# HG changeset patch # User chaieb # Date 1181552838 -7200 # Node ID e36fc1bcb8c681754578a08ca1a0d2cd274b98b4 # Parent 2274edb9a8b2132039803165c7f2fd5d2e096100 Generated reflected QE procedure for Presburger Arithmetic-- Cooper's Algorithm -- see HOL/ex/Reflected_Presburger.thy diff -r 2274edb9a8b2 -r e36fc1bcb8c6 src/HOL/Tools/Presburger/generated_cooper.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Tools/Presburger/generated_cooper.ML Mon Jun 11 11:07:18 2007 +0200 @@ -0,0 +1,1693 @@ +structure GeneratedCooper = +struct + +fun nat i = if i < 0 then 0 else i; + +val one_def0 : int = (0 + 1); + +datatype num = C of int | Bound of int | CX of int * num | Neg of num + | Add of num * num | Sub of num * num | Mul of int * num; + +fun snd (a, b) = b; + +fun negateSnd x = (fn (q, r) => (q, ~ r)) x; + +fun minus_def2 z w = (z + ~ w); + +fun adjust b = + (fn (q, r) => + (if (0 <= minus_def2 r b) then (((2 * q) + 1), minus_def2 r b) + else ((2 * q), r))); + +fun negDivAlg a b = + (if ((0 <= (a + b)) orelse (b <= 0)) then (~1, (a + b)) + else adjust b (negDivAlg a (2 * b))); + +fun posDivAlg a b = + (if ((a < b) orelse (b <= 0)) then (0, a) + else adjust b (posDivAlg a (2 * b))); + +fun divAlg x = + (fn (a, b) => + (if (0 <= a) + then (if (0 <= b) then posDivAlg a b + else (if (a = 0) then (0, 0) + else negateSnd (negDivAlg (~ a) (~ b)))) + else (if (0 < b) then negDivAlg a b + else negateSnd (posDivAlg (~ a) (~ b))))) + x; + +fun mod_def1 a b = snd (divAlg (a, b)); + +fun dvd m n = (mod_def1 n m = 0); + +fun abs i = (if (i < 0) then ~ i else i); + +fun less_def3 m n = ((m) < (n)); + +fun less_eq_def3 m n = Bool.not (less_def3 n m); + +fun numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (c2, Bound n2), r2)) = + (if (n1 = n2) + then let val c = (c1 + c2) + in (if (c = 0) then numadd (r1, r2) + else Add (Mul (c, Bound n1), numadd (r1, r2))) + end + else (if less_eq_def3 n1 n2 + then Add (Mul (c1, Bound n1), + numadd (r1, Add (Mul (c2, Bound n2), r2))) + else Add (Mul (c2, Bound n2), + numadd (Add (Mul (c1, Bound n1), r1), r2)))) + | numadd (Add (Mul (c1, Bound n1), r1), C afq) = + Add (Mul (c1, Bound n1), numadd (r1, C afq)) + | numadd (Add (Mul (c1, Bound n1), r1), Bound afr) = + Add (Mul (c1, Bound n1), numadd (r1, Bound afr)) + | numadd (Add (Mul (c1, Bound n1), r1), CX (afs, aft)) = + Add (Mul (c1, Bound n1), numadd (r1, CX (afs, aft))) + | numadd (Add (Mul (c1, Bound n1), r1), Neg afu) = + Add (Mul (c1, Bound n1), numadd (r1, Neg afu)) + | numadd (Add (Mul (c1, Bound n1), r1), Add (C agx, afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (C agx, afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Bound agy, afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Bound agy, afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (CX (agz, aha), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (CX (agz, aha), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Neg ahb, afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Neg ahb, afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Add (ahc, ahd), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Add (ahc, ahd), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Sub (ahe, ahf), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Sub (ahe, ahf), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, C aie), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, C aie), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, CX (aig, aih)), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, CX (aig, aih)), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Neg aii), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Neg aii), afw))) + | numadd + (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Add (aij, aik)), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Add (aij, aik)), afw))) + | numadd + (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Sub (ail, aim)), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Sub (ail, aim)), afw))) + | numadd + (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Mul (ain, aio)), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Mul (ain, aio)), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Sub (afx, afy)) = + Add (Mul (c1, Bound n1), numadd (r1, Sub (afx, afy))) + | numadd (Add (Mul (c1, Bound n1), r1), Mul (afz, aga)) = + Add (Mul (c1, Bound n1), numadd (r1, Mul (afz, aga))) + | numadd (C w, Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (C w, r2)) + | numadd (Bound x, Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Bound x, r2)) + | numadd (CX (y, z), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (CX (y, z), r2)) + | numadd (Neg ab, Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Neg ab, r2)) + | numadd (Add (C li, ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (C li, ad), r2)) + | numadd (Add (Bound lj, ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Bound lj, ad), r2)) + | numadd (Add (CX (lk, ll), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (CX (lk, ll), ad), r2)) + | numadd (Add (Neg lm, ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Neg lm, ad), r2)) + | numadd (Add (Add (ln, lo), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Add (ln, lo), ad), r2)) + | numadd (Add (Sub (lp, lq), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Sub (lp, lq), ad), r2)) + | numadd (Add (Mul (lr, C abv), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, C abv), ad), r2)) + | numadd (Add (Mul (lr, CX (abx, aby)), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, CX (abx, aby)), ad), r2)) + | numadd (Add (Mul (lr, Neg abz), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Neg abz), ad), r2)) + | numadd (Add (Mul (lr, Add (aca, acb)), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Add (aca, acb)), ad), r2)) + | numadd (Add (Mul (lr, Sub (acc, acd)), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Sub (acc, acd)), ad), r2)) + | numadd (Add (Mul (lr, Mul (ace, acf)), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Mul (ace, acf)), ad), r2)) + | numadd (Sub (ae, af), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Sub (ae, af), r2)) + | numadd (Mul (ag, ah), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Mul (ag, ah), r2)) + | numadd (C b1, C b2) = C (b1 + b2) + | numadd (C ai, Bound bf) = Add (C ai, Bound bf) + | numadd (C ai, CX (bg, bh)) = Add (C ai, CX (bg, bh)) + | numadd (C ai, Neg bi) = Add (C ai, Neg bi) + | numadd (C ai, Add (C ca, bk)) = Add (C ai, Add (C ca, bk)) + | numadd (C ai, Add (Bound cb, bk)) = Add (C ai, Add (Bound cb, bk)) + | numadd (C ai, Add (CX (cc, cd), bk)) = Add (C ai, Add (CX (cc, cd), bk)) + | numadd (C ai, Add (Neg ce, bk)) = Add (C ai, Add (Neg ce, bk)) + | numadd (C ai, Add (Add (cf, cg), bk)) = Add (C ai, Add (Add (cf, cg), bk)) + | numadd (C ai, Add (Sub (ch, ci), bk)) = Add (C ai, Add (Sub (ch, ci), bk)) + | numadd (C ai, Add (Mul (cj, C cw), bk)) = + Add (C ai, Add (Mul (cj, C cw), bk)) + | numadd (C ai, Add (Mul (cj, CX (cy, cz)), bk)) = + Add (C ai, Add (Mul (cj, CX (cy, cz)), bk)) + | numadd (C ai, Add (Mul (cj, Neg da), bk)) = + Add (C ai, Add (Mul (cj, Neg da), bk)) + | numadd (C ai, Add (Mul (cj, Add (db, dc)), bk)) = + Add (C ai, Add (Mul (cj, Add (db, dc)), bk)) + | numadd (C ai, Add (Mul (cj, Sub (dd, de)), bk)) = + Add (C ai, Add (Mul (cj, Sub (dd, de)), bk)) + | numadd (C ai, Add (Mul (cj, Mul (df, dg)), bk)) = + Add (C ai, Add (Mul (cj, Mul (df, dg)), bk)) + | numadd (C ai, Sub (bl, bm)) = Add (C ai, Sub (bl, bm)) + | numadd (C ai, Mul (bn, bo)) = Add (C ai, Mul (bn, bo)) + | numadd (Bound aj, C ds) = Add (Bound aj, C ds) + | numadd (Bound aj, Bound dt) = Add (Bound aj, Bound dt) + | numadd (Bound aj, CX (du, dv)) = Add (Bound aj, CX (du, dv)) + | numadd (Bound aj, Neg dw) = Add (Bound aj, Neg dw) + | numadd (Bound aj, Add (C eo, dy)) = Add (Bound aj, Add (C eo, dy)) + | numadd (Bound aj, Add (Bound ep, dy)) = Add (Bound aj, Add (Bound ep, dy)) + | numadd (Bound aj, Add (CX (eq, er), dy)) = + Add (Bound aj, Add (CX (eq, er), dy)) + | numadd (Bound aj, Add (Neg es, dy)) = Add (Bound aj, Add (Neg es, dy)) + | numadd (Bound aj, Add (Add (et, eu), dy)) = + Add (Bound aj, Add (Add (et, eu), dy)) + | numadd (Bound aj, Add (Sub (ev, ew), dy)) = + Add (Bound aj, Add (Sub (ev, ew), dy)) + | numadd (Bound aj, Add (Mul (ex, C fk), dy)) = + Add (Bound aj, Add (Mul (ex, C fk), dy)) + | numadd (Bound aj, Add (Mul (ex, CX (fm, fn')), dy)) = + Add (Bound aj, Add (Mul (ex, CX (fm, fn')), dy)) + | numadd (Bound aj, Add (Mul (ex, Neg fo), dy)) = + Add (Bound aj, Add (Mul (ex, Neg fo), dy)) + | numadd (Bound aj, Add (Mul (ex, Add (fp, fq)), dy)) = + Add (Bound aj, Add (Mul (ex, Add (fp, fq)), dy)) + | numadd (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy)) = + Add (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy)) + | numadd (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) = + Add (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) + | numadd (Bound aj, Sub (dz, ea)) = Add (Bound aj, Sub (dz, ea)) + | numadd (Bound aj, Mul (eb, ec)) = Add (Bound aj, Mul (eb, ec)) + | numadd (CX (ak, al), C gg) = Add (CX (ak, al), C gg) + | numadd (CX (ak, al), Bound gh) = Add (CX (ak, al), Bound gh) + | numadd (CX (ak, al), CX (gi, gj)) = Add (CX (ak, al), CX (gi, gj)) + | numadd (CX (ak, al), Neg gk) = Add (CX (ak, al), Neg gk) + | numadd (CX (ak, al), Add (C hc, gm)) = Add (CX (ak, al), Add (C hc, gm)) + | numadd (CX (ak, al), Add (Bound hd, gm)) = + Add (CX (ak, al), Add (Bound hd, gm)) + | numadd (CX (ak, al), Add (CX (he, hf), gm)) = + Add (CX (ak, al), Add (CX (he, hf), gm)) + | numadd (CX (ak, al), Add (Neg hg, gm)) = Add (CX (ak, al), Add (Neg hg, gm)) + | numadd (CX (ak, al), Add (Add (hh, hi), gm)) = + Add (CX (ak, al), Add (Add (hh, hi), gm)) + | numadd (CX (ak, al), Add (Sub (hj, hk), gm)) = + Add (CX (ak, al), Add (Sub (hj, hk), gm)) + | numadd (CX (ak, al), Add (Mul (hl, C hy), gm)) = + Add (CX (ak, al), Add (Mul (hl, C hy), gm)) + | numadd (CX (ak, al), Add (Mul (hl, CX (ia, ib)), gm)) = + Add (CX (ak, al), Add (Mul (hl, CX (ia, ib)), gm)) + | numadd (CX (ak, al), Add (Mul (hl, Neg ic), gm)) = + Add (CX (ak, al), Add (Mul (hl, Neg ic), gm)) + | numadd (CX (ak, al), Add (Mul (hl, Add (id, ie)), gm)) = + Add (CX (ak, al), Add (Mul (hl, Add (id, ie)), gm)) + | numadd (CX (ak, al), Add (Mul (hl, Sub (if', ig)), gm)) = + Add (CX (ak, al), Add (Mul (hl, Sub (if', ig)), gm)) + | numadd (CX (ak, al), Add (Mul (hl, Mul (ih, ii)), gm)) = + Add (CX (ak, al), Add (Mul (hl, Mul (ih, ii)), gm)) + | numadd (CX (ak, al), Sub (gn, go)) = Add (CX (ak, al), Sub (gn, go)) + | numadd (CX (ak, al), Mul (gp, gq)) = Add (CX (ak, al), Mul (gp, gq)) + | numadd (Neg am, C iu) = Add (Neg am, C iu) + | numadd (Neg am, Bound iv) = Add (Neg am, Bound iv) + | numadd (Neg am, CX (iw, ix)) = Add (Neg am, CX (iw, ix)) + | numadd (Neg am, Neg iy) = Add (Neg am, Neg iy) + | numadd (Neg am, Add (C jq, ja)) = Add (Neg am, Add (C jq, ja)) + | numadd (Neg am, Add (Bound jr, ja)) = Add (Neg am, Add (Bound jr, ja)) + | numadd (Neg am, Add (CX (js, jt), ja)) = Add (Neg am, Add (CX (js, jt), ja)) + | numadd (Neg am, Add (Neg ju, ja)) = Add (Neg am, Add (Neg ju, ja)) + | numadd (Neg am, Add (Add (jv, jw), ja)) = + Add (Neg am, Add (Add (jv, jw), ja)) + | numadd (Neg am, Add (Sub (jx, jy), ja)) = + Add (Neg am, Add (Sub (jx, jy), ja)) + | numadd (Neg am, Add (Mul (jz, C km), ja)) = + Add (Neg am, Add (Mul (jz, C km), ja)) + | numadd (Neg am, Add (Mul (jz, CX (ko, kp)), ja)) = + Add (Neg am, Add (Mul (jz, CX (ko, kp)), ja)) + | numadd (Neg am, Add (Mul (jz, Neg kq), ja)) = + Add (Neg am, Add (Mul (jz, Neg kq), ja)) + | numadd (Neg am, Add (Mul (jz, Add (kr, ks)), ja)) = + Add (Neg am, Add (Mul (jz, Add (kr, ks)), ja)) + | numadd (Neg am, Add (Mul (jz, Sub (kt, ku)), ja)) = + Add (Neg am, Add (Mul (jz, Sub (kt, ku)), ja)) + | numadd (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) = + Add (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) + | numadd (Neg am, Sub (jb, jc)) = Add (Neg am, Sub (jb, jc)) + | numadd (Neg am, Mul (jd, je)) = Add (Neg am, Mul (jd, je)) + | numadd (Add (C lt, ao), C mp) = Add (Add (C lt, ao), C mp) + | numadd (Add (C lt, ao), Bound mq) = Add (Add (C lt, ao), Bound mq) + | numadd (Add (C lt, ao), CX (mr, ms)) = Add (Add (C lt, ao), CX (mr, ms)) + | numadd (Add (C lt, ao), Neg mt) = Add (Add (C lt, ao), Neg mt) + | numadd (Add (C lt, ao), Add (C nl, mv)) = + Add (Add (C lt, ao), Add (C nl, mv)) + | numadd (Add (C lt, ao), Add (Bound nm, mv)) = + Add (Add (C lt, ao), Add (Bound nm, mv)) + | numadd (Add (C lt, ao), Add (CX (nn, no), mv)) = + Add (Add (C lt, ao), Add (CX (nn, no), mv)) + | numadd (Add (C lt, ao), Add (Neg np, mv)) = + Add (Add (C lt, ao), Add (Neg np, mv)) + | numadd (Add (C lt, ao), Add (Add (nq, nr), mv)) = + Add (Add (C lt, ao), Add (Add (nq, nr), mv)) + | numadd (Add (C lt, ao), Add (Sub (ns, nt), mv)) = + Add (Add (C lt, ao), Add (Sub (ns, nt), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, C oh), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, C oh), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, CX (oj, ok)), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, CX (oj, ok)), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, Neg ol), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Neg ol), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, Sub (oo, op')), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Sub (oo, op')), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv)) + | numadd (Add (C lt, ao), Sub (mw, mx)) = Add (Add (C lt, ao), Sub (mw, mx)) + | numadd (Add (C lt, ao), Mul (my, mz)) = Add (Add (C lt, ao), Mul (my, mz)) + | numadd (Add (Bound lu, ao), C pd) = Add (Add (Bound lu, ao), C pd) + | numadd (Add (Bound lu, ao), Bound pe) = Add (Add (Bound lu, ao), Bound pe) + | numadd (Add (Bound lu, ao), CX (pf, pg)) = + Add (Add (Bound lu, ao), CX (pf, pg)) + | numadd (Add (Bound lu, ao), Neg ph) = Add (Add (Bound lu, ao), Neg ph) + | numadd (Add (Bound lu, ao), Add (C pz, pj)) = + Add (Add (Bound lu, ao), Add (C pz, pj)) + | numadd (Add (Bound lu, ao), Add (Bound qa, pj)) = + Add (Add (Bound lu, ao), Add (Bound qa, pj)) + | numadd (Add (Bound lu, ao), Add (CX (qb, qc), pj)) = + Add (Add (Bound lu, ao), Add (CX (qb, qc), pj)) + | numadd (Add (Bound lu, ao), Add (Neg qd, pj)) = + Add (Add (Bound lu, ao), Add (Neg qd, pj)) + | numadd (Add (Bound lu, ao), Add (Add (qe, qf), pj)) = + Add (Add (Bound lu, ao), Add (Add (qe, qf), pj)) + | numadd (Add (Bound lu, ao), Add (Sub (qg, qh), pj)) = + Add (Add (Bound lu, ao), Add (Sub (qg, qh), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, C qv), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, C qv), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, CX (qx, qy)), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, CX (qx, qy)), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj)) + | numadd (Add (Bound lu, ao), Sub (pk, pl)) = + Add (Add (Bound lu, ao), Sub (pk, pl)) + | numadd (Add (Bound lu, ao), Mul (pm, pn)) = + Add (Add (Bound lu, ao), Mul (pm, pn)) + | numadd (Add (CX (lv, lw), ao), C rr) = Add (Add (CX (lv, lw), ao), C rr) + | numadd (Add (CX (lv, lw), ao), Bound rs) = + Add (Add (CX (lv, lw), ao), Bound rs) + | numadd (Add (CX (lv, lw), ao), CX (rt, ru)) = + Add (Add (CX (lv, lw), ao), CX (rt, ru)) + | numadd (Add (CX (lv, lw), ao), Neg rv) = Add (Add (CX (lv, lw), ao), Neg rv) + | numadd (Add (CX (lv, lw), ao), Add (C sn, rx)) = + Add (Add (CX (lv, lw), ao), Add (C sn, rx)) + | numadd (Add (CX (lv, lw), ao), Add (Bound so, rx)) = + Add (Add (CX (lv, lw), ao), Add (Bound so, rx)) + | numadd (Add (CX (lv, lw), ao), Add (CX (sp, sq), rx)) = + Add (Add (CX (lv, lw), ao), Add (CX (sp, sq), rx)) + | numadd (Add (CX (lv, lw), ao), Add (Neg sr, rx)) = + Add (Add (CX (lv, lw), ao), Add (Neg sr, rx)) + | numadd (Add (CX (lv, lw), ao), Add (Add (ss, st), rx)) = + Add (Add (CX (lv, lw), ao), Add (Add (ss, st), rx)) + | numadd (Add (CX (lv, lw), ao), Add (Sub (su, sv), rx)) = + Add (Add (CX (lv, lw), ao), Add (Sub (su, sv), rx)) + | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, C tj), rx)) = + Add (Add (CX (lv, lw), ao), Add (Mul (sw, C tj), rx)) + | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, CX (tl, tm)), rx)) = + Add (Add (CX (lv, lw), ao), Add (Mul (sw, CX (tl, tm)), rx)) + | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Neg tn), rx)) = + Add (Add (CX (lv, lw), ao), Add (Mul (sw, Neg tn), rx)) + | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx)) = + Add (Add (CX (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx)) + | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx)) = + Add (Add (CX (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx)) + | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx)) = + Add (Add (CX (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx)) + | numadd (Add (CX (lv, lw), ao), Sub (ry, rz)) = + Add (Add (CX (lv, lw), ao), Sub (ry, rz)) + | numadd (Add (CX (lv, lw), ao), Mul (sa, sb)) = + Add (Add (CX (lv, lw), ao), Mul (sa, sb)) + | numadd (Add (Neg lx, ao), C uf) = Add (Add (Neg lx, ao), C uf) + | numadd (Add (Neg lx, ao), Bound ug) = Add (Add (Neg lx, ao), Bound ug) + | numadd (Add (Neg lx, ao), CX (uh, ui)) = Add (Add (Neg lx, ao), CX (uh, ui)) + | numadd (Add (Neg lx, ao), Neg uj) = Add (Add (Neg lx, ao), Neg uj) + | numadd (Add (Neg lx, ao), Add (C vb, ul)) = + Add (Add (Neg lx, ao), Add (C vb, ul)) + | numadd (Add (Neg lx, ao), Add (Bound vc, ul)) = + Add (Add (Neg lx, ao), Add (Bound vc, ul)) + | numadd (Add (Neg lx, ao), Add (CX (vd, ve), ul)) = + Add (Add (Neg lx, ao), Add (CX (vd, ve), ul)) + | numadd (Add (Neg lx, ao), Add (Neg vf, ul)) = + Add (Add (Neg lx, ao), Add (Neg vf, ul)) + | numadd (Add (Neg lx, ao), Add (Add (vg, vh), ul)) = + Add (Add (Neg lx, ao), Add (Add (vg, vh), ul)) + | numadd (Add (Neg lx, ao), Add (Sub (vi, vj), ul)) = + Add (Add (Neg lx, ao), Add (Sub (vi, vj), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, C vx), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, C vx), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, CX (vz, wa)), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, CX (vz, wa)), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul)) + | numadd (Add (Neg lx, ao), Sub (um, un)) = + Add (Add (Neg lx, ao), Sub (um, un)) + | numadd (Add (Neg lx, ao), Mul (uo, up)) = + Add (Add (Neg lx, ao), Mul (uo, up)) + | numadd (Add (Add (ly, lz), ao), C wt) = Add (Add (Add (ly, lz), ao), C wt) + | numadd (Add (Add (ly, lz), ao), Bound wu) = + Add (Add (Add (ly, lz), ao), Bound wu) + | numadd (Add (Add (ly, lz), ao), CX (wv, ww)) = + Add (Add (Add (ly, lz), ao), CX (wv, ww)) + | numadd (Add (Add (ly, lz), ao), Neg wx) = + Add (Add (Add (ly, lz), ao), Neg wx) + | numadd (Add (Add (ly, lz), ao), Add (C xp, wz)) = + Add (Add (Add (ly, lz), ao), Add (C xp, wz)) + | numadd (Add (Add (ly, lz), ao), Add (Bound xq, wz)) = + Add (Add (Add (ly, lz), ao), Add (Bound xq, wz)) + | numadd (Add (Add (ly, lz), ao), Add (CX (xr, xs), wz)) = + Add (Add (Add (ly, lz), ao), Add (CX (xr, xs), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Neg xt, wz)) = + Add (Add (Add (ly, lz), ao), Add (Neg xt, wz)) + | numadd (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz)) = + Add (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz)) = + Add (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, CX (yn, yo)), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, CX (yn, yo)), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz)) + | numadd (Add (Add (ly, lz), ao), Sub (xa, xb)) = + Add (Add (Add (ly, lz), ao), Sub (xa, xb)) + | numadd (Add (Add (ly, lz), ao), Mul (xc, xd)) = + Add (Add (Add (ly, lz), ao), Mul (xc, xd)) + | numadd (Add (Sub (ma, mb), ao), C zh) = Add (Add (Sub (ma, mb), ao), C zh) + | numadd (Add (Sub (ma, mb), ao), Bound zi) = + Add (Add (Sub (ma, mb), ao), Bound zi) + | numadd (Add (Sub (ma, mb), ao), CX (zj, zk)) = + Add (Add (Sub (ma, mb), ao), CX (zj, zk)) + | numadd (Add (Sub (ma, mb), ao), Neg zl) = + Add (Add (Sub (ma, mb), ao), Neg zl) + | numadd (Add (Sub (ma, mb), ao), Add (C aad, zn)) = + Add (Add (Sub (ma, mb), ao), Add (C aad, zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Bound aae, zn)) = + Add (Add (Sub (ma, mb), ao), Add (Bound aae, zn)) + | numadd (Add (Sub (ma, mb), ao), Add (CX (aaf, aag), zn)) = + Add (Add (Sub (ma, mb), ao), Add (CX (aaf, aag), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Neg aah, zn)) = + Add (Add (Sub (ma, mb), ao), Add (Neg aah, zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, CX (abb, abc)), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, CX (abb, abc)), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn)) + | numadd (Add (Sub (ma, mb), ao), Sub (zo, zp)) = + Add (Add (Sub (ma, mb), ao), Sub (zo, zp)) + | numadd (Add (Sub (ma, mb), ao), Mul (zq, zr)) = + Add (Add (Sub (ma, mb), ao), Mul (zq, zr)) + | numadd (Add (Mul (mc, C acg), ao), C adc) = + Add (Add (Mul (mc, C acg), ao), C adc) + | numadd (Add (Mul (mc, C acg), ao), Bound add) = + Add (Add (Mul (mc, C acg), ao), Bound add) + | numadd (Add (Mul (mc, C acg), ao), CX (ade, adf)) = + Add (Add (Mul (mc, C acg), ao), CX (ade, adf)) + | numadd (Add (Mul (mc, C acg), ao), Neg adg) = + Add (Add (Mul (mc, C acg), ao), Neg adg) + | numadd (Add (Mul (mc, C acg), ao), Add (C ady, adi)) = + Add (Add (Mul (mc, C acg), ao), Add (C ady, adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Bound adz, adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Bound adz, adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (CX (aea, aeb), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (CX (aea, aeb), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Neg aec, adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Neg aec, adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, CX (aew, aex)), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, CX (aew, aex)), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi)) + | numadd (Add (Mul (mc, C acg), ao), Sub (adj, adk)) = + Add (Add (Mul (mc, C acg), ao), Sub (adj, adk)) + | numadd (Add (Mul (mc, C acg), ao), Mul (adl, adm)) = + Add (Add (Mul (mc, C acg), ao), Mul (adl, adm)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), C ajl) = + Add (Add (Mul (mc, CX (aci, acj)), ao), C ajl) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Bound ajm) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Bound ajm) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), CX (ajn, ajo)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), CX (ajn, ajo)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Neg ajp) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Neg ajp) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (C akh, ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (C akh, ajr)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Bound aki, ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Bound aki, ajr)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (CX (akj, akk), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (CX (akj, akk), ajr)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Neg akl, ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Neg akl, ajr)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Add (akm, akn), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Add (akm, akn), ajr)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Sub (ako, akp), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Sub (ako, akp), ajr)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, C ald), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, C ald), ajr)) + | numadd + (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, CX (alf, alg)), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, CX (alf, alg)), ajr)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr)) + | numadd + (Add (Mul (mc, CX (aci, acj)), ao), + Add (Mul (akq, Add (ali, alj)), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), + Add (Mul (akq, Add (ali, alj)), ajr)) + | numadd + (Add (Mul (mc, CX (aci, acj)), ao), + Add (Mul (akq, Sub (alk, all)), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), + Add (Mul (akq, Sub (alk, all)), ajr)) + | numadd + (Add (Mul (mc, CX (aci, acj)), ao), + Add (Mul (akq, Mul (alm, aln)), ajr)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), + Add (Mul (akq, Mul (alm, aln)), ajr)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Sub (ajs, ajt)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Sub (ajs, ajt)) + | numadd (Add (Mul (mc, CX (aci, acj)), ao), Mul (aju, ajv)) = + Add (Add (Mul (mc, CX (aci, acj)), ao), Mul (aju, ajv)) + | numadd (Add (Mul (mc, Neg ack), ao), C alz) = + Add (Add (Mul (mc, Neg ack), ao), C alz) + | numadd (Add (Mul (mc, Neg ack), ao), Bound ama) = + Add (Add (Mul (mc, Neg ack), ao), Bound ama) + | numadd (Add (Mul (mc, Neg ack), ao), CX (amb, amc)) = + Add (Add (Mul (mc, Neg ack), ao), CX (amb, amc)) + | numadd (Add (Mul (mc, Neg ack), ao), Neg amd) = + Add (Add (Mul (mc, Neg ack), ao), Neg amd) + | numadd (Add (Mul (mc, Neg ack), ao), Add (C amv, amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (C amv, amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (CX (amx, amy), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (CX (amx, amy), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, and'), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, and'), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, CX (ant, anu)), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, CX (ant, anu)), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Sub (amg, amh)) = + Add (Add (Mul (mc, Neg ack), ao), Sub (amg, amh)) + | numadd (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) = + Add (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), C aon) = + Add (Add (Mul (mc, Add (acl, acm)), ao), C aon) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), CX (aop, aoq)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), CX (aop, aoq)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (CX (apl, apm), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (CX (apl, apm), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot)) + | numadd + (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, CX (aqh, aqi)), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, CX (aqh, aqi)), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot)) + | numadd + (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Add (aqk, aql)), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Add (aqk, aql)), aot)) + | numadd + (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Sub (aqm, aqn)), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Sub (aqm, aqn)), aot)) + | numadd + (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Mul (aqo, aqp)), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Mul (aqo, aqp)), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), C arb) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), C arb) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), CX (ard, are)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), CX (ard, are)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (CX (arz, asa), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (CX (arz, asa), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh)) + | numadd + (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, CX (asv, asw)), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, CX (asv, asw)), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh)) + | numadd + (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Add (asy, asz)), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Add (asy, asz)), arh)) + | numadd + (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Sub (ata, atb)), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Sub (ata, atb)), arh)) + | numadd + (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Mul (atc, atd)), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Mul (atc, atd)), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), C atp) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), C atp) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), CX (atr, ats)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), CX (atr, ats)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (CX (aun, auo), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (CX (aun, auo), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv)) + | numadd + (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, CX (avj, avk)), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, CX (avj, avk)), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv)) + | numadd + (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, Add (avm, avn)), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, Add (avm, avn)), atv)) + | numadd + (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, Sub (avo, avp)), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, Sub (avo, avp)), atv)) + | numadd + (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, Mul (avq, avr)), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, Mul (avq, avr)), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) + | numadd (Sub (ap, aq), C awd) = Add (Sub (ap, aq), C awd) + | numadd (Sub (ap, aq), Bound awe) = Add (Sub (ap, aq), Bound awe) + | numadd (Sub (ap, aq), CX (awf, awg)) = Add (Sub (ap, aq), CX (awf, awg)) + | numadd (Sub (ap, aq), Neg awh) = Add (Sub (ap, aq), Neg awh) + | numadd (Sub (ap, aq), Add (C awz, awj)) = + Add (Sub (ap, aq), Add (C awz, awj)) + | numadd (Sub (ap, aq), Add (Bound axa, awj)) = + Add (Sub (ap, aq), Add (Bound axa, awj)) + | numadd (Sub (ap, aq), Add (CX (axb, axc), awj)) = + Add (Sub (ap, aq), Add (CX (axb, axc), awj)) + | numadd (Sub (ap, aq), Add (Neg axd, awj)) = + Add (Sub (ap, aq), Add (Neg axd, awj)) + | numadd (Sub (ap, aq), Add (Add (axe, axf), awj)) = + Add (Sub (ap, aq), Add (Add (axe, axf), awj)) + | numadd (Sub (ap, aq), Add (Sub (axg, axh), awj)) = + Add (Sub (ap, aq), Add (Sub (axg, axh), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, C axv), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, C axv), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, CX (axx, axy)), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, CX (axx, axy)), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, Neg axz), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Neg axz), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj)) + | numadd (Sub (ap, aq), Sub (awk, awl)) = Add (Sub (ap, aq), Sub (awk, awl)) + | numadd (Sub (ap, aq), Mul (awm, awn)) = Add (Sub (ap, aq), Mul (awm, awn)) + | numadd (Mul (ar, as'), C ayr) = Add (Mul (ar, as'), C ayr) + | numadd (Mul (ar, as'), Bound ays) = Add (Mul (ar, as'), Bound ays) + | numadd (Mul (ar, as'), CX (ayt, ayu)) = Add (Mul (ar, as'), CX (ayt, ayu)) + | numadd (Mul (ar, as'), Neg ayv) = Add (Mul (ar, as'), Neg ayv) + | numadd (Mul (ar, as'), Add (C azn, ayx)) = + Add (Mul (ar, as'), Add (C azn, ayx)) + | numadd (Mul (ar, as'), Add (Bound azo, ayx)) = + Add (Mul (ar, as'), Add (Bound azo, ayx)) + | numadd (Mul (ar, as'), Add (CX (azp, azq), ayx)) = + Add (Mul (ar, as'), Add (CX (azp, azq), ayx)) + | numadd (Mul (ar, as'), Add (Neg azr, ayx)) = + Add (Mul (ar, as'), Add (Neg azr, ayx)) + | numadd (Mul (ar, as'), Add (Add (azs, azt), ayx)) = + Add (Mul (ar, as'), Add (Add (azs, azt), ayx)) + | numadd (Mul (ar, as'), Add (Sub (azu, azv), ayx)) = + Add (Mul (ar, as'), Add (Sub (azu, azv), ayx)) + | numadd (Mul (ar, as'), Add (Mul (azw, C baj), ayx)) = + Add (Mul (ar, as'), Add (Mul (azw, C baj), ayx)) + | numadd (Mul (ar, as'), Add (Mul (azw, CX (bal, bam)), ayx)) = + Add (Mul (ar, as'), Add (Mul (azw, CX (bal, bam)), ayx)) + | numadd (Mul (ar, as'), Add (Mul (azw, Neg ban), ayx)) = + Add (Mul (ar, as'), Add (Mul (azw, Neg ban), ayx)) + | numadd (Mul (ar, as'), Add (Mul (azw, Add (bao, bap)), ayx)) = + Add (Mul (ar, as'), Add (Mul (azw, Add (bao, bap)), ayx)) + | numadd (Mul (ar, as'), Add (Mul (azw, Sub (baq, bar)), ayx)) = + Add (Mul (ar, as'), Add (Mul (azw, Sub (baq, bar)), ayx)) + | numadd (Mul (ar, as'), Add (Mul (azw, Mul (bas, bat)), ayx)) = + Add (Mul (ar, as'), Add (Mul (azw, Mul (bas, bat)), ayx)) + | numadd (Mul (ar, as'), Sub (ayy, ayz)) = Add (Mul (ar, as'), Sub (ayy, ayz)) + | numadd (Mul (ar, as'), Mul (aza, azb)) = + Add (Mul (ar, as'), Mul (aza, azb)); + +fun nummul (C j) = (fn i => C (i * j)) + | nummul (Add (a, b)) = (fn i => numadd (nummul a i, nummul b i)) + | nummul (Mul (c, t)) = (fn i => nummul t (i * c)) + | nummul (Bound v) = (fn i => Mul (i, Bound v)) + | nummul (CX (w, x)) = (fn i => Mul (i, CX (w, x))) + | nummul (Neg y) = (fn i => Mul (i, Neg y)) + | nummul (Sub (ac, ad)) = (fn i => Mul (i, Sub (ac, ad))); + +fun numneg t = nummul t (~ 1); + +fun numsub s t = (if (s = t) then C 0 else numadd (s, numneg t)); + +fun simpnum (C j) = C j + | simpnum (Bound n) = Add (Mul (1, Bound n), C 0) + | simpnum (Neg t) = numneg (simpnum t) + | simpnum (Add (t, s)) = numadd (simpnum t, simpnum s) + | simpnum (Sub (t, s)) = numsub (simpnum t) (simpnum s) + | simpnum (Mul (i, t)) = (if (i = 0) then C 0 else nummul (simpnum t) i) + | simpnum (CX (w, x)) = CX (w, x); + +datatype fm = T | F | Lt of num | Le of num | Gt of num | Ge of num | Eq of num + | NEq of num | Dvd of int * num | NDvd of int * num | NOT of fm + | And of fm * fm | Or of fm * fm | Imp of fm * fm | Iff of fm * fm | E of fm + | A of fm | Closed of int | NClosed of int; + +fun not (NOT p) = p + | not T = F + | not F = T + | not (Lt u) = NOT (Lt u) + | not (Le v) = NOT (Le v) + | not (Gt w) = NOT (Gt w) + | not (Ge x) = NOT (Ge x) + | not (Eq y) = NOT (Eq y) + | not (NEq z) = NOT (NEq z) + | not (Dvd (aa, ab)) = NOT (Dvd (aa, ab)) + | not (NDvd (ac, ad)) = NOT (NDvd (ac, ad)) + | not (And (af, ag)) = NOT (And (af, ag)) + | not (Or (ah, ai)) = NOT (Or (ah, ai)) + | not (Imp (aj, ak)) = NOT (Imp (aj, ak)) + | not (Iff (al, am)) = NOT (Iff (al, am)) + | not (E an) = NOT (E an) + | not (A ao) = NOT (A ao) + | not (Closed ap) = NOT (Closed ap) + | not (NClosed aq) = NOT (NClosed aq); + +fun iff p q = + (if (p = q) then T + else (if ((p = not q) orelse (not p = q)) then F + else (if (p = F) then not q + else (if (q = F) then not p + else (if (p = T) then q + else (if (q = T) then p else Iff (p, q))))))); + +fun imp p q = + (if ((p = F) orelse (q = T)) then T + else (if (p = T) then q else (if (q = F) then not p else Imp (p, q)))); + +fun disj p q = + (if ((p = T) orelse (q = T)) then T + else (if (p = F) then q else (if (q = F) then p else Or (p, q)))); + +fun conj p q = + (if ((p = F) orelse (q = F)) then F + else (if (p = T) then q else (if (q = T) then p else And (p, q)))); + +fun simpfm (And (p, q)) = conj (simpfm p) (simpfm q) + | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q) + | simpfm (Imp (p, q)) = imp (simpfm p) (simpfm q) + | simpfm (Iff (p, q)) = iff (simpfm p) (simpfm q) + | simpfm (NOT p) = not (simpfm p) + | simpfm (Lt a) = + let val a' = simpnum a + in (case a' of C x => (if (x < 0) then T else F) | Bound x => Lt a' + | CX (x, xa) => Lt a' | Neg x => Lt a' | Add (x, xa) => Lt a' + | Sub (x, xa) => Lt a' | Mul (x, xa) => Lt a') + end + | simpfm (Le a) = + let val a' = simpnum a + in (case a' of C x => (if (x <= 0) then T else F) | Bound x => Le a' + | CX (x, xa) => Le a' | Neg x => Le a' | Add (x, xa) => Le a' + | Sub (x, xa) => Le a' | Mul (x, xa) => Le a') + end + | simpfm (Gt a) = + let val a' = simpnum a + in (case a' of C x => (if (0 < x) then T else F) | Bound x => Gt a' + | CX (x, xa) => Gt a' | Neg x => Gt a' | Add (x, xa) => Gt a' + | Sub (x, xa) => Gt a' | Mul (x, xa) => Gt a') + end + | simpfm (Ge a) = + let val a' = simpnum a + in (case a' of C x => (if (0 <= x) then T else F) | Bound x => Ge a' + | CX (x, xa) => Ge a' | Neg x => Ge a' | Add (x, xa) => Ge a' + | Sub (x, xa) => Ge a' | Mul (x, xa) => Ge a') + end + | simpfm (Eq a) = + let val a' = simpnum a + in (case a' of C x => (if (x = 0) then T else F) | Bound x => Eq a' + | CX (x, xa) => Eq a' | Neg x => Eq a' | Add (x, xa) => Eq a' + | Sub (x, xa) => Eq a' | Mul (x, xa) => Eq a') + end + | simpfm (NEq a) = + let val a' = simpnum a + in (case a' of C x => (if Bool.not (x = 0) then T else F) + | Bound x => NEq a' | CX (x, xa) => NEq a' | Neg x => NEq a' + | Add (x, xa) => NEq a' | Sub (x, xa) => NEq a' + | Mul (x, xa) => NEq a') + end + | simpfm (Dvd (i, a)) = + (if (i = 0) then simpfm (Eq a) + else (if (abs i = 1) then T + else let val a' = simpnum a + in (case a' of C x => (if dvd i x then T else F) + | Bound x => Dvd (i, a') | CX (x, xa) => Dvd (i, a') + | Neg x => Dvd (i, a') | Add (x, xa) => Dvd (i, a') + | Sub (x, xa) => Dvd (i, a') + | Mul (x, xa) => Dvd (i, a')) + end)) + | simpfm (NDvd (i, a)) = + (if (i = 0) then simpfm (NEq a) + else (if (abs i = 1) then F + else let val a' = simpnum a + in (case a' of C x => (if Bool.not (dvd i x) then T else F) + | Bound x => NDvd (i, a') | CX (x, xa) => NDvd (i, a') + | Neg x => NDvd (i, a') | Add (x, xa) => NDvd (i, a') + | Sub (x, xa) => NDvd (i, a') + | Mul (x, xa) => NDvd (i, a')) + end)) + | simpfm T = T + | simpfm F = F + | simpfm (E ao) = E ao + | simpfm (A ap) = A ap + | simpfm (Closed aq) = Closed aq + | simpfm (NClosed ar) = NClosed ar; + +fun foldr f [] a = a + | foldr f (x :: xs) a = f x (foldr f xs a); + +fun djf f p q = + (if (q = T) then T + else (if (q = F) then f p + else let val fp = f p + in (case fp of T => T | F => q | Lt x => Or (f p, q) + | Le x => Or (f p, q) | Gt x => Or (f p, q) + | Ge x => Or (f p, q) | Eq x => Or (f p, q) + | NEq x => Or (f p, q) | Dvd (x, xa) => Or (f p, q) + | NDvd (x, xa) => Or (f p, q) | NOT x => Or (f p, q) + | And (x, xa) => Or (f p, q) | Or (x, xa) => Or (f p, q) + | Imp (x, xa) => Or (f p, q) | Iff (x, xa) => Or (f p, q) + | E x => Or (f p, q) | A x => Or (f p, q) + | Closed x => Or (f p, q) | NClosed x => Or (f p, q)) + end)); + +fun evaldjf f ps = foldr (djf f) ps F; + +fun append [] ys = ys + | append (x :: xs) ys = (x :: append xs ys); + +fun disjuncts (Or (p, q)) = append (disjuncts p) (disjuncts q) + | disjuncts F = [] + | disjuncts T = [T] + | disjuncts (Lt u) = [Lt u] + | disjuncts (Le v) = [Le v] + | disjuncts (Gt w) = [Gt w] + | disjuncts (Ge x) = [Ge x] + | disjuncts (Eq y) = [Eq y] + | disjuncts (NEq z) = [NEq z] + | disjuncts (Dvd (aa, ab)) = [Dvd (aa, ab)] + | disjuncts (NDvd (ac, ad)) = [NDvd (ac, ad)] + | disjuncts (NOT ae) = [NOT ae] + | disjuncts (And (af, ag)) = [And (af, ag)] + | disjuncts (Imp (aj, ak)) = [Imp (aj, ak)] + | disjuncts (Iff (al, am)) = [Iff (al, am)] + | disjuncts (E an) = [E an] + | disjuncts (A ao) = [A ao] + | disjuncts (Closed ap) = [Closed ap] + | disjuncts (NClosed aq) = [NClosed aq]; + +fun DJ f p = evaldjf f (disjuncts p); + +fun qelim (E p) = (fn qe => DJ qe (qelim p qe)) + | qelim (A p) = (fn qe => not (qe (qelim (NOT p) qe))) + | qelim (NOT p) = (fn qe => not (qelim p qe)) + | qelim (And (p, q)) = (fn qe => conj (qelim p qe) (qelim q qe)) + | qelim (Or (p, q)) = (fn qe => disj (qelim p qe) (qelim q qe)) + | qelim (Imp (p, q)) = (fn qe => imp (qelim p qe) (qelim q qe)) + | qelim (Iff (p, q)) = (fn qe => iff (qelim p qe) (qelim q qe)) + | qelim T = (fn y => simpfm T) + | qelim F = (fn y => simpfm F) + | qelim (Lt u) = (fn y => simpfm (Lt u)) + | qelim (Le v) = (fn y => simpfm (Le v)) + | qelim (Gt w) = (fn y => simpfm (Gt w)) + | qelim (Ge x) = (fn y => simpfm (Ge x)) + | qelim (Eq y) = (fn ya => simpfm (Eq y)) + | qelim (NEq z) = (fn y => simpfm (NEq z)) + | qelim (Dvd (aa, ab)) = (fn y => simpfm (Dvd (aa, ab))) + | qelim (NDvd (ac, ad)) = (fn y => simpfm (NDvd (ac, ad))) + | qelim (Closed ap) = (fn y => simpfm (Closed ap)) + | qelim (NClosed aq) = (fn y => simpfm (NClosed aq)); + +fun minus_def1 m n = nat (minus_def2 (m) (n)); + +fun decrnum (Bound n) = Bound (minus_def1 n one_def0) + | decrnum (Neg a) = Neg (decrnum a) + | decrnum (Add (a, b)) = Add (decrnum a, decrnum b) + | decrnum (Sub (a, b)) = Sub (decrnum a, decrnum b) + | decrnum (Mul (c, a)) = Mul (c, decrnum a) + | decrnum (C u) = C u + | decrnum (CX (w, x)) = CX (w, x); + +fun decr (Lt a) = Lt (decrnum a) + | decr (Le a) = Le (decrnum a) + | decr (Gt a) = Gt (decrnum a) + | decr (Ge a) = Ge (decrnum a) + | decr (Eq a) = Eq (decrnum a) + | decr (NEq a) = NEq (decrnum a) + | decr (Dvd (i, a)) = Dvd (i, decrnum a) + | decr (NDvd (i, a)) = NDvd (i, decrnum a) + | decr (NOT p) = NOT (decr p) + | decr (And (p, q)) = And (decr p, decr q) + | decr (Or (p, q)) = Or (decr p, decr q) + | decr (Imp (p, q)) = Imp (decr p, decr q) + | decr (Iff (p, q)) = Iff (decr p, decr q) + | decr T = T + | decr F = F + | decr (E ao) = E ao + | decr (A ap) = A ap + | decr (Closed aq) = Closed aq + | decr (NClosed ar) = NClosed ar; + +fun map f [] = [] + | map f (x :: xs) = (f x :: map f xs); + +fun allpairs f [] ys = [] + | allpairs f (x :: xs) ys = append (map (f x) ys) (allpairs f xs ys); + +fun numsubst0 t (C c) = C c + | numsubst0 t (Bound n) = (if (n = 0) then t else Bound n) + | numsubst0 t (CX (i, a)) = Add (Mul (i, t), numsubst0 t a) + | numsubst0 t (Neg a) = Neg (numsubst0 t a) + | numsubst0 t (Add (a, b)) = Add (numsubst0 t a, numsubst0 t b) + | numsubst0 t (Sub (a, b)) = Sub (numsubst0 t a, numsubst0 t b) + | numsubst0 t (Mul (i, a)) = Mul (i, numsubst0 t a); + +fun subst0 t T = T + | subst0 t F = F + | subst0 t (Lt a) = Lt (numsubst0 t a) + | subst0 t (Le a) = Le (numsubst0 t a) + | subst0 t (Gt a) = Gt (numsubst0 t a) + | subst0 t (Ge a) = Ge (numsubst0 t a) + | subst0 t (Eq a) = Eq (numsubst0 t a) + | subst0 t (NEq a) = NEq (numsubst0 t a) + | subst0 t (Dvd (i, a)) = Dvd (i, numsubst0 t a) + | subst0 t (NDvd (i, a)) = NDvd (i, numsubst0 t a) + | subst0 t (NOT p) = NOT (subst0 t p) + | subst0 t (And (p, q)) = And (subst0 t p, subst0 t q) + | subst0 t (Or (p, q)) = Or (subst0 t p, subst0 t q) + | subst0 t (Imp (p, q)) = Imp (subst0 t p, subst0 t q) + | subst0 t (Iff (p, q)) = Iff (subst0 t p, subst0 t q) + | subst0 t (Closed P) = Closed P + | subst0 t (NClosed P) = NClosed P; + +fun minusinf (And (p, q)) = And (minusinf p, minusinf q) + | minusinf (Or (p, q)) = Or (minusinf p, minusinf q) + | minusinf (Eq (CX (c, e))) = F + | minusinf (NEq (CX (c, e))) = T + | minusinf (Lt (CX (c, e))) = T + | minusinf (Le (CX (c, e))) = T + | minusinf (Gt (CX (c, e))) = F + | minusinf (Ge (CX (c, e))) = F + | minusinf T = T + | minusinf F = F + | minusinf (Lt (C bo)) = Lt (C bo) + | minusinf (Lt (Bound bp)) = Lt (Bound bp) + | minusinf (Lt (Neg bs)) = Lt (Neg bs) + | minusinf (Lt (Add (bt, bu))) = Lt (Add (bt, bu)) + | minusinf (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw)) + | minusinf (Lt (Mul (bx, by))) = Lt (Mul (bx, by)) + | minusinf (Le (C ck)) = Le (C ck) + | minusinf (Le (Bound cl)) = Le (Bound cl) + | minusinf (Le (Neg co)) = Le (Neg co) + | minusinf (Le (Add (cp, cq))) = Le (Add (cp, cq)) + | minusinf (Le (Sub (cr, cs))) = Le (Sub (cr, cs)) + | minusinf (Le (Mul (ct, cu))) = Le (Mul (ct, cu)) + | minusinf (Gt (C dg)) = Gt (C dg) + | minusinf (Gt (Bound dh)) = Gt (Bound dh) + | minusinf (Gt (Neg dk)) = Gt (Neg dk) + | minusinf (Gt (Add (dl, dm))) = Gt (Add (dl, dm)) + | minusinf (Gt (Sub (dn, do'))) = Gt (Sub (dn, do')) + | minusinf (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq)) + | minusinf (Ge (C ec)) = Ge (C ec) + | minusinf (Ge (Bound ed)) = Ge (Bound ed) + | minusinf (Ge (Neg eg)) = Ge (Neg eg) + | minusinf (Ge (Add (eh, ei))) = Ge (Add (eh, ei)) + | minusinf (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek)) + | minusinf (Ge (Mul (el, em))) = Ge (Mul (el, em)) + | minusinf (Eq (C ey)) = Eq (C ey) + | minusinf (Eq (Bound ez)) = Eq (Bound ez) + | minusinf (Eq (Neg fc)) = Eq (Neg fc) + | minusinf (Eq (Add (fd, fe))) = Eq (Add (fd, fe)) + | minusinf (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg)) + | minusinf (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi)) + | minusinf (NEq (C fu)) = NEq (C fu) + | minusinf (NEq (Bound fv)) = NEq (Bound fv) + | minusinf (NEq (Neg fy)) = NEq (Neg fy) + | minusinf (NEq (Add (fz, ga))) = NEq (Add (fz, ga)) + | minusinf (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc)) + | minusinf (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge)) + | minusinf (Dvd (aa, ab)) = Dvd (aa, ab) + | minusinf (NDvd (ac, ad)) = NDvd (ac, ad) + | minusinf (NOT ae) = NOT ae + | minusinf (Imp (aj, ak)) = Imp (aj, ak) + | minusinf (Iff (al, am)) = Iff (al, am) + | minusinf (E an) = E an + | minusinf (A ao) = A ao + | minusinf (Closed ap) = Closed ap + | minusinf (NClosed aq) = NClosed aq; + +fun iupt (i, j) = (if (j < i) then [] else (i :: iupt ((i + 1), j))); + +fun mirror (And (p, q)) = And (mirror p, mirror q) + | mirror (Or (p, q)) = Or (mirror p, mirror q) + | mirror (Eq (CX (c, e))) = Eq (CX (c, Neg e)) + | mirror (NEq (CX (c, e))) = NEq (CX (c, Neg e)) + | mirror (Lt (CX (c, e))) = Gt (CX (c, Neg e)) + | mirror (Le (CX (c, e))) = Ge (CX (c, Neg e)) + | mirror (Gt (CX (c, e))) = Lt (CX (c, Neg e)) + | mirror (Ge (CX (c, e))) = Le (CX (c, Neg e)) + | mirror (Dvd (i, CX (c, e))) = Dvd (i, CX (c, Neg e)) + | mirror (NDvd (i, CX (c, e))) = NDvd (i, CX (c, Neg e)) + | mirror T = T + | mirror F = F + | mirror (Lt (C bo)) = Lt (C bo) + | mirror (Lt (Bound bp)) = Lt (Bound bp) + | mirror (Lt (Neg bs)) = Lt (Neg bs) + | mirror (Lt (Add (bt, bu))) = Lt (Add (bt, bu)) + | mirror (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw)) + | mirror (Lt (Mul (bx, by))) = Lt (Mul (bx, by)) + | mirror (Le (C ck)) = Le (C ck) + | mirror (Le (Bound cl)) = Le (Bound cl) + | mirror (Le (Neg co)) = Le (Neg co) + | mirror (Le (Add (cp, cq))) = Le (Add (cp, cq)) + | mirror (Le (Sub (cr, cs))) = Le (Sub (cr, cs)) + | mirror (Le (Mul (ct, cu))) = Le (Mul (ct, cu)) + | mirror (Gt (C dg)) = Gt (C dg) + | mirror (Gt (Bound dh)) = Gt (Bound dh) + | mirror (Gt (Neg dk)) = Gt (Neg dk) + | mirror (Gt (Add (dl, dm))) = Gt (Add (dl, dm)) + | mirror (Gt (Sub (dn, do'))) = Gt (Sub (dn, do')) + | mirror (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq)) + | mirror (Ge (C ec)) = Ge (C ec) + | mirror (Ge (Bound ed)) = Ge (Bound ed) + | mirror (Ge (Neg eg)) = Ge (Neg eg) + | mirror (Ge (Add (eh, ei))) = Ge (Add (eh, ei)) + | mirror (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek)) + | mirror (Ge (Mul (el, em))) = Ge (Mul (el, em)) + | mirror (Eq (C ey)) = Eq (C ey) + | mirror (Eq (Bound ez)) = Eq (Bound ez) + | mirror (Eq (Neg fc)) = Eq (Neg fc) + | mirror (Eq (Add (fd, fe))) = Eq (Add (fd, fe)) + | mirror (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg)) + | mirror (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi)) + | mirror (NEq (C fu)) = NEq (C fu) + | mirror (NEq (Bound fv)) = NEq (Bound fv) + | mirror (NEq (Neg fy)) = NEq (Neg fy) + | mirror (NEq (Add (fz, ga))) = NEq (Add (fz, ga)) + | mirror (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc)) + | mirror (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge)) + | mirror (Dvd (aa, C gq)) = Dvd (aa, C gq) + | mirror (Dvd (aa, Bound gr)) = Dvd (aa, Bound gr) + | mirror (Dvd (aa, Neg gu)) = Dvd (aa, Neg gu) + | mirror (Dvd (aa, Add (gv, gw))) = Dvd (aa, Add (gv, gw)) + | mirror (Dvd (aa, Sub (gx, gy))) = Dvd (aa, Sub (gx, gy)) + | mirror (Dvd (aa, Mul (gz, ha))) = Dvd (aa, Mul (gz, ha)) + | mirror (NDvd (ac, C hm)) = NDvd (ac, C hm) + | mirror (NDvd (ac, Bound hn)) = NDvd (ac, Bound hn) + | mirror (NDvd (ac, Neg hq)) = NDvd (ac, Neg hq) + | mirror (NDvd (ac, Add (hr, hs))) = NDvd (ac, Add (hr, hs)) + | mirror (NDvd (ac, Sub (ht, hu))) = NDvd (ac, Sub (ht, hu)) + | mirror (NDvd (ac, Mul (hv, hw))) = NDvd (ac, Mul (hv, hw)) + | mirror (NOT ae) = NOT ae + | mirror (Imp (aj, ak)) = Imp (aj, ak) + | mirror (Iff (al, am)) = Iff (al, am) + | mirror (E an) = E an + | mirror (A ao) = A ao + | mirror (Closed ap) = Closed ap + | mirror (NClosed aq) = NClosed aq; + +fun plus_def0 m n = nat ((m) + (n)); + +fun size_def9 [] = 0 + | size_def9 (a :: list) = plus_def0 (size_def9 list) (0 + 1); + +fun alpha (And (p, q)) = append (alpha p) (alpha q) + | alpha (Or (p, q)) = append (alpha p) (alpha q) + | alpha (Eq (CX (c, e))) = [Add (C ~1, e)] + | alpha (NEq (CX (c, e))) = [e] + | alpha (Lt (CX (c, e))) = [e] + | alpha (Le (CX (c, e))) = [Add (C ~1, e)] + | alpha (Gt (CX (c, e))) = [] + | alpha (Ge (CX (c, e))) = [] + | alpha T = [] + | alpha F = [] + | alpha (Lt (C bo)) = [] + | alpha (Lt (Bound bp)) = [] + | alpha (Lt (Neg bs)) = [] + | alpha (Lt (Add (bt, bu))) = [] + | alpha (Lt (Sub (bv, bw))) = [] + | alpha (Lt (Mul (bx, by))) = [] + | alpha (Le (C ck)) = [] + | alpha (Le (Bound cl)) = [] + | alpha (Le (Neg co)) = [] + | alpha (Le (Add (cp, cq))) = [] + | alpha (Le (Sub (cr, cs))) = [] + | alpha (Le (Mul (ct, cu))) = [] + | alpha (Gt (C dg)) = [] + | alpha (Gt (Bound dh)) = [] + | alpha (Gt (Neg dk)) = [] + | alpha (Gt (Add (dl, dm))) = [] + | alpha (Gt (Sub (dn, do'))) = [] + | alpha (Gt (Mul (dp, dq))) = [] + | alpha (Ge (C ec)) = [] + | alpha (Ge (Bound ed)) = [] + | alpha (Ge (Neg eg)) = [] + | alpha (Ge (Add (eh, ei))) = [] + | alpha (Ge (Sub (ej, ek))) = [] + | alpha (Ge (Mul (el, em))) = [] + | alpha (Eq (C ey)) = [] + | alpha (Eq (Bound ez)) = [] + | alpha (Eq (Neg fc)) = [] + | alpha (Eq (Add (fd, fe))) = [] + | alpha (Eq (Sub (ff, fg))) = [] + | alpha (Eq (Mul (fh, fi))) = [] + | alpha (NEq (C fu)) = [] + | alpha (NEq (Bound fv)) = [] + | alpha (NEq (Neg fy)) = [] + | alpha (NEq (Add (fz, ga))) = [] + | alpha (NEq (Sub (gb, gc))) = [] + | alpha (NEq (Mul (gd, ge))) = [] + | alpha (Dvd (aa, ab)) = [] + | alpha (NDvd (ac, ad)) = [] + | alpha (NOT ae) = [] + | alpha (Imp (aj, ak)) = [] + | alpha (Iff (al, am)) = [] + | alpha (E an) = [] + | alpha (A ao) = [] + | alpha (Closed ap) = [] + | alpha (NClosed aq) = []; + +fun memberl x [] = false + | memberl x (y :: ys) = ((x = y) orelse memberl x ys); + +fun remdups [] = [] + | remdups (x :: xs) = + (if memberl x xs then remdups xs else (x :: remdups xs)); + +fun beta (And (p, q)) = append (beta p) (beta q) + | beta (Or (p, q)) = append (beta p) (beta q) + | beta (Eq (CX (c, e))) = [Sub (C ~1, e)] + | beta (NEq (CX (c, e))) = [Neg e] + | beta (Lt (CX (c, e))) = [] + | beta (Le (CX (c, e))) = [] + | beta (Gt (CX (c, e))) = [Neg e] + | beta (Ge (CX (c, e))) = [Sub (C ~1, e)] + | beta T = [] + | beta F = [] + | beta (Lt (C bo)) = [] + | beta (Lt (Bound bp)) = [] + | beta (Lt (Neg bs)) = [] + | beta (Lt (Add (bt, bu))) = [] + | beta (Lt (Sub (bv, bw))) = [] + | beta (Lt (Mul (bx, by))) = [] + | beta (Le (C ck)) = [] + | beta (Le (Bound cl)) = [] + | beta (Le (Neg co)) = [] + | beta (Le (Add (cp, cq))) = [] + | beta (Le (Sub (cr, cs))) = [] + | beta (Le (Mul (ct, cu))) = [] + | beta (Gt (C dg)) = [] + | beta (Gt (Bound dh)) = [] + | beta (Gt (Neg dk)) = [] + | beta (Gt (Add (dl, dm))) = [] + | beta (Gt (Sub (dn, do'))) = [] + | beta (Gt (Mul (dp, dq))) = [] + | beta (Ge (C ec)) = [] + | beta (Ge (Bound ed)) = [] + | beta (Ge (Neg eg)) = [] + | beta (Ge (Add (eh, ei))) = [] + | beta (Ge (Sub (ej, ek))) = [] + | beta (Ge (Mul (el, em))) = [] + | beta (Eq (C ey)) = [] + | beta (Eq (Bound ez)) = [] + | beta (Eq (Neg fc)) = [] + | beta (Eq (Add (fd, fe))) = [] + | beta (Eq (Sub (ff, fg))) = [] + | beta (Eq (Mul (fh, fi))) = [] + | beta (NEq (C fu)) = [] + | beta (NEq (Bound fv)) = [] + | beta (NEq (Neg fy)) = [] + | beta (NEq (Add (fz, ga))) = [] + | beta (NEq (Sub (gb, gc))) = [] + | beta (NEq (Mul (gd, ge))) = [] + | beta (Dvd (aa, ab)) = [] + | beta (NDvd (ac, ad)) = [] + | beta (NOT ae) = [] + | beta (Imp (aj, ak)) = [] + | beta (Iff (al, am)) = [] + | beta (E an) = [] + | beta (A ao) = [] + | beta (Closed ap) = [] + | beta (NClosed aq) = []; + +fun fst (a, b) = a; + +fun div_def1 a b = fst (divAlg (a, b)); + +fun div_def0 m n = nat (div_def1 (m) (n)); + +fun mod_def0 m n = nat (mod_def1 (m) (n)); + +fun gcd (m, n) = (if (n = 0) then m else gcd (n, mod_def0 m n)); + +fun times_def0 m n = nat ((m) * (n)); + +fun lcm x = (fn (m, n) => div_def0 (times_def0 m n) (gcd (m, n))) x; + +fun ilcm x = (fn j => (lcm (nat (abs x), nat (abs j)))); + +fun delta (And (p, q)) = ilcm (delta p) (delta q) + | delta (Or (p, q)) = ilcm (delta p) (delta q) + | delta (Dvd (i, CX (c, e))) = i + | delta (NDvd (i, CX (c, e))) = i + | delta T = 1 + | delta F = 1 + | delta (Lt u) = 1 + | delta (Le v) = 1 + | delta (Gt w) = 1 + | delta (Ge x) = 1 + | delta (Eq y) = 1 + | delta (NEq z) = 1 + | delta (Dvd (aa, C bo)) = 1 + | delta (Dvd (aa, Bound bp)) = 1 + | delta (Dvd (aa, Neg bs)) = 1 + | delta (Dvd (aa, Add (bt, bu))) = 1 + | delta (Dvd (aa, Sub (bv, bw))) = 1 + | delta (Dvd (aa, Mul (bx, by))) = 1 + | delta (NDvd (ac, C ck)) = 1 + | delta (NDvd (ac, Bound cl)) = 1 + | delta (NDvd (ac, Neg co)) = 1 + | delta (NDvd (ac, Add (cp, cq))) = 1 + | delta (NDvd (ac, Sub (cr, cs))) = 1 + | delta (NDvd (ac, Mul (ct, cu))) = 1 + | delta (NOT ae) = 1 + | delta (Imp (aj, ak)) = 1 + | delta (Iff (al, am)) = 1 + | delta (E an) = 1 + | delta (A ao) = 1 + | delta (Closed ap) = 1 + | delta (NClosed aq) = 1; + +fun a_beta (And (p, q)) = (fn k => And (a_beta p k, a_beta q k)) + | a_beta (Or (p, q)) = (fn k => Or (a_beta p k, a_beta q k)) + | a_beta (Eq (CX (c, e))) = (fn k => Eq (CX (1, Mul (div_def1 k c, e)))) + | a_beta (NEq (CX (c, e))) = (fn k => NEq (CX (1, Mul (div_def1 k c, e)))) + | a_beta (Lt (CX (c, e))) = (fn k => Lt (CX (1, Mul (div_def1 k c, e)))) + | a_beta (Le (CX (c, e))) = (fn k => Le (CX (1, Mul (div_def1 k c, e)))) + | a_beta (Gt (CX (c, e))) = (fn k => Gt (CX (1, Mul (div_def1 k c, e)))) + | a_beta (Ge (CX (c, e))) = (fn k => Ge (CX (1, Mul (div_def1 k c, e)))) + | a_beta (Dvd (i, CX (c, e))) = + (fn k => Dvd ((div_def1 k c * i), CX (1, Mul (div_def1 k c, e)))) + | a_beta (NDvd (i, CX (c, e))) = + (fn k => NDvd ((div_def1 k c * i), CX (1, Mul (div_def1 k c, e)))) + | a_beta T = (fn k => T) + | a_beta F = (fn k => F) + | a_beta (Lt (C bo)) = (fn k => Lt (C bo)) + | a_beta (Lt (Bound bp)) = (fn k => Lt (Bound bp)) + | a_beta (Lt (Neg bs)) = (fn k => Lt (Neg bs)) + | a_beta (Lt (Add (bt, bu))) = (fn k => Lt (Add (bt, bu))) + | a_beta (Lt (Sub (bv, bw))) = (fn k => Lt (Sub (bv, bw))) + | a_beta (Lt (Mul (bx, by))) = (fn k => Lt (Mul (bx, by))) + | a_beta (Le (C ck)) = (fn k => Le (C ck)) + | a_beta (Le (Bound cl)) = (fn k => Le (Bound cl)) + | a_beta (Le (Neg co)) = (fn k => Le (Neg co)) + | a_beta (Le (Add (cp, cq))) = (fn k => Le (Add (cp, cq))) + | a_beta (Le (Sub (cr, cs))) = (fn k => Le (Sub (cr, cs))) + | a_beta (Le (Mul (ct, cu))) = (fn k => Le (Mul (ct, cu))) + | a_beta (Gt (C dg)) = (fn k => Gt (C dg)) + | a_beta (Gt (Bound dh)) = (fn k => Gt (Bound dh)) + | a_beta (Gt (Neg dk)) = (fn k => Gt (Neg dk)) + | a_beta (Gt (Add (dl, dm))) = (fn k => Gt (Add (dl, dm))) + | a_beta (Gt (Sub (dn, do'))) = (fn k => Gt (Sub (dn, do'))) + | a_beta (Gt (Mul (dp, dq))) = (fn k => Gt (Mul (dp, dq))) + | a_beta (Ge (C ec)) = (fn k => Ge (C ec)) + | a_beta (Ge (Bound ed)) = (fn k => Ge (Bound ed)) + | a_beta (Ge (Neg eg)) = (fn k => Ge (Neg eg)) + | a_beta (Ge (Add (eh, ei))) = (fn k => Ge (Add (eh, ei))) + | a_beta (Ge (Sub (ej, ek))) = (fn k => Ge (Sub (ej, ek))) + | a_beta (Ge (Mul (el, em))) = (fn k => Ge (Mul (el, em))) + | a_beta (Eq (C ey)) = (fn k => Eq (C ey)) + | a_beta (Eq (Bound ez)) = (fn k => Eq (Bound ez)) + | a_beta (Eq (Neg fc)) = (fn k => Eq (Neg fc)) + | a_beta (Eq (Add (fd, fe))) = (fn k => Eq (Add (fd, fe))) + | a_beta (Eq (Sub (ff, fg))) = (fn k => Eq (Sub (ff, fg))) + | a_beta (Eq (Mul (fh, fi))) = (fn k => Eq (Mul (fh, fi))) + | a_beta (NEq (C fu)) = (fn k => NEq (C fu)) + | a_beta (NEq (Bound fv)) = (fn k => NEq (Bound fv)) + | a_beta (NEq (Neg fy)) = (fn k => NEq (Neg fy)) + | a_beta (NEq (Add (fz, ga))) = (fn k => NEq (Add (fz, ga))) + | a_beta (NEq (Sub (gb, gc))) = (fn k => NEq (Sub (gb, gc))) + | a_beta (NEq (Mul (gd, ge))) = (fn k => NEq (Mul (gd, ge))) + | a_beta (Dvd (aa, C gq)) = (fn k => Dvd (aa, C gq)) + | a_beta (Dvd (aa, Bound gr)) = (fn k => Dvd (aa, Bound gr)) + | a_beta (Dvd (aa, Neg gu)) = (fn k => Dvd (aa, Neg gu)) + | a_beta (Dvd (aa, Add (gv, gw))) = (fn k => Dvd (aa, Add (gv, gw))) + | a_beta (Dvd (aa, Sub (gx, gy))) = (fn k => Dvd (aa, Sub (gx, gy))) + | a_beta (Dvd (aa, Mul (gz, ha))) = (fn k => Dvd (aa, Mul (gz, ha))) + | a_beta (NDvd (ac, C hm)) = (fn k => NDvd (ac, C hm)) + | a_beta (NDvd (ac, Bound hn)) = (fn k => NDvd (ac, Bound hn)) + | a_beta (NDvd (ac, Neg hq)) = (fn k => NDvd (ac, Neg hq)) + | a_beta (NDvd (ac, Add (hr, hs))) = (fn k => NDvd (ac, Add (hr, hs))) + | a_beta (NDvd (ac, Sub (ht, hu))) = (fn k => NDvd (ac, Sub (ht, hu))) + | a_beta (NDvd (ac, Mul (hv, hw))) = (fn k => NDvd (ac, Mul (hv, hw))) + | a_beta (NOT ae) = (fn k => NOT ae) + | a_beta (Imp (aj, ak)) = (fn k => Imp (aj, ak)) + | a_beta (Iff (al, am)) = (fn k => Iff (al, am)) + | a_beta (E an) = (fn k => E an) + | a_beta (A ao) = (fn k => A ao) + | a_beta (Closed ap) = (fn k => Closed ap) + | a_beta (NClosed aq) = (fn k => NClosed aq); + +fun zeta (And (p, q)) = ilcm (zeta p) (zeta q) + | zeta (Or (p, q)) = ilcm (zeta p) (zeta q) + | zeta (Eq (CX (c, e))) = c + | zeta (NEq (CX (c, e))) = c + | zeta (Lt (CX (c, e))) = c + | zeta (Le (CX (c, e))) = c + | zeta (Gt (CX (c, e))) = c + | zeta (Ge (CX (c, e))) = c + | zeta (Dvd (i, CX (c, e))) = c + | zeta (NDvd (i, CX (c, e))) = c + | zeta T = 1 + | zeta F = 1 + | zeta (Lt (C bo)) = 1 + | zeta (Lt (Bound bp)) = 1 + | zeta (Lt (Neg bs)) = 1 + | zeta (Lt (Add (bt, bu))) = 1 + | zeta (Lt (Sub (bv, bw))) = 1 + | zeta (Lt (Mul (bx, by))) = 1 + | zeta (Le (C ck)) = 1 + | zeta (Le (Bound cl)) = 1 + | zeta (Le (Neg co)) = 1 + | zeta (Le (Add (cp, cq))) = 1 + | zeta (Le (Sub (cr, cs))) = 1 + | zeta (Le (Mul (ct, cu))) = 1 + | zeta (Gt (C dg)) = 1 + | zeta (Gt (Bound dh)) = 1 + | zeta (Gt (Neg dk)) = 1 + | zeta (Gt (Add (dl, dm))) = 1 + | zeta (Gt (Sub (dn, do'))) = 1 + | zeta (Gt (Mul (dp, dq))) = 1 + | zeta (Ge (C ec)) = 1 + | zeta (Ge (Bound ed)) = 1 + | zeta (Ge (Neg eg)) = 1 + | zeta (Ge (Add (eh, ei))) = 1 + | zeta (Ge (Sub (ej, ek))) = 1 + | zeta (Ge (Mul (el, em))) = 1 + | zeta (Eq (C ey)) = 1 + | zeta (Eq (Bound ez)) = 1 + | zeta (Eq (Neg fc)) = 1 + | zeta (Eq (Add (fd, fe))) = 1 + | zeta (Eq (Sub (ff, fg))) = 1 + | zeta (Eq (Mul (fh, fi))) = 1 + | zeta (NEq (C fu)) = 1 + | zeta (NEq (Bound fv)) = 1 + | zeta (NEq (Neg fy)) = 1 + | zeta (NEq (Add (fz, ga))) = 1 + | zeta (NEq (Sub (gb, gc))) = 1 + | zeta (NEq (Mul (gd, ge))) = 1 + | zeta (Dvd (aa, C gq)) = 1 + | zeta (Dvd (aa, Bound gr)) = 1 + | zeta (Dvd (aa, Neg gu)) = 1 + | zeta (Dvd (aa, Add (gv, gw))) = 1 + | zeta (Dvd (aa, Sub (gx, gy))) = 1 + | zeta (Dvd (aa, Mul (gz, ha))) = 1 + | zeta (NDvd (ac, C hm)) = 1 + | zeta (NDvd (ac, Bound hn)) = 1 + | zeta (NDvd (ac, Neg hq)) = 1 + | zeta (NDvd (ac, Add (hr, hs))) = 1 + | zeta (NDvd (ac, Sub (ht, hu))) = 1 + | zeta (NDvd (ac, Mul (hv, hw))) = 1 + | zeta (NOT ae) = 1 + | zeta (Imp (aj, ak)) = 1 + | zeta (Iff (al, am)) = 1 + | zeta (E an) = 1 + | zeta (A ao) = 1 + | zeta (Closed ap) = 1 + | zeta (NClosed aq) = 1; + +fun split x = (fn p => x (fst p) (snd p)); + +fun zsplit0 (C c) = (0, C c) + | zsplit0 (Bound n) = (if (n = 0) then (1, C 