(* Title: HOL/Tools/Presburger/cooper.ML
ID: $Id$
Author: Amine Chaieb, TU Muenchen
*)
signature COOPER =
sig
val cooper_conv : Proof.context -> Conv.conv
exception COOPER of string * exn
end;
structure Cooper: COOPER =
struct
open Conv;
open Normalizer;
structure Integertab = TableFun(type key = integer val ord = Integer.cmp);
exception COOPER of string * exn;
val simp_thms_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms);
fun C f x y = f y x;
val FWD = C (fold (C implies_elim));
val true_tm = @{cterm "True"};
val false_tm = @{cterm "False"};
val zdvd1_eq = @{thm "zdvd1_eq"};
val presburger_ss = @{simpset} addsimps [zdvd1_eq];
val lin_ss = presburger_ss addsimps (@{thm "dvd_eq_mod_eq_0"}::zdvd1_eq::@{thms zadd_ac});
(* Some types and constants *)
val iT = HOLogic.intT
val bT = HOLogic.boolT;
val dest_numeral = HOLogic.dest_number #> snd;
val [miconj, midisj, mieq, mineq, milt, mile, migt, mige, midvd, mindvd, miP] =
map(instantiate' [SOME @{ctyp "int"}] []) @{thms "minf"};
val [infDconj, infDdisj, infDdvd,infDndvd,infDP] =
map(instantiate' [SOME @{ctyp "int"}] []) @{thms "inf_period"};
val [piconj, pidisj, pieq,pineq,pilt,pile,pigt,pige,pidvd,pindvd,piP] =
map (instantiate' [SOME @{ctyp "int"}] []) @{thms "pinf"};
val [miP, piP] = map (instantiate' [SOME @{ctyp "bool"}] []) [miP, piP];
val infDP = instantiate' (map SOME [@{ctyp "int"}, @{ctyp "bool"}]) [] infDP;
val [[asetconj, asetdisj, aseteq, asetneq, asetlt, asetle,
asetgt, asetge, asetdvd, asetndvd,asetP],
[bsetconj, bsetdisj, bseteq, bsetneq, bsetlt, bsetle,
bsetgt, bsetge, bsetdvd, bsetndvd,bsetP]] = [@{thms "aset"}, @{thms "bset"}];
val [miex, cpmi, piex, cppi] = [@{thm "minusinfinity"}, @{thm "cpmi"},
@{thm "plusinfinity"}, @{thm "cppi"}];
val unity_coeff_ex = instantiate' [SOME @{ctyp "int"}] [] @{thm "unity_coeff_ex"};
val [zdvd_mono,simp_from_to,all_not_ex] =
[@{thm "zdvd_mono"}, @{thm "simp_from_to"}, @{thm "all_not_ex"}];
val [dvd_uminus, dvd_uminus'] = @{thms "uminus_dvd_conv"};
val eval_ss = presburger_ss addsimps [simp_from_to] delsimps [insert_iff,bex_triv];
val eval_conv = Simplifier.rewrite eval_ss;
(* recongnising cterm without moving to terms *)
datatype fm = And of cterm*cterm| Or of cterm*cterm| Eq of cterm | NEq of cterm
| Lt of cterm | Le of cterm | Gt of cterm | Ge of cterm
| Dvd of cterm*cterm | NDvd of cterm*cterm | Nox
fun whatis x ct =
( case (term_of ct) of
Const("op &",_)$_$_ => And (Thm.dest_binop ct)
| Const ("op |",_)$_$_ => Or (Thm.dest_binop ct)
| Const ("op =",ty)$y$_ => if term_of x aconv y then Eq (Thm.dest_arg ct) else Nox
| Const("Not",_) $ (Const ("op =",_)$y$_) =>
if term_of x aconv y then NEq (funpow 2 Thm.dest_arg ct) else Nox
| Const ("Orderings.ord_class.less",_)$y$z =>
if term_of x aconv y then Lt (Thm.dest_arg ct)
else if term_of x aconv z then Gt (Thm.dest_arg1 ct) else Nox
| Const ("Orderings.ord_class.less_eq",_)$y$z =>
if term_of x aconv y then Le (Thm.dest_arg ct)
else if term_of x aconv z then Ge (Thm.dest_arg1 ct) else Nox
| Const ("Divides.dvd",_)$_$(Const(@{const_name "HOL.plus"},_)$y$_) =>
if term_of x aconv y then Dvd (Thm.dest_binop ct ||> Thm.dest_arg) else Nox
| Const("Not",_) $ (Const ("Divides.dvd",_)$_$(Const(@{const_name "HOL.plus"},_)$y$_)) =>
if term_of x aconv y then
NDvd (Thm.dest_binop (Thm.dest_arg ct) ||> Thm.dest_arg) else Nox
