(* Title: HOL/IMP/Natural.thy
ID: $Id$
Author: Tobias Nipkow & Robert Sandner, TUM
Isar Version: Gerwin Klein, 2001
Copyright 1996 TUM
*)
header "Natural Semantics of Commands"
theory Natural = Com:
subsection "Execution of commands"
consts evalc :: "(com \<times> state \<times> state) set"
syntax "_evalc" :: "[com,state,state] \<Rightarrow> bool" ("<_,_>/ -c-> _" [0,0,60] 60)
syntax (xsymbols)
"_evalc" :: "[com,state,state] \<Rightarrow> bool" ("\<langle>_,_\<rangle>/ \<longrightarrow>\<^sub>c _" [0,0,60] 60)
syntax (HTML output)
"_evalc" :: "[com,state,state] \<Rightarrow> bool" ("\<langle>_,_\<rangle>/ \<longrightarrow>\<^sub>c _" [0,0,60] 60)
text {*
We write @{text "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c s'"} for \emph{Statement @{text c}, started
in state @{text s}, terminates in state @{text s'}}. Formally,
@{text "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c s'"} is just another form of saying \emph{the tuple
@{text "(c,s,s')"} is part of the relation @{text evalc}}:
*}
translations "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c s'" == "(c,s,s') \<in> evalc"
constdefs
update :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)" ("_/[_ ::= /_]" [900,0,0] 900)
"update == fun_upd"
syntax (xsymbols)
update :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)" ("_/[_ \<mapsto> /_]" [900,0,0] 900)
text {*
The big-step execution relation @{text evalc} is defined inductively:
*}
inductive evalc
intros
Skip: "\<langle>\<SKIP>,s\<rangle> \<longrightarrow>\<^sub>c s"
Assign: "\<langle>x :== a,s\<rangle> \<longrightarrow>\<^sub>c s[x\<mapsto>a s]"
Semi: "\<langle>c0,s\<rangle> \<longrightarrow>\<^sub>c s'' \<Longrightarrow> \<langle>c1,s''\<rangle> \<longrightarrow>\<^sub>c s' \<Longrightarrow> \<langle>c0; c1, s\<rangle> \<longrightarrow>\<^sub>c s'"
IfTrue: "b s \<Longrightarrow> \<langle>c0,s\<rangle> \<longrightarrow>\<^sub>c s' \<Longrightarrow> \<langle>\<IF> b \<THEN> c0 \<ELSE> c1, s\<rangle> \<longrightarrow>\<^sub>c s'"
IfFalse: "\<not>b s \<Longrightarrow> \<langle>c1,s\<rangle> \<longrightarrow>\<^sub>c s' \<Longrightarrow> \<langle>\<IF> b \<THEN> c0 \<ELSE> c1, s\<rangle> \<longrightarrow>\<^sub>c s'"
WhileFalse: "\<not>b s \<Longrightarrow> \<langle>\<WHILE> b \<DO> c,s\<rangle> \<longrightarrow>\<^sub>c s"
WhileTrue: "b s \<Longrightarrow> \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c s'' \<Longrightarrow> \<langle>\<WHILE> b \<DO> c, s''\<rangle> \<longrightarrow>\<^sub>c s'
\<Longrightarrow> \<langle>\<WHILE> b \<DO> c, s\<rangle> \<longrightarrow>\<^sub>c s'"
lemmas evalc.intros [intro] -- "use those rules in automatic proofs"
text {*
The induction principle induced by this definition looks like this:
@{thm [display] evalc.induct [no_vars]}
(@{text "\<And>"} and @{text "\<Longrightarrow>"} are Isabelle's
meta symbols for @{text "\<forall>"} and @{text "\<longrightarrow>"})
*}
text {*
The rules of @{text evalc} are syntax directed, i.e.~for each
syntactic category there is always only one rule applicable. That
means we can use the rules in both directions. The proofs for this
are all the same: one direction is trivial, the other one is shown
