header "Small-Step Semantics of Commands"
theory Small_Step imports Star Big_Step begin
subsection "The transition relation"
inductive
small_step :: "com * state \<Rightarrow> com * state \<Rightarrow> bool" (infix "\<rightarrow>" 55)
where
Assign: "(x ::= a, s) \<rightarrow> (SKIP, s(x := aval a s))" |
Semi1: "(SKIP;c\<^isub>2,s) \<rightarrow> (c\<^isub>2,s)" |
Semi2: "(c\<^isub>1,s) \<rightarrow> (c\<^isub>1',s') \<Longrightarrow> (c\<^isub>1;c\<^isub>2,s) \<rightarrow> (c\<^isub>1';c\<^isub>2,s')" |
IfTrue: "bval b s \<Longrightarrow> (IF b THEN c\<^isub>1 ELSE c\<^isub>2,s) \<rightarrow> (c\<^isub>1,s)" |
IfFalse: "\<not>bval b s \<Longrightarrow> (IF b THEN c\<^isub>1 ELSE c\<^isub>2,s) \<rightarrow> (c\<^isub>2,s)" |
While: "(WHILE b DO c,s) \<rightarrow> (IF b THEN c; WHILE b DO c ELSE SKIP,s)"
abbreviation small_steps :: "com * state \<Rightarrow> com * state \<Rightarrow> bool" (infix "\<rightarrow>*" 55)
where "x \<rightarrow>* y == star small_step x y"
subsection{* Executability *}
code_pred small_step .
values "{(c',map t [''x'',''y'',''z'']) |c' t.
(''x'' ::= V ''z''; ''y'' ::= V ''x'',
lookup[(''x'',3),(''y'',7),(''z'',5)]) \<rightarrow>* (c',t)}"
subsection{* Proof infrastructure *}
subsubsection{* Induction rules *}
text{* The default induction rule @{thm[source] small_step.induct} only works
for lemmas of the form @{text"a \<rightarrow> b \<Longrightarrow> \<dots>"} where @{text a} and @{text b} are
not already pairs @{text"(DUMMY,DUMMY)"}. We can generate a suitable variant
of @{thm[source] small_step.induct} for pairs by ``splitting'' the arguments
@{text"\<rightarrow>"} into pairs: *}
lemmas small_step_induct = small_step.induct[split_format(complete)]
subsubsection{* Proof automation *}
declare small_step.intros[simp,intro]
text{* So called transitivity rules. See below. *}
declare step[trans] step1[trans]
lemma step2[trans]:
"cs \<rightarrow> cs' \<Longrightarrow> cs' \<rightarrow> cs'' \<Longrightarrow> cs \<rightarrow>* cs''"
by(metis refl step)
declare star_trans[trans]
text{* Rule inversion: *}
inductive_cases SkipE[elim!]: "(SKIP,s) \<rightarrow> ct"
thm SkipE
inductive_cases AssignE[elim!]: "(x::=a,s) \<rightarrow> ct"
thm AssignE
inductive_cases SemiE[elim]: "(c1;c2,s) \<rightarrow> ct"
thm SemiE
inductive_cases IfE[elim!]: "(IF b THEN c1 ELSE c2,s) \<rightarrow> ct"
inductive_cases WhileE[elim]: "(WHILE b DO c, s) \<rightarrow> ct"
text{* A simple property: *}
lemma deterministic:
"cs \<rightarrow> cs' \<Longrightarrow> cs \<rightarrow> cs'' \<Longrightarrow> cs'' = cs'"
apply(induct arbitrary: cs'' rule: small_step.induct)
apply blast+
done
subsection "Equivalence with big-step semantics"
lemma star_semi2: "(c1,s) \<rightarrow>* (c1',s') \<Longrightarrow> (c1;c2,s) \<rightarrow>* (c1';c2,s')"
proof(induct rule: star_induct)
case refl thus ?case by simp
next
case step
thus ?case by (metis Semi2 star.step)
qed
lemma semi_comp:
"\<lbrakk> (c1,s1) \<rightarrow>* (SKIP,s2); (c2,s2) \<rightarrow>* (SKIP,s3) \<rbrakk>
\<Longrightarrow> (c1;c2, s1) \<rightarrow>* (SKIP,s3)"
by(blast intro: star.step star_semi2 star_trans)
text{* The following proof corresponds to one on the board where one would
show chains of @{text "\<rightarrow>"} and @{text "\<rightarrow>*"} steps. This is what the
also/finally proof steps do: they compose chains, implicitly using the rules
declared with attribute [trans] above. *}
lemma big_to_small:
"cs \<Rightarrow> t \<Longrightarrow> cs \<rightarrow>* (SKIP,t)"
proof (induct rule: big_step.induct)
fix s show "(SKIP,s) \<rightarrow>* (SKIP,s)" by simp
next
fix x a s show "(x ::= a,s) \<rightarrow>* (SKIP, s(x := aval a s))" by auto
next
fix c1 c2 s1 s2 s3
assume "(c1,s1) \<rightarrow>* (SKIP,s2)" and "(c2,s2) \<rightarrow>* (SKIP,s3)"
thus "(c1;c2, s1) \<rightarrow>* (SKIP,s3)" by (rule semi_comp)
next
fix s::state and b c0 c1 t
assume "bval b s"
hence "(IF b THEN c0 ELSE c1,s) \<rightarrow> (c0,s)" by simp
also assume "(c0,s) \<rightarrow>* (SKIP,t)"
finally show "(IF b THEN c0 ELSE c1,s) \<rightarrow>* (SKIP,t)" . --"= by assumption"
next
fix s::state and b c0 c1 t
assume "\<not>bval b s"
hence "(IF b THEN c0 ELSE c1,s) \<rightarrow> (c1,s)" by simp
also assume "(c1,s) \<rightarrow>* (SKIP,t)"
finally show "(IF b THEN c0 ELSE c1,s) \<rightarrow>* (SKIP,t)" .