| _ => Nox)
handle CTERM _ => Nox;
fun get_pmi_term t =
let val (x,eq) =
(Thm.dest_abs NONE o Thm.dest_arg o snd o Thm.dest_abs NONE o Thm.dest_arg)
(Thm.dest_arg t)
in (Thm.cabs x o Thm.dest_arg o Thm.dest_arg) eq end;
val get_pmi = get_pmi_term o cprop_of;
val p_v' = @{cpat "?P' :: int => bool"};
val q_v' = @{cpat "?Q' :: int => bool"};
val p_v = @{cpat "?P:: int => bool"};
val q_v = @{cpat "?Q:: int => bool"};
fun myfwd (th1, th2, th3) p q
[(th_1,th_2,th_3), (th_1',th_2',th_3')] =
let
val (mp', mq') = (get_pmi th_1, get_pmi th_1')
val mi_th = FWD (instantiate ([],[(p_v,p),(q_v,q), (p_v',mp'),(q_v',mq')]) th1)
[th_1, th_1']
val infD_th = FWD (instantiate ([],[(p_v,mp'), (q_v, mq')]) th3) [th_3,th_3']
val set_th = FWD (instantiate ([],[(p_v,p), (q_v,q)]) th2) [th_2, th_2']
in (mi_th, set_th, infD_th)
end;
val inst' = fn cts => instantiate' [] (map SOME cts);
val infDTrue = instantiate' [] [SOME true_tm] infDP;
val infDFalse = instantiate' [] [SOME false_tm] infDP;
val cadd = @{cterm "op + :: int => _"}
val cmulC = @{cterm "op * :: int => _"}
val cminus = @{cterm "op - :: int => _"}
val cone = @{cterm "1:: int"}
val cneg = @{cterm "uminus :: int => _"}
val [addC, mulC, subC, negC] = map term_of [cadd, cmulC, cminus, cneg]
val [zero, one] = [@{term "0::int"}, @{term "1::int"}];
val is_numeral = can dest_numeral;
fun numeral1 f n = HOLogic.mk_number iT (f (dest_numeral n));
fun numeral2 f m n = HOLogic.mk_number iT (f (dest_numeral m) (dest_numeral n));
val [minus1,plus1] =
map (fn c => fn t => Thm.capply (Thm.capply c t) cone) [cminus,cadd];
fun decomp_pinf x dvd inS [aseteq, asetneq, asetlt, asetle,
asetgt, asetge,asetdvd,asetndvd,asetP,
infDdvd, infDndvd, asetconj,
asetdisj, infDconj, infDdisj] cp =
case (whatis x cp) of
And (p,q) => ([p,q], myfwd (piconj, asetconj, infDconj) (Thm.cabs x p) (Thm.cabs x q))
| Or (p,q) => ([p,q], myfwd (pidisj, asetdisj, infDdisj) (Thm.cabs x p) (Thm.cabs x q))
| Eq t => ([], K (inst' [t] pieq, FWD (inst' [t] aseteq) [inS (plus1 t)], infDFalse))
| NEq t => ([], K (inst' [t] pineq, FWD (inst' [t] asetneq) [inS t], infDTrue))
| Lt t => ([], K (inst' [t] pilt, FWD (inst' [t] asetlt) [inS t], infDFalse))
| Le t => ([], K (inst' [t] pile, FWD (inst' [t] asetle) [inS (plus1 t)], infDFalse))
| Gt t => ([], K (inst' [t] pigt, (inst' [t] asetgt), infDTrue))
| Ge t => ([], K (inst' [t] pige, (inst' [t] asetge), infDTrue))
| Dvd (d,s) =>
([],let val dd = dvd d
in K (inst' [d,s] pidvd, FWD (inst' [d,s] asetdvd) [dd],FWD (inst' [d,s] infDdvd) [dd]) end)
| NDvd(d,s) => ([],let val dd = dvd d
in K (inst' [d,s] pindvd, FWD (inst' [d,s] asetndvd) [dd], FWD (inst' [d,s] infDndvd) [dd]) end)
| _ => ([], K (inst' [cp] piP, inst' [cp] asetP, inst' [cp] infDP));
fun decomp_minf x dvd inS [bseteq,bsetneq,bsetlt, bsetle, bsetgt,
bsetge,bsetdvd,bsetndvd,bsetP,
infDdvd, infDndvd, bsetconj,
bsetdisj, infDconj, infDdisj] cp =
case (whatis x cp) of
And (p,q) => ([p,q], myfwd (miconj, bsetconj, infDconj) (Thm.cabs x p) (Thm.cabs x q))
| Or (p,q) => ([p,q], myfwd (midisj, bsetdisj, infDdisj) (Thm.cabs x p) (Thm.cabs x q))
| Eq t => ([], K (inst' [t] mieq, FWD (inst' [t] bseteq) [inS (minus1 t)], infDFalse))
| NEq t => ([], K (inst' [t] mineq, FWD (inst' [t] bsetneq) [inS t], infDTrue))
| Lt t => ([], K (inst' [t] milt, (inst' [t] bsetlt), infDTrue))
