src/HOL/Induct/Com.thy
 author berghofe Fri, 24 Jul 1998 13:19:38 +0200 changeset 5184 9b8547a9496a parent 5102 8c782c25a11e child 10759 994877ee68cb permissions -rw-r--r--
```
(*  Title:      HOL/Induct/Com
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Example of Mutual Induction via Iteratived Inductive Definitions: Commands
*)

Com = Datatype +

types loc
state = "loc => nat"
n2n2n = "nat => nat => nat"

arities loc :: term

(*To avoid a mutually recursive datatype declaration, expressions and commands
are combined to form a single datatype.*)
datatype
exp = N nat
| X loc
| Op n2n2n exp exp
| valOf exp exp          ("VALOF _ RESULTIS _"  60)
| SKIP
| ":="  loc exp          (infixl  60)
| Semi  exp exp          ("_;;_"  [60, 60] 60)
| Cond  exp exp exp      ("IF _ THEN _ ELSE _"  60)
| While exp exp          ("WHILE _ DO _"  60)

(** Execution of commands **)
consts  exec    :: "((exp*state) * (nat*state)) set => ((exp*state)*state)set"
"@exec"  :: "((exp*state) * (nat*state)) set =>
[exp*state,state] => bool"     ("_/ -[_]-> _" [50,0,50] 50)

translations  "csig -[eval]-> s" == "(csig,s) : exec eval"

syntax  eval'   :: "[exp*state,nat*state] =>
((exp*state) * (nat*state)) set => bool"
("_/ -|[_]-> _" [50,0,50] 50)
translations
"esig -|[eval]-> ns" => "(esig,ns) : eval"

constdefs assign :: [state,nat,loc] => state    ("_[_'/_]" [95,0,0] 95)
"s[m/x] ==  (%y. if y=x then m else s y)"

(*Command execution.  Natural numbers represent Booleans: 0=True, 1=False*)
inductive "exec eval"
intrs
Skip    "(SKIP,s) -[eval]-> s"

Assign  "(e,s) -|[eval]-> (v,s') ==> (x := e, s) -[eval]-> s'[v/x]"

Semi    "[| (c0,s) -[eval]-> s2; (c1,s2) -[eval]-> s1 |]
==> (c0 ;; c1, s) -[eval]-> s1"

IfTrue "[| (e,s) -|[eval]-> (0,s');  (c0,s') -[eval]-> s1 |]
==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"

IfFalse "[| (e,s) -|[eval]->  (1,s');  (c1,s') -[eval]-> s1 |]
==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"

WhileFalse "(e,s) -|[eval]-> (1,s1) ==> (WHILE e DO c, s) -[eval]-> s1"

WhileTrue  "[| (e,s) -|[eval]-> (0,s1);
(c,s1) -[eval]-> s2;  (WHILE e DO c, s2) -[eval]-> s3 |]
==> (WHILE e DO c, s) -[eval]-> s3"

constdefs
Unique   :: "['a, 'b, ('a*'b) set] => bool"
"Unique x y r == ALL y'. (x,y'): r --> y = y'"

Function :: "('a*'b) set => bool"
"Function r == ALL x y. (x,y): r --> Unique x y r"
end
```