# HG changeset patch # User haftmann # Date 1233676240 -3600 # Node ID 23bf900a21db45c1e9f62dafd479bbdf3f9045bc # Parent 84a3f86441ebcd4293052bba909dc075e969fdb9 regenerated presburger code diff -r 84a3f86441eb -r 23bf900a21db src/HOL/Tools/Qelim/cooper.ML --- a/src/HOL/Tools/Qelim/cooper.ML Tue Feb 03 16:50:40 2009 +0100 +++ b/src/HOL/Tools/Qelim/cooper.ML Tue Feb 03 16:50:40 2009 +0100 @@ -558,11 +558,11 @@ | Const(@{const_name Ring_and_Field.dvd},_)$t1$t2 => (Dvd(HOLogic.dest_number t1 |> snd, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd") (* FIXME avoid handle _ *) | @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2)) - | @{term "op = :: bool => _ "}$t1$t2 => Iffa(qf_of_term ps vs t1,qf_of_term ps 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 => Impa(qf_of_term ps vs t1,qf_of_term ps vs t2) - | Const (@{const_name Not},_)$t' => Nota(qf_of_term ps vs t') + | Const("op -->",_)$t1$t2 => Imp(qf_of_term ps vs t1,qf_of_term ps vs t2) + | Const (@{const_name Not},_)$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), 0) :: (map (fn(v,n) => (v,1+ n)) vs) @@ -608,7 +608,7 @@ | Sub (t1, t2) => @{term "op - :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2 | Mul (i, t2) => @{term "op * :: int => _"} $ HOLogic.mk_number HOLogic.intT i $ term_of_i vs t2 - | Cx (i, t') => term_of_i vs (Add (Mul (i, Bound 0), t')); + | Cn (n, i, t') => term_of_i vs (Add (Mul (i, Bound n), t')); fun term_of_qf ps vs t = case t of @@ -619,18 +619,18 @@ | 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 (Nota (Eq t')) + | NEq t' => term_of_qf ps vs (Not (Eq t')) | Dvd(i,t') => @{term "op dvd :: int => _ "} $ HOLogic.mk_number HOLogic.intT i $ term_of_i vs t' - | NDvd(i,t')=> term_of_qf ps vs (Nota(Dvd(i,t'))) - | Nota t' => HOLogic.Not$(term_of_qf ps vs t') + | NDvd(i,t')=> term_of_qf ps vs (Not(Dvd(i,t'))) + | Not t' => HOLogic.Not$(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) - | Impa(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2) - | Iffa(t1,t2) => @{term "op = :: bool => _"} $ 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) => @{term "op = :: bool => _"} $ term_of_qf ps vs t1 $ term_of_qf ps vs t2 | Closed n => the (myassoc2 ps n) - | NClosed n => term_of_qf ps vs (Nota (Closed n)) - | _ => cooper "If this is raised, Isabelle/HOL or generate_code is inconsistent!"; + | NClosed n => term_of_qf ps vs (Not (Closed n)) + | _ => cooper "If this is raised, Isabelle/HOL or code generator is inconsistent!"; fun cooper_oracle ct = let diff -r 84a3f86441eb -r 23bf900a21db src/HOL/Tools/Qelim/generated_cooper.ML --- a/src/HOL/Tools/Qelim/generated_cooper.ML Tue Feb 03 16:50:40 2009 +0100 +++ b/src/HOL/Tools/Qelim/generated_cooper.ML Tue Feb 03 16:50:40 2009 +0100 @@ -1,7 +1,6 @@ (* Title: HOL/Tools/Qelim/generated_cooper.ML - ID: $Id$ -This file is generated from HOL/ex/Reflected_Presburger.thy. DO NOT EDIT. +This file is generated from HOL/Reflection/Cooper.thy. DO NOT EDIT. *) structure GeneratedCooper = @@ -10,7 +9,687 @@ type 'a eq = {eq : 'a -> 'a -> bool}; fun eq (A_:'a eq) = #eq A_; -datatype bit = B0 | B1; +val eq_nat = {eq = (fn a => fn b => ((a : IntInf.int) = b))} : IntInf.int eq; + +fun eqop A_ a b = eq A_ a b; + +fun divmod n m = (if eqop eq_nat m 0 then (0, n) else IntInf.divMod (n, m)); + +fun snd (a, y) = y; + +fun mod_nat m n = snd (divmod m n); + +fun gcd m n = (if eqop eq_nat n 0 then m else gcd n (mod_nat m n)); + +fun fst (y, b) = y; + +fun div_nat m n = fst (divmod m n); + +fun lcm m n = div_nat (IntInf.* (m, n)) (gcd m n); + +fun leta s f = f s; + +fun suc n = IntInf.+ (n, 1); + +datatype num = Mul of IntInf.int * num | Sub of num * num | Add of num * num | + Neg of num | Cn of IntInf.int * IntInf.int * num | Bound of IntInf.int | + C of IntInf.int; + +datatype fm = NClosed of IntInf.int | Closed of IntInf.int | A of fm | E of fm | + Iff of fm * fm | Imp of fm * fm | Or of fm * fm | And of fm * fm | Not of fm | + NDvd of IntInf.int * num | Dvd of IntInf.int * num | NEq of num | Eq of num | + Ge of num | Gt of num | Le of num | Lt of num | F | T; + +fun abs_int i = (if IntInf.< (i, (0 : IntInf.int)) then IntInf.~ i else i); + +fun zlcm i j = + (lcm (IntInf.max (0, (abs_int i))) (IntInf.max (0, (abs_int j)))); + +fun map f [] = [] + | map f (x :: xs) = f x :: map f xs; + +fun append [] y = y + | 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 fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (NClosed nat) = f19 nat + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Closed nat) = f18 nat + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (A fm) = f17 fm + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (E fm) = f16 fm + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Iff (fm1, fm2)) = f15 fm1 fm2 + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Imp (fm1, fm2)) = f14 fm1 fm2 + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Or (fm1, fm2)) = f13 fm1 fm2 + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (And (fm1, fm2)) = f12 fm1 fm2 + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Not fm) = f11 fm + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (NDvd (inta, num)) = f10 inta num + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Dvd (inta, num)) = f9 inta num + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (NEq num) = f8 num + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Eq num) = f7 num + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Ge num) = f6 num + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Gt num) = f5 num + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Le num) = f4 num + | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 + (Lt num) = f3 num + | fm_case f1 y f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 F + = y + | fm_case y f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 T + = y; + +fun eq_num (Mul (cb, dc)) (Sub (ae, be)) = false + | eq_num (Mul (cb, dc)) (Add (ae, be)) = false + | eq_num (Sub (cc, dc)) (Add (ae, be)) = false + | eq_num (Mul (bd, cc)) (Neg ae) = false + | eq_num (Sub (be, cc)) (Neg ae) = false + | eq_num (Add (be, cc)) (Neg ae) = false + | eq_num (Mul (db, ea)) (Cn (ac, bd, cc)) = false + | eq_num (Sub (dc, ea)) (Cn (ac, bd, cc)) = false + | eq_num (Add (dc, ea)) (Cn (ac, bd, cc)) = false + | eq_num (Neg dc) (Cn (ac, bd, cc)) = false + | eq_num (Mul (bd, cc)) (Bound ac) = false + | eq_num (Sub (be, cc)) (Bound ac) = false + | eq_num (Add (be, cc)) (Bound ac) = false + | eq_num (Neg be) (Bound ac) = false + | eq_num (Cn (bc, cb, dc)) (Bound ac) = false + | eq_num (Mul (bd, cc)) (C ad) = false + | eq_num (Sub (be, cc)) (C ad) = false + | eq_num (Add (be, cc)) (C ad) = false + | eq_num (Neg be) (C ad) = false + | eq_num (Cn (bc, cb, dc)) (C ad) = false + | eq_num (Bound bc) (C ad) = false + | eq_num (Sub (ab, bb)) (Mul (c, da)) = false + | eq_num (Add (ab, bb)) (Mul (c, da)) = false + | eq_num (Add (ab, bb)) (Sub (ca, da)) = false + | eq_num (Neg ab) (Mul (ba, ca)) = false + | eq_num (Neg ab) (Sub (bb, ca)) = false + | eq_num (Neg ab) (Add (bb, ca)) = false + | eq_num (Cn (a, ba, ca)) (Mul (d, e)) = false + | eq_num (Cn (a, ba, ca)) (Sub (da, e)) = false + | eq_num (Cn (a, ba, ca)) (Add (da, e)) = false + | eq_num (Cn (a, ba, ca)) (Neg da) = false + | eq_num (Bound a) (Mul (ba, ca)) = false + | eq_num (Bound a) (Sub (bb, ca)) = false + | eq_num (Bound a) (Add (bb, ca)) = false + | eq_num (Bound a) (Neg bb) = false + | eq_num (Bound a) (Cn (b, c, da)) = false + | eq_num (C aa) (Mul (ba, ca)) = false + | eq_num (C aa) (Sub (bb, ca)) = false + | eq_num (C aa) (Add (bb, ca)) = false + | eq_num (C aa) (Neg bb) = false + | eq_num (C aa) (Cn (b, c, da)) = false + | eq_num (C aa) (Bound b) = false + | eq_num (Mul (inta, num)) (Mul (int', num')) = + ((inta : IntInf.int) = int') andalso eq_num num num' + | eq_num (Sub (num1, num2)) (Sub (num1', num2')) = + eq_num num1 num1' andalso eq_num num2 num2' + | eq_num (Add (num1, num2)) (Add (num1', num2')) = + eq_num num1 num1' andalso eq_num num2 num2' + | eq_num (Neg num) (Neg num') = eq_num num num' + | eq_num (Cn (nat, inta, num)) (Cn (nat', int', num')) = + ((nat : IntInf.int) = nat') andalso + (((inta : IntInf.int) = int') andalso eq_num num num') + | eq_num (Bound nat) (Bound nat') = ((nat : IntInf.int) = nat') + | eq_num (C inta) (C int') = ((inta : IntInf.int) = int'); + +fun eq_fm (NClosed bd) (Closed ad) = false + | eq_fm (NClosed bd) (A af) = false + | eq_fm (Closed bd) (A af) = false + | eq_fm (NClosed bd) (E af) = false + | eq_fm (Closed bd) (E af) = false + | eq_fm (A bf) (E af) = false + | eq_fm (NClosed cd) (Iff (af, bf)) = false + | eq_fm (Closed cd) (Iff (af, bf)) = false + | eq_fm (A cf) (Iff (af, bf)) = false + | eq_fm (E cf) (Iff (af, bf)) = false + | eq_fm (NClosed cd) (Imp (af, bf)) = false + | eq_fm (Closed cd) (Imp (af, bf)) = false + | eq_fm (A cf) (Imp (af, bf)) = false + | eq_fm (E cf) (Imp (af, bf)) = false + | eq_fm (Iff (cf, db)) (Imp (af, bf)) = false + | eq_fm (NClosed cd) (Or (af, bf)) = false + | eq_fm (Closed cd) (Or (af, bf)) = false + | eq_fm (A cf) (Or (af, bf)) = false + | eq_fm (E cf) (Or (af, bf)) = false + | eq_fm (Iff (cf, db)) (Or (af, bf)) = false + | eq_fm (Imp (cf, db)) (Or (af, bf)) = false + | eq_fm (NClosed cd) (And (af, bf)) = false + | eq_fm (Closed cd) (And (af, bf)) = false + | eq_fm (A cf) (And (af, bf)) = false + | eq_fm (E cf) (And (af, bf)) = false + | eq_fm (Iff (cf, db)) (And (af, bf)) = false + | eq_fm (Imp (cf, db)) (And (af, bf)) = false + | eq_fm (Or (cf, db)) (And (af, bf)) = false + | eq_fm (NClosed bd) (Not af) = false + | eq_fm (Closed bd) (Not af) = false + | eq_fm (A bf) (Not af) = false + | eq_fm (E bf) (Not af) = false + | eq_fm (Iff (bf, cf)) (Not af) = false + | eq_fm (Imp (bf, cf)) (Not af) = false + | eq_fm (Or (bf, cf)) (Not af) = false + | eq_fm (And (bf, cf)) (Not af) = false + | eq_fm (NClosed cd) (NDvd (ae, bg)) = false + | eq_fm (Closed cd) (NDvd (ae, bg)) = false + | eq_fm (A cf) (NDvd (ae, bg)) = false + | eq_fm (E cf) (NDvd (ae, bg)) = false + | eq_fm (Iff (cf, db)) (NDvd (ae, bg)) = false + | eq_fm (Imp (cf, db)) (NDvd (ae, bg)) = false + | eq_fm (Or (cf, db)) (NDvd (ae, bg)) = false + | eq_fm (And (cf, db)) (NDvd (ae, bg)) = false + | eq_fm (Not cf) (NDvd (ae, bg)) = false + | eq_fm (NClosed cd) (Dvd (ae, bg)) = false + | eq_fm (Closed cd) (Dvd (ae, bg)) = false + | eq_fm (A cf) (Dvd (ae, bg)) = false + | eq_fm (E cf) (Dvd (ae, bg)) = false + | eq_fm (Iff (cf, db)) (Dvd (ae, bg)) = false + | eq_fm (Imp (cf, db)) (Dvd (ae, bg)) = false + | eq_fm (Or (cf, db)) (Dvd (ae, bg)) = false + | eq_fm (And (cf, db)) (Dvd (ae, bg)) = false + | eq_fm (Not cf) (Dvd (ae, bg)) = false + | eq_fm (NDvd (ce, dc)) (Dvd (ae, bg)) = false + | eq_fm (NClosed bd) (NEq ag) = false + | eq_fm (Closed bd) (NEq ag) = false + | eq_fm (A bf) (NEq ag) = false + | eq_fm (E bf) (NEq ag) = false + | eq_fm (Iff (bf, cf)) (NEq ag) = false + | eq_fm (Imp (bf, cf)) (NEq ag) = false + | eq_fm (Or (bf, cf)) (NEq ag) = false + | eq_fm (And (bf, cf)) (NEq ag) = false + | eq_fm (Not bf) (NEq ag) = false + | eq_fm (NDvd (be, cg)) (NEq ag) = false + | eq_fm (Dvd (be, cg)) (NEq ag) = false + | eq_fm (NClosed bd) (Eq ag) = false + | eq_fm (Closed bd) (Eq ag) = false + | eq_fm (A bf) (Eq ag) = false + | eq_fm (E bf) (Eq ag) = false + | eq_fm (Iff (bf, cf)) (Eq ag) = false + | eq_fm (Imp (bf, cf)) (Eq ag) = false + | eq_fm (Or (bf, cf)) (Eq ag) = false + | eq_fm (And (bf, cf)) (Eq ag) = false + | eq_fm (Not bf) (Eq ag) = false + | eq_fm (NDvd (be, cg)) (Eq ag) = false + | eq_fm (Dvd (be, cg)) (Eq ag) = false + | eq_fm (NEq bg) (Eq ag) = false + | eq_fm (NClosed bd) (Ge ag) = false + | eq_fm (Closed bd) (Ge ag) = false + | eq_fm (A bf) (Ge ag) = false + | eq_fm (E bf) (Ge ag) = false + | eq_fm (Iff (bf, cf)) (Ge ag) = false + | eq_fm (Imp (bf, cf)) (Ge ag) = false + | eq_fm (Or (bf, cf)) (Ge ag) = false + | eq_fm (And (bf, cf)) (Ge ag) = false + | eq_fm (Not bf) (Ge ag) = false + | eq_fm (NDvd (be, cg)) (Ge ag) = false + | eq_fm (Dvd (be, cg)) (Ge ag) = false + | eq_fm (NEq bg) (Ge ag) = false + | eq_fm (Eq bg) (Ge ag) = false + | eq_fm (NClosed bd) (Gt ag) = false + | eq_fm (Closed bd) (Gt ag) = false + | eq_fm (A bf) (Gt ag) = false + | eq_fm (E bf) (Gt ag) = false + | eq_fm (Iff (bf, cf)) (Gt ag) = false + | eq_fm (Imp (bf, cf)) (Gt ag) = false + | eq_fm (Or (bf, cf)) (Gt ag) = false + | eq_fm (And (bf, cf)) (Gt ag) = false + | eq_fm (Not bf) (Gt ag) = false + | eq_fm (NDvd (be, cg)) (Gt ag) = false + | eq_fm (Dvd (be, cg)) (Gt ag) = false + | eq_fm (NEq bg) (Gt ag) = false + | eq_fm (Eq bg) (Gt ag) = false + | eq_fm (Ge bg) (Gt ag) = false + | eq_fm (NClosed bd) (Le ag) = false + | eq_fm (Closed bd) (Le ag) = false + | eq_fm (A bf) (Le ag) = false + | eq_fm (E bf) (Le ag) = false + | eq_fm (Iff (bf, cf)) (Le ag) = false + | eq_fm (Imp (bf, cf)) (Le ag) = false + | eq_fm (Or (bf, cf)) (Le ag) = false + | eq_fm (And (bf, cf)) (Le ag) = false + | eq_fm (Not bf) (Le ag) = false + | eq_fm (NDvd (be, cg)) (Le ag) = false + | eq_fm (Dvd (be, cg)) (Le ag) = false + | eq_fm (NEq bg) (Le ag) = false + | eq_fm (Eq bg) (Le ag) = false + | eq_fm (Ge bg) (Le ag) = false + | eq_fm (Gt bg) (Le ag) = false + | eq_fm (NClosed bd) (Lt ag) = false + | eq_fm (Closed bd) (Lt ag) = false + | eq_fm (A bf) (Lt ag) = false + | eq_fm (E bf) (Lt ag) = false + | eq_fm (Iff (bf, cf)) (Lt ag) = false + | eq_fm (Imp (bf, cf)) (Lt ag) = false + | eq_fm (Or (bf, cf)) (Lt ag) = false + | eq_fm (And (bf, cf)) (Lt ag) = false + | eq_fm (Not bf) (Lt ag) = false + | eq_fm (NDvd (be, cg)) (Lt ag) = false + | eq_fm (Dvd (be, cg)) (Lt ag) = false + | eq_fm (NEq bg) (Lt ag) = false + | eq_fm (Eq bg) (Lt ag) = false + | eq_fm (Ge bg) (Lt ag) = false + | eq_fm (Gt bg) (Lt ag) = false + | eq_fm (Le bg) (Lt ag) = false + | eq_fm (NClosed ad) F = false + | eq_fm (Closed ad) F = false + | eq_fm (A af) F = false + | eq_fm (E af) F = false + | eq_fm (Iff (af, bf)) F = false + | eq_fm (Imp (af, bf)) F = false + | eq_fm (Or (af, bf)) F = false + | eq_fm (And (af, bf)) F = false + | eq_fm (Not af) F = false + | eq_fm (NDvd (ae, bg)) F = false + | eq_fm (Dvd (ae, bg)) F = false + | eq_fm (NEq ag) F = false + | eq_fm (Eq ag) F = false + | eq_fm (Ge ag) F = false + | eq_fm (Gt ag) F = false + | eq_fm (Le ag) F = false + | eq_fm (Lt ag) F = false + | eq_fm (NClosed ad) T = false + | eq_fm (Closed ad) T = false + | eq_fm (A af) T = false + | eq_fm (E af) T = false + | eq_fm (Iff (af, bf)) T = false + | eq_fm (Imp (af, bf)) T = false + | eq_fm (Or (af, bf)) T = false + | eq_fm (And (af, bf)) T = false + | eq_fm (Not af) T = false + | eq_fm (NDvd (ae, bg)) T = false + | eq_fm (Dvd (ae, bg)) T = false + | eq_fm (NEq ag) T = false + | eq_fm (Eq ag) T = false + | eq_fm (Ge ag) T = false + | eq_fm (Gt ag) T = false + | eq_fm (Le ag) T = false + | eq_fm (Lt ag) T = false + | eq_fm F T = false + | eq_fm (Closed a) (NClosed b) = false + | eq_fm (A ab) (NClosed b) = false + | eq_fm (A ab) (Closed b) = false + | eq_fm (E ab) (NClosed b) = false + | eq_fm (E ab) (Closed b) = false + | eq_fm (E ab) (A bb) = false + | eq_fm (Iff (ab, bb)) (NClosed c) = false + | eq_fm (Iff (ab, bb)) (Closed c) = false + | eq_fm (Iff (ab, bb)) (A cb) = false + | eq_fm (Iff (ab, bb)) (E cb) = false + | eq_fm (Imp (ab, bb)) (NClosed c) = false + | eq_fm (Imp (ab, bb)) (Closed c) = false + | eq_fm (Imp (ab, bb)) (A cb) = false + | eq_fm (Imp (ab, bb)) (E cb) = false + | eq_fm (Imp (ab, bb)) (Iff (cb, d)) = false + | eq_fm (Or (ab, bb)) (NClosed c) = false + | eq_fm (Or (ab, bb)) (Closed c) = false + | eq_fm (Or (ab, bb)) (A cb) = false + | eq_fm (Or (ab, bb)) (E cb) = false + | eq_fm (Or (ab, bb)) (Iff (cb, d)) = false + | eq_fm (Or (ab, bb)) (Imp (cb, d)) = false + | eq_fm (And (ab, bb)) (NClosed c) = false + | eq_fm (And (ab, bb)) (Closed c) = false + | eq_fm (And (ab, bb)) (A cb) = false + | eq_fm (And (ab, bb)) (E cb) = false + | eq_fm (And (ab, bb)) (Iff (cb, d)) = false + | eq_fm (And (ab, bb)) (Imp (cb, d)) = false + | eq_fm (And (ab, bb)) (Or (cb, d)) = false + | eq_fm (Not ab) (NClosed b) = false + | eq_fm (Not ab) (Closed b) = false + | eq_fm (Not ab) (A bb) = false + | eq_fm (Not ab) (E bb) = false + | eq_fm (Not ab) (Iff (bb, cb)) = false + | eq_fm (Not ab) (Imp (bb, cb)) = false + | eq_fm (Not ab) (Or (bb, cb)) = false + | eq_fm (Not ab) (And (bb, cb)) = false + | eq_fm (NDvd (aa, bc)) (NClosed c) = false + | eq_fm (NDvd (aa, bc)) (Closed c) = false + | eq_fm (NDvd (aa, bc)) (A cb) = false + | eq_fm (NDvd (aa, bc)) (E cb) = false + | eq_fm (NDvd (aa, bc)) (Iff (cb, d)) = false + | eq_fm (NDvd (aa, bc)) (Imp (cb, d)) = false + | eq_fm (NDvd (aa, bc)) (Or (cb, d)) = false + | eq_fm (NDvd (aa, bc)) (And (cb, d)) = false + | eq_fm (NDvd (aa, bc)) (Not cb) = false + | eq_fm (Dvd (aa, bc)) (NClosed c) = false + | eq_fm (Dvd (aa, bc)) (Closed c) = false + | eq_fm (Dvd (aa, bc)) (A cb) = false + | eq_fm (Dvd (aa, bc)) (E cb) = false + | eq_fm (Dvd (aa, bc)) (Iff (cb, d)) = false + | eq_fm (Dvd (aa, bc)) (Imp (cb, d)) = false + | eq_fm (Dvd (aa, bc)) (Or (cb, d)) = false + | eq_fm (Dvd (aa, bc)) (And (cb, d)) = false + | eq_fm (Dvd (aa, bc)) (Not cb) = false + | eq_fm (Dvd (aa, bc)) (NDvd (ca, da)) = false + | eq_fm (NEq ac) (NClosed b) = false + | eq_fm (NEq ac) (Closed b) = false + | eq_fm (NEq ac) (A bb) = false + | eq_fm (NEq ac) (E bb) = false + | eq_fm (NEq ac) (Iff (bb, cb)) = false + | eq_fm (NEq ac) (Imp (bb, cb)) = false + | eq_fm (NEq ac) (Or (bb, cb)) = false + | eq_fm (NEq ac) (And (bb, cb)) = false + | eq_fm (NEq ac) (Not bb) = false + | eq_fm (NEq ac) (NDvd (ba, cc)) = false + | eq_fm (NEq ac) (Dvd (ba, cc)) = false + | eq_fm (Eq ac) (NClosed b) = false + | eq_fm (Eq ac) (Closed b) = false + | eq_fm (Eq ac) (A bb) = false + | eq_fm (Eq ac) (E bb) = false + | eq_fm (Eq ac) (Iff (bb, cb)) = false + | eq_fm (Eq ac) (Imp (bb, cb)) = false + | eq_fm (Eq ac) (Or (bb, cb)) = false + | eq_fm (Eq ac) (And (bb, cb)) = false + | eq_fm (Eq ac) (Not bb) = false + | eq_fm (Eq ac) (NDvd (ba, cc)) = false + | eq_fm (Eq ac) (Dvd (ba, cc)) = false + | eq_fm (Eq ac) (NEq bc) = false + | eq_fm (Ge ac) (NClosed b) = false + | eq_fm (Ge ac) (Closed b) = false + | eq_fm (Ge ac) (A bb) = false + | eq_fm (Ge ac) (E bb) = false + | eq_fm (Ge ac) (Iff (bb, cb)) = false + | eq_fm (Ge ac) (Imp (bb, cb)) = false + | eq_fm (Ge ac) (Or (bb, cb)) = false + | eq_fm (Ge ac) (And (bb, cb)) = false + | eq_fm (Ge ac) (Not bb) = false + | eq_fm (Ge ac) (NDvd (ba, cc)) = false + | eq_fm (Ge ac) (Dvd (ba, cc)) = false + | eq_fm (Ge ac) (NEq bc) = false + | eq_fm (Ge ac) (Eq bc) = false + | eq_fm (Gt ac) (NClosed b) = false + | eq_fm (Gt ac) (Closed b) = false + | eq_fm (Gt ac) (A bb) = false + | eq_fm (Gt ac) (E bb) = false + | eq_fm (Gt ac) (Iff (bb, cb)) = false + | eq_fm (Gt ac) (Imp (bb, cb)) = false + | eq_fm (Gt ac) (Or (bb, cb)) = false + | eq_fm (Gt ac) (And (bb, cb)) = false + | eq_fm (Gt ac) (Not bb) = false + | eq_fm (Gt ac) (NDvd (ba, cc)) = false + | eq_fm (Gt ac) (Dvd (ba, cc)) = false + | eq_fm (Gt ac) (NEq bc) = false + | eq_fm (Gt ac) (Eq bc) = false + | eq_fm (Gt ac) (Ge bc) = false + | eq_fm (Le ac) (NClosed b) = false + | eq_fm (Le ac) (Closed b) = false + | eq_fm (Le ac) (A bb) = false + | eq_fm (Le ac) (E bb) = false + | eq_fm (Le ac) (Iff (bb, cb)) = false + | eq_fm (Le ac) (Imp (bb, cb)) = false + | eq_fm (Le ac) (Or (bb, cb)) = false + | eq_fm (Le ac) (And (bb, cb)) = false + | eq_fm (Le ac) (Not bb) = false + | eq_fm (Le ac) (NDvd (ba, cc)) = false + | eq_fm (Le ac) (Dvd (ba, cc)) = false + | eq_fm (Le ac) (NEq bc) = false + | eq_fm (Le ac) (Eq bc) = false + | eq_fm (Le ac) (Ge bc) = false + | eq_fm (Le ac) (Gt bc) = false + | eq_fm (Lt ac) (NClosed b) = false + | eq_fm (Lt ac) (Closed b) = false + | eq_fm (Lt ac) (A bb) = false + | eq_fm (Lt ac) (E bb) = false + | eq_fm (Lt ac) (Iff (bb, cb)) = false + | eq_fm (Lt ac) (Imp (bb, cb)) = false + | eq_fm (Lt ac) (Or (bb, cb)) = false + | eq_fm (Lt ac) (And (bb, cb)) = false + | eq_fm (Lt ac) (Not bb) = false + | eq_fm (Lt ac) (NDvd (ba, cc)) = false + | eq_fm (Lt ac) (Dvd (ba, cc)) = false + | eq_fm (Lt ac) (NEq bc) = false + | eq_fm (Lt ac) (Eq bc) = false + | eq_fm (Lt ac) (Ge bc) = false + | eq_fm (Lt