(* Title: HOL/Tools/svc_funcs.ML
ID: $Id$
Author: Lawrence C Paulson
Copyright 1999 University of Cambridge
Translation and abstraction 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 =
bracketed_expr of expr
| ref_def_expr of string * expr
| ref_expr of string
| typed_expr of Type * expr
| buildin_expr of string * expr list
| interp_expr of string * expr list
| uninterp_expr of string * expr list
| false_expr
| true_expr
| distinct_expr of string
| int_expr of int
| rat_expr of int * int
and Type =
simple_type of string
| array_type of Type * Type
| record_type of (expr * Type) list
| bitvec_type of int;
open BasisLibrary
fun toString t =
let fun signedInt i =
if i < 0 then "-" ^ Int.toString (~i)
else Int.toString i
fun ut(simple_type s) = s ^ " "
| ut(array_type(t1, t2)) = "ARRAY " ^ (ut t1) ^ (ut t2)
| ut(record_type fl) =
"RECORD" ^
(foldl (fn (a, (d, t)) => a ^ (ue d) ^ (ut t)) (" ", fl))
| ut(bitvec_type n) = "BITVEC " ^ (Int.toString n) ^ " "
and ue(bracketed_expr e) = "(" ^ (ue e) ^ ") "
| ue(ref_def_expr(r, e)) = "$" ^ r ^ ":" ^ (ue e)
| ue(ref_expr r) = "$" ^ r ^ " "
| ue(typed_expr(t, e)) = (ut t) ^ (ue e)
| ue(buildin_expr(s, l)) =
"(" ^ s ^ (foldl (fn (a, b) => a ^ " " ^ (ue b)) ("", l)) ^ ") "
| ue(interp_expr(s, l)) =
"{" ^ s ^ (foldl (fn (a, b) => a ^ " " ^ (ue b)) ("", l)) ^ "} "
| ue(uninterp_expr(s, l)) =
"(" ^ s ^ (foldl (fn (a, b) => a ^ " " ^ (ue b)) ("", l)) ^ ") "
| ue(false_expr) = "FALSE "
| ue(true_expr) = "TRUE "
| ue(distinct_expr s) = "@" ^ s ^ " "
| ue(int_expr i) = (signedInt i) ^ " "
| ue(rat_expr(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 writeln ("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 = File.read svc_output_file
handle _ => error "SVC returned no output"
val _ = if !trace then writeln ("SVC Returns:\n" ^ svc_output) else ()
in
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;
fun is_intnat T = T = HOLogic.intT orelse T = HOLogic.natT;
(*Determining whether the biconditionals must be unfoled: 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 (*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_expr("=>", [buildin_expr("<=", [int_expr 0,
uninterp_expr (a, [])]),
e]);
(*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) =
(if T = HOLogic.natT then nat_vars := (a ins_string (!nat_vars))
else ();
uninterp_expr (a, []))
fun var (Free(a,T)) = trans_var ("F_" ^ a, T)
| var (Var((a, 0), T)) = trans_var (a, T)
| var (Bound i) =
let val (a,T) = List.nth (params, i)
in trans_var ("B_" ^ a, T) end
| var (t $ Bound _) = var t (*removing a parameter from a Var*)
| var t = raise OracleExn t;
(*translation of a literal*)
fun lit (Const("Numeral.number_of", _) $ w) = NumeralSyntax.dest_bin w
| lit (Const("0", _)) = 0
| lit (Const("RealDef.0r", _)) = 0
| lit (Const("RealDef.1r", _)) = 1
(*translation of a literal expression [no variables]*)
fun litExp (Const("op +", T) $ x $ y) = (litExp x) + (litExp y)
| litExp (Const("op -", T) $ x $ y) = (litExp x) - (litExp y)
| litExp (Const("op *", T) $ x $ y) = (litExp x) * (litExp y)
| litExp (Const("uminus", _) $ x) = ~(litExp x)
| litExp t = lit t
handle Match => raise OracleExn t
(*translation of a real/rational expression*)
fun suc t = interp_expr("+", [int_expr 1, t])
fun tm (Const("Suc", T) $ x) = suc (tm x)
| tm (Const("op +", T) $ x $ y) = interp_expr("+", [tm x, tm y])
| tm (Const("op -", _) $ x $ y) =
interp_expr("+", [tm x, interp_expr("*", [int_expr ~1, tm y])])
| tm (Const("op *", _) $ x $ y) = interp_expr("*", [tm x, tm y])
| tm (Const("op /", _) $ x $ y) =
interp_expr("*", [tm x, rat_expr(1, litExp y)])
| tm (Const("uminus", _) $ x) = interp_expr("*", [int_expr ~1, tm x])
| tm t = int_expr (lit t)
handle Match => var t
(*translation of a formula*)
and fm pos (Const("op &", _) $ p $ q) =
buildin_expr("AND", [fm pos p, fm pos q])
| fm pos (Const("op |", _) $ p $ q) =
buildin_expr("OR", [fm pos p, fm pos q])
| fm pos (Const("op -->", _) $ p $ q) =
buildin_expr("=>", [fm (not pos) p, fm pos q])
| fm pos (Const("Not", _) $ p) =
buildin_expr("NOT", [fm (not pos) p])
| fm pos (Const("True", _)) = true_expr
| fm pos (Const("False", _)) = false_expr
| fm pos (Const("SVC_Oracle.iff_keep", _) $ p $ q) =
(*polarity doesn't matter*)
buildin_expr("=", [fm pos p, fm pos q])
| fm pos (Const("SVC_Oracle.iff_unfold", _) $ p $ q) =
buildin_expr("AND", (*unfolding uses both polarities*)
[buildin_expr("=>", [fm (not pos) p, fm pos q]),
buildin_expr("=>", [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_expr("=", [tx, ty])
else if is_intnat T then
buildin_expr("AND",
[buildin_expr("<", [tx, suc ty]),
buildin_expr("<", [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_expr("<", [tm x, tm y])
else if is_intnat T then
buildin_expr("<=", [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_expr("<=", [tm x, tm y])
else if is_intnat T then
buildin_expr("<", [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_expr("=>", [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;
(*Generalize an Isabelle formula, replacing by Vars
all subterms not intelligible to SVC.