| Le t => ([], K (inst' [t] mile, (inst' [t] bsetle), infDTrue))
| Gt t => ([], K (inst' [t] migt, FWD (inst' [t] bsetgt) [inS t], infDFalse))
| Ge t => ([], K (inst' [t] mige,FWD (inst' [t] bsetge) [inS (minus1 t)], infDFalse))
| Dvd (d,s) => ([],let val dd = dvd d
in K (inst' [d,s] midvd, FWD (inst' [d,s] bsetdvd) [dd] , FWD (inst' [d,s] infDdvd) [dd]) end)
| NDvd (d,s) => ([],let val dd = dvd d
in K (inst' [d,s] mindvd, FWD (inst' [d,s] bsetndvd) [dd], FWD (inst' [d,s] infDndvd) [dd]) end)
| _ => ([], K (inst' [cp] miP, inst' [cp] bsetP, inst' [cp] infDP))
(* Canonical linear form for terms, formulae etc.. *)
fun provelin ctxt t = Goal.prove ctxt [] [] t
(fn _ => EVERY [simp_tac lin_ss 1, TRY (simple_arith_tac 1)]);
fun linear_cmul 0 tm = zero
| linear_cmul n tm =
case tm of
Const("HOL.plus_class.plus",_)$a$b => addC$(linear_cmul n a)$(linear_cmul n b)
| Const ("HOL.times_class.times",_)$c$x => mulC$(numeral1 (Integer.mult n) c)$x
| Const("HOL.minus_class.minus",_)$a$b => subC$(linear_cmul n a)$(linear_cmul n b)
| (m as Const("HOL.minus_class.uminus",_))$a => m$(linear_cmul n a)
| _ => numeral1 (Integer.mult n) tm;
fun earlier [] x y = false
| earlier (h::t) x y =
if h aconv y then false else if h aconv x then true else earlier t x y;
fun linear_add vars tm1 tm2 =
case (tm1,tm2) of
(Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c1$x1)$r1,
Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c2$x2)$r2) =>
if x1 = x2 then
let val c = numeral2 Integer.add c1 c2
in if c = zero then linear_add vars r1 r2
else addC$(mulC$c$x1)$(linear_add vars r1 r2)
end
else if earlier vars x1 x2 then addC$(mulC$ c1 $ x1)$(linear_add vars r1 tm2)
else addC$(mulC$c2$x2)$(linear_add vars tm1 r2)
| (Const("HOL.plus_class.plus",_) $ (Const("HOL.times_class.times",_)$c1$x1)$r1 ,_) =>
addC$(mulC$c1$x1)$(linear_add vars r1 tm2)
| (_, Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c2$x2)$r2) =>
addC$(mulC$c2$x2)$(linear_add vars tm1 r2)
| (_,_) => numeral2 Integer.add tm1 tm2;
fun linear_neg tm = linear_cmul ~1 tm;
fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2);
fun lint vars tm =
if is_numeral tm then tm
else case tm of
Const("HOL.minus_class.uminus",_)$t => linear_neg (lint vars t)
| Const("HOL.plus_class.plus",_) $ s $ t => linear_add vars (lint vars s) (lint vars t)
| Const("HOL.minus_class.minus",_) $ s $ t => linear_sub vars (lint vars s) (lint vars t)
| Const ("HOL.times_class.times",_) $ s $ t =>
let val s' = lint vars s
val t' = lint vars t
in if is_numeral s' then (linear_cmul (dest_numeral s') t')
else if is_numeral t' then (linear_cmul (dest_numeral t') s')
else raise COOPER ("Cooper Failed", TERM ("lint: not linear",[tm]))
end
| _ => addC$(mulC$one$tm)$zero;
fun lin (vs as x::_) (Const("Not",_)$(Const("Orderings.ord_class.less",T)$s$t)) =
lin vs (Const("Orderings.ord_class.less_eq",T)$t$s)
| lin (vs as x::_) (Const("Not",_)$(Const("Orderings.ord_class.less_eq",T)$s$t)) =
lin vs (Const("Orderings.ord_class.less",T)$t$s)
| lin vs (Const ("Not",T)$t) = Const ("Not",T)$ (lin vs t)
| lin (vs as x::_) (Const("Divides.dvd",_)$d$t) =
HOLogic.mk_binrel "Divides.dvd" (numeral1 abs d, lint vs t)
| lin (vs as x::_) ((b as Const("op =",_))$s$t) =
(case lint vs (subC$t$s) of
(t as a$(m$c$y)$r) =>
if x <> y then b$zero$t
else if dest_numeral c < 0 then b$(m$(numeral1 ~ c)$y)$r