0) else (0, Bound n)) + | zsplit0 (CX (i, a)) = split (fn i' => (fn x => ((i + i'), x))) (zsplit0 a) + | zsplit0 (Neg a) = (fn (i', a') => (~ i', Neg a')) (zsplit0 a) + | zsplit0 (Add (a, b)) = + (fn (ia, a') => (fn (ib, b') => ((ia + ib), Add (a', b'))) (zsplit0 b)) + (zsplit0 a) + | zsplit0 (Sub (a, b)) = + (fn (ia, a') => + (fn (ib, b') => (minus_def2 ia ib, Sub (a', b'))) (zsplit0 b)) + (zsplit0 a) + | zsplit0 (Mul (i, a)) = (fn (i', a') => ((i * i'), Mul (i, a'))) (zsplit0 a); + +fun zlfm (And (p, q)) = And (zlfm p, zlfm q) + | zlfm (Or (p, q)) = Or (zlfm p, zlfm q) + | zlfm (Imp (p, q)) = Or (zlfm (NOT p), zlfm q) + | zlfm (Iff (p, q)) = + Or (And (zlfm p, zlfm q), And (zlfm (NOT p), zlfm (NOT q))) + | zlfm (Lt a) = + let val x = zsplit0 a + in (fn (c, r) => + (if (c = 0) then Lt r + else (if (0 < c) then Lt (CX (c, r)) else Gt (CX (~ c, Neg r))))) + x + end + | zlfm (Le a) = + let val x = zsplit0 a + in (fn (c, r) => + (if (c = 0) then Le r + else (if (0 < c) then Le (CX (c, r)) else Ge (CX (~ c, Neg r))))) + x + end + | zlfm (Gt a) = + let val x = zsplit0 a + in (fn (c, r) => + (if (c = 0) then Gt r + else (if (0 < c) then Gt (CX (c, r)) else Lt (CX (~ c, Neg r))))) + x + end + | zlfm (Ge a) = + let val x = zsplit0 a + in (fn (c, r) => + (if (c = 0) then Ge r + else (if (0 < c) then Ge (CX (c, r)) else Le (CX (~ c, Neg r))))) + x + end + | zlfm (Eq a) = + let val x = zsplit0 a + in (fn (c, r) => + (if (c = 0) then Eq r + else (if (0 < c) then Eq (CX (c, r)) else Eq (CX (~ c, Neg r))))) + x + end + | zlfm (NEq a) = + let val x = zsplit0 a + in (fn (c, r) => + (if (c = 0) then NEq r + else (if (0 < c) then NEq (CX (c, r)) else NEq (CX (~ c, Neg r))))) + x + end + | zlfm (Dvd (i, a)) = + (if (i = 0) then zlfm (Eq a) + else let val x = zsplit0 a + in (fn (c, r) => + (if (c = 0) then Dvd (abs i, r) + else (if (0 < c) then Dvd (abs i, CX (c, r)) + else Dvd (abs i, CX (~ c, Neg r))))) + x + end) + | zlfm (NDvd (i, a)) = + (if (i = 0) then zlfm (NEq a) + else let val x = zsplit0 a + in (fn (c, r) => + (if (c = 0) then NDvd (abs i, r) + else (if (0 < c) then NDvd (abs i, CX (c, r)) + else NDvd (abs i, CX (~ c, Neg r))))) + x + end) + | zlfm (NOT (And (p, q))) = Or (zlfm (NOT p), zlfm (NOT q)) + | zlfm (NOT (Or (p, q))) = And (zlfm (NOT p), zlfm (NOT q)) + | zlfm (NOT (Imp (p, q))) = And (zlfm p, zlfm (NOT q)) + | zlfm (NOT (Iff (p, q))) = + Or (And (zlfm p, zlfm (NOT q)), And (zlfm (NOT p), zlfm q)) + | zlfm (NOT (NOT p)) = zlfm p + | zlfm (NOT T) = F + | zlfm (NOT F) = T + | zlfm (NOT (Lt a)) = zlfm (Ge a) + | zlfm (NOT (Le a)) = zlfm (Gt a) + | zlfm (NOT (Gt a)) = zlfm (Le a) + | zlfm (NOT (Ge a)) = zlfm (Lt a) + | zlfm (NOT (Eq a)) = zlfm (NEq a) + | zlfm (NOT (NEq a)) = zlfm (Eq a) + | zlfm (NOT (Dvd (i, a))) = zlfm (NDvd (i, a)) + | zlfm (NOT (NDvd (i, a))) = zlfm (Dvd (i, a)) + | zlfm (NOT (Closed P)) = NClosed P + | zlfm (NOT (NClosed P)) = Closed P + | zlfm T = T + | zlfm F = F + | zlfm (NOT (E ci)) = NOT (E ci) + | zlfm (NOT (A cj)) = NOT (A cj) + | zlfm (E ao) = E ao + | zlfm (A ap) = A ap + | zlfm (Closed aq) = Closed aq + | zlfm (NClosed ar) = NClosed ar; + +fun unit p = + let val p' = zlfm p; val l = zeta p'; + val q = And (Dvd (l, CX (1, C 0)), a_beta p' l); val d = delta q; + val B = remdups (map simpnum (beta q)); + val a = remdups (map simpnum (alpha q)) + in (if less_eq_def3 (size_def9 B) (size_def9 a) then (q, (B, d)) + else (mirror q, (a, d))) + end; + +fun cooper p = + let val (q, (B, d)) = unit p; val js = iupt (1, d); + val mq = simpfm (minusinf q); + val md = evaldjf (fn j => simpfm (subst0 (C j) mq)) js + in (if (md = T) then T + else let val qd = + evaldjf (fn (b, j) => simpfm (subst0 (Add (b, C j)) q)) + (allpairs (fn x => fn xa => (x, xa)) B js) + in decr (disj md qd) end) + end; + +fun prep (E T) = T + | prep (E F) = F + | prep (E (Or (p, q))) = Or (prep (E p), prep (E q)) + | prep (E (Imp (p, q))) = Or (prep (E (NOT p)), prep (E q)) + | prep (E (Iff (p, q))) = + Or (prep (E (And (p, q))), prep (E (And (NOT p, NOT q)))) + | prep (E (NOT (And (p, q)))) = Or (prep (E (NOT p)), prep (E (NOT q))) + | prep (E (NOT (Imp (p, q)))) = prep (E (And (p, NOT q))) + | prep (E (NOT (Iff (p, q)))) = + Or (prep (E (And (p, NOT q))), prep (E (And (NOT p, q)))) + | prep (E (Lt ef)) = E (prep (Lt ef)) + | prep (E (Le eg)) = E (prep (Le eg)) + | prep (E (Gt eh)) = E (prep (Gt eh)) + | prep (E (Ge ei)) = E (prep (Ge ei)) + | prep (E (Eq ej)) = E (prep (Eq ej)) + | prep (E (NEq ek)) = E (prep (NEq ek)) + | prep (E (Dvd (el, em))) = E (prep (Dvd (el, em))) + | prep (E (NDvd (en, eo))) = E (prep (NDvd (en, eo))) + | prep (E (NOT T)) = E (prep (NOT T)) + | prep (E (NOT F)) = E (prep (NOT F)) + | prep (E (NOT (Lt gw))) = E (prep (NOT (Lt gw))) + | prep (E (NOT (Le gx))) = E (prep (NOT (Le gx))) + | prep (E (NOT (Gt gy))) = E (prep (NOT (Gt gy))) + | prep (E (NOT (Ge gz))) = E (prep (NOT (Ge gz))) + | prep (E (NOT (Eq ha))) = E (prep (NOT (Eq ha))) + | prep (E (NOT (NEq hb))) = E (prep (NOT (NEq hb))) + | prep (E (NOT (Dvd (hc, hd)))) = E (prep (NOT (Dvd (hc, hd)))) + | prep (E (NOT (NDvd (he, hf)))) = E (prep (NOT (NDvd (he, hf)))) + | prep (E (NOT (NOT hg))) = E (prep (NOT (NOT hg))) + | prep (E (NOT (Or (hj, hk)))) = E (prep (NOT (Or (hj, hk)))) + | prep (E (NOT (E hp))) = E (prep (NOT (E hp))) + | prep (E (NOT (A hq))) = E (prep (NOT (A hq))) + | prep (E (NOT (Closed hr))) = E (prep (NOT (Closed hr))) + | prep (E (NOT (NClosed hs))) = E (prep (NOT (NClosed hs))) + | prep (E (And (eq, er))) = E (prep (And (eq, er))) + | prep (E (E ey)) = E (prep (E ey)) + | prep (E (A ez)) = E (prep (A ez)) + | prep (E (Closed fa)) = E (prep (Closed fa)) + | prep (E (NClosed fb)) = E (prep (NClosed fb)) + | prep (A (And (p, q))) = And (prep (A p), prep (A q)) + | prep (A T) = prep (NOT (E (NOT T))) + | prep (A F) = prep (NOT (E (NOT F))) + | prep (A (Lt jn)) = prep (NOT (E (NOT (Lt jn)))) + | prep (A (Le jo)) = prep (NOT (E (NOT (Le jo)))) + | prep (A (Gt jp)) = prep (NOT (E (NOT (Gt jp)))) + | prep (A (Ge jq)) = prep (NOT (E (NOT (Ge jq)))) + | prep (A (Eq jr)) = prep (NOT (E (NOT (Eq jr)))) + | prep (A (NEq js)) = prep (NOT (E (NOT (NEq js)))) + | prep (A (Dvd (jt, ju))) = prep (NOT (E (NOT (Dvd (jt, ju))))) + | prep (A (NDvd (jv, jw))) = prep (NOT (E (NOT (NDvd (jv, jw))))) + | prep (A (NOT jx)) = prep (NOT (E (NOT (NOT jx)))) + | prep (A (Or (ka, kb))) = prep (NOT (E (NOT (Or (ka, kb))))) + | prep (A (Imp (kc, kd))) = prep (NOT (E (NOT (Imp (kc, kd))))) + | prep (A (Iff (ke, kf))) = prep (NOT (E (NOT (Iff (ke, kf))))) + | prep (A (E kg)) = prep (NOT (E (NOT (E kg)))) + | prep (A (A kh)) = prep (NOT (E (NOT (A kh)))) + | prep (A (Closed ki)) = prep (NOT (E (NOT (Closed ki)))) + | prep (A (NClosed kj)) = prep (NOT (E (NOT (NClosed kj)))) + | prep (NOT (NOT p)) = prep p + | prep (NOT (And (p, q))) = Or (prep (NOT p), prep (NOT q)) + | prep (NOT (A p)) = prep (E (NOT p)) + | prep (NOT (Or (p, q))) = And (prep (NOT p), prep (NOT q)) + | prep (NOT (Imp (p, q))) = And (prep p, prep (NOT q)) + | prep (NOT (Iff (p, q))) = Or (prep (And (p, NOT q)), prep (And (NOT p, q))) + | prep (NOT T) = NOT (prep T) + | prep (NOT F) = NOT (prep F) + | prep (NOT (Lt bo)) = NOT (prep (Lt bo)) + | prep (NOT (Le bp)) = NOT (prep (Le bp)) + | prep (NOT (Gt bq)) = NOT (prep (Gt bq)) + | prep (NOT (Ge br)) = NOT (prep (Ge br)) + | prep (NOT (Eq bs)) = NOT (prep (Eq bs)) + | prep (NOT (NEq bt)) = NOT (prep (NEq bt)) + | prep (NOT (Dvd (bu, bv))) = NOT (prep (Dvd (bu, bv))) + | prep (NOT (NDvd (bw, bx))) = NOT (prep (NDvd (bw, bx))) + | prep (NOT (E ch)) = NOT (prep (E ch)) + | prep (NOT (Closed cj)) = NOT (prep (Closed cj)) + | prep (NOT (NClosed ck)) = NOT (prep (NClosed ck)) + | prep (Or (p, q)) = Or (prep p, prep q) + | prep (And (p, q)) = And (prep p, prep q) + | prep (Imp (p, q)) = prep (Or (NOT p, q)) + | prep (Iff (p, q)) = Or (prep (And (p, q)), prep (And (NOT p, NOT q))) + | prep T = T + | prep F = F + | prep (Lt u) = Lt u + | prep (Le v) = Le v + | prep (Gt w) = Gt w + | prep (Ge x) = Ge x + | prep (Eq y) = Eq y + | prep (NEq z) = NEq z + | prep (Dvd (aa, ab)) = Dvd (aa, ab) + | prep (NDvd (ac, ad)) = NDvd (ac, ad) + | prep (Closed ap) = Closed ap + | prep (NClosed aq) = NClosed aq; + +fun pa x = qelim (prep x) cooper; + +val pa = (fn x => pa x); + +val test = + (fn x => + pa (E (A (Imp (Ge (Sub (Bound 0, Bound one_def0)), + E (E (Eq (Sub (Add (Mul (3, Bound one_def0), + Mul (5, Bound 0)), + Bound (nat 2)))))))))); + +end;