header "A Typed Language"
theory Types imports Star Complex_Main begin
subsection "Arithmetic Expressions"
datatype val = Iv int | Rv real
type_synonym name = string
type_synonym state = "name \<Rightarrow> val"
datatype aexp = Ic int | Rc real | V name | Plus aexp aexp
inductive taval :: "aexp \<Rightarrow> state \<Rightarrow> val \<Rightarrow> bool" where
"taval (Ic i) s (Iv i)" |
"taval (Rc r) s (Rv r)" |
"taval (V x) s (s x)" |
"taval a1 s (Iv i1) \<Longrightarrow> taval a2 s (Iv i2)
\<Longrightarrow> taval (Plus a1 a2) s (Iv(i1+i2))" |
"taval a1 s (Rv r1) \<Longrightarrow> taval a2 s (Rv r2)
\<Longrightarrow> taval (Plus a1 a2) s (Rv(r1+r2))"
inductive_cases [elim!]:
"taval (Ic i) s v" "taval (Rc i) s v"
"taval (V x) s v"
"taval (Plus a1 a2) s v"
subsection "Boolean Expressions"
datatype bexp = Bc bool | Not bexp | And bexp bexp | Less aexp aexp
inductive tbval :: "bexp \<Rightarrow> state \<Rightarrow> bool \<Rightarrow> bool" where
"tbval (Bc v) s v" |
"tbval b s bv \<Longrightarrow> tbval (Not b) s (\<not> bv)" |
"tbval b1 s bv1 \<Longrightarrow> tbval b2 s bv2 \<Longrightarrow> tbval (And b1 b2) s (bv1 & bv2)" |
"taval a1 s (Iv i1) \<Longrightarrow> taval a2 s (Iv i2) \<Longrightarrow> tbval (Less a1 a2) s (i1 < i2)" |
"taval a1 s (Rv r1) \<Longrightarrow> taval a2 s (Rv r2) \<Longrightarrow> tbval (Less a1 a2) s (r1 < r2)"
subsection "Syntax of Commands"
(* a copy of Com.thy - keep in sync! *)
datatype
com = SKIP
| Assign name aexp ("_ ::= _" [1000, 61] 61)
| Semi com com ("_; _" [60, 61] 60)
| If bexp com com ("IF _ THEN _ ELSE _" [0, 0, 61] 61)
| While bexp com ("WHILE _ DO _" [0, 61] 61)
subsection "Small-Step Semantics of Commands"
inductive
small_step :: "(com \<times> state) \<Rightarrow> (com \<times> state) \<Rightarrow> bool" (infix "\<rightarrow>" 55)
where
Assign: "taval a s v \<Longrightarrow> (x ::= a, s) \<rightarrow> (SKIP, s(x := v))" |
Semi1: "(SKIP;c,s) \<rightarrow> (c,s)" |
Semi2: "(c1,s) \<rightarrow> (c1',s') \<Longrightarrow> (c1;c2,s) \<rightarrow> (c1';c2,s')" |
IfTrue: "tbval b s True \<Longrightarrow> (IF b THEN c1 ELSE c2,s) \<rightarrow> (c1,s)" |
IfFalse: "tbval b s False \<Longrightarrow> (IF b THEN c1 ELSE c2,s) \<rightarrow> (c2,s)" |
While: "(WHILE b DO c,s) \<rightarrow> (IF b THEN c; WHILE b DO c ELSE SKIP,s)"
lemmas small_step_induct = small_step.induct[split_format(complete)]
subsection "The Type System"
datatype ty = Ity | Rty
type_synonym tyenv = "name \<Rightarrow> ty"
inductive atyping :: "tyenv \<Rightarrow> aexp \<Rightarrow> ty \<Rightarrow> bool"
("(1_/ \<turnstile>/ (_ :/ _))" [50,0,50] 50)
where
Ic_ty: "\<Gamma> \<turnstile> Ic i : Ity" |
Rc_ty: "\<Gamma> \<turnstile> Rc r : Rty" |
V_ty: "\<Gamma> \<turnstile> V x : \<Gamma> x" |
Plus_ty: "\<Gamma> \<turnstile> a1 : \<tau> \<Longrightarrow> \<Gamma> \<turnstile> a2 : \<tau> \<Longrightarrow> \<Gamma> \<turnstile> Plus a1 a2 : \<tau>"
text{* Warning: the ``:'' notation leads to syntactic ambiguities,
i.e. multiple parse trees, because ``:'' also stands for set membership.