next
fix b c and s::state
assume b: "\<not>bval b s"
let ?if = "IF b THEN c; WHILE b DO c ELSE SKIP"
have "(WHILE b DO c,s) \<rightarrow> (?if, s)" by blast
also have "(?if,s) \<rightarrow> (SKIP, s)" by (simp add: b)
finally show "(WHILE b DO c,s) \<rightarrow>* (SKIP,s)" by auto
next
fix b c s s' t
let ?w = "WHILE b DO c"
let ?if = "IF b THEN c; ?w ELSE SKIP"
assume w: "(?w,s') \<rightarrow>* (SKIP,t)"
assume c: "(c,s) \<rightarrow>* (SKIP,s')"
assume b: "bval b s"
have "(?w,s) \<rightarrow> (?if, s)" by blast
also have "(?if, s) \<rightarrow> (c; ?w, s)" by (simp add: b)
also have "(c; ?w,s) \<rightarrow>* (SKIP,t)" by(rule semi_comp[OF c w])
finally show "(WHILE b DO c,s) \<rightarrow>* (SKIP,t)" by auto
qed
text{* Each case of the induction can be proved automatically: *}
lemma "cs \<Rightarrow> t \<Longrightarrow> cs \<rightarrow>* (SKIP,t)"
proof (induct rule: big_step.induct)
case Skip show ?case by blast
next
case Assign show ?case by blast
next
case Semi thus ?case by (blast intro: semi_comp)
next
case IfTrue thus ?case by (blast intro: step)
next
case IfFalse thus ?case by (blast intro: step)
next
case WhileFalse thus ?case
by (metis step step1 small_step.IfFalse small_step.While)
next
case WhileTrue
thus ?case
by(metis While semi_comp small_step.IfTrue step[of small_step])
qed
lemma small1_big_continue:
"cs \<rightarrow> cs' \<Longrightarrow> cs' \<Rightarrow> t \<Longrightarrow> cs \<Rightarrow> t"
apply (induct arbitrary: t rule: small_step.induct)
apply auto
done
lemma small_big_continue:
"cs \<rightarrow>* cs' \<Longrightarrow> cs' \<Rightarrow> t \<Longrightarrow> cs \<Rightarrow> t"
apply (induct rule: star.induct)
apply (auto intro: small1_big_continue)
done
lemma small_to_big: "cs \<rightarrow>* (SKIP,t) \<Longrightarrow> cs \<Rightarrow> t"
by (metis small_big_continue Skip)
text {*
Finally, the equivalence theorem:
*}
theorem big_iff_small:
"cs \<Rightarrow> t = cs \<rightarrow>* (SKIP,t)"
by(metis big_to_small small_to_big)
subsection "Final configurations and infinite reductions"
definition "final cs \<longleftrightarrow> \<not>(EX cs'. cs \<rightarrow> cs')"
lemma finalD: "final (c,s) \<Longrightarrow> c = SKIP"
apply(simp add: final_def)
apply(induct c)
apply blast+
done
lemma final_iff_SKIP: "final (c,s) = (c = SKIP)"
by (metis SkipE finalD final_def)
text{* Now we can show that @{text"\<Rightarrow>"} yields a final state iff @{text"\<rightarrow>"}
terminates: *}
lemma big_iff_small_termination:
"(EX t. cs \<Rightarrow> t) \<longleftrightarrow> (EX cs'. cs \<rightarrow>* cs' \<and> final cs')"
by(simp add: big_iff_small final_iff_SKIP)
text{* This is the same as saying that the absence of a big step result is
equivalent with absence of a terminating small step sequence, i.e.\ with
nontermination. Since @{text"\<rightarrow>"} is determininistic, there is no difference
between may and must terminate. *}
end