(* Title: HOL/Tools/svc_funcs.ML
ID: $Id$
Author: Lawrence C Paulson
Copyright 1999 University of Cambridge
Translation functions for the interface to SVC
Based upon the work of Søren T. Heilmann
Integers and naturals are translated as follows:
In a positive context, replace x<y by x+1<=y
In a negative context, replace x<=y by x<y+1
In a negative context, replace x=y by x<y+1 & y<x+1
Biconditionals (if-and-only-iff) are expanded if they require such translations
in either operand.
For each variable of type nat, an assumption is added that it is non-negative.
*)
structure Svc =
struct
val trace = ref false;
datatype expr =
Buildin of string * expr list
| Interp of string * expr list
| UnInterp of string * expr list
| FalseExpr
| TrueExpr
| Int of int
| Rat of int * int;
open BasisLibrary
fun signedInt i =
if i < 0 then "-" ^ Int.toString (~i)
else Int.toString i;
fun is_intnat T = T = HOLogic.intT orelse T = HOLogic.natT;
fun is_numeric T = is_intnat T orelse T = HOLogic.realT;
fun is_numeric_op T = is_numeric (domain_type T);
fun toString t =
let fun ue (Buildin(s, l)) =
"(" ^ s ^ (foldl (fn (a, b) => a ^ " " ^ (ue b)) ("", l)) ^ ") "
| ue (Interp(s, l)) =
"{" ^ s ^ (foldl (fn (a, b) => a ^ " " ^ (ue b)) ("", l)) ^ "} "
| ue (UnInterp(s, l)) =
"(" ^ s ^ (foldl (fn (a, b) => a ^ " " ^ (ue b)) ("", l)) ^ ") "
| ue (FalseExpr) = "FALSE "
| ue (TrueExpr) = "TRUE "
| ue (Int i) = (signedInt i) ^ " "
| ue (Rat(i, j)) = (signedInt i) ^ "|" ^ (signedInt j) ^ " "
in
ue t
end;
fun valid e =
let val svc_home = getenv "SVC_HOME"
val svc_machine = getenv "SVC_MACHINE"
val check_valid = if svc_home = ""
then error "Environment variable SVC_HOME not set"
else if svc_machine = ""
then error "Environment variable SVC_MACHINE not set"
else svc_home ^ "/" ^ svc_machine ^ "/bin/check_valid"
val svc_input = toString e
val _ = if !trace then tracing ("Calling SVC:\n" ^ svc_input) else ()
val svc_input_file = File.tmp_path (Path.basic "SVM_in");
val svc_output_file = File.tmp_path (Path.basic "SVM_out");
val _ = (File.write svc_input_file svc_input;
execute (check_valid ^ " -dump-result " ^
File.sysify_path svc_output_file ^
" " ^ File.sysify_path svc_input_file ^
"> /dev/null 2>&1"))
val svc_output =
(case Library.try File.read svc_output_file of
Some out => out
| None => error "SVC returned no output");
in
if ! trace then tracing ("SVC Returns:\n" ^ svc_output)
else (File.rm svc_input_file; File.rm svc_output_file);
String.isPrefix "VALID" svc_output
end
(*New exception constructor for passing arguments to the oracle*)
exception OracleExn of term;
fun apply c args =
let val (ts, bs) = ListPair.unzip args
in (list_comb(c,ts), exists I bs) end;
(*Determining whether the biconditionals must be unfolded: if there are
int or nat comparisons below*)
val iff_tag =
let fun tag t =
let val (c,ts) = strip_comb t
in case c of
Const("op &", _) => apply c (map tag ts)
| Const("op |", _) => apply c (map tag ts)
| Const("op -->", _) => apply c (map tag ts)
| Const("Not", _) => apply c (map tag ts)
| Const("True", _) => (c, false)
| Const("False", _) => (c, false)
| Const("op =", Type ("fun", [T,_])) =>
if T = HOLogic.boolT then
(*biconditional: with int/nat comparisons below?*)
let val [t1,t2] = ts
val (u1,b1) = tag t1
and (u2,b2) = tag t2
val cname = if b1 orelse b2 then "unfold" else "keep"
in
(Const ("SVC_Oracle.iff_" ^ cname, dummyT) $ u1 $ u2,
b1 orelse b2)
end
else (*might be numeric equality*) (t, is_intnat T)
| Const("op <", Type ("fun", [T,_])) => (t, is_intnat T)
| Const("op <=", Type ("fun", [T,_])) => (t, is_intnat T)
| _ => (t, false)
end
in #1 o tag end;
(*Map expression e to 0<=a --> e, where "a" is the name of a nat variable*)
fun add_nat_var (a, e) =
Buildin("=>", [Buildin("<=", [Int 0, UnInterp (a, [])]),
e]);
fun param_string [] = ""
| param_string is = "_" ^ space_implode "_" (map string_of_int is)
(*Translate an Isabelle formula into an SVC expression
pos ["positive"]: true if an assumption, false if a goal*)
fun expr_of pos t =
let
val params = rev (rename_wrt_term t (Term.strip_all_vars t))
and body = Term.strip_all_body t
val nat_vars = ref ([] : string list)
(*translation of a variable: record all natural numbers*)