else b$(m$c$y)$(linear_neg r)
| t => b$zero$t)
| lin (vs as x::_) (b$s$t) =
(case lint vs (subC$t$s) of
(t as a$(m$c$y)$r) =>
if x <> y then b$zero$t
else if dest_numeral c < 0 then b$(m$(numeral1 ~ c)$y)$r
else b$(linear_neg r)$(m$c$y)
| t => b$zero$t)
| lin vs fm = fm;
fun lint_conv ctxt vs ct =
let val t = term_of ct
in (provelin ctxt ((HOLogic.eq_const iT)$t$(lint vs t) |> HOLogic.mk_Trueprop))
RS eq_reflection
end;
fun is_intrel (b$_$_) = domain_type (fastype_of b) = HOLogic.intT
| is_intrel (@{term "Not"}$(b$_$_)) = domain_type (fastype_of b) = HOLogic.intT
| is_intrel _ = false;
fun linearize_conv ctxt vs ct =
case (term_of ct) of
Const("Divides.dvd",_)$d$t =>
let
val th = binop_conv (lint_conv ctxt vs) ct
val (d',t') = Thm.dest_binop (Thm.rhs_of th)
val (dt',tt') = (term_of d', term_of t')
in if is_numeral dt' andalso is_numeral tt'
then Conv.fconv_rule (arg_conv (Simplifier.rewrite presburger_ss)) th
else
let
val dth =
((if dest_numeral (term_of d') < 0 then
Conv.fconv_rule (arg_conv (arg1_conv (lint_conv ctxt vs)))
(Thm.transitive th (inst' [d',t'] dvd_uminus))
else th) handle TERM _ => th)
val d'' = Thm.rhs_of dth |> Thm.dest_arg1
in
case tt' of
Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$_)$_ =>
let val x = dest_numeral c
in if x < 0 then Conv.fconv_rule (arg_conv (arg_conv (lint_conv ctxt vs)))
(Thm.transitive dth (inst' [d'',t'] dvd_uminus'))
else dth end
| _ => dth
end
end
| Const("Not",_)$(Const("Divides.dvd",_)$_$_) => arg_conv (linearize_conv ctxt vs) ct
| t => if is_intrel t
then (provelin ctxt ((HOLogic.eq_const bT)$t$(lin vs t) |> HOLogic.mk_Trueprop))
RS eq_reflection
else reflexive ct;
val dvdc = @{cterm "op dvd :: int => _"};
fun unify ctxt q =
let
val (e,(cx,p)) = q |> Thm.dest_comb ||> Thm.dest_abs NONE
val x = term_of cx
val ins = insert (op = : integer*integer -> bool)
fun h (acc,dacc) t =
case (term_of t) of
Const(s,_)$(Const("HOL.times_class.times",_)$c$y)$ _ =>
if x aconv y
andalso s mem ["op =", "Orderings.ord_class.less", "Orderings.ord_class.less_eq"]
then (ins (dest_numeral c) acc,dacc) else (acc,dacc)
| Const(s,_)$_$(Const("HOL.times_class.times",_)$c$y) =>
if x aconv y
andalso s mem ["Orderings.ord_class.less", "Orderings.ord_class.less_eq"]
then (ins (dest_numeral c) acc, dacc) else (acc,dacc)
| Const("Divides.dvd",_)$_$(Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_) =>
if x aconv y then (acc,ins (dest_numeral c) dacc) else (acc,dacc)
| Const("op &",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
| Const("op |",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
| Const("Not",_)$_ => h (acc,dacc) (Thm.dest_arg t)
| _ => (acc, dacc)
val (cs,ds) = h ([],[]) p
val l = fold (curry lcm) (cs union ds) 1
fun cv k ct =
let val (tm as b$s$t) = term_of ct
in ((HOLogic.eq_const bT)$tm$(b$(linear_cmul k s)$(linear_cmul k t))
|> HOLogic.mk_Trueprop |> provelin ctxt) RS eq_reflection end
fun nzprop x =
let
val th =
Simplifier.rewrite lin_ss
(Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"}
(Thm.capply (Thm.capply @{cterm "op = :: int => _"} (mk_cnumber @{ctyp "int"} x))
@{cterm "0::int"})))
in equal_elim (Thm.symmetric th) TrueI end;
val notz = let val tab = fold Integertab.update
(ds ~~ (map (fn x => nzprop (Integer.div l x)) ds)) Integertab.empty
in
(fn ct => (valOf (Integertab.lookup tab (ct |> term_of |> dest_numeral))