ac) (Gt bc) = false + | eq_fm (Lt ac) (Le bc) = false + | eq_fm F (NClosed a) = false + | eq_fm F (Closed a) = false + | eq_fm F (A ab) = false + | eq_fm F (E ab) = false + | eq_fm F (Iff (ab, bb)) = false + | eq_fm F (Imp (ab, bb)) = false + | eq_fm F (Or (ab, bb)) = false + | eq_fm F (And (ab, bb)) = false + | eq_fm F (Not ab) = false + | eq_fm F (NDvd (aa, bc)) = false + | eq_fm F (Dvd (aa, bc)) = false + | eq_fm F (NEq ac) = false + | eq_fm F (Eq ac) = false + | eq_fm F (Ge ac) = false + | eq_fm F (Gt ac) = false + | eq_fm F (Le ac) = false + | eq_fm F (Lt ac) = false + | eq_fm T (NClosed a) = false + | eq_fm T (Closed a) = false + | eq_fm T (A ab) = false + | eq_fm T (E ab) = false + | eq_fm T (Iff (ab, bb)) = false + | eq_fm T (Imp (ab, bb)) = false + | eq_fm T (Or (ab, bb)) = false + | eq_fm T (And (ab, bb)) = false + | eq_fm T (Not ab) = false + | eq_fm T (NDvd (aa, bc)) = false + | eq_fm T (Dvd (aa, bc)) = false + | eq_fm T (NEq ac) = false + | eq_fm T (Eq ac) = false + | eq_fm T (Ge ac) = false + | eq_fm T (Gt ac) = false + | eq_fm T (Le ac) = false + | eq_fm T (Lt ac) = false + | eq_fm T F = false + | eq_fm (NClosed nat) (NClosed nat') = ((nat : IntInf.int) = nat') + | eq_fm (Closed nat) (Closed nat') = ((nat : IntInf.int) = nat') + | eq_fm (A fm) (A fm') = eq_fm fm fm' + | eq_fm (E fm) (E fm') = eq_fm fm fm' + | eq_fm (Iff (fm1, fm2)) (Iff (fm1', fm2')) = + eq_fm fm1 fm1' andalso eq_fm fm2 fm2' + | eq_fm (Imp (fm1, fm2)) (Imp (fm1', fm2')) = + eq_fm fm1 fm1' andalso eq_fm fm2 fm2' + | eq_fm (Or (fm1, fm2)) (Or (fm1', fm2')) = + eq_fm fm1 fm1' andalso eq_fm fm2 fm2' + | eq_fm (And (fm1, fm2)) (And (fm1', fm2')) = + eq_fm fm1 fm1' andalso eq_fm fm2 fm2' + | eq_fm (Not fm) (Not fm') = eq_fm fm fm' + | eq_fm (NDvd (inta, num)) (NDvd (int', num')) = + ((inta : IntInf.int) = int') andalso eq_num num num' + | eq_fm (Dvd (inta, num)) (Dvd (int', num')) = + ((inta : IntInf.int) = int') andalso eq_num num num' + | eq_fm (NEq num) (NEq num') = eq_num num num' + | eq_fm (Eq num) (Eq num') = eq_num num num' + | eq_fm (Ge num) (Ge num') = eq_num num num' + | eq_fm (Gt num) (Gt num') = eq_num num num' + | eq_fm (Le num) (Le num') = eq_num num num' + | eq_fm (Lt num) (Lt num') = eq_num num num' + | eq_fm F F = true + | eq_fm T T = true; + +val eq_fma = {eq = eq_fm} : fm eq; + +fun djf f p q = + (if eqop eq_fma q T then T + else (if eqop eq_fma q F then f p + else let + val a = f p; + in + (case a of T => T | F => q | Lt num => Or (f p, q) + | Le num => Or (f p, q) | Gt num => Or (f p, q) + | Ge num => Or (f p, q) | Eq num => Or (f p, q) + | NEq num => Or (f p, q) | Dvd (inta, num) => Or (f p, q) + | NDvd (inta, num) => Or (f p, q) | Not fm => Or (f p, q) + | And (fm1, fm2) => Or (f p, q) + | Or (fm1, fm2) => Or (f p, q) + | Imp (fm1, fm2) => Or (f p, q) + | Iff (fm1, fm2) => Or (f p, q) | E fm => Or (f p, q) + | A fm => Or (f p, q) | Closed nat => Or (f p, q) + | NClosed nat => Or (f p, q)) + end)); + +fun foldr f [] y = y + | foldr f (x :: xs) a = f x (foldr f xs a); + +fun evaldjf f ps = foldr (djf f) ps F; + +fun dj f p = evaldjf f (disjuncts p); + +fun disj p q = + (if eqop eq_fma p T orelse eqop eq_fma q T then T + else (if eqop eq_fma p F then q + else (if eqop eq_fma q F then p else Or (p, q)))); + +fun minus_nat n m = IntInf.max (0, (IntInf.- (n, m))); + +fun decrnum (Bound n) = Bound (minus_nat n 1) + | 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 (Cn (n, i, a)) = Cn (minus_nat n 1, i, decrnum a) + | decrnum (C u) = C u; + +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 concat [] = [] + | concat (x :: xs) = append x (concat xs); + +fun split f (a, b) = f a b; + +fun numsubst0 t (C c) = C c + | numsubst0 t (Bound n) = (if eqop eq_nat n 0 then t else Bound n) + | 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) + | numsubst0 ta (Cn (v, ia, aa)) = + (if eqop eq_nat v 0 then Add (Mul (ia, ta), numsubst0 ta aa) + else Cn (suc (minus_nat v 1), ia, numsubst0 ta aa)); + +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 T = T + | minusinf F = F + | minusinf (Lt (C bo)) = Lt (C bo) + | minusinf (Lt (Bound bp)) = Lt (Bound bp) + | minusinf (Lt (Neg bt)) = Lt (Neg bt) + | minusinf (Lt (Add (bu, bv))) = Lt (Add (bu, bv)) + | minusinf (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx)) + | minusinf (Lt (Mul (by, bz))) = Lt (Mul (by, bz)) + | minusinf (Le (C co)) = Le (C co) + | minusinf (Le (Bound cp)) = Le (Bound cp) + | minusinf (Le (Neg ct)) = Le (Neg ct) + | minusinf (Le (Add (cu, cv))) = Le (Add (cu, cv)) + | minusinf (Le (Sub (cw, cx))) = Le (Sub (cw, cx)) + | minusinf (Le (Mul (cy, cz))) = Le (Mul (cy, cz)) + | minusinf (Gt (C doa)) = Gt (C doa) + | minusinf (Gt (Bound dp)) = Gt (Bound dp) + | minusinf (Gt (Neg dt)) = Gt (Neg dt) + | minusinf (Gt (Add (du, dv))) = Gt (Add (du, dv)) + | minusinf (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx)) + | minusinf (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz)) + | minusinf (Ge (C eo)) = Ge (C eo) + | minusinf (Ge (Bound ep)) = Ge (Bound ep) + | minusinf (Ge (Neg et)) = Ge (Neg et) + | minusinf (Ge (Add (eu, ev))) = Ge (Add (eu, ev)) + | minusinf (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex)) + | minusinf (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez)) + | minusinf (Eq (C fo)) = Eq (C fo) + | minusinf (Eq (Bound fp)) = Eq (Bound fp) + | minusinf (Eq (Neg ft)) = Eq (Neg ft) + | minusinf (Eq (Add (fu, fv))) = Eq (Add (fu, fv)) + | minusinf (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx)) + | minusinf (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz)) + | minusinf (NEq (C go)) = NEq (C go) + | minusinf (NEq (Bound gp)) = NEq (Bound gp) + | minusinf (NEq (Neg gt)) = NEq (Neg gt) + | minusinf (NEq (Add (gu, gv))) = NEq (Add (gu, gv)) + | minusinf (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx)) + | minusinf (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz)) + | 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 + | minusinf (Lt (Cn (cm, c, e))) = + (if eqop eq_nat cm 0 then T else Lt (Cn (suc (minus_nat cm 1), c, e))) + | minusinf (Le (Cn (dm, c, e))) = + (if eqop eq_nat dm 0 then T else Le (Cn (suc (minus_nat dm 1), c, e))) + | minusinf (Gt (Cn (em, c, e))) = + (if eqop eq_nat em 0 then F else Gt (Cn (suc (minus_nat em 1), c, e))) + | minusinf (Ge (Cn (fm, c, e))) = + (if eqop eq_nat fm 0 then F else Ge (Cn (suc (minus_nat fm 1), c, e))) + | minusinf (Eq (Cn (gm, c, e))) = + (if eqop eq_nat gm 0 then F else Eq (Cn (suc (minus_nat gm 1), c, e))) + | minusinf (NEq (Cn (hm, c, e))) = + (if eqop eq_nat hm 0 then T else NEq (Cn (suc (minus_nat hm 1), c, e))); fun adjust b = (fn a as (q, r) => @@ -25,2216 +704,991 @@ then ((~1 : IntInf.int), IntInf.+ (a, b)) else adjust b (negDivAlg a (IntInf.* ((2 : IntInf.int), b)))); -val negateSnd : IntInf.int * IntInf.int -> IntInf.int * IntInf.int = - (fn a as (q, r) => (q, IntInf.~ r)); +fun apsnd f (x, y) = (x, f y); + +val eq_int = {eq = (fn a => fn b => ((a : IntInf.int) = b))} : IntInf.int eq; fun posDivAlg a b = (if IntInf.< (a, b) orelse IntInf.<= (b, (0 : IntInf.int)) then ((0 : IntInf.int), a) else adjust b (posDivAlg a (IntInf.* ((2 : IntInf.int), b)))); -val divAlg : IntInf.int * IntInf.int -> IntInf.int * IntInf.int = - (fn a as (aa, b) => - (if IntInf.<= ((0 : IntInf.int), aa) - then (if IntInf.<= ((0 : IntInf.int), b) then posDivAlg aa b - else (if ((aa : IntInf.int) = (0 : IntInf.int)) - then ((0 : IntInf.int), (0 : IntInf.int)) - else negateSnd (negDivAlg (IntInf.~ aa) (IntInf.~ b)))) - else (if IntInf.< ((0 : IntInf.int), b) then negDivAlg aa b - else negateSnd (posDivAlg (IntInf.~ aa) (IntInf.~ b))))); +fun divmoda a b = + (if IntInf.<= ((0 : IntInf.int), a) + then (if IntInf.<= ((0 : IntInf.int), b) then posDivAlg a b + else (if eqop eq_int a (0 : IntInf.int) + then ((0 : IntInf.int), (0 : IntInf.int)) + else apsnd IntInf.~ (negDivAlg (IntInf.~ a) (IntInf.~ b)))) + else (if IntInf.< ((0 : IntInf.int), b) then negDivAlg a b + else apsnd IntInf.~ (posDivAlg (IntInf.~ a) (IntInf.~ b)))); -fun snd (a, y) = y; - -fun mod_nat m k = (snd (divAlg (m, k))); - -val zero_nat : IntInf.int = (0 : IntInf.int); - -fun gcd (m, n) = - (if ((n : IntInf.int) = zero_nat) then m else gcd (n, mod_nat m n)); - -fun fst (y, b) = y; +fun mod_int a b = snd (divmoda a b); -fun div_nat m k = (fst (divAlg (m, k))); - -val lcm : IntInf.int * IntInf.int -> IntInf.int = - (fn a as (m, n) => div_nat (IntInf.* (m, n)) (gcd (m, n))); - -fun suc n = (IntInf.+ (n, (1 : IntInf.int))); - -fun abs_int i = (if IntInf.< (i, (0 : IntInf.int)) then IntInf.~ i else i); - -fun nat k = (if IntInf.< (k, (0 : IntInf.int)) then zero_nat else k); +fun num_case f1 f2 f3 f4 f5 f6 f7 (Mul (inta, num)) = f7 inta num + | num_case f1 f2 f3 f4 f5 f6 f7 (Sub (num1, num2)) = f6 num1 num2 + | num_case f1 f2 f3 f4 f5 f6 f7 (Add (num1, num2)) = f5 num1 num2 + | num_case f1 f2 f3 f4 f5 f6 f7 (Neg num) = f4 num + | num_case f1 f2 f3 f4 f5 f6 f7 (Cn (nat, inta, num)) = f3 nat inta num + | num_case f1 f2 f3 f4 f5 f6 f7 (Bound nat) = f2 nat + | num_case f1 f2 f3 f4 f5 f6 f7 (C inta) = f1 inta; -fun int_aux i n = - (if ((n : IntInf.int) = (0 : IntInf.int)) then i - else int_aux (IntInf.+ (i, (1 : IntInf.int))) - (IntInf.- (n, (1 : IntInf.int)))); +fun nummul i (C j) = C (IntInf.* (i, j)) + | nummul i (Cn (n, c, t)) = Cn (n, IntInf.* (c, i), nummul i t) + | nummul i (Bound v) = Mul (i, Bound v) + | nummul i (Neg v) = Mul (i, Neg v) + | nummul i (Add (v, va)) = Mul (i, Add (v, va)) + | nummul i (Sub (v, va)) = Mul (i, Sub (v, va)) + | nummul i (Mul (v, va)) = Mul (i, Mul (v, va)); -val ilcm : IntInf.int -> IntInf.int -> IntInf.int = - (fn i => fn j => - int_aux (0 : IntInf.int) (lcm (nat (abs_int i), nat (abs_int j)))); - -type 'a zero = {zero : 'a}; -fun zero (A_:'a zero) = #zero A_; +fun numneg t = nummul (IntInf.