fun trans_var (a,T,is) =
(if T = HOLogic.natT then nat_vars := (a ins_string (!nat_vars))
else ();
UnInterp (a ^ param_string is, []))
(*A variable, perhaps applied to a series of parameters*)
fun var (Free(a,T), is) = trans_var ("F_" ^ a, T, is)
| var (Var((a, 0), T), is) = trans_var (a, T, is)
| var (Bound i, is) =
let val (a,T) = List.nth (params, i)
in trans_var ("B_" ^ a, T, is) end
| var (t $ Bound i, is) = var(t,i::is)
(*removing a parameter from a Var: the bound var index will
become part of the Var's name*)
| var (t,_) = raise OracleExn t;
(*translation of a literal*)
fun lit (Const("Numeral.number_of", _) $ w) =
(HOLogic.dest_binum w handle TERM _ => raise Match)
| lit (Const("0", _)) = 0
| lit (Const("1", _)) = 1
(*translation of a literal expression [no variables]*)
fun litExp (Const("op +", T) $ x $ y) =
if is_numeric_op T then (litExp x) + (litExp y)
else raise OracleExn t
| litExp (Const("op -", T) $ x $ y) =
if is_numeric_op T then (litExp x) - (litExp y)
else raise OracleExn t
| litExp (Const("op *", T) $ x $ y) =
if is_numeric_op T then (litExp x) * (litExp y)
else raise OracleExn t
| litExp (Const("uminus", T) $ x) =
if is_numeric_op T then ~(litExp x)
else raise OracleExn t
| litExp t = lit t
handle Match => raise OracleExn t
(*translation of a real/rational expression*)
fun suc t = Interp("+", [Int 1, t])
fun tm (Const("Suc", T) $ x) = suc (tm x)
| tm (Const("op +", T) $ x $ y) =
if is_numeric_op T then Interp("+", [tm x, tm y])
else raise OracleExn t
| tm (Const("op -", T) $ x $ y) =
if is_numeric_op T then
Interp("+", [tm x, Interp("*", [Int ~1, tm y])])
else raise OracleExn t
| tm (Const("op *", T) $ x $ y) =
if is_numeric_op T then Interp("*", [tm x, tm y])
else raise OracleExn t
| tm (Const("RealDef.rinv", T) $ x) =
if domain_type T = HOLogic.realT then
Rat(1, litExp x)
else raise OracleExn t
| tm (Const("uminus", T) $ x) =
if is_numeric_op T then Interp("*", [Int ~1, tm x])
else raise OracleExn t
| tm t = Int (lit t)
handle Match => var (t,[])
(*translation of a formula*)
and fm pos (Const("op &", _) $ p $ q) =
Buildin("AND", [fm pos p, fm pos q])
| fm pos (Const("op |", _) $ p $ q) =
Buildin("OR", [fm pos p, fm pos q])
| fm pos (Const("op -->", _) $ p $ q) =
Buildin("=>", [fm (not pos) p, fm pos q])
| fm pos (Const("Not", _) $ p) =
Buildin("NOT", [fm (not pos) p])
| fm pos (Const("True", _)) = TrueExpr
| fm pos (Const("False", _)) = FalseExpr
| fm pos (Const("SVC_Oracle.iff_keep", _) $ p $ q) =
(*polarity doesn't matter*)
Buildin("=", [fm pos p, fm pos q])
| fm pos (Const("SVC_Oracle.iff_unfold", _) $ p $ q) =
Buildin("AND", (*unfolding uses both polarities*)
[Buildin("=>", [fm (not pos) p, fm pos q]),
Buildin("=>", [fm (not pos) q, fm pos p])])
| fm pos (t as Const("op =", Type ("fun", [T,_])) $ x $ y) =
let val tx = tm x and ty = tm y
in if pos orelse T = HOLogic.realT then
Buildin("=", [tx, ty])
else if is_intnat T then
Buildin("AND",
[Buildin("<", [tx, suc ty]),
Buildin("<", [ty, suc tx])])
else raise OracleExn t
end
(*inequalities: possible types are nat, int, real*)
| fm pos (t as Const("op <", Type ("fun", [T,_])) $ x $ y) =
if not pos orelse T = HOLogic.realT then
Buildin("<", [tm x, tm y])
else if is_intnat T then
Buildin("<=", [suc (tm x), tm y])
else raise OracleExn t
| fm pos (t as Const("op <=", Type ("fun", [T,_])) $ x $ y) =
if pos orelse T = HOLogic.realT then
Buildin("<=", [tm x, tm y])
else if is_intnat T then
Buildin("<", [tm x, suc (tm y)])
else raise OracleExn t
| fm pos t = var(t,[]);
(*entry point, and translation of a meta-formula*)
fun mt pos ((c as Const("Trueprop", _)) $ p) = fm pos (iff_tag p)
| mt pos ((c as Const("==>", _)) $ p $ q) =
Buildin("=>", [mt (not pos) p, mt pos q])
| mt pos t = fm pos (iff_tag t) (*it might be a formula*)
val body_e = mt pos body (*evaluate now to assign into !nat_vars*)
in
foldr add_nat_var (!nat_vars, body_e)
end;
(*The oracle proves the given formula t, if possible*)
fun oracle (sign, OracleExn t) =
let val dummy = if !trace then tracing ("Subgoal abstracted to\n" ^
Sign.string_of_term sign t)
else ()
in
if valid (expr_of false t) then t
else raise OracleExn t
end;
end;