by using the @{text evalc} rules backwards:
*}
lemma skip:
"\<langle>\<SKIP>,s\<rangle> \<longrightarrow>\<^sub>c s' = (s' = s)"
by (rule, erule evalc.elims) auto
lemma assign:
"\<langle>x :== a,s\<rangle> \<longrightarrow>\<^sub>c s' = (s' = s[x\<mapsto>a s])"
by (rule, erule evalc.elims) auto
lemma semi:
"\<langle>c0; c1, s\<rangle> \<longrightarrow>\<^sub>c s' = (\<exists>s''. \<langle>c0,s\<rangle> \<longrightarrow>\<^sub>c s'' \<and> \<langle>c1,s''\<rangle> \<longrightarrow>\<^sub>c s')"
by (rule, erule evalc.elims) auto
lemma ifTrue:
"b s \<Longrightarrow> \<langle>\<IF> b \<THEN> c0 \<ELSE> c1, s\<rangle> \<longrightarrow>\<^sub>c s' = \<langle>c0,s\<rangle> \<longrightarrow>\<^sub>c s'"
by (rule, erule evalc.elims) auto
lemma ifFalse:
"\<not>b s \<Longrightarrow> \<langle>\<IF> b \<THEN> c0 \<ELSE> c1, s\<rangle> \<longrightarrow>\<^sub>c s' = \<langle>c1,s\<rangle> \<longrightarrow>\<^sub>c s'"
by (rule, erule evalc.elims) auto
lemma whileFalse:
"\<not> b s \<Longrightarrow> \<langle>\<WHILE> b \<DO> c,s\<rangle> \<longrightarrow>\<^sub>c s' = (s' = s)"
by (rule, erule evalc.elims) auto
lemma whileTrue:
"b s \<Longrightarrow>
\<langle>\<WHILE> b \<DO> c, s\<rangle> \<longrightarrow>\<^sub>c s' =
(\<exists>s''. \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c s'' \<and> \<langle>\<WHILE> b \<DO> c, s''\<rangle> \<longrightarrow>\<^sub>c s')"
by (rule, erule evalc.elims) auto
text "Again, Isabelle may use these rules in automatic proofs:"
lemmas evalc_cases [simp] = skip assign ifTrue ifFalse whileFalse semi whileTrue
subsection "Equivalence of statements"
text {*
We call two statements @{text c} and @{text c'} equivalent wrt.~the
big-step semantics when \emph{@{text c} started in @{text s} terminates
in @{text s'} iff @{text c'} started in the same @{text s} also terminates
in the same @{text s'}}. Formally:
*}
constdefs
equiv_c :: "com \<Rightarrow> com \<Rightarrow> bool" ("_ \<sim> _")
"c \<sim> c' \<equiv> \<forall>s s'. \<langle>c, s\<rangle> \<longrightarrow>\<^sub>c s' = \<langle>c', s\<rangle> \<longrightarrow>\<^sub>c s'"
text {*
Proof rules telling Isabelle to unfold the definition
if there is something to be proved about equivalent
statements: *}
lemma equivI [intro!]:
"(\<And>s s'. \<langle>c, s\<rangle> \<longrightarrow>\<^sub>c s' = \<langle>c', s\<rangle> \<longrightarrow>\<^sub>c s') \<Longrightarrow> c \<sim> c'"
by (unfold equiv_c_def) blast
lemma equivD1:
"c \<sim> c' \<Longrightarrow> \<langle>c, s\<rangle> \<longrightarrow>\<^sub>c s' \<Longrightarrow> \<langle>c', s\<rangle> \<longrightarrow>\<^sub>c s'"
by (unfold equiv_c_def) blast
lemma equivD2:
"c \<sim> c' \<Longrightarrow> \<langle>c', s\<rangle> \<longrightarrow>\<^sub>c s' \<Longrightarrow> \<langle>c, s\<rangle> \<longrightarrow>\<^sub>c s'"
by (unfold equiv_c_def) blast
text {*
As an example, we show that loop unfolding is an equivalence
transformation on programs:
*}
lemma unfold_while:
"(\<WHILE> b \<DO> c) \<sim> (\<IF> b \<THEN> c; \<WHILE> b \<DO> c \<ELSE> \<SKIP>)" (is "?w \<sim> ?if")
proof -
-- "to show the equivalence, we look at the derivation tree for"
-- "each side and from that construct a derivation tree for the other side"
