exec(B)"
apply (unfold exec.defs )
apply (rule lfp_mono)
apply (assumption | rule basic_monos)+
done
ML {*
Unify.trace_bound := 30;
Unify.search_bound := 60;
*}
text{*Command execution is functional (deterministic) provided evaluation is*}
theorem single_valued_exec: "single_valued ev ==> single_valued(exec ev)"
apply (simp add: single_valued_def)
apply (intro allI)
apply (rule impI)
apply (erule exec.induct)
apply (blast elim: exec_WHILE_case)+
done
subsection {* Expressions *}
text{* Evaluation of arithmetic expressions *}
consts eval :: "((exp*state) * (nat*state)) set"
"-|->" :: "[exp*state,nat*state] => bool" (infixl 50)
translations
"esig -|-> (n,s)" <= "(esig,n,s) \<in> eval"
"esig -|-> ns" == "(esig,ns ) \<in> eval"
inductive eval
intros
N [intro!]: "(N(n),s) -|-> (n,s)"
X [intro!]: "(X(x),s) -|-> (s(x),s)"
Op [intro]: "[| (e0,s) -|-> (n0,s0); (e1,s0) -|-> (n1,s1) |]
==> (Op f e0 e1, s) -|-> (f n0 n1, s1)"
valOf [intro]: "[| (c,s) -[eval]-> s0; (e,s0) -|-> (n,s1) |]
==> (VALOF c RESULTIS e, s) -|-> (n, s1)"
monos exec_mono
inductive_cases
[elim!]: "(N(n),sigma) -|-> (n',s')"
and [elim!]: "(X(x),sigma) -|-> (n,s')"
and [elim!]: "(Op f a1 a2,sigma) -|-> (n,s')"
and [elim!]: "(VALOF c RESULTIS e, s) -|-> (n, s1)"
lemma var_assign_eval [intro!]: "(X x, s(x:=n)) -|-> (n, s(x:=n))"
by (rule fun_upd_same [THEN subst], fast)
text{* Make the induction rule look nicer -- though eta_contract makes the new
version look worse than it is...*}
lemma split_lemma:
"{((e,s),(n,s')). P e s n s'} = Collect (split (%v. split (split P v)))"
by auto
text{*New induction rule. Note the form of the VALOF induction hypothesis*}
lemma eval_induct:
"[| (e,s) -|-> (n,s');
!!n s. P (N n) s n s;
!!s x. P (X x) s (s x) s;
!!e0 e1 f n0 n1 s s0 s1.
[| (e0,s) -|-> (n0,s0); P e0 s n0 s0;
(e1,s0) -|-> (n1,s1); P e1 s0 n1 s1
|] ==> P (Op f e0 e1) s (f n0 n1) s1;
!!c e n s s0 s1.
[| (c,s) -[eval Int {((e,s),(n,s')). P e s n s'}]-> s0;
(c,s) -[eval]-> s0;
(e,s0) -|-> (n,s1); P e s0 n s1 |]
==> P (VALOF c RESULTIS e) s n s1
|] ==> P e s n s'"
apply (erule eval.induct, blast)
apply blast
apply blast
apply (frule Int_lower1 [THEN exec_mono, THEN subsetD])
apply (auto simp add: split_lemma)
done
text{*Lemma for Function_eval. The major premise is that (c,s) executes to s1
using eval restricted to its functional part. Note that the execution
(c,s) -[eval]-> s2 can use unrestricted eval! The reason is that
the execution (c,s) -[eval Int {...}]-> s1 assures us that execution is
functional on the argument (c,s).