~ (1 : IntInf.int)) t; -fun map f (x :: xs) = f x :: map f xs - | map f [] = []; - -type 'a times = {times : 'a -> 'a -> 'a}; -fun times (A_:'a times) = #times A_; - -fun foldr f (x :: xs) a = f x (foldr f xs a) - | foldr f [] y = y; - -type 'a diva = {div : 'a -> 'a -> 'a, mod : 'a -> 'a -> 'a}; -fun diva (A_:'a diva) = #div A_; -fun moda (A_:'a diva) = #mod A_; - -type 'a dvd_mod = - {Divides__dvd_mod_div : 'a diva, Divides__dvd_mod_times : 'a times, - Divides__dvd_mod_zero : 'a zero}; -fun dvd_mod_div (A_:'a dvd_mod) = #Divides__dvd_mod_div A_; -fun dvd_mod_times (A_:'a dvd_mod) = #Divides__dvd_mod_times A_; -fun dvd_mod_zero (A_:'a dvd_mod) = #Divides__dvd_mod_zero A_; - -fun dvd (A1_, A2_) x y = - eq A2_ (moda (dvd_mod_div A1_) y x) (zero (dvd_mod_zero A1_)); +fun numadd (Cn (n1, c1, r1), Cn (n2, c2, r2)) = + (if eqop eq_nat n1 n2 + then let + val c = IntInf.+ (c1, c2); + in + (if eqop eq_int c (0 : IntInf.int) then numadd (r1, r2) + else Cn (n1, c, numadd (r1, r2))) + end + else (if IntInf.<= (n1, n2) + then Cn (n1, c1, numadd (r1, Add (Mul (c2, Bound n2), r2))) + else Cn (n2, c2, numadd (Add (Mul (c1, Bound n1), r1), r2)))) + | numadd (Cn (n1, c1, r1), C dd) = Cn (n1, c1, numadd (r1, C dd)) + | numadd (Cn (n1, c1, r1), Bound de) = Cn (n1, c1, numadd (r1, Bound de)) + | numadd (Cn (n1, c1, r1), Neg di) = Cn (n1, c1, numadd (r1, Neg di)) + | numadd (Cn (n1, c1, r1), Add (dj, dk)) = + Cn (n1, c1, numadd (r1, Add (dj, dk))) + | numadd (Cn (n1, c1, r1), Sub (dl, dm)) = + Cn (n1, c1, numadd (r1, Sub (dl, dm))) + | numadd (Cn (n1, c1, r1), Mul (dn, doa)) = + Cn (n1, c1, numadd (r1, Mul (dn, doa))) + | numadd (C w, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (C w, r2)) + | numadd (Bound x, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Bound x, r2)) + | numadd (Neg ac, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Neg ac, r2)) + | numadd (Add (ad, ae), Cn (n2, c2, r2)) = + Cn (n2, c2, numadd (Add (ad, ae), r2)) + | numadd (Sub (af, ag), Cn (n2, c2, r2)) = + Cn (n2, c2, numadd (Sub (af, ag), r2)) + | numadd (Mul (ah, ai), Cn (n2, c2, r2)) = + Cn (n2, c2, numadd (Mul (ah, ai), r2)) + | numadd (C b1, C b2) = C (IntInf.+ (b1, b2)) + | numadd (C aj, Bound bi) = Add (C aj, Bound bi) + | numadd (C aj, Neg bm) = Add (C aj, Neg bm) + | numadd (C aj, Add (bn, bo)) = Add (C aj, Add (bn, bo)) + | numadd (C aj, Sub (bp, bq)) = Add (C aj, Sub (bp, bq)) + | numadd (C aj, Mul (br, bs)) = Add (C aj, Mul (br, bs)) + | numadd (Bound ak, C cf) = Add (Bound ak, C cf) + | numadd (Bound ak, Bound cg) = Add (Bound ak, Bound cg) + | numadd (Bound ak, Neg ck) = Add (Bound ak, Neg ck) + | numadd (Bound ak, Add (cl, cm)) = Add (Bound ak, Add (cl, cm)) + | numadd (Bound ak, Sub (cn, co)) = Add (Bound ak, Sub (cn, co)) + | numadd (Bound ak, Mul (cp, cq)) = Add (Bound ak, Mul (cp, cq)) + | numadd (Neg ao, C en) = Add (Neg ao, C en) + | numadd (Neg ao, Bound eo) = Add (Neg ao, Bound eo) + | numadd (Neg ao, Neg es) = Add (Neg ao, Neg es) + | numadd (Neg ao, Add (et, eu)) = Add (Neg ao, Add (et, eu)) + | numadd (Neg ao, Sub (ev, ew)) = Add (Neg ao, Sub (ev, ew)) + | numadd (Neg ao, Mul (ex, ey)) = Add (Neg ao, Mul (ex, ey)) + | numadd (Add (ap, aq), C fl) = Add (Add (ap, aq), C fl) + | numadd (Add (ap, aq), Bound fm) = Add (Add (ap, aq), Bound fm) + | numadd (Add (ap, aq), Neg fq) = Add (Add (ap, aq), Neg fq) + | numadd (Add (ap, aq), Add (fr, fs)) = Add (Add (ap, aq), Add (fr, fs)) + | numadd (Add (ap, aq), Sub (ft, fu)) = Add (Add (ap, aq), Sub (ft, fu)) + | numadd (Add (ap, aq), Mul (fv, fw)) = Add (Add (ap, aq), Mul (fv, fw)) + | numadd (Sub (ar, asa), C gj) = Add (Sub (ar, asa), C gj) + | numadd (Sub (ar, asa), Bound gk) = Add (Sub (ar, asa), Bound gk) + | numadd (Sub (ar, asa), Neg go) = Add (Sub (ar, asa), Neg go) + | numadd (Sub (ar, asa), Add (gp, gq)) = Add (Sub (ar, asa), Add (gp, gq)) + | numadd (Sub (ar, asa), Sub (gr, gs)) = Add (Sub (ar, asa), Sub (gr, gs)) + | numadd (Sub (ar, asa), Mul (gt, gu)) = Add (Sub (ar, asa), Mul (gt, gu)) + | numadd (Mul (at, au), C hh) = Add (Mul (at, au), C hh) + | numadd (Mul (at, au), Bound hi) = Add (Mul (at, au), Bound hi) + | numadd (Mul (at, au), Neg hm) = Add (Mul (at, au), Neg hm) + | numadd (Mul (at, au), Add (hn, ho)) = Add (Mul (at, au), Add (hn, ho)) + | numadd (Mul (at, au), Sub (hp, hq)) = Add (Mul (at, au), Sub (hp, hq)) + | numadd (Mul (at, au), Mul (hr, hs)) = Add (Mul (at, au), Mul (hr, hs)); -fun append (x :: xs) ys = x :: append xs ys - | append [] y = y; +val eq_numa = {eq = eq_num} : num eq; -fun memberl A_ x (y :: ys) = eq A_ x y orelse memberl A_ x ys - | memberl A_ x [] = false; +fun numsub s t = + (if eqop eq_numa s t then C (0 : IntInf.int) else numadd (s, numneg t)); -fun remdups A_ (x :: xs) = - (if memberl A_ x xs then remdups A_ xs else x :: remdups A_ xs) - | remdups A_ [] = []; - -fun mod_int a b = snd (divAlg (a, b)); +fun simpnum (C j) = C j + | simpnum (Bound n) = Cn (n, (1 : IntInf.int), C (0 : IntInf.int)) + | 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 eqop eq_int i (0 : IntInf.int) then C (0 : IntInf.int) + else nummul i (simpnum t)) + | simpnum (Cn (v, va, vb)) = Cn (v, va, vb); -fun div_int a b = fst (divAlg (a, b)); - -val div_inta = {div = div_int, mod = mod_int} : IntInf.int diva; +fun nota (Not y) = y + | nota T = F + | nota F = T + | nota (Lt vc) = Not (Lt vc) + | nota (Le vc) = Not (Le vc) + | nota (Gt vc) = Not (Gt vc) + | nota (Ge vc) = Not (Ge vc) + | nota (Eq vc) = Not (Eq vc) + | nota (NEq vc) = Not (NEq vc) + | nota (Dvd (va, vab)) = Not (Dvd (va, vab)) + | nota (NDvd (va, vab)) = Not (NDvd (va, vab)) + | nota (And (vb, vaa)) = Not (And (vb, vaa)) + | nota (Or (vb, vaa)) = Not (Or (vb, vaa)) + | nota (Imp (vb, vaa)) = Not (Imp (vb, vaa)) + | nota (Iff (vb, vaa)) = Not (Iff (vb, vaa)) + | nota (E vb) = Not (E vb) + | nota (A vb) = Not (A vb) + | nota (Closed v) = Not (Closed v) + | nota (NClosed v) = Not (NClosed v); -fun allpairs f (x :: xs) ys = append (map (f x) ys) (allpairs f xs ys) - | allpairs f [] ys = []; +fun iffa p q = + (if eqop eq_fma p q then T + else (if eqop eq_fma p (nota q) orelse eqop eq_fma (nota p) q then F + else (if eqop eq_fma p F then nota q + else (if eqop eq_fma q F then nota p + else (if eqop eq_fma p T then q + else (if eqop eq_fma q T then p + else Iff (p, q))))))); -val eq_int = {eq = (fn a => fn b => ((a : IntInf.int) = b))} : IntInf.int eq; +fun impa p q = + (if eqop eq_fma p F orelse eqop eq_fma q T then T + else (if eqop eq_fma p T then q + else (if eqop eq_fma q F then nota p else Imp (p, q)))); -val zero_int : IntInf.int = (0 : IntInf.int); - -val zero_inta = {zero = zero_int} : IntInf.int zero; +fun conj p q = + (if eqop eq_fma p F orelse eqop eq_fma q F then F + else (if eqop eq_fma p T then q + else (if eqop eq_fma q T then p else And (p, q)))); -fun size_list (a :: lista) = (IntInf.+ ((size_list lista), (suc zero_nat))) - | size_list [] = zero_nat; - -fun eq_bit B0 B0 = true - | eq_bit B1 B1 = true - | eq_bit B0 B1 = false - | eq_bit B1 B0 = false; - -val times_int = {times = (fn a => fn b => IntInf.* (a, b))} : IntInf.int times; - -val dvd_mod_int = - {Divides__dvd_mod_div = div_inta, Divides__dvd_mod_times = times_int, - Divides__dvd_mod_zero = zero_inta} - : IntInf.int dvd_mod; - -datatype num = C of IntInf.int | Bound of IntInf.int | Cx of IntInf.int * num | - Neg of num | Add of num * num | Sub of num * num | Mul of IntInf.int * num; - -datatype fm = T | F | Lt of num | Le of num | Gt of num | Ge of num | Eq of num - | NEq of num | Dvd of IntInf.int * num | NDvd of IntInf.int * num | Nota of fm - | And of fm * fm | Or of fm * fm | Impa of fm * fm | Iffa of fm * fm | E of fm - | A of fm | Closed of IntInf.int | NClosed of IntInf.int; - -fun disjuncts (NClosed aq) = [NClosed aq] - | disjuncts (Closed ap) = [Closed ap] - | disjuncts (A ao) = [A ao] - | disjuncts (E an) = [E an] - | disjuncts (Iffa (al, am)) = [Iffa (al, am)] - | disjuncts (Impa (aj, ak)) = [Impa (aj, ak)] - | disjuncts (And (af, ag)) = [And (af, ag)] - | disjuncts (Nota ae) = [Nota ae] - | disjuncts (NDvd (ac, ad)) = [NDvd (ac, ad)] - | disjuncts (Dvd (aa, ab)) = [Dvd (aa, ab)] - | disjuncts (NEq z) = [NEq z] - | disjuncts (Eq y) = [Eq y] - | disjuncts (Ge x) = [Ge x] - | disjuncts (Gt w) = [Gt w] - | disjuncts (Le v) = [Le v] - | disjuncts (Lt u) = [Lt u] - | disjuncts T = [T] - | disjuncts F = [] - | disjuncts (Or (p, q)) = append (disjuncts p) (disjuncts q); +fun simpfm (And (p, q)) = conj (simpfm p) (simpfm q) + | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q) + | simpfm (Imp (p, q)) = impa (simpfm p) (simpfm q) + | simpfm (Iff (p, q)) = iffa (simpfm p) (simpfm q) + | simpfm (Not p) = nota (simpfm p) + | simpfm (Lt a) = + let + val a' = simpnum a; + in + (case a' of C v => (if IntInf.< (v, (0 : IntInf.int)) then T else F) + | Bound nat => Lt a' | Cn (nat, inta, num) => Lt a' | Neg num => Lt a' + | Add (num1, num2) => Lt a' | Sub (num1, num2) => Lt a' + | Mul (inta, num) => Lt a') + end + | simpfm (Le a) = + let + val a' = simpnum a; + in + (case a' of C v => (if IntInf.<= (v, (0 : IntInf.int)) then T else F) + | Bound nat => Le a' | Cn (nat, inta, num) => Le a' | Neg num => Le a' + | Add (num1, num2) => Le a' | Sub (num1, num2) => Le a' + | Mul (inta, num) => Le a') + end + | simpfm (Gt a) = + let + val a' = simpnum a; + in + (case a' of C v => (if IntInf.< ((0 : IntInf.int), v) then T else F) + | Bound nat => Gt a' | Cn (nat, inta, num) => Gt a' | Neg num => Gt a' + | Add (num1, num2) => Gt a' | Sub (num1, num2) => Gt a' + | Mul (inta, num) => Gt a') + end + | simpfm (Ge a) = + let + val a' = simpnum a; + in + (case a' of C v => (if IntInf.