src/HOL/Induct/Com.thy
author berghofe
Fri, 24 Jul 1998 13:39:47 +0200
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Renamed '$' to 'Scons' because of clashes with constants of the same name in theories using datatypes.
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(*  Title:      HOL/Induct/Com
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1997  University of Cambridge
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Example of Mutual Induction via Iteratived Inductive Definitions: Commands
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*)
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Com = Datatype +
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types loc
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      state = "loc => nat"
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      n2n2n = "nat => nat => nat"
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arities loc :: term
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(*To avoid a mutually recursive datatype declaration, expressions and commands
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  are combined to form a single datatype.*)
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datatype
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  exp = N nat
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      | X loc
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      | Op n2n2n exp exp
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      | valOf exp exp          ("VALOF _ RESULTIS _"  60)
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      | SKIP
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      | ":="  loc exp          (infixl  60)
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      | Semi  exp exp          ("_;;_"  [60, 60] 60)
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      | Cond  exp exp exp      ("IF _ THEN _ ELSE _"  60)
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      | While exp exp          ("WHILE _ DO _"  60)
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(** Execution of commands **)
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consts  exec    :: "((exp*state) * (nat*state)) set => ((exp*state)*state)set"
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       "@exec"  :: "((exp*state) * (nat*state)) set => 
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                    [exp*state,state] => bool"     ("_/ -[_]-> _" [50,0,50] 50)
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translations  "csig -[eval]-> s" == "(csig,s) : exec eval"
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syntax  eval'   :: "[exp*state,nat*state] => 
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		    ((exp*state) * (nat*state)) set => bool"
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						   ("_/ -|[_]-> _" [50,0,50] 50)
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translations
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    "esig -|[eval]-> ns" => "(esig,ns) : eval"
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constdefs assign :: [state,nat,loc] => state    ("_[_'/_]" [95,0,0] 95)
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  "s[m/x] ==  (%y. if y=x then m else s y)"
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(*Command execution.  Natural numbers represent Booleans: 0=True, 1=False*)
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inductive "exec eval"
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  intrs
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    Skip    "(SKIP,s) -[eval]-> s"
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    Assign  "(e,s) -|[eval]-> (v,s') ==> (x := e, s) -[eval]-> s'[v/x]"
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    Semi    "[| (c0,s) -[eval]-> s2; (c1,s2) -[eval]-> s1 |] 
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            ==> (c0 ;; c1, s) -[eval]-> s1"
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    IfTrue "[| (e,s) -|[eval]-> (0,s');  (c0,s') -[eval]-> s1 |] 
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            ==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"
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    IfFalse "[| (e,s) -|[eval]->  (1,s');  (c1,s') -[eval]-> s1 |] 
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             ==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"
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    WhileFalse "(e,s) -|[eval]-> (1,s1) ==> (WHILE e DO c, s) -[eval]-> s1"
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    WhileTrue  "[| (e,s) -|[eval]-> (0,s1);
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                (c,s1) -[eval]-> s2;  (WHILE e DO c, s2) -[eval]-> s3 |] 
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                ==> (WHILE e DO c, s) -[eval]-> s3"
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constdefs
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    Unique   :: "['a, 'b, ('a*'b) set] => bool"
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    "Unique x y r == ALL y'. (x,y'): r --> y = y'"
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    Function :: "('a*'b) set => bool"
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    "Function r == ALL x y. (x,y): r --> Unique x y r"
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end