*}
lemma com_Unique:
"(c,s) -[eval Int {((e,s),(n,t)). \<forall>nt'. (e,s) -|-> nt' --> (n,t)=nt'}]-> s1
==> \<forall>s2. (c,s) -[eval]-> s2 --> s2=s1"
apply (erule exec.induct, simp_all)
apply blast
apply force
apply blast
apply blast
apply blast
apply (blast elim: exec_WHILE_case)
apply (erule_tac V = "(?c,s2) -[?ev]-> s3" in thin_rl)
apply clarify
apply (erule exec_WHILE_case, blast+)
done
text{*Expression evaluation is functional, or deterministic*}
theorem single_valued_eval: "single_valued eval"
apply (unfold single_valued_def)
apply (intro allI, rule impI)
apply (simp (no_asm_simp) only: split_tupled_all)
apply (erule eval_induct)
apply (drule_tac [4] com_Unique)
apply (simp_all (no_asm_use))
apply blast+
done
lemma eval_N_E_lemma: "(e,s) -|-> (v,s') ==> (e = N n) --> (v=n & s'=s)"
by (erule eval_induct, simp_all)
lemmas eval_N_E [dest!] = eval_N_E_lemma [THEN mp, OF _ refl]
text{*This theorem says that "WHILE TRUE DO c" cannot terminate*}
lemma while_true_E [rule_format]:
"(c', s) -[eval]-> t ==> (c' = WHILE (N 0) DO c) --> False"
by (erule exec.induct, auto)
subsection{* Equivalence of IF e THEN c;;(WHILE e DO c) ELSE SKIP and
WHILE e DO c *}
lemma while_if1 [rule_format]:
"(c',s) -[eval]-> t
==> (c' = WHILE e DO c) -->
(IF e THEN c;;c' ELSE SKIP, s) -[eval]-> t"
by (erule exec.induct, auto)
lemma while_if2 [rule_format]:
"(c',s) -[eval]-> t
==> (c' = IF e THEN c;;(WHILE e DO c) ELSE SKIP) -->
(WHILE e DO c, s) -[eval]-> t"
by (erule exec.induct, auto)
theorem while_if:
"((IF e THEN c;;(WHILE e DO c) ELSE SKIP, s) -[eval]-> t) =
((WHILE e DO c, s) -[eval]-> t)"
by (blast intro: while_if1 while_if2)
subsection{* Equivalence of (IF e THEN c1 ELSE c2);;c
and IF e THEN (c1;;c) ELSE (c2;;c) *}
lemma if_semi1 [rule_format]:
"(c',s) -[eval]-> t
==> (c' = (IF e THEN c1 ELSE c2);;c) -->
(IF e THEN (c1;;c) ELSE (c2;;c), s) -[eval]-> t"
by (erule exec.induct, auto)
lemma if_semi2 [rule_format]:
"(c',s) -[eval]-> t
==> (c' = IF e THEN (c1;;c) ELSE (c2;;c)) -->
((IF e THEN c1 ELSE c2);;c, s) -[eval]-> t"
by (erule exec.induct, auto)
theorem if_semi: "(((IF e THEN c1 ELSE c2);;c, s) -[eval]-> t) =
((IF e THEN (c1;;c) ELSE (c2;;c), s) -[eval]-> t)"
by (blast intro: if_semi1 if_semi2)
subsection{* Equivalence of VALOF c1 RESULTIS (VALOF c2 RESULTIS e)
and VALOF c1;;c2 RESULTIS e
*}
lemma valof_valof1 [rule_format]:
"(e',s) -|-> (v,s')
==> (e' = VALOF c1 RESULTIS (VALOF c2 RESULTIS e)) -->
(VALOF c1;;c2 RESULTIS e, s) -|-> (v,s')"
by (erule eval_induct, auto)
lemma valof_valof2 [rule_format]:
"(e',s) -|-> (v,s')
==> (e' = VALOF c1;;c2 RESULTIS e) -->
(VALOF c1 RESULTIS (VALOF c2 RESULTIS e), s) -|-> (v,s')"
by (erule eval_induct, auto)
theorem valof_valof:
"((VALOF c1 RESULTIS (VALOF c2 RESULTIS e), s) -|-> (v,s')) =
((VALOF c1;;c2 RESULTIS e, s) -|-> (v,s'))"
by (blast intro: valof_valof1 valof_valof2)
subsection{* Equivalence of VALOF SKIP RESULTIS e and e *}
lemma valof_skip1 [rule_format]:
"(e',s) -|-> (v,s')
==> (e' = VALOF SKIP RESULTIS e) -->
(e, s) -|-> (v,s')"
by (erule eval_induct, auto)
lemma valof_skip2:
"(e,s) -|-> (v,s') ==> (VALOF SKIP RESULTIS e, s) -|-> (v,s')"
by blast
theorem valof_skip:
"((VALOF SKIP RESULTIS e, s) -|-> (v,s')) = ((e, s) -|-> (v,s'))"
by (blast intro: valof_skip1 valof_skip2)
subsection{* Equivalence of VALOF x:=e RESULTIS x and e *}
lemma valof_assign1 [rule_format]:
"(e',s) -|-> (v,s'')
==> (e' = VALOF x:=e RESULTIS X x) -->
(\<exists>s'. (e, s) -|-> (v,s') & (s'' = s'(x:=v)))"
apply (erule eval_induct)
apply (simp_all del: fun_upd_apply, clarify, auto)
done
lemma valof_assign2:
"(e,s) -|-> (v,s') ==> (VALOF x:=e RESULTIS X x, s) -|-> (v,s'(x:=v))"
by blast
end