In most situations Isabelle's type system will reject all but one parse tree,
but will still inform you of the potential ambiguity. *}
inductive btyping :: "tyenv \<Rightarrow> bexp \<Rightarrow> bool" (infix "\<turnstile>" 50)
where
B_ty: "\<Gamma> \<turnstile> Bc v" |
Not_ty: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> Not b" |
And_ty: "\<Gamma> \<turnstile> b1 \<Longrightarrow> \<Gamma> \<turnstile> b2 \<Longrightarrow> \<Gamma> \<turnstile> And b1 b2" |
Less_ty: "\<Gamma> \<turnstile> a1 : \<tau> \<Longrightarrow> \<Gamma> \<turnstile> a2 : \<tau> \<Longrightarrow> \<Gamma> \<turnstile> Less a1 a2"
inductive ctyping :: "tyenv \<Rightarrow> com \<Rightarrow> bool" (infix "\<turnstile>" 50) where
Skip_ty: "\<Gamma> \<turnstile> SKIP" |
Assign_ty: "\<Gamma> \<turnstile> a : \<Gamma>(x) \<Longrightarrow> \<Gamma> \<turnstile> x ::= a" |
Semi_ty: "\<Gamma> \<turnstile> c1 \<Longrightarrow> \<Gamma> \<turnstile> c2 \<Longrightarrow> \<Gamma> \<turnstile> c1;c2" |
If_ty: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> c1 \<Longrightarrow> \<Gamma> \<turnstile> c2 \<Longrightarrow> \<Gamma> \<turnstile> IF b THEN c1 ELSE c2" |
While_ty: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> WHILE b DO c"
inductive_cases [elim!]:
"\<Gamma> \<turnstile> x ::= a" "\<Gamma> \<turnstile> c1;c2"
"\<Gamma> \<turnstile> IF b THEN c1 ELSE c2"
"\<Gamma> \<turnstile> WHILE b DO c"
subsection "Well-typed Programs Do Not Get Stuck"
fun type :: "val \<Rightarrow> ty" where
"type (Iv i) = Ity" |
"type (Rv r) = Rty"
lemma [simp]: "type v = Ity \<longleftrightarrow> (\<exists>i. v = Iv i)"
by (cases v) simp_all
lemma [simp]: "type v = Rty \<longleftrightarrow> (\<exists>r. v = Rv r)"
by (cases v) simp_all
definition styping :: "tyenv \<Rightarrow> state \<Rightarrow> bool" (infix "\<turnstile>" 50)
where "\<Gamma> \<turnstile> s \<longleftrightarrow> (\<forall>x. type (s x) = \<Gamma> x)"
lemma apreservation:
"\<Gamma> \<turnstile> a : \<tau> \<Longrightarrow> taval a s v \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> type v = \<tau>"
apply(induction arbitrary: v rule: atyping.induct)
apply (fastforce simp: styping_def)+
done
lemma aprogress: "\<Gamma> \<turnstile> a : \<tau> \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<exists>v. taval a s v"
proof(induction rule: atyping.induct)
case (Plus_ty \<Gamma> a1 t a2)
then obtain v1 v2 where v: "taval a1 s v1" "taval a2 s v2" by blast
show ?case
proof (cases v1)
case Iv
with Plus_ty v show ?thesis
by(fastforce intro: taval.intros(4) dest!: apreservation)
next
case Rv
with Plus_ty v show ?thesis
by(fastforce intro: taval.intros(5) dest!: apreservation)
qed
qed (auto intro: taval.intros)
lemma bprogress: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<exists>v. tbval b s v"
proof(induction rule: btyping.induct)
case (Less_ty \<Gamma> a1 t a2)
then obtain v1 v2 where v: "taval a1 s v1" "taval a2 s v2"
by (metis aprogress)
show ?case
proof (cases v1)
case Iv
with Less_ty v show ?thesis
by (fastforce intro!: tbval.intros(4) dest!:apreservation)
next
case Rv
with Less_ty v show ?thesis
by (fastforce intro!: tbval.intros(5) dest!:apreservation)
qed
qed (auto intro: tbval.intros)
theorem progress:
"\<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> c \<noteq> SKIP \<Longrightarrow> \<exists>cs'. (c,s) \<rightarrow> cs'"
proof(induction rule: ctyping.induct)
case Skip_ty thus ?case by simp
next
case Assign_ty
thus ?case by (metis Assign aprogress)
next
case Semi_ty thus ?case by simp (metis Semi1 Semi2)
next
case (If_ty \<Gamma> b c1 c2)
then obtain bv where "tbval b s bv" by (metis bprogress)
show ?case
proof(cases bv)
assume "bv"
with `tbval b s bv` show ?case by simp (metis IfTrue)
next
assume "\<not>bv"
with `tbval b s bv` show ?case by simp (metis IfFalse)
qed
next
case While_ty show ?case by (metis While)
qed
theorem styping_preservation:
"(c,s) \<rightarrow> (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<Gamma> \<turnstile> s'"
proof(induction rule: small_step_induct)
case Assign thus ?case
by (auto simp: styping_def) (metis Assign(1,3) apreservation)
qed auto
theorem ctyping_preservation:
"(c,s) \<rightarrow> (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> c'"
by (induct rule: small_step_induct) (auto simp: ctyping.intros)
abbreviation small_steps :: "com * state \<Rightarrow> com * state \<Rightarrow> bool" (infix "\<rightarrow>*" 55)
where "x \<rightarrow>* y == star small_step x y"
theorem type_sound:
"(c,s) \<rightarrow>* (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> c' \<noteq> SKIP
\<Longrightarrow> \<exists>cs''. (c',s') \<rightarrow> cs''"
apply(induction rule:star_induct)
apply (metis progress)
by (metis styping_preservation ctyping_preservation)
end