handle Option => (writeln "noz: Theorems-Table contains no entry for";
print_cterm ct ; raise Option)))
end
fun unit_conv t =
case (term_of t) of
Const("op &",_)$_$_ => binop_conv unit_conv t
| Const("op |",_)$_$_ => binop_conv unit_conv t
| Const("Not",_)$_ => arg_conv unit_conv t
| Const(s,_)$(Const("HOL.times_class.times",_)$c$y)$ _ =>
if x=y andalso s mem ["op =", "Orderings.ord_class.less", "Orderings.ord_class.less_eq"]
then cv (Integer.div l (dest_numeral c)) t else Thm.reflexive t
| Const(s,_)$_$(Const("HOL.times_class.times",_)$c$y) =>
if x=y andalso s mem ["Orderings.ord_class.less", "Orderings.ord_class.less_eq"]
then cv (Integer.div l (dest_numeral c)) t else Thm.reflexive t
| Const("Divides.dvd",_)$d$(r as (Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_)) =>
if x=y then
let
val k = Integer.div l (dest_numeral c)
val kt = HOLogic.mk_number iT k
val th1 = inst' [Thm.dest_arg1 t, Thm.dest_arg t]
((Thm.dest_arg t |> funpow 2 Thm.dest_arg1 |> notz) RS zdvd_mono)
val (d',t') = (mulC$kt$d, mulC$kt$r)
val thc = (provelin ctxt ((HOLogic.eq_const iT)$d'$(lint [] d') |> HOLogic.mk_Trueprop))
RS eq_reflection
val tht = (provelin ctxt ((HOLogic.eq_const iT)$t'$(linear_cmul k r) |> HOLogic.mk_Trueprop))
RS eq_reflection
in Thm.transitive th1 (Thm.combination (Drule.arg_cong_rule dvdc thc) tht) end
else Thm.reflexive t
| _ => Thm.reflexive t
val uth = unit_conv p
val clt = mk_cnumber @{ctyp "int"} l
val ltx = Thm.capply (Thm.capply cmulC clt) cx
val th = Drule.arg_cong_rule e (Thm.abstract_rule (fst (dest_Free x )) cx uth)
val th' = inst' [Thm.cabs ltx (Thm.rhs_of uth), clt] unity_coeff_ex
val thf = transitive th
(transitive (symmetric (beta_conversion true (cprop_of th' |> Thm.dest_arg1))) th')
val (lth,rth) = Thm.dest_comb (cprop_of thf) |>> Thm.dest_arg |>> Thm.beta_conversion true
||> beta_conversion true |>> Thm.symmetric
in transitive (transitive lth thf) rth end;
val emptyIS = @{cterm "{}::int set"};
val insert_tm = @{cterm "insert :: int => _"};
val mem_tm = Const("op :",[iT , HOLogic.mk_setT iT] ---> bT);
fun mkISet cts = fold_rev (Thm.capply insert_tm #> Thm.capply) cts emptyIS;
val cTrp = @{cterm "Trueprop"};
val eqelem_imp_imp = (thm"eqelem_imp_iff") RS iffD1;
val [A_tm,B_tm] = map (fn th => cprop_of th |> funpow 2 Thm.dest_arg |> Thm.dest_abs NONE |> snd |> Thm.dest_arg1 |> Thm.dest_arg
|> Thm.dest_abs NONE |> snd |> Thm.dest_fun |> Thm.dest_arg)
[asetP,bsetP];
val D_tm = @{cpat "?D::int"};
val int_eq = (op =):integer*integer -> bool;
fun cooperex_conv ctxt vs q =
let
val uth = unify ctxt q
val (x,p) = Thm.dest_abs NONE (Thm.dest_arg (Thm.rhs_of uth))
val ins = insert (op aconvc)
fun h t (bacc,aacc,dacc) =
case (whatis x t) of
And (p,q) => h q (h p (bacc,aacc,dacc))
| Or (p,q) => h q (h p (bacc,aacc,dacc))
| Eq t => (ins (minus1 t) bacc,
ins (plus1 t) aacc,dacc)
| NEq t => (ins t bacc,
ins t aacc, dacc)
| Lt t => (bacc, ins t aacc, dacc)
| Le t => (bacc, ins (plus1 t) aacc,dacc)
| Gt t => (ins t bacc, aacc,dacc)
| Ge t => (ins (minus1 t) bacc, aacc,dacc)
| Dvd (d,s) => (bacc,aacc,insert int_eq (term_of d |> dest_numeral) dacc)
| NDvd (d,s) => (bacc,aacc,insert int_eq (term_of d|> dest_numeral) dacc)
| _ => (bacc, aacc, dacc)
val (b0,a0,ds) = h p ([],[],[])
val d = fold (curry lcm) ds 1
val cd = mk_cnumber @{ctyp "int"} d