{ fix s s' assume w: "\<langle>?w, s\<rangle> \<longrightarrow>\<^sub>c s'"
-- "as a first thing we note that, if @{text b} is @{text False} in state @{text s},"
-- "then both statements do nothing:"
hence "\<not>b s \<Longrightarrow> s = s'" by simp
hence "\<not>b s \<Longrightarrow> \<langle>?if, s\<rangle> \<longrightarrow>\<^sub>c s'" by simp
moreover
-- "on the other hand, if @{text b} is @{text True} in state @{text s},"
-- {* then only the @{text WhileTrue} rule can have been used to derive @{text "\<langle>?w, s\<rangle> \<longrightarrow>\<^sub>c s'"} *}
{ assume b: "b s"
with w obtain s'' where
"\<langle>c, s\<rangle> \<longrightarrow>\<^sub>c s''" and "\<langle>?w, s''\<rangle> \<longrightarrow>\<^sub>c s'" by (cases set: evalc) auto
-- "now we can build a derivation tree for the @{text \<IF>}"
-- "first, the body of the True-branch:"
hence "\<langle>c; ?w, s\<rangle> \<longrightarrow>\<^sub>c s'" by (rule Semi)
-- "then the whole @{text \<IF>}"
with b have "\<langle>?if, s\<rangle> \<longrightarrow>\<^sub>c s'" by (rule IfTrue)
}
ultimately
-- "both cases together give us what we want:"
have "\<langle>?if, s\<rangle> \<longrightarrow>\<^sub>c s'" by blast
}
moreover
-- "now the other direction:"
{ fix s s' assume if: "\<langle>?if, s\<rangle> \<longrightarrow>\<^sub>c s'"
-- "again, if @{text b} is @{text False} in state @{text s}, then the False-branch"
-- "of the @{text \<IF>} is executed, and both statements do nothing:"
hence "\<not>b s \<Longrightarrow> s = s'" by simp
hence "\<not>b s \<Longrightarrow> \<langle>?w, s\<rangle> \<longrightarrow>\<^sub>c s'" by simp
moreover
-- "on the other hand, if @{text b} is @{text True} in state @{text s},"
-- {* then this time only the @{text IfTrue} rule can have be used *}
{ assume b: "b s"
with if have "\<langle>c; ?w, s\<rangle> \<longrightarrow>\<^sub>c s'" by (cases set: evalc) auto
-- "and for this, only the Semi-rule is applicable:"
then obtain s'' where
"\<langle>c, s\<rangle> \<longrightarrow>\<^sub>c s''" and "\<langle>?w, s''\<rangle> \<longrightarrow>\<^sub>c s'" by (cases set: evalc) auto
-- "with this information, we can build a derivation tree for the @{text \<WHILE>}"
with b
have "\<langle>?w, s\<rangle> \<longrightarrow>\<^sub>c s'" by (rule WhileTrue)
}
ultimately
-- "both cases together again give us what we want:"
have "\<langle>?w, s\<rangle> \<longrightarrow>\<^sub>c s'" by blast
}
ultimately
show ?thesis by blast
qed
subsection "Execution is deterministic"
text {*
The following proof presents all the details:
*}
theorem com_det: "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t \<and> \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c u \<longrightarrow> u=t"
proof clarify -- "transform the goal into canonical form"
assume "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t"
thus "\<And>u. \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u=t"
proof (induct set: evalc)
fix s u assume "\<langle>\<SKIP>,s\<rangle> \<longrightarrow>\<^sub>c u"
thus "u = s" by simp
next
fix a s x u assume "\<langle>x :== a,s\<rangle> \<longrightarrow>\<^sub>c u"
thus "u = s[x \<mapsto> a s]" by simp
next
fix c0 c1 s s1 s2 u