<= ((0 : IntInf.int), v) then T else F) + | Bound nat => Ge a' | Cn (nat, inta, num) => Ge a' | Neg num => Ge a' + | Add (num1, num2) => Ge a' | Sub (num1, num2) => Ge a' + | Mul (inta, num) => Ge a') + end + | simpfm (Eq a) = + let + val a' = simpnum a; + in + (case a' of C v => (if eqop eq_int v (0 : IntInf.int) then T else F) + | Bound nat => Eq a' | Cn (nat, inta, num) => Eq a' | Neg num => Eq a' + | Add (num1, num2) => Eq a' | Sub (num1, num2) => Eq a' + | Mul (inta, num) => Eq a') + end + | simpfm (NEq a) = + let + val a' = simpnum a; + in + (case a' of C v => (if not (eqop eq_int v (0 : IntInf.int)) then T else F) + | Bound nat => NEq a' | Cn (nat, inta, num) => NEq a' + | Neg num => NEq a' | Add (num1, num2) => NEq a' + | Sub (num1, num2) => NEq a' | Mul (inta, num) => NEq a') + end + | simpfm (Dvd (i, a)) = + (if eqop eq_int i (0 : IntInf.int) then simpfm (Eq a) + else (if eqop eq_int (abs_int i) (1 : IntInf.int) then T + else let + val a' = simpnum a; + in + (case a' + of C v => + (if eqop eq_int (mod_int v i) (0 : IntInf.int) then T + else F) + | Bound nat => Dvd (i, a') + | Cn (nat, inta, num) => Dvd (i, a') + | Neg num => Dvd (i, a') + | Add (num1, num2) => Dvd (i, a') + | Sub (num1, num2) => Dvd (i, a') + | Mul (inta, num) => Dvd (i, a')) + end)) + | simpfm (NDvd (i, a)) = + (if eqop eq_int i (0 : IntInf.int) then simpfm (NEq a) + else (if eqop eq_int (abs_int i) (1 : IntInf.int) then F + else let + val a' = simpnum a; + in + (case a' + of C v => + (if not (eqop eq_int (mod_int v i) (0 : IntInf.int)) + then T else F) + | Bound nat => NDvd (i, a') + | Cn (nat, inta, num) => NDvd (i, a') + | Neg num => NDvd (i, a') + | Add (num1, num2) => NDvd (i, a') + | Sub (num1, num2) => NDvd (i, a') + | Mul (inta, num) => NDvd (i, a')) + end)) + | simpfm T = T + | simpfm F = F + | simpfm (E v) = E v + | simpfm (A v) = A v + | simpfm (Closed v) = Closed v + | simpfm (NClosed v) = NClosed v; -fun eq_num (C int) (C int') = ((int : IntInf.int) = int') - | eq_num (Bound nat) (Bound nat') = ((nat : IntInf.int) = nat') - | eq_num (Cx (int, num)) (Cx (int', num')) = - ((int : IntInf.int) = int') andalso eq_num num num' - | eq_num (Neg num) (Neg num') = eq_num num num' - | eq_num (Add (num1, num2)) (Add (num1', num2')) = - eq_num num1 num1' andalso eq_num num2 num2' - | eq_num (Sub (num1, num2)) (Sub (num1', num2')) = - eq_num num1 num1' andalso eq_num num2 num2' - | eq_num (Mul (int, num)) (Mul (int', num')) = - ((int : IntInf.int) = int') andalso eq_num num num' - | eq_num (C a) (Bound b) = false - | eq_num (C a) (Cx (b, c)) = false - | eq_num (C a) (Neg b) = false - | eq_num (C a) (Add (b, c)) = false - | eq_num (C a) (Sub (b, c)) = false - | eq_num (C a) (Mul (b, c)) = false - | eq_num (Bound a) (Cx (b, c)) = false - | eq_num (Bound a) (Neg b) = false - | eq_num (Bound a) (Add (b, c)) = false - | eq_num (Bound a) (Sub (b, c)) = false - | eq_num (Bound a) (Mul (b, c)) = false - | eq_num (Cx (a, b)) (Neg c) = false - | eq_num (Cx (a, b)) (Add (c, d)) = false - | eq_num (Cx (a, b)) (Sub (c, d)) = false - | eq_num (Cx (a, b)) (Mul (c, d)) = false - | eq_num (Neg a) (Add (b, c)) = false - | eq_num (Neg a) (Sub (b, c)) = false - | eq_num (Neg a) (Mul (b, c)) = false - | eq_num (Add (a, b)) (Sub (c, d)) = false - | eq_num (Add (a, b)) (Mul (c, d)) = false - | eq_num (Sub (a, b)) (Mul (c, d)) = false - | eq_num (Bound b) (C a) = false - | eq_num (Cx (b, c)) (C a) = false - | eq_num (Neg b) (C a) = false - | eq_num (Add (b, c)) (C a) = false - | eq_num (Sub (b, c)) (C a) = false - | eq_num (Mul (b, c)) (C a) = false - | eq_num (Cx (b, c)) (Bound a) = false - | eq_num (Neg b) (Bound a) = false - | eq_num (Add (b, c)) (Bound a) = false - | eq_num (Sub (b, c)) (Bound a) = false - | eq_num (Mul (b, c)) (Bound a) = false - | eq_num (Neg c) (Cx (a, b)) = false - | eq_num (Add (c, d)) (Cx (a, b)) = false - | eq_num (Sub (c, d)) (Cx (a, b)) = false - | eq_num (Mul (c, d)) (Cx (a, b)) = false - | eq_num (Add (b, c)) (Neg a) = false - | eq_num (Sub (b, c)) (Neg a) = false - | eq_num (Mul (b, c)) (Neg a) = false - | eq_num (Sub (c, d)) (Add (a, b)) = false - | eq_num (Mul (c, d)) (Add (a, b)) = false - | eq_num (Mul (c, d)) (Sub (a, b)) = false; +fun iupt i j = + (if IntInf.< (j, i) then [] + else i :: iupt (IntInf.+ (i, (1 : IntInf.int))) j); + +fun mirror (And (p, q)) = And (mirror p, mirror q) + | mirror (Or (p, q)) = Or (mirror p, mirror q) + | mirror T = T + | mirror F = F + | mirror (Lt (C bo)) = Lt (C bo) + | mirror (Lt (Bound bp)) = Lt (Bound bp) + | mirror (Lt (Neg bt)) = Lt (Neg bt) + | mirror (Lt (Add (bu, bv))) = Lt (Add (bu, bv)) + | mirror (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx)) + | mirror (Lt (Mul (by, bz))) = Lt (Mul (by, bz)) + | mirror (Le (C co)) = Le (C co) + | mirror (Le (Bound cp)) = Le (Bound cp) + | mirror (Le (Neg ct)) = Le (Neg ct) + | mirror (Le (Add (cu, cv))) = Le (Add (cu, cv)) + | mirror (Le (Sub (cw, cx))) = Le (Sub (cw, cx)) + | mirror (Le (Mul (cy, cz))) = Le (Mul (cy, cz)) + | mirror (Gt (C doa)) = Gt (C doa) + | mirror (Gt (Bound dp)) = Gt (Bound dp) + | mirror (Gt (Neg dt)) = Gt (Neg dt) + | mirror (Gt (Add (du, dv))) = Gt (Add (du, dv)) + | mirror (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx)) + | mirror (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz)) + | mirror (Ge (C eo)) = Ge (C eo) + | mirror (Ge (Bound ep)) = Ge (Bound ep) + | mirror (Ge (Neg et)) = Ge (Neg et) + | mirror (Ge (Add (eu, ev))) = Ge (Add (eu, ev)) + | mirror (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex)) + | mirror (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez)) + | mirror (Eq (C fo)) = Eq (C fo) + | mirror (Eq (Bound fp)) = Eq (Bound fp) + | mirror (Eq (Neg ft)) = Eq (Neg ft) + | mirror (Eq (Add (fu, fv))) = Eq (Add (fu, fv)) + | mirror (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx)) + | mirror (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz)) + | mirror (NEq (C go)) = NEq (C go) + | mirror (NEq (Bound gp)) = NEq (Bound gp) + | mirror (NEq (Neg gt)) = NEq (Neg gt) + | mirror (NEq (Add (gu, gv))) = NEq (Add (gu, gv)) + | mirror (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx)) + | mirror (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz)) + | mirror (Dvd (aa, C ho)) = Dvd (aa, C ho) + | mirror (Dvd (aa, Bound hp)) = Dvd (aa, Bound hp) + | mirror (Dvd (aa, Neg ht)) = Dvd (aa, Neg ht) + | mirror (Dvd (aa, Add (hu, hv))) = Dvd (aa, Add (hu, hv)) + | mirror (Dvd (aa, Sub (hw, hx))) = Dvd (aa, Sub (hw, hx)) + | mirror (Dvd (aa, Mul (hy, hz))) = Dvd (aa, Mul (hy, hz)) + | mirror (NDvd (ac, C io)) = NDvd (ac, C io) + | mirror (NDvd (ac, Bound ip)) = NDvd (ac, Bound ip) + | mirror (NDvd (ac, Neg it)) = NDvd (ac, Neg it) + | mirror (NDvd (ac, Add (iu, iv))) = NDvd (ac, Add (iu, iv)) + | mirror (NDvd (ac, Sub (iw, ix))) = NDvd (ac, Sub (iw, ix)) + | mirror (NDvd (ac, Mul (iy, iz))) = NDvd (ac, Mul (iy, iz)) + | 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 + | mirror (Lt (Cn (cm, c, e))) = + (if eqop eq_nat cm 0 then Gt (Cn (0, c, Neg e)) + else Lt (Cn (suc (minus_nat cm 1), c, e))) + | mirror (Le (Cn (dm, c, e))) = + (if eqop eq_nat dm 0 then Ge (Cn (0, c, Neg e)) + else Le (Cn (suc (minus_nat dm 1), c, e))) + | mirror (Gt (Cn (em, c, e))) = + (if eqop eq_nat em 0 then Lt (Cn (0, c, Neg e)) + else Gt (Cn (suc (minus_nat em 1), c, e))) + | mirror (Ge (Cn (fm, c, e))) = + (if eqop eq_nat fm 0 then Le (Cn (0, c, Neg e)) + else Ge (Cn (suc (minus_nat fm 1), c, e))) + | mirror (Eq (Cn (gm, c, e))) = + (if eqop eq_nat gm 0 then Eq (Cn (0, c, Neg e)) + else Eq (Cn (suc (minus_nat gm 1), c, e))) + | mirror (NEq (Cn (hm, c, e))) = + (if eqop eq_nat hm 0 then NEq (Cn (0, c, Neg e)) + else NEq (Cn (suc (minus_nat hm 1), c, e))) + | mirror (Dvd (i, Cn (im, c, e))) = + (if eqop eq_nat im 0 then Dvd (i, Cn (0, c, Neg e)) + else Dvd (i, Cn (suc (minus_nat im 1), c, e))) + | mirror (NDvd (i, Cn (jm, c, e))) = + (if eqop eq_nat jm 0 then NDvd (i, Cn (0, c, Neg e)) + else NDvd (i, Cn (suc (minus_nat jm 1), c, e))); + +fun size_list [] = 0 + | size_list (a :: lista) = IntInf.+ (size_list lista, suc 0); -fun eq_fm T T = true - | eq_fm F F = true - | eq_fm (Lt num) (Lt num') = eq_num num num' - | eq_fm (Le num) (Le num') = eq_num num num' - | eq_fm (Gt num) (Gt num') = eq_num num num' - | eq_fm (Ge num) (Ge num') = eq_num num num' - | eq_fm (Eq num) (Eq num') = eq_num num num' - | eq_fm (NEq num) (NEq num') = eq_num num num' - | eq_fm (Dvd (int, num)) (Dvd (int', num')) = - ((int : IntInf.int) = int') andalso eq_num num num' - | eq_fm (NDvd (int, num)) (NDvd (int', num')) = - ((int : IntInf.int) = int') andalso eq_num num num' - | eq_fm (Nota fm) (Nota fm') = eq_fm fm fm' - | eq_fm (And (fm1, fm2)) (And (fm1', fm2')) = - eq_fm fm1 fm1' andalso eq_fm fm2 fm2' - | eq_fm (Or (fm1, fm2)) (Or (fm1', fm2')) = - eq_fm fm1 fm1' andalso eq_fm fm2 fm2' - | eq_fm (Impa (fm1, fm2)) (Impa (fm1', fm2')) = - eq_fm fm1 fm1' andalso eq_fm fm2 fm2' - | eq_fm (Iffa (fm1, fm2)) (Iffa (fm1', fm2')) = - eq_fm fm1 fm1' andalso eq_fm fm2 fm2' - | eq_fm (E fm) (E fm') = eq_fm fm fm' - | eq_fm (A fm) (A fm') = eq_fm fm fm' - | eq_fm (Closed nat) (Closed nat') = ((nat : IntInf.int) = nat') - | eq_fm (NClosed nat) (NClosed nat') = ((nat : IntInf.int) = nat') - | eq_fm T F = false - | eq_fm T (Lt a) = false - | eq_fm T (Le a) = false - | eq_fm T (Gt a) = false - | eq_fm T (Ge a) = false - | eq_fm T (Eq a) = false - | eq_fm T (NEq a) = false - | eq_fm T (Dvd (a, b)) = false - | eq_fm T (NDvd (a, b)) = false - | eq_fm T (Nota a) = false - | eq_fm T (And (a, b)) = false - | eq_fm T (Or (a, b)) = false - | eq_fm T (Impa (a, b)) = false - | eq_fm T (Iffa (a, b)) = false - | eq_fm T (E a) = false - | eq_fm T (A a) = false - | eq_fm T (Closed a) = false - | eq_fm T (NClosed a) = false - | eq_fm F (Lt a) = false - | eq_fm F (Le a) = false - | eq_fm F (Gt a) = false - | eq_fm F (Ge a) = false - | eq_fm F (Eq a) = false - | eq_fm F (NEq a) = false - | eq_fm F (Dvd (a, b)) = false - | eq_fm F (NDvd (a, b)) = false - | eq_fm F (Nota a) = false - | eq_fm F (And (a, b)) = false - | eq_fm F (Or (a, b)) = false - | eq_fm F (Impa (a, b)) = false - | eq_fm F (Iffa (a, b)) = false - | eq_fm F (E a) = false - | eq_fm F (A a) = false - | eq_fm F (Closed a) = false - | eq_fm F (NClosed a) = false - | eq_fm (Lt a) (Le b) = false - | eq_fm (Lt a) (Gt b) = false - | eq_fm (Lt a) (Ge b) = false - | eq_fm (Lt a) (Eq b) = false - | eq_fm (Lt a) (NEq b) = false - | eq_fm (Lt a) (Dvd (b, c)) = false - | eq_fm (Lt a) (NDvd (b, c)) = false - | eq_fm (Lt a) (Nota b) = false - | eq_fm (Lt a) (And (b, c)) = false - | eq_fm (Lt a) (Or (b, c)) = false - | eq_fm (Lt a) (Impa (b, c)) = false - | eq_fm (Lt a) (Iffa (b, c)) = false - | eq_fm (Lt a) (E b) = false - | eq_fm (Lt a) (A b) = false - | eq_fm (Lt a) (Closed b) = false - | eq_fm (Lt a) (NClosed b) = false - | eq_fm (Le a) (Gt b) = false - | eq_fm (Le a) (Ge b) = false - | eq_fm (Le a) (Eq b) = false - | eq_fm (Le a) (NEq b) = false - | eq_fm (Le a) (Dvd (b, c)) = false - | eq_fm (Le a) (NDvd (b, c)) = false - | eq_fm (Le a) (Nota b) = false - | eq_fm (Le a) (And (b, c)) = false - | eq_fm (Le a) (Or (b, c)) = false - | eq_fm (Le a) (Impa (b, c)) = false - | eq_fm (Le a) (Iffa (b, c)) = false - | eq_fm (Le a) (E b) = false - | eq_fm (Le a) (A b) = false - | eq_fm (Le a) (Closed b) = false - | eq_fm (Le a) (NClosed b) = false - | eq_fm (Gt a) (Ge b) = false - | eq_fm (Gt a) (Eq b) = false - | eq_fm (Gt a) (NEq b) = false - | eq_fm (Gt a) (Dvd (b, c)) = false - | eq_fm (Gt a) (NDvd (b, c)) = false - | eq_fm (Gt a) (Nota b) = false - | eq_fm (Gt a) (And (b, c)) = false - | eq_fm (Gt a) (Or (b, c)) = false - | eq_fm (Gt a) (Impa (b, c)) = false - | eq_fm (Gt a) (Iffa (b, c)) = false - | eq_fm (Gt a) (E b) = false - | eq_fm (Gt a) (A b) = false - | eq_fm (Gt a) (Closed b) = false - | eq_fm (Gt a) (NClosed b) = false - | eq_fm (Ge a) (Eq b) = false - | eq_fm (Ge a) (NEq b) = false - | eq_fm (Ge a) (Dvd (b, c)) = false - | eq_fm (Ge a) (NDvd (b, c)) = false - | eq_fm (Ge a) (Nota b) = false - | eq_fm (Ge a) (And (b, c)) = false - | eq_fm (Ge a) (Or (b, c)) = false - | eq_fm (Ge a) (Impa (b, c)) = false - | eq_fm (Ge a) (Iffa (b, c)) = false - | eq_fm (Ge a) (E b) = false - | eq_fm (Ge a) (A b) = false - | eq_fm (Ge a) (Closed b) = false - | eq_fm (Ge a) (NClosed b) = false - | eq_fm (Eq a) (NEq b) = false - | eq_fm (Eq a) (Dvd (b, c)) = false - | eq_fm (Eq a) (NDvd (b, c)) = false - | eq_fm (Eq a) (Nota b) = false - | eq_fm (Eq a) (And (b, c)) = false - | eq_fm (Eq a) (Or (b, c)) = false - | eq_fm (Eq a) (Impa (b, c)) = false - | eq_fm (Eq a) (Iffa (b, c)) = false - | eq_fm (Eq a) (E b) = false - | eq_fm (Eq a) (A b) = false - | eq_fm (Eq a) (Closed b) = false - | eq_fm (Eq a) (NClosed b) = false - | eq_fm (NEq a) (Dvd (b, c)) = false - | eq_fm (NEq a) (NDvd (b, c)) = false - | eq_fm (NEq a) (Nota b) = false - | eq_fm (NEq a) (And (b, c)) = false - | eq_fm (NEq a) (Or (b, c)) = false - | eq_fm (NEq a) (Impa (b, c)) = false - | eq_fm (NEq a) (Iffa (b, c)) = false - | eq_fm (NEq a) (E b) = false - | eq_fm (NEq a) (A b) = false - | eq_fm (NEq a) (Closed b) = false - | eq_fm (NEq a) (NClosed b) = false - | eq_fm (Dvd (a, b)) (NDvd (c, d)) = false - | eq_fm (Dvd (a, b)) (Nota c) = false - | eq_fm (Dvd (a, b)) (And (c, d)) = false - | eq_fm (Dvd (a, b)) (Or (c, d)) = false - | eq_fm (Dvd (a, b)) (Impa (c, d)) = false - | eq_fm (Dvd (a, b)) (Iffa (c, d)) = false - | eq_fm (Dvd (a, b)) (E c) = false - | eq_fm (Dvd (a, b)) (A c) = false - | eq_fm (Dvd (a, b)) (Closed c) = false - | eq_fm (Dvd (a, b)) (NClosed c) = false - | eq_fm (NDvd (a, b)) (Nota c) = false - | eq_fm (NDvd (a, b)) (And (c, d)) = false - | eq_fm (NDvd (a, b)) (Or (c, d)) = false - | eq_fm (NDvd (a, b)) (Impa (c, d)) = false - | eq_fm (NDvd (a, b)) (Iffa (c, d)) = false - | eq_fm (NDvd (a, b)) (E c) = false - | eq_fm (NDvd (a, b)) (A c) = false - | eq_fm (NDvd (a, b)) (Closed c) = false - | eq_fm (NDvd (a, b)) (NClosed c) = false - | eq_fm (Nota a) (And (b, c)) = false - | eq_fm (Nota a) (Or (b, c)) = false - | eq_fm (Nota a) (Impa (b, c)) = false - | eq_fm (Nota a) (Iffa (b, c)) = false - | eq_fm (Nota a) (E b) = false - | eq_fm (Nota a) (A b) = false - | eq_fm (Nota a) (Closed b) = false - | eq_fm (Nota a) (NClosed b) = false - | eq_fm (And (a, b)) (Or (c, d)) = false - | eq_fm (And (a, b)) (Impa (c, d)) = false - | eq_fm (And (a, b)) (Iffa (c, d)) = false - | eq_fm (And (a, b)) (E c) = false - | eq_fm (And (a, b)) (A c) = false - | eq_fm (And (a, b)) (Closed c) = false - | eq_fm (And (a, b)) (NClosed c) = false - | eq_fm (Or (a, b)) (Impa (c, d)) = false - | eq_fm (Or (a, b)) (Iffa (c, d)) = false - | eq_fm (Or (a, b)) (E c) = false - | eq_fm (Or (a, b)) (A c) = false - | eq_fm (Or (a, b)) (Closed c) = false - | eq_fm (Or (a, b)) (NClosed c) = false - | eq_fm (Impa (a, b)) (Iffa (c, d)) = false - | eq_fm (Impa (a, b)) (E c) = false - | eq_fm (Impa (a, b)) (A c) = false - | eq_fm (Impa (a, b)) (Closed c) = false - | eq_fm (Impa (a, b)) (NClosed c) = false - | eq_fm (Iffa (a, b)) (E c) = false - | eq_fm (Iffa (a, b)) (A c) = false - | eq_fm (Iffa (a, b)) (Closed c) = false - | eq_fm (Iffa (a, b)) (NClosed c) = false - | eq_fm (E a) (A b) = false - | eq_fm (E a) (Closed b) = false - | eq_fm (E a) (NClosed b) = false - | eq_fm (A a) (Closed b) = false - | eq_fm (A a) (NClosed b) = false - | eq_fm (Closed a) (NClosed b) = false - | eq_fm F T = false - | eq_fm (Lt a) T = false - | eq_fm (Le a) T = false - | eq_fm (Gt a) T = false - | eq_fm (Ge a) T = false - | eq_fm (Eq a) T = false - | eq_fm (NEq a) T = false - | eq_fm (Dvd (a, b)) T = false - | eq_fm (NDvd (a, b)) T = false - | eq_fm (Nota a) T = false - | eq_fm (And (a, b)) T = false - | eq_fm (Or (a, b)) T = false - | eq_fm (Impa (a, b)) T = false - | eq_fm (Iffa (a, b)) T = false - | eq_fm (E a) T = false - | eq_fm (A a) T = false - | eq_fm (Closed a) T = false - | eq_fm (NClosed a) T = false - | eq_fm (Lt a) F = false - | eq_fm (Le a) F = false - | eq_fm (Gt a) F = false - | eq_fm (Ge a) F = false - | eq_fm (Eq a) F = false - | eq_fm (NEq a) F = false - | eq_fm (Dvd (a, b)) F = false - | eq_fm (NDvd (a, b)) F = false - | eq_fm (Nota a) F = false - | eq_fm (And (a, b)) F = false - | eq_fm (Or (a, b)) F = false - | eq_fm (Impa (a, b)) F = false - | eq_fm (Iffa (a, b)) F = false - | eq_fm (E a) F = false - | eq_fm (A a) F = false - | eq_fm (Closed a) F = false - | eq_fm (NClosed a) F = false - | eq_fm (Le b) (Lt a) = false - | eq_fm (Gt b) (Lt a) = false - | eq_fm (Ge b) (Lt a) = false - | eq_fm (Eq b) (Lt a) = false - | eq_fm (NEq b) (Lt a) = false - | eq_fm (Dvd (b, c)) (Lt a) = false - | eq_fm (NDvd (b, c)) (Lt a) = false - | eq_fm (Nota b) (Lt a) = false - | eq_fm (And (b, c)) (Lt a) = false - | eq_fm (Or (b, c)) (Lt a) = false - | eq_fm (Impa (b, c)) (Lt a) = false - | eq_fm (Iffa (b, c)) (Lt a) = false - | eq_fm (E b) (Lt a) = false - | eq_fm (A b) (Lt a) = false - | eq_fm (Closed b) (Lt a) = false - | eq_fm (NClosed b) (Lt a) = false - | eq_fm (Gt b) (Le a) = false - | eq_fm (Ge b) (Le a) = false - | eq_fm (Eq b) (Le a) = false - | eq_fm (NEq b) (Le a) = false - | eq_fm (Dvd (b, c)) (Le a) = false - | eq_fm (NDvd (b, c)) (Le a) = false - | eq_fm (Nota b) (Le a) = false - | eq_fm (And (b, c)) (Le a) = false - | eq_fm (Or (b, c)) (Le a) = false - | eq_fm (Impa (b, c)) (Le a) = false - | eq_fm (Iffa (b, c)) (Le a) = false - | eq_fm (E b) (Le a) = false - | eq_fm (A b) (Le a) = false - | eq_fm (Closed b) (Le a) = false - | eq_fm (NClosed b) (Le a) = false - | eq_fm (Ge b) (Gt a) = false - | eq_fm (Eq b) (Gt a) = false - | eq_fm (NEq b) (Gt a) = false - | eq_fm (Dvd (b, c)) (Gt a) = false - | eq_fm (NDvd (b, c)) (Gt a) = false - | eq_fm (Nota b) (Gt a) = false - | eq_fm (And (b, c)) (Gt a) = false - | eq_fm (Or (b, c)) (Gt a) = false - | eq_fm (Impa (b, c)) (Gt a) = false - | eq_fm (Iffa (b, c)) (Gt a) = false - | eq_fm (E b) (Gt a) = false - | eq_fm (A b) (Gt a) = false - | eq_fm (Closed b) (Gt a) = false - | eq_fm (NClosed b) (Gt a) = false - | eq_fm (Eq b) (Ge a) = false - | eq_fm (NEq b) (Ge a) = false - | eq_fm (Dvd (b, c)) (Ge a) = false - | eq_fm (NDvd (b, c)) (Ge a) = false - | eq_fm (Nota b) (Ge a) = false - | eq_fm (And (b, c)) (Ge a) = false - | eq_fm (Or (b, c)) (Ge a) = false - | eq_fm (Impa (b, c)) (Ge a) = false - | eq_fm (Iffa (b, c)) (Ge a) = false - | eq_fm (E b) (Ge a) = false - | eq_fm (A b) (Ge a) = false - | eq_fm (Closed b) (Ge a) = false - | eq_fm (NClosed b) (Ge a) = false - | eq_fm (NEq b) (Eq a) = false - | eq_fm (Dvd (b, c)) (Eq a) = false - | eq_fm (NDvd (b, c)) (Eq a) = false - | eq_fm (Nota b) (Eq a) = false - | eq_fm (And (b, c)) (Eq a) = false - | eq_fm (Or (b, c)) (Eq a) = false - | eq_fm (Impa (b, c)) (Eq a) = false - | eq_fm (Iffa (b, c)) (Eq a) = false - | eq_fm (E b) (Eq a) = false - | eq_fm (A b) (Eq a) = false - | eq_fm (Closed b) (Eq a) = false - | eq_fm (NClosed b) (Eq a) = false - | eq_fm (Dvd (b, c)) (NEq a) = false - | eq_fm (NDvd (b, c)) (NEq a) = false - | eq_fm (Nota b) (NEq a) = false - | eq_fm (And (b, c)) (NEq a) = false - | eq_fm (Or (b, c)) (NEq a) = false - | eq_fm (Impa (b, c)) (NEq a) = false - | eq_fm (Iffa (b, c)) (NEq a) = false - | eq_fm (E b) (NEq a) = false - | eq_fm (A b) (NEq a) = false - | eq_fm (Closed b) (NEq a) = false - | eq_fm (NClosed b) (NEq a) = false - | eq_fm (NDvd (c, d)) (Dvd (a, b)) = false - | eq_fm (Nota c) (Dvd (a, b)) = false - | eq_fm (And (c, d)) (Dvd (a, b)) = false - | eq_fm (Or (c, d)) (Dvd (a, b)) = false - | eq_fm (Impa (c, d)) (Dvd (a, b)) = false - | eq_fm (Iffa (c, d)) (Dvd (a, b)) = false - | eq_fm (E c) (Dvd (a, b)) = false - | eq_fm (A c) (Dvd (a, b)) = false - | eq_fm (Closed c) (Dvd (a, b)) = false - | eq_fm (NClosed c) (Dvd (a, b)) = false - | eq_fm (Nota c) (NDvd (a, b)) = false - | eq_fm (And (c, d)) (NDvd (a, b)) = false - | eq_fm (Or (c, d)) (NDvd (a, b)) = false - | eq_fm (Impa (c, d)) (NDvd (a, b)) = false - | eq_fm (Iffa (c, d)) (NDvd (a, b)) = false - | eq_fm (E c) (NDvd (a, b)) = false - | eq_fm (A c) (NDvd (a, b)) = false - | eq_fm (Closed c) (NDvd (a, b)) = false - | eq_fm (NClosed c) (NDvd (a, b)) = false - | eq_fm (And (b, c)) (Nota a) = false - | eq_fm (Or (b, c)) (Nota a) = false - | eq_fm (Impa (b, c)) (Nota a) = false - | eq_fm (Iffa (b, c)) (Nota a) = false - | eq_fm (E b) (Nota a) = false - | eq_fm (A b) (Nota a) = false - | eq_fm (Closed b) (Nota a) = false - | eq_fm (NClosed b) (Nota a) = false - | eq_fm (Or (c, d)) (And (a, b)) = false - | eq_fm (Impa (c, d)) (And (a, b)) = false - | eq_fm (Iffa (c, d)) (And (a, b)) = false - | eq_fm (E c) (And (a, b)) = false - | eq_fm (A c) (And (a, b)) = false - | eq_fm (Closed c) (And (a, b)) = false - | eq_fm (NClosed c) (And (a, b)) = false - | eq_fm (Impa (c, d)) (Or (a, b)) = false - | eq_fm (Iffa (c, d)) (Or (a, b)) = false - | eq_fm (E c) (Or (a, b)) = false - | eq_fm (A c) (Or (a, b)) = false - | eq_fm (Closed