val dt = term_of cd
fun divprop x =
let
val th =
Simplifier.rewrite lin_ss
(Thm.capply @{cterm Trueprop}
(Thm.capply (Thm.capply dvdc (mk_cnumber @{ctyp "int"} x)) cd))
in equal_elim (Thm.symmetric th) TrueI end;
val dvd = let val tab = fold Integertab.update
(ds ~~ (map divprop ds)) Integertab.empty in
(fn ct => (valOf (Integertab.lookup tab (term_of ct |> dest_numeral))
handle Option => (writeln "dvd: Theorems-Table contains no entry for";
print_cterm ct ; raise Option)))
end
val dp =
let val th = Simplifier.rewrite lin_ss
(Thm.capply @{cterm Trueprop}
(Thm.capply (Thm.capply @{cterm "op < :: int => _"} @{cterm "0::int"}) cd))
in equal_elim (Thm.symmetric th) TrueI end;
(* A and B set *)
local
val insI1 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI1"}
val insI2 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI2"}
in
fun provein x S =
case term_of S of
Const("{}",_) => error "Unexpected error in Cooper please email Amine Chaieb"
| Const("insert",_)$y$_ =>
let val (cy,S') = Thm.dest_binop S
in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1
else implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2)
(provein x S')
end
end
val al = map (lint vs o term_of) a0
val bl = map (lint vs o term_of) b0
val (sl,s0,f,abths,cpth) =
if length (distinct (op aconv) bl) <= length (distinct (op aconv) al)
then
(bl,b0,decomp_minf,
fn B => (map (fn th => implies_elim (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)]) th) dp)
[bseteq,bsetneq,bsetlt, bsetle, bsetgt,bsetge])@
(map (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)]))
[bsetdvd,bsetndvd,bsetP,infDdvd, infDndvd,bsetconj,
bsetdisj,infDconj, infDdisj]),
cpmi)
else (al,a0,decomp_pinf,fn A =>
(map (fn th => implies_elim (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)]) th) dp)
[aseteq,asetneq,asetlt, asetle, asetgt,asetge])@
(map (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)]))
[asetdvd,asetndvd, asetP, infDdvd, infDndvd,asetconj,
asetdisj,infDconj, infDdisj]),cppi)
val cpth =
let
val sths = map (fn (tl,t0) =>
if tl = term_of t0
then instantiate' [SOME @{ctyp "int"}] [SOME t0] refl
else provelin ctxt ((HOLogic.eq_const iT)$tl$(term_of t0)
|> HOLogic.mk_Trueprop))
(sl ~~ s0)
val csl = distinct (op aconvc) (map (cprop_of #> Thm.dest_arg #> Thm.dest_arg1) sths)
val S = mkISet csl
val inStab = fold (fn ct => fn tab => Termtab.update (term_of ct, provein ct S) tab)
csl Termtab.empty
val eqelem_th = instantiate' [SOME @{ctyp "int"}] [NONE,NONE, SOME S] eqelem_imp_imp
val inS =
let
fun transmem th0 th1 =
Thm.equal_elim
(Drule.arg_cong_rule cTrp (Drule.fun_cong_rule (Drule.arg_cong_rule
((Thm.dest_fun o Thm.dest_fun o Thm.dest_arg o cprop_of) th1) th0) S)) th1
val tab = fold Termtab.update
(map (fn eq =>
let val (s,t) = cprop_of eq |> Thm.dest_arg |> Thm.dest_binop
val th = if term_of s = term_of t
then valOf(Termtab.lookup inStab (term_of s))
else FWD (instantiate' [] [SOME s, SOME t] eqelem_th)
[eq, valOf(Termtab.lookup inStab (term_of s))]
in (term_of t, th) end)
sths) Termtab.empty
in fn ct =>
(valOf (Termtab.lookup tab (term_of ct))
handle Option => (writeln "inS: No theorem for " ; print_cterm ct ; raise Option))
end
val (inf, nb, pd) = divide_and_conquer (f x dvd inS (abths S)) p
in [dp, inf, nb, pd] MRS cpth
end
val cpth' = Thm.transitive uth (cpth RS eq_reflection)