Do not present "raw" terms to expr_of; the translation could be unsound!*)
fun abstract t =
let
val params = Term.strip_all_vars t
and body = Term.strip_all_body t
val Us = map #2 params
val nPar = length params
val vname = ref "V_a"
val pairs = ref ([] : (term*term) list)
fun insert t =
let val T = fastype_of t
val v = Unify.combound (Var ((!vname,0), Us--->T),
0, nPar)
in vname := bump_string (!vname);
pairs := (t, v) :: !pairs;
v
end;
fun replace t =
case t of
Free _ => t (*but not existing Vars, lest the names clash*)
| Bound _ => t
| _ => (case gen_assoc (op aconv) (!pairs, t) of
Some v => v
| None => insert t)
(*abstraction of a real/rational expression*)
fun rat ((c as Const("op +", _)) $ x $ y) = c $ (rat x) $ (rat y)
| rat ((c as Const("op -", _)) $ x $ y) = c $ (rat x) $ (rat y)
| rat ((c as Const("op /", _)) $ x $ y) = c $ (rat x) $ (rat y)
| rat ((c as Const("op *", _)) $ x $ y) = c $ (rat x) $ (rat y)
| rat ((c as Const("uminus", _)) $ x) = c $ (rat x)
| rat ((c as Const("RealDef.0r", _))) = c
| rat ((c as Const("RealDef.1r", _))) = c
| rat (t as Const("Numeral.number_of", _) $ w) = t
| rat t = replace t
(*abstraction of an integer expression: no div, mod*)
fun int ((c as Const("op +", _)) $ x $ y) = c $ (int x) $ (int y)
| int ((c as Const("op -", _)) $ x $ y) = c $ (int x) $ (int y)
| int ((c as Const("op *", _)) $ x $ y) = c $ (int x) $ (int y)
| int ((c as Const("uminus", _)) $ x) = c $ (int x)
| int (t as Const("Numeral.number_of", _) $ w) = t
| int t = replace t
(*abstraction of a natural number expression: no minus*)
fun nat ((c as Const("op +", _)) $ x $ y) = c $ (nat x) $ (nat y)
| nat ((c as Const("op *", _)) $ x $ y) = c $ (nat x) $ (nat y)
| nat ((c as Const("Suc", _)) $ x) = c $ (nat x)
| nat (t as Const("0", _)) = t
| nat (t as Const("Numeral.number_of", _) $ w) = t
| nat t = replace t
(*abstraction of a relation: =, <, <=*)
fun rel (T, c $ x $ y) =
if T = HOLogic.realT then c $ (rat x) $ (rat y)
else if T = HOLogic.intT then c $ (int x) $ (int y)
else if T = HOLogic.natT then c $ (nat x) $ (nat y)
else if T = HOLogic.boolT then c $ (fm x) $ (fm y)
else replace (c $ x $ y) (*non-numeric comparison*)
(*abstraction of a formula*)
and fm ((c as Const("op &", _)) $ p $ q) = c $ (fm p) $ (fm q)
| fm ((c as Const("op |", _)) $ p $ q) = c $ (fm p) $ (fm q)
| fm ((c as Const("op -->", _)) $ p $ q) = c $ (fm p) $ (fm q)
| fm ((c as Const("Not", _)) $ p) = c $ (fm p)
| fm ((c as Const("True", _))) = c
| fm ((c as Const("False", _))) = c
| fm (t as Const("op =", Type ("fun", [T,_])) $ x $ y) = rel (T, t)
| fm (t as Const("op <", Type ("fun", [T,_])) $ x $ y) = rel (T, t)
| fm (t as Const("op <=", Type ("fun", [T,_])) $ x $ y) = rel (T, t)
| fm t = replace t
(*entry point, and abstraction of a meta-formula*)
fun mt ((c as Const("Trueprop", _)) $ p) = c $ (fm p)
| mt ((c as Const("==>", _)) $ p $ q) = c $ (mt p) $ (mt q)
| mt t = fm t (*it might be a formula*)
in (list_all (params, mt body), !pairs) end;
(*The oracle proves not the original formula but the abstracted version*)
fun oracle (sign, OracleExn P) =
let val (absP, _) = abstract P
val dummy = if !trace then writeln ("Subgoal abstracted to\n" ^
Sign.string_of_term sign absP)
else ()
in
if valid (expr_of false absP) then absP
else raise OracleExn P
end;
end;