c) (Or (a, b)) = false - | eq_fm (NClosed c) (Or (a, b)) = false - | eq_fm (Iffa (c, d)) (Impa (a, b)) = false - | eq_fm (E c) (Impa (a, b)) = false - | eq_fm (A c) (Impa (a, b)) = false - | eq_fm (Closed c) (Impa (a, b)) = false - | eq_fm (NClosed c) (Impa (a, b)) = false - | eq_fm (E c) (Iffa (a, b)) = false - | eq_fm (A c) (Iffa (a, b)) = false - | eq_fm (Closed c) (Iffa (a, b)) = false - | eq_fm (NClosed c) (Iffa (a, b)) = false - | eq_fm (A b) (E a) = false - | eq_fm (Closed b) (E a) = false - | eq_fm (NClosed b) (E a) = false - | eq_fm (Closed b) (A a) = false - | eq_fm (NClosed b) (A a) = false - | eq_fm (NClosed b) (Closed a) = false; +fun alpha (And (p, q)) = append (alpha p) (alpha q) + | alpha (Or (p, q)) = append (alpha p) (alpha q) + | alpha T = [] + | alpha F = [] + | alpha (Lt (C bo)) = [] + | alpha (Lt (Bound bp)) = [] + | alpha (Lt (Neg bt)) = [] + | alpha (Lt (Add (bu, bv))) = [] + | alpha (Lt (Sub (bw, bx))) = [] + | alpha (Lt (Mul (by, bz))) = [] + | alpha (Le (C co)) = [] + | alpha (Le (Bound cp)) = [] + | alpha (Le (Neg ct)) = [] + | alpha (Le (Add (cu, cv))) = [] + | alpha (Le (Sub (cw, cx))) = [] + | alpha (Le (Mul (cy, cz))) = [] + | alpha (Gt (C doa)) = [] + | alpha (Gt (Bound dp)) = [] + | alpha (Gt (Neg dt)) = [] + | alpha (Gt (Add (du, dv))) = [] + | alpha (Gt (Sub (dw, dx))) = [] + | alpha (Gt (Mul (dy, dz))) = [] + | alpha (Ge (C eo)) = [] + | alpha (Ge (Bound ep)) = [] + | alpha (Ge (Neg et)) = [] + | alpha (Ge (Add (eu, ev))) = [] + | alpha (Ge (Sub (ew, ex))) = [] + | alpha (Ge (Mul (ey, ez))) = [] + | alpha (Eq (C fo)) = [] + | alpha (Eq (Bound fp)) = [] + | alpha (Eq (Neg ft)) = [] + | alpha (Eq (Add (fu, fv))) = [] + | alpha (Eq (Sub (fw, fx))) = [] + | alpha (Eq (Mul (fy, fz))) = [] + | alpha (NEq (C go)) = [] + | alpha (NEq (Bound gp)) = [] + | alpha (NEq (Neg gt)) = [] + | alpha (NEq (Add (gu, gv))) = [] + | alpha (NEq (Sub (gw, gx))) = [] + | alpha (NEq (Mul (gy, gz))) = [] + | 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) = [] + | alpha (Lt (Cn (cm, c, e))) = (if eqop eq_nat cm 0 then [e] else []) + | alpha (Le (Cn (dm, c, e))) = + (if eqop eq_nat dm 0 then [Add (C (~1 : IntInf.int), e)] else []) + | alpha (Gt (Cn (em, c, e))) = (if eqop eq_nat em 0 then [] else []) + | alpha (Ge (Cn (fm, c, e))) = (if eqop eq_nat fm 0 then [] else []) + | alpha (Eq (Cn (gm, c, e))) = + (if eqop eq_nat gm 0 then [Add (C (~1 : IntInf.int), e)] else []) + | alpha (NEq (Cn (hm, c, e))) = (if eqop eq_nat hm 0 then [e] else []); + +fun beta (And (p, q)) = append (beta p) (beta q) + | beta (Or (p, q)) = append (beta p) (beta q) + | beta T = [] + | beta F = [] + | beta (Lt (C bo)) = [] + | beta (Lt (Bound bp)) = [] + | beta (Lt (Neg bt)) = [] + | beta (Lt (Add (bu, bv))) = [] + | beta (Lt (Sub (bw, bx))) = [] + | beta (Lt (Mul (by, bz))) = [] + | beta (Le (C co)) = [] + | beta (Le (Bound cp)) = [] + | beta (Le (Neg ct)) = [] + | beta (Le (Add (cu, cv))) = [] + | beta (Le (Sub (cw, cx))) = [] + | beta (Le (Mul (cy, cz))) = [] + | beta (Gt (C doa)) = [] + | beta (Gt (Bound dp)) = [] + | beta (Gt (Neg dt)) = [] + | beta (Gt (Add (du, dv))) = [] + | beta (Gt (Sub (dw, dx))) = [] + | beta (Gt (Mul (dy, dz))) = [] + | beta (Ge (C eo)) = [] + | beta (Ge (Bound ep)) = [] + | beta (Ge (Neg et)) = [] + | beta (Ge (Add (eu, ev))) = [] + | beta (Ge (Sub (ew, ex))) = [] + | beta (Ge (Mul (ey, ez))) = [] + | beta (Eq (C fo)) = [] + | beta (Eq (Bound fp)) = [] + | beta (Eq (Neg ft)) = [] + | beta (Eq (Add (fu, fv))) = [] + | beta (Eq (Sub (fw, fx))) = [] + | beta (Eq (Mul (fy, fz))) = [] + | beta (NEq (C go)) = [] + | beta (NEq (Bound gp)) = [] + | beta (NEq (Neg gt)) = [] + | beta (NEq (Add (gu, gv))) = [] + | beta (NEq (Sub (gw, gx))) = [] + | beta (NEq (Mul (gy, gz))) = [] + | 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) = [] + | beta (Lt (Cn (cm, c, e))) = (if eqop eq_nat cm 0 then [] else []) + | beta (Le (Cn (dm, c, e))) = (if eqop eq_nat dm 0 then [] else []) + | beta (Gt (Cn (em, c, e))) = (if eqop eq_nat em 0 then [Neg e] else []) + | beta (Ge (Cn (fm, c, e))) = + (if eqop eq_nat fm 0 then [Sub (C (~1 : IntInf.int), e)] else []) + | beta (Eq (Cn (gm, c, e))) = + (if eqop eq_nat gm 0 then [Sub (C (~1 : IntInf.int), e)] else []) + | beta (NEq (Cn (hm, c, e))) = (if eqop eq_nat hm 0 then [Neg e] else []); + +fun member A_ x [] = false + | member A_ x (y :: ys) = eqop A_ x y orelse member A_ x ys; + +fun remdups A_ [] = [] + | remdups A_ (x :: xs) = + (if member A_ x xs then remdups A_ xs else x :: remdups A_ xs); + +fun delta (And (p, q)) = zlcm (delta p) (delta q) + | delta (Or (p, q)) = zlcm (delta p) (delta q) + | delta T = (1 : IntInf.int) + | delta F = (1 : IntInf.int) + | delta (Lt u) = (1 : IntInf.int) + | delta (Le v) = (1 : IntInf.int) + | delta (Gt w) = (1 : IntInf.int) + | delta (Ge x) = (1 : IntInf.int) + | delta (Eq ya) = (1 : IntInf.int) + | delta (NEq z) = (1 : IntInf.int) + | delta (Dvd (aa, C bo)) = (1 : IntInf.int) + | delta (Dvd (aa, Bound bp)) = (1 : IntInf.int) + | delta (Dvd (aa, Neg bt)) = (1 : IntInf.int) + | delta (Dvd (aa, Add (bu, bv))) = (1 : IntInf.int) + | delta (Dvd (aa, Sub (bw, bx))) = (1 : IntInf.int) + | delta (Dvd (aa, Mul (by, bz))) = (1 : IntInf.int) + | delta (NDvd (ac, C co)) = (1 : IntInf.int) + | delta (NDvd (ac, Bound cp)) = (1 : IntInf.int) + | delta (NDvd (ac, Neg ct)) = (1 : IntInf.int) + | delta (NDvd (ac, Add (cu, cv))) = (1 : IntInf.int) + | delta (NDvd (ac, Sub (cw, cx))) = (1 : IntInf.int) + | delta (NDvd (ac, Mul (cy, cz))) = (1 : IntInf.int) + | delta (Not ae) = (1 : IntInf.int) + | delta (Imp (aj, ak)) = (1 : IntInf.int) + | delta (Iff (al, am)) = (1 : IntInf.int) + | delta (E an) = (1 : IntInf.int) + | delta (A ao) = (1 : IntInf.int) + | delta (Closed ap) = (1 : IntInf.int) + | delta (NClosed aq) = (1 : IntInf.int) + | delta (Dvd (b, Cn (cm, c, e))) = + (if eqop eq_nat cm 0 then b else (1 : IntInf.int)) + | delta (NDvd (b, Cn (dm, c, e))) = + (if eqop eq_nat dm 0 then b else (1 : IntInf.int)); + +fun div_int a b = fst (divmoda a b); -fun djf f p q = - (if eq_fm q T then T - else (if eq_fm q F then f p - else (case f p of T => T | F => q | Lt num => Or (f p, q) - | Le num => Or (f p, q) | Gt num => Or (f p, q) - | Ge num => Or (f p, q) | Eq num => Or (f p, q) - | NEq num => Or (f p, q) | Dvd (int, num) => Or (f p, q) - | NDvd (int, num) => Or (f p, q) | Nota fm => Or (f p, q) - | And (fm1, fm2) => Or (f p, q) - | Or (fm1, fm2) => Or (f p, q) - | Impa (fm1, fm2) => Or (f p, q) - | Iffa (fm1, fm2) => Or (f p, q) | E fm => Or (f p, q) - | A fm => Or (f p, q) | Closed nat => Or (f p, q) - | NClosed nat => Or (f p, q)))); +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 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 bt)) = (fn k => Lt (Neg bt)) + | a_beta (Lt (Add (bu, bv))) = (fn k => Lt (Add (bu, bv))) + | a_beta (Lt (Sub (bw, bx))) = (fn k => Lt (Sub (bw, bx))) + | a_beta (Lt (Mul (by, bz))) = (fn k => Lt (Mul (by, bz))) + | a_beta (Le (C co)) = (fn k => Le (C co)) + | a_beta (Le (Bound cp)) = (fn k => Le (Bound cp)) + | a_beta (Le (Neg ct)) = (fn k => Le (Neg ct)) + | a_beta (Le (Add (cu, cv))) = (fn k => Le (Add (cu, cv))) + | a_beta (Le (Sub (cw, cx))) = (fn k => Le (Sub (cw, cx))) + | a_beta (Le (Mul (cy, cz))) = (fn k => Le (Mul (cy, cz))) + | a_beta (Gt (C doa)) = (fn k => Gt (C doa)) + | a_beta (Gt (Bound dp)) = (fn k => Gt (Bound dp)) + | a_beta (Gt (Neg dt)) = (fn k => Gt (Neg dt)) + | a_beta (Gt (Add (du, dv))) = (fn k => Gt (Add (du, dv))) + | a_beta (Gt (Sub (dw, dx))) = (fn k => Gt (Sub (dw, dx))) + | a_beta (Gt (Mul (dy, dz))) = (fn k => Gt (Mul (dy, dz))) + | a_beta (Ge (C eo)) = (fn k => Ge (C eo)) + | a_beta (Ge (Bound ep)) = (fn k => Ge (Bound ep)) + | a_beta (Ge (Neg et)) = (fn k => Ge (Neg et)) + | a_beta (Ge (Add (eu, ev))) = (fn k => Ge (Add (eu, ev))) + | a_beta (Ge (Sub (ew, ex))) = (fn k => Ge (Sub (ew, ex))) + | a_beta (Ge (Mul (ey, ez))) = (fn k => Ge (Mul (ey, ez))) + | a_beta (Eq (C fo)) = (fn k => Eq (C fo)) + | a_beta (Eq (Bound fp)) = (fn k => Eq (Bound fp)) + | a_beta (Eq (Neg ft)) = (fn k => Eq (Neg ft)) + | a_beta (Eq (Add (fu, fv))) = (fn k => Eq (Add (fu, fv))) + | a_beta (Eq (Sub (fw, fx))) = (fn k => Eq (Sub (fw, fx))) + | a_beta (Eq (Mul (fy, fz))) = (fn k => Eq (Mul (fy, fz))) + | a_beta (NEq (C go)) = (fn k => NEq (C go)) + | a_beta (NEq (Bound gp)) = (fn k => NEq (Bound gp)) + | a_beta (NEq (Neg gt)) = (fn k => NEq (Neg gt)) + | a_beta (NEq (Add (gu, gv))) = (fn k => NEq (Add (gu, gv))) + | a_beta (NEq (Sub (gw, gx))) = (fn k => NEq (Sub (gw, gx))) + | a_beta (NEq (Mul (gy, gz))) = (fn k => NEq (Mul (gy, gz))) + | a_beta (Dvd (aa, C ho)) = (fn k => Dvd (aa, C ho)) + | a_beta (Dvd (aa, Bound hp)) = (fn k => Dvd (aa, Bound hp)) + | a_beta (Dvd (aa, Neg ht)) = (fn k => Dvd (aa, Neg ht)) + | a_beta (Dvd (aa, Add (hu, hv))) = (fn k => Dvd (aa, Add (hu, hv))) + | a_beta (Dvd (aa, Sub (hw, hx))) = (fn k => Dvd (aa, Sub (hw, hx))) + | a_beta (Dvd (aa, Mul (hy, hz))) = (fn k => Dvd (aa, Mul (hy, hz))) + | a_beta (NDvd (ac, C io)) = (fn k => NDvd (ac, C io)) + | a_beta (NDvd (ac, Bound ip)) = (fn k => NDvd (ac, Bound ip)) + | a_beta (NDvd (ac, Neg it)) = (fn k => NDvd (ac, Neg it)) + | a_beta (NDvd (ac, Add (iu, iv))) = (fn k => NDvd (ac, Add (iu, iv))) + | a_beta (NDvd (ac, Sub (iw, ix))) = (fn k => NDvd (ac, Sub (iw, ix))) + | a_beta (NDvd (ac, Mul (iy, iz))) = (fn k => NDvd (ac, Mul (iy, iz))) + | 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) + | a_beta (Lt (Cn (cm, c, e))) = + (if eqop eq_nat cm 0 + then (fn k => Lt (Cn (0, (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn k => Lt (Cn (suc (minus_nat cm 1), c, e)))) + | a_beta (Le (Cn (dm, c, e))) = + (if eqop eq_nat dm 0 + then (fn k => Le (Cn (0, (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn k => Le (Cn (suc (minus_nat dm 1), c, e)))) + | a_beta (Gt (Cn (em, c, e))) = + (if eqop eq_nat em 0 + then (fn k => Gt (Cn (0, (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn k => Gt (Cn (suc (minus_nat em 1), c, e)))) + | a_beta (Ge (Cn (fm, c, e))) = + (if eqop eq_nat fm 0 + then (fn k => Ge (Cn (0, (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn k => Ge (Cn (suc (minus_nat fm 1), c, e)))) + | a_beta (Eq (Cn (gm, c, e))) = + (if eqop eq_nat gm 0 + then (fn k => Eq (Cn (0, (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn k => Eq (Cn (suc (minus_nat gm 1), c, e)))) + | a_beta (NEq (Cn (hm, c, e))) = + (if eqop eq_nat hm 0 + then (fn k => NEq (Cn (0, (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn k => NEq (Cn (suc (minus_nat hm 1), c, e)))) + | a_beta (Dvd (i, Cn (im, c, e))) = + (if eqop eq_nat im 0 + then (fn k => + Dvd (IntInf.