in Thm.transitive cpth' ((simp_thms_conv then_conv eval_conv) (Thm.rhs_of cpth'))
end;
fun literals_conv bops uops env cv =
let fun h t =
case (term_of t) of
b$_$_ => if member (op aconv) bops b then binop_conv h t else cv env t
| u$_ => if member (op aconv) uops u then arg_conv h t else cv env t
| _ => cv env t
in h end;
fun integer_nnf_conv ctxt env =
nnf_conv then_conv literals_conv [HOLogic.conj, HOLogic.disj] [] env (linearize_conv ctxt);
(* val my_term = ref (@{cterm "NOTHING"}); *)
local
val pcv = Simplifier.rewrite
(HOL_basic_ss addsimps (simp_thms @ (List.take(ex_simps,4))
@ [not_all,all_not_ex, ex_disj_distrib]))
val postcv = Simplifier.rewrite presburger_ss
fun conv ctxt p =
let val _ = () (* my_term := p *)
in
Qelim.gen_qelim_conv ctxt pcv postcv pcv (cons o term_of)
(term_frees (term_of p)) (linearize_conv ctxt) (integer_nnf_conv ctxt)
(cooperex_conv ctxt) p
end
handle CTERM s => raise COOPER ("Cooper Failed", CTERM s)
| THM s => raise COOPER ("Cooper Failed", THM s)
in val cooper_conv = conv
end;
end;
structure Coopereif =
struct
open GeneratedCooper;
fun cooper s = raise Cooper.COOPER ("Cooper Oracle Failed", ERROR s);
fun i_of_term vs t =
case t of
Free(xn,xT) => (case AList.lookup (op aconv) vs t of
NONE => cooper "Variable not found in the list!!"
| SOME n => Bound n)
| @{term "0::int"} => C 0
| @{term "1::int"} => C 1
| Term.Bound i => Bound i
| Const(@{const_name "HOL.uminus"},_)$t' => Neg (i_of_term vs t')
| Const(@{const_name "HOL.plus"},_)$t1$t2 => Add (i_of_term vs t1,i_of_term vs t2)
| Const(@{const_name "HOL.minus"},_)$t1$t2 => Sub (i_of_term vs t1,i_of_term vs t2)
| Const(@{const_name "HOL.times"},_)$t1$t2 =>
(Mul (HOLogic.dest_number t1 |> snd |> Integer.machine_int,i_of_term vs t2)
handle TERM _ =>
(Mul (HOLogic.dest_number t2 |> snd |> Integer.machine_int,i_of_term vs t1)
handle TERM _ => cooper "Reification: Unsupported kind of multiplication"))
| _ => (C (HOLogic.dest_number t |> snd |> Integer.machine_int)
handle TERM _ => cooper "Reification: unknown term");
fun qf_of_term ps vs t =
case t of
Const("True",_) => T
| Const("False",_) => F
| Const(@{const_name "Orderings.less"},_)$t1$t2 => Lt (Sub (i_of_term vs t1,i_of_term vs t2))
| Const(@{const_name "Orderings.less_eq"},_)$t1$t2 => Le (Sub(i_of_term vs t1,i_of_term vs t2))
| Const(@{const_name "Divides.dvd"},_)$t1$t2 =>
(Dvd(HOLogic.dest_number t1 |> snd |> Integer.machine_int, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd")
| @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2))
| @{term "op = :: bool => _ "}$t1$t2 => Iff(qf_of_term ps vs t1,qf_of_term ps vs t2)
| Const("op &",_)$t1$t2 => And(qf_of_term ps vs t1,qf_of_term ps vs t2)
| Const("op |",_)$t1$t2 => Or(qf_of_term ps vs t1,qf_of_term ps vs t2)
| Const("op -->",_)$t1$t2 => Imp(qf_of_term ps vs t1,qf_of_term ps vs t2)
| Const("Not",_)$t' => NOT(qf_of_term ps vs t')
| Const("Ex",_)$Abs(xn,xT,p) =>
let val (xn',p') = variant_abs (xn,xT,p)
val vs' = (Free (xn',xT), nat 0) :: (map (fn(v,n) => (v,1+ n)) vs)
in E (qf_of_term ps vs' p')
end
| Const("All",_)$Abs(xn,xT,p) =>
let val (xn',p') = variant_abs (xn,xT,p)
val vs' = (Free (xn',xT), nat 0) :: (map (fn(v,n) => (v,1+ n)) vs)
in A (qf_of_term ps vs' p')
end
| _ =>(case AList.lookup (op aconv) ps t of
NONE => cooper "Reification: unknown term!"
