(* Title: ZF/IMP/Com.thy
Author: Heiko Loetzbeyer and Robert Sandner, TU München
*)
header {* Arithmetic expressions, boolean expressions, commands *}
theory Com imports Main begin
subsection {* Arithmetic expressions *}
consts
loc :: i
aexp :: i
datatype \<subseteq> "univ(loc \<union> (nat -> nat) \<union> ((nat \<times> nat) -> nat))"
aexp = N ("n \<in> nat")
| X ("x \<in> loc")
| Op1 ("f \<in> nat -> nat", "a \<in> aexp")
| Op2 ("f \<in> (nat \<times> nat) -> nat", "a0 \<in> aexp", "a1 \<in> aexp")
consts evala :: i
abbreviation
evala_syntax :: "[i, i] => o" (infixl "-a->" 50)
where "p -a-> n == <p,n> \<in> evala"
inductive
domains "evala" \<subseteq> "(aexp \<times> (loc -> nat)) \<times> nat"
intros
N: "[| n \<in> nat; sigma \<in> loc->nat |] ==> <N(n),sigma> -a-> n"
X: "[| x \<in> loc; sigma \<in> loc->nat |] ==> <X(x),sigma> -a-> sigma`x"
Op1: "[| <e,sigma> -a-> n; f \<in> nat -> nat |] ==> <Op1(f,e),sigma> -a-> f`n"
Op2: "[| <e0,sigma> -a-> n0; <e1,sigma> -a-> n1; f \<in> (nat\<times>nat) -> nat |]
==> <Op2(f,e0,e1),sigma> -a-> f`<n0,n1>"
type_intros aexp.intros apply_funtype
subsection {* Boolean expressions *}
consts bexp :: i
datatype \<subseteq> "univ(aexp \<union> ((nat \<times> nat)->bool))"
bexp = true
| false
| ROp ("f \<in> (nat \<times> nat)->bool", "a0 \<in> aexp", "a1 \<in> aexp")
| noti ("b \<in> bexp")
| andi ("b0 \<in> bexp", "b1 \<in> bexp") (infixl "andi" 60)
| ori ("b0 \<in> bexp", "b1 \<in> bexp") (infixl "ori" 60)
consts evalb :: i
abbreviation
evalb_syntax :: "[i,i] => o" (infixl "-b->" 50)
where "p -b-> b == <p,b> \<in> evalb"
inductive
domains "evalb" \<subseteq> "(bexp \<times> (loc -> nat)) \<times> bool"
intros
true: "[| sigma \<in> loc -> nat |] ==> <true,sigma> -b-> 1"
false: "[| sigma \<in> loc -> nat |] ==> <false,sigma> -b-> 0"
ROp: "[| <a0,sigma> -a-> n0; <a1,sigma> -a-> n1; f \<in> (nat*nat)->bool |]
==> <ROp(f,a0,a1),sigma> -b-> f`<n0,n1> "
noti: "[| <b,sigma> -b-> w |] ==> <noti(b),sigma> -b-> not(w)"
andi: "[| <b0,sigma> -b-> w0; <b1,sigma> -b-> w1 |]
==> <b0 andi b1,sigma> -b-> (w0 and w1)"
ori: "[| <b0,sigma> -b-> w0; <b1,sigma> -b-> w1 |]
==> <b0 ori b1,sigma> -b-> (w0 or w1)"
type_intros bexp.intros
apply_funtype and_type or_type bool_1I bool_0I not_type
type_elims evala.dom_subset [THEN subsetD, elim_format]
subsection {* Commands *}
consts com :: i
datatype com =
skip ("\<SKIP>" [])
| assignment ("x \<in> loc", "a \<in> aexp") (infixl "\<ASSN>" 60)
| semicolon ("c0 \<in> com", "c1 \<in> com") ("_\<SEQ> _" [60, 60] 10)
| while ("b \<in> bexp", "c \<in> com") ("\<WHILE> _ \<DO> _" 60)
| "if" ("b \<in> bexp", "c0 \<in> com", "c1 \<in> com") ("\<IF> _ \<THEN> _ \<ELSE> _" 60)
consts evalc :: i
abbreviation
evalc_syntax :: "[i, i] => o" (infixl "-c->" 50)
where "p -c-> s == <p,s> \<in> evalc"
inductive
domains "evalc" \<subseteq> "(com \<times> (loc -> nat)) \<times> (loc -> nat)"
intros
skip: "[| sigma \<in> loc -> nat |] ==> <\<SKIP>,sigma> -c-> sigma"
assign: "[| m \<in> nat; x \<in> loc; <a,sigma> -a-> m |]
==> <x \<ASSN> a,sigma> -c-> sigma(x:=m)"
semi: "[| <c0,sigma> -c-> sigma2; <c1,sigma2> -c-> sigma1 |]
==> <c0\<SEQ> c1, sigma> -c-> sigma1"
if1: "[| b \<in> bexp; c1 \<in> com; sigma \<in> loc->nat;
<b,sigma> -b-> 1; <c0,sigma> -c-> sigma1 |]
==> <\<IF> b \<THEN> c0 \<ELSE> c1, sigma> -c-> sigma1"
if0: "[| b \<in> bexp; c0 \<in> com; sigma \<in> loc->nat;
<b,sigma> -b-> 0; <c1,sigma> -c-> sigma1 |]
==> <\<IF> b \<THEN> c0 \<ELSE> c1, sigma> -c-> sigma1"
while0: "[| c \<in> com; <b, sigma> -b-> 0 |]
==> <\<WHILE> b \<DO> c,sigma> -c-> sigma"
while1: "[| c \<in> com; <b,sigma> -b-> 1; <c,sigma> -c-> sigma2;
<\<WHILE> b \<DO> c, sigma2> -c-> sigma1 |]
==> <\<WHILE> b \<DO> c, sigma> -c-> sigma1"
type_intros com.intros update_type
type_elims evala.dom_subset [THEN subsetD, elim_format]
evalb.dom_subset [THEN subsetD, elim_format]
subsection {* Misc lemmas *}
lemmas evala_1 [simp] = evala.dom_subset [THEN subsetD, THEN SigmaD1, THEN SigmaD1]
and evala_2 [simp] = evala.dom_subset [THEN subsetD, THEN SigmaD1, THEN SigmaD2]
and evala_3 [simp] = evala.dom_subset [THEN subsetD, THEN SigmaD2]
lemmas evalb_1 [simp] = evalb.dom_subset [THEN subsetD, THEN SigmaD1, THEN SigmaD1]
and evalb_2 [simp] = evalb.dom_subset [THEN subsetD, THEN SigmaD1, THEN SigmaD2]
and evalb_3 [simp] = evalb.dom_subset [THEN subsetD, THEN SigmaD2]
lemmas evalc_1 [simp] = evalc.dom_subset [THEN subsetD, THEN SigmaD1, THEN SigmaD1]
and evalc_2 [simp] = evalc.dom_subset [THEN subsetD, THEN SigmaD1, THEN SigmaD2]
and evalc_3 [simp] = evalc.dom_subset [THEN subsetD, THEN SigmaD2]
inductive_cases
evala_N_E [elim!]: "<N(n),sigma> -a-> i"
and evala_X_E [elim!]: "<X(x),sigma> -a-> i"
and evala_Op1_E [elim!]: "<Op1(f,e),sigma> -a-> i"
and evala_Op2_E [elim!]: "<Op2(f,a1,a2),sigma> -a-> i"
end