assume IH0: "\<And>u. \<langle>c0,s\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u = s2"
assume IH1: "\<And>u. \<langle>c1,s2\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u = s1"
assume "\<langle>c0;c1, s\<rangle> \<longrightarrow>\<^sub>c u"
then obtain s' where
c0: "\<langle>c0,s\<rangle> \<longrightarrow>\<^sub>c s'" and
c1: "\<langle>c1,s'\<rangle> \<longrightarrow>\<^sub>c u"
by auto
from c0 IH0 have "s'=s2" by blast
with c1 IH1 show "u=s1" by blast
next
fix b c0 c1 s s1 u
assume IH: "\<And>u. \<langle>c0,s\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u = s1"
assume "b s" and "\<langle>\<IF> b \<THEN> c0 \<ELSE> c1,s\<rangle> \<longrightarrow>\<^sub>c u"
hence "\<langle>c0, s\<rangle> \<longrightarrow>\<^sub>c u" by simp
with IH show "u = s1" by blast
next
fix b c0 c1 s s1 u
assume IH: "\<And>u. \<langle>c1,s\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u = s1"
assume "\<not>b s" and "\<langle>\<IF> b \<THEN> c0 \<ELSE> c1,s\<rangle> \<longrightarrow>\<^sub>c u"
hence "\<langle>c1, s\<rangle> \<longrightarrow>\<^sub>c u" by simp
with IH show "u = s1" by blast
next
fix b c s u
assume "\<not>b s" and "\<langle>\<WHILE> b \<DO> c,s\<rangle> \<longrightarrow>\<^sub>c u"
thus "u = s" by simp
next
fix b c s s1 s2 u
assume "IH\<^sub>c": "\<And>u. \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u = s2"
assume "IH\<^sub>w": "\<And>u. \<langle>\<WHILE> b \<DO> c,s2\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u = s1"
assume "b s" and "\<langle>\<WHILE> b \<DO> c,s\<rangle> \<longrightarrow>\<^sub>c u"
then obtain s' where
c: "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c s'" and
w: "\<langle>\<WHILE> b \<DO> c,s'\<rangle> \<longrightarrow>\<^sub>c u"
by auto
from c "IH\<^sub>c" have "s' = s2" by blast
with w "IH\<^sub>w" show "u = s1" by blast
qed
qed
text {*
This is the proof as you might present it in a lecture. The remaining
cases are simple enough to be proved automatically:
*}
theorem "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t \<and> \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c u \<longrightarrow> u=t"
proof clarify
assume "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t"
thus "\<And>u. \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u=t"
proof (induct set: evalc)
-- "the simple @{text \<SKIP>} case for demonstration:"
fix s u assume "\<langle>\<SKIP>,s\<rangle> \<longrightarrow>\<^sub>c u"
thus "u = s" by simp
next
-- "and the only really interesting case, @{text \<WHILE>}:"
fix b c s s1 s2 u
assume "IH\<^sub>c": "\<And>u. \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u = s2"
assume "IH\<^sub>w": "\<And>u. \<langle>\<WHILE> b \<DO> c,s2\<rangle> \<longrightarrow>\<^sub>c u \<Longrightarrow> u = s1"
assume "b s" and "\<langle>\<WHILE> b \<DO> c,s\<rangle> \<longrightarrow>\<^sub>c u"
then obtain s' where
c: "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c s'" and
w: "\<langle>\<WHILE> b \<DO> c,s'\<rangle> \<longrightarrow>\<^sub>c u"
by auto
from c "IH\<^sub>c" have "s' = s2" by blast
with w "IH\<^sub>w" show "u = s1" by blast
qed (best dest: evalc_cases [THEN iffD1])+ -- "prove the rest automatically"
qed
end