| SOME n => Closed n);
local
val ops = [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"},
@{term "op = :: int => _"}, @{term "op < :: int => _"},
@{term "op <= :: int => _"}, @{term "Not"}, @{term "All:: (int => _) => _"},
@{term "Ex:: (int => _) => _"}, @{term "True"}, @{term "False"}]
fun ty t = Bool.not (fastype_of t = HOLogic.boolT)
in
fun term_bools acc t =
case t of
(l as f $ a) $ b => if ty t orelse f mem ops then term_bools (term_bools acc l)b
else insert (op aconv) t acc
| f $ a => if ty t orelse f mem ops then term_bools (term_bools acc f) a
else insert (op aconv) t acc
| Abs p => term_bools acc (snd (variant_abs p))
| _ => if ty t orelse t mem ops then acc else insert (op aconv) t acc
end;
fun start_vs t =
let
val fs = term_frees t
val ps = term_bools [] t
in (fs ~~ (0 upto (length fs - 1)), ps ~~ (0 upto (length ps - 1)))
end ;
val iT = HOLogic.intT;
val bT = HOLogic.boolT;
fun myassoc2 l v =
case l of
[] => NONE
| (x,v')::xs => if v = v' then SOME x
else myassoc2 xs v;
fun term_of_i vs t =
case t of
C i => HOLogic.mk_number HOLogic.intT (Integer.int i)
| Bound n => valOf (myassoc2 vs n)
| Neg t' => @{term "uminus :: int => _"}$(term_of_i vs t')
| Add(t1,t2) => @{term "op +:: int => _"}$ (term_of_i vs t1)$(term_of_i vs t2)
| Sub(t1,t2) => Const(@{const_name "HOL.minus"},[iT,iT] ---> iT)$
(term_of_i vs t1)$(term_of_i vs t2)
| Mul(i,t2) => Const(@{const_name "HOL.times"},[iT,iT] ---> iT)$
(HOLogic.mk_number HOLogic.intT (Integer.int i))$(term_of_i vs t2)
| CX(i,t')=> term_of_i vs (Add(Mul (i,Bound (nat 0)),t'));
fun term_of_qf ps vs t =
case t of
T => HOLogic.true_const
| F => HOLogic.false_const
| Lt t' => @{term "op < :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
| Le t' => @{term "op <= :: int => _ "}$ term_of_i vs t' $ @{term "0::int"}
| Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
| Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
| Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
| NEq t' => term_of_qf ps vs (NOT(Eq t'))
| Dvd(i,t') => @{term "op dvd :: int => _ "}$
(HOLogic.mk_number HOLogic.intT (Integer.int i))$(term_of_i vs t')
| NDvd(i,t')=> term_of_qf ps vs (NOT(Dvd(i,t')))
| NOT t' => HOLogic.Not$(term_of_qf ps vs t')
| And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
| Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
| Imp(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
| Iff(t1,t2) => (HOLogic.eq_const bT)$(term_of_qf ps vs t1)$ (term_of_qf ps vs t2)
| Closed n => valOf (myassoc2 ps n)
| NClosed n => term_of_qf ps vs (NOT (Closed n))
| _ => cooper "If this is raised, Isabelle/HOL or generate_code is inconsistent!";
(* The oracle *)
fun cooper_oracle thy t =
let val (vs,ps) = start_vs t
in (equals propT) $ (HOLogic.mk_Trueprop t) $
(HOLogic.mk_Trueprop (term_of_qf ps vs (pa (qf_of_term ps vs t))))
end;
end;