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(* Title: Inductive definition of termination
Author: Tobias Nipkow, 2001/2006
Maintainer: Tobias Nipkow
*)
theory PTermi imports PLang begin
subsection\<open>Termination\<close>
inductive
termi :: "com \<Rightarrow> state \<Rightarrow> bool" (infixl "\<down>" 50)
where
Do[iff]: "f s \<noteq> {} \<Longrightarrow> Do f \<down> s"
| Semi[intro!]: "\<lbrakk> c1 \<down> s0; \<And>s1. s0 -c1\<rightarrow> s1 \<Longrightarrow> c2 \<down> s1 \<rbrakk>
\<Longrightarrow> (c1;c2) \<down> s0"
| IfTrue[intro,simp]: "\<lbrakk> b s; c1 \<down> s \<rbrakk> \<Longrightarrow> IF b THEN c1 ELSE c2 \<down> s"
| IfFalse[intro,simp]: "\<lbrakk> \<not>b s; c2 \<down> s \<rbrakk> \<Longrightarrow> IF b THEN c1 ELSE c2 \<down> s"
| WhileFalse: "\<not>b s \<Longrightarrow> WHILE b DO c \<down> s"
| WhileTrue: "\<lbrakk> b s; c \<down> s; \<And>t. s -c\<rightarrow> t \<Longrightarrow> WHILE b DO c \<down> t \<rbrakk>
\<Longrightarrow> WHILE b DO c \<down> s"
| "body \<down> s \<Longrightarrow> CALL \<down> s"
| Local: "c \<down> f s \<Longrightarrow> LOCAL f;c;g \<down> s"
lemma [iff]: "(Do f \<down> s) = (f s \<noteq> {})"
apply(rule iffI)
prefer 2
apply(best intro:termi.intros)
apply(erule termi.cases)
apply blast+
done
lemma [iff]: "((c1;c2) \<down> s0) = (c1 \<down> s0 \<and> (\<forall>s1. s0 -c1\<rightarrow> s1 \<longrightarrow> c2 \<down> s1))"
apply(rule iffI)
prefer 2
apply(best intro:termi.intros)
apply(erule termi.cases)
apply blast+
done
lemma [iff]: "(IF b THEN c1 ELSE c2 \<down> s) =
((if b s then c1 else c2) \<down> s)"
apply simp
apply(rule conjI)
apply(rule impI)
apply(rule iffI)
prefer 2
apply(blast intro:termi.intros)
apply(erule termi.cases)
apply blast+
apply(rule impI)
apply(rule iffI)
prefer 2
apply(blast intro:termi.intros)
apply(erule termi.cases)
apply blast+
done
lemma [iff]: "(CALL \<down> s) = (body \<down> s)"
by(fast elim: termi.cases intro:termi.intros)
lemma [iff]: "(LOCAL f;c;g \<down> s) = (c \<down> f s)"
by(fast elim: termi.cases intro:termi.intros)
lemma termi_while_lemma[rule_format]:
"w\<down>fk \<Longrightarrow>
(\<forall>k b c. fk = f k \<and> w = WHILE b DO c \<and> (\<forall>i. f i -c\<rightarrow> f(Suc i))
\<longrightarrow> (\<exists>i. \<not>b(f i)))"
apply(erule termi.induct)
apply simp_all
apply blast
apply blast
done
lemma termi_while:
"\<lbrakk> (WHILE b DO c) \<down> f k; \<forall>i. f i -c\<rightarrow> f(Suc i) \<rbrakk> \<Longrightarrow> \<exists>i. \<not>b(f i)"
by(blast intro:termi_while_lemma)
lemma wf_termi: "wf {(t,s). WHILE b DO c \<down> s \<and> b s \<and> s -c\<rightarrow> t}"
apply(subst wf_iff_no_infinite_down_chain)
apply(rule notI)
apply clarsimp
apply(insert termi_while)
apply blast
done
end
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