* (div_int k c, i), + Cn (0, (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn k => Dvd (i, Cn (suc (minus_nat im 1), c, e)))) + | a_beta (NDvd (i, Cn (jm, c, e))) = + (if eqop eq_nat jm 0 + then (fn k => + NDvd (IntInf.* (div_int k c, i), + Cn (0, (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn k => NDvd (i, Cn (suc (minus_nat jm 1), c, e)))); -fun evaldjf f ps = foldr (djf f) ps F; - -fun dj f p = evaldjf f (disjuncts p); +fun zeta (And (p, q)) = zlcm (zeta p) (zeta q) + | zeta (Or (p, q)) = zlcm (zeta p) (zeta q) + | zeta T = (1 : IntInf.int) + | zeta F = (1 : IntInf.int) + | zeta (Lt (C bo)) = (1 : IntInf.int) + | zeta (Lt (Bound bp)) = (1 : IntInf.int) + | zeta (Lt (Neg bt)) = (1 : IntInf.int) + | zeta (Lt (Add (bu, bv))) = (1 : IntInf.int) + | zeta (Lt (Sub (bw, bx))) = (1 : IntInf.int) + | zeta (Lt (Mul (by, bz))) = (1 : IntInf.int) + | zeta (Le (C co)) = (1 : IntInf.int) + | zeta (Le (Bound cp)) = (1 : IntInf.int) + | zeta (Le (Neg ct)) = (1 : IntInf.int) + | zeta (Le (Add (cu, cv))) = (1 : IntInf.int) + | zeta (Le (Sub (cw, cx))) = (1 : IntInf.int) + | zeta (Le (Mul (cy, cz))) = (1 : IntInf.int) + | zeta (Gt (C doa)) = (1 : IntInf.int) + | zeta (Gt (Bound dp)) = (1 : IntInf.int) + | zeta (Gt (Neg dt)) = (1 : IntInf.int) + | zeta (Gt (Add (du, dv))) = (1 : IntInf.int) + | zeta (Gt (Sub (dw, dx))) = (1 : IntInf.int) + | zeta (Gt (Mul (dy, dz))) = (1 : IntInf.int) + | zeta (Ge (C eo)) = (1 : IntInf.int) + | zeta (Ge (Bound ep)) = (1 : IntInf.int) + | zeta (Ge (Neg et)) = (1 : IntInf.int) + | zeta (Ge (Add (eu, ev))) = (1 : IntInf.int) + | zeta (Ge (Sub (ew, ex))) = (1 : IntInf.int) + | zeta (Ge (Mul (ey, ez))) = (1 : IntInf.int) + | zeta (Eq (C fo)) = (1 : IntInf.int) + | zeta (Eq (Bound fp)) = (1 : IntInf.int) + | zeta (Eq (Neg ft)) = (1 : IntInf.int) + | zeta (Eq (Add (fu, fv))) = (1 : IntInf.int) + | zeta (Eq (Sub (fw, fx))) = (1 : IntInf.int) + | zeta (Eq (Mul (fy, fz))) = (1 : IntInf.int) + | zeta (NEq (C go)) = (1 : IntInf.int) + | zeta (NEq (Bound gp)) = (1 : IntInf.int) + | zeta (NEq (Neg gt)) = (1 : IntInf.int) + | zeta (NEq (Add (gu, gv))) = (1 : IntInf.int) + | zeta (NEq (Sub (gw, gx))) = (1 : IntInf.int) + | zeta (NEq (Mul (gy, gz))) = (1 : IntInf.int) + | zeta (Dvd (aa, C ho)) = (1 : IntInf.int) + | zeta (Dvd (aa, Bound hp)) = (1 : IntInf.int) + | zeta (Dvd (aa, Neg ht)) = (1 : IntInf.int) + | zeta (Dvd (aa, Add (hu, hv))) = (1 : IntInf.int) + | zeta (Dvd (aa, Sub (hw, hx))) = (1 : IntInf.int) + | zeta (Dvd (aa, Mul (hy, hz))) = (1 : IntInf.int) + | zeta (NDvd (ac, C io)) = (1 : IntInf.int) + | zeta (NDvd (ac, Bound ip)) = (1 : IntInf.int) + | zeta (NDvd (ac, Neg it)) = (1 : IntInf.int) + | zeta (NDvd (ac, Add (iu, iv))) = (1 : IntInf.int) + | zeta (NDvd (ac, Sub (iw, ix))) = (1 : IntInf.int) + | zeta (NDvd (ac, Mul (iy, iz))) = (1 : IntInf.int) + | zeta (Not ae) = (1 : IntInf.int) + | zeta (Imp (aj, ak)) = (1 : IntInf.int) + | zeta (Iff (al, am)) = (1 : IntInf.int) + | zeta (E an) = (1 : IntInf.int) + | zeta (A ao) = (1 : IntInf.int) + | zeta (Closed ap) = (1 : IntInf.int) + | zeta (NClosed aq) = (1 : IntInf.int) + | zeta (Lt (Cn (cm, b, e))) = + (if eqop eq_nat cm 0 then b else (1 : IntInf.int)) + | zeta (Le (Cn (dm, b, e))) = + (if eqop eq_nat dm 0 then b else (1 : IntInf.int)) + | zeta (Gt (Cn (em, b, e))) = + (if eqop eq_nat em 0 then b else (1 : IntInf.int)) + | zeta (Ge (Cn (fm, b, e))) = + (if eqop eq_nat fm 0 then b else (1 : IntInf.int)) + | zeta (Eq (Cn (gm, b, e))) = + (if eqop eq_nat gm 0 then b else (1 : IntInf.int)) + | zeta (NEq (Cn (hm, b, e))) = + (if eqop eq_nat hm 0 then b else (1 : IntInf.int)) + | zeta (Dvd (i, Cn (im, b, e))) = + (if eqop eq_nat im 0 then b else (1 : IntInf.int)) + | zeta (NDvd (i, Cn (jm, b, e))) = + (if eqop eq_nat jm 0 then b else (1 : IntInf.int)); -fun zsplit0 (Mul (i, a)) = - let - val (i', a') = zsplit0 a; - in - (IntInf.* (i, i'), Mul (i, a')) - end +fun zsplit0 (C c) = ((0 : IntInf.int), C c) + | zsplit0 (Bound n) = + (if eqop eq_nat n 0 then ((1 : IntInf.int), C (0 : IntInf.int)) + else ((0 : IntInf.int), Bound n)) + | zsplit0 (Cn (n, i, a)) = + let + val aa = zsplit0 a; + val (i', a') = aa; + in + (if eqop eq_nat n 0 then (IntInf.+ (i, i'), a') else (i', Cn (n, i, a'))) + end + | zsplit0 (Neg a) = + let + val aa = zsplit0 a; + val (i', a') = aa; + in + (IntInf.~ i', Neg a') + end + | zsplit0 (Add (a, b)) = + let + val aa = zsplit0 a; + val (ia, a') = aa; + val ab = zsplit0 b; + val (ib, b') = ab; + in + (IntInf.+ (ia, ib), Add (a', b')) + end | zsplit0 (Sub (a, b)) = let - val (ia, a') = zsplit0 a; - val (ib, b') = zsplit0 b; + val aa = zsplit0 a; + val (ia, a') = aa; + val ab = zsplit0 b; + val (ib, b') = ab; in (IntInf.- (ia, ib), Sub (a', b')) end - | zsplit0 (Add (a, b)) = + | zsplit0 (Mul (i, a)) = let - val (ia, a') = zsplit0 a; - val (ib, b') = zsplit0 b; - in - (IntInf.+ (ia, ib), Add (a', b')) - end - | zsplit0 (Neg a) = - let - val (i', a') = zsplit0 a; + val aa = zsplit0 a; + val (i', a') = aa; in - (IntInf.~ i', Neg a') - end - | zsplit0 (Cx (i, a)) = - let - val (i', aa) = zsplit0 a; - in - (IntInf.+ (i, i'), aa) - end - | zsplit0 (Bound n) = - (if ((n : IntInf.int) = zero_nat) - then ((1 : IntInf.int), C (0 : IntInf.int)) - else ((0 : IntInf.int), Bound n)) - | zsplit0 (C c) = ((0 : IntInf.int), C c); + (IntInf.* (i, i'), Mul (i, a')) + end; -fun zlfm (NClosed ar) = NClosed ar - | zlfm (Closed aq) = Closed aq - | zlfm (A ap) = A ap - | zlfm (E ao) = E ao - | zlfm (Nota (A cj)) = Nota (A cj) - | zlfm (Nota (E ci)) = Nota (E ci) - | zlfm F = F - | zlfm T = T - | zlfm (Nota (NClosed p)) = Closed p - | zlfm (Nota (Closed p)) = NClosed p - | zlfm (Nota (NDvd (i, a))) = zlfm (Dvd (i, a)) - | zlfm (Nota (Dvd (i, a))) = zlfm (NDvd (i, a)) - | zlfm (Nota (NEq a)) = zlfm (Eq a) - | zlfm (Nota (Eq a)) = zlfm (NEq a) - | zlfm (Nota (Ge a)) = zlfm (Lt a) - | zlfm (Nota (Gt a)) = zlfm (Le a) - | zlfm (Nota (Le a)) = zlfm (Gt a) - | zlfm (Nota (Lt a)) = zlfm (Ge a) - | zlfm (Nota F) = T - | zlfm (Nota T) = F - | zlfm (Nota (Nota p)) = zlfm p - | zlfm (Nota (Iffa (p, q))) = - Or (And (zlfm p, zlfm (Nota q)), And (zlfm (Nota p), zlfm q)) - | zlfm (Nota (Impa (p, q))) = And (zlfm p, zlfm (Nota q)) - | zlfm (Nota (Or (p, q))) = And (zlfm (Nota p), zlfm (Nota q)) - | zlfm (Nota (And (p, q))) = Or (zlfm (Nota p), zlfm (Nota q)) - | zlfm (NDvd (i, a)) = - (if ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (NEq a) - else let - val (c, r) = zsplit0 a; - in - (if ((c : IntInf.int) = (0 : IntInf.int)) then NDvd (abs_int i, r) - else (if IntInf.< ((0 : IntInf.int), c) - then NDvd (abs_int i, Cx (c, r)) - else NDvd (abs_int i, Cx (IntInf.~ c, Neg r)))) - end) - | zlfm (Dvd (i, a)) = - (if ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (Eq a) - else let - val (c, r) = zsplit0 a; - in - (if ((c : IntInf.int) = (0 : IntInf.int)) then Dvd (abs_int i, r) - else (if IntInf.< ((0 : IntInf.int), c) - then Dvd (abs_int i, Cx (c, r)) - else Dvd (abs_int i, Cx (IntInf.~ c, Neg r)))) - end) - | zlfm (NEq 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 (c, r) = zsplit0 a; - in - (if ((c : IntInf.int) = (0 : IntInf.int)) then NEq r - else (if IntInf.< ((0 : IntInf.int), c) then NEq (Cx (c, r)) - else NEq (Cx (IntInf.~ c, Neg r)))) - end - | zlfm (Eq a) = - let - val (c, r) = zsplit0 a; + val aa = zsplit0 a; + val (c, r) = aa; in - (if ((c : IntInf.int) = (0 : IntInf.int)) then Eq r - else (if IntInf.< ((0 : IntInf.int), c) then Eq (Cx (c, r)) - else Eq (Cx (IntInf.~ c, Neg r)))) - end - | zlfm (Ge a) = - let - val (c, r) = zsplit0 a; - in - (if ((c : IntInf.int) = (0 : IntInf.int)) then Ge r - else (if IntInf.< ((0 : IntInf.int), c) then Ge (Cx (c, r)) - else Le (Cx (IntInf.~ c, Neg r)))) - end - | zlfm (Gt a) = - let - val (c, r) = zsplit0 a; - in - (if ((c : IntInf.int) = (0 : IntInf.int)) then Gt r - else (if IntInf.< ((0 : IntInf.int), c) then Gt (Cx (c, r)) - else Lt (Cx (IntInf.~ c, Neg r)))) + (if eqop eq_int c (0 : IntInf.int) then Lt r + else (if IntInf.< ((0 : IntInf.int), c) then Lt (Cn (0, c, r)) + else Gt (Cn (0, IntInf.~ c, Neg r)))) end | zlfm (Le a) = let - val (c, r) = zsplit0 a; + val aa = zsplit0 a; + val (c, r) = aa; in - (if ((c : IntInf.int) = (0 : IntInf.int)) then Le r - else (if IntInf.< ((0 : IntInf.int), c) then Le (Cx (c, r)) - else Ge (Cx (IntInf.~ c, Neg r)))) + (if eqop eq_int c (0 : IntInf.int) then Le r + else (if IntInf.< ((0 : IntInf.int), c) then Le (Cn (0, c, r)) + else Ge (Cn (0, IntInf.~ c, Neg r)))) + end + | zlfm (Gt a) = + let + val aa = zsplit0 a; + val (c, r) = aa; + in + (if eqop eq_int c (0 : IntInf.int) then Gt r + else (if IntInf.< ((0 : IntInf.int), c) then Gt (Cn (0, c, r)) + else Lt (Cn (0, IntInf.~ c, Neg r)))) end - | zlfm (Lt a) = + | zlfm (Ge a) = let - val (c, r) = zsplit0 a; + val aa = zsplit0 a; + val (c, r) = aa; in - (if ((c : IntInf.int) = (0 : IntInf.int)) then Lt r - else (if IntInf.< ((0 : IntInf.int), c) then Lt (Cx (c, r)) - else Gt (Cx (IntInf.~ c, Neg r)))) + (if eqop eq_int c (0 : IntInf.int) then Ge r + else (if IntInf.< ((0 : IntInf.int), c) then Ge (Cn (0, c, r)) + else Le (Cn (0, IntInf.~ c, Neg r)))) + end + | zlfm (Eq a) = + let + val aa = zsplit0 a; + val (c, r) = aa; + in + (if eqop eq_int c (0 : IntInf.int) then Eq r + else (if IntInf.< ((0 : IntInf.int), c) then Eq (Cn (0, c, r)) + else Eq (Cn (0, IntInf.~ c, Neg r)))) + end + | zlfm (NEq a) = + let + val aa = zsplit0 a; + val (c, r) = aa; + in + (if eqop eq_int c (0 : IntInf.int) then NEq r + else (if IntInf.< ((0 : IntInf.int), c) then NEq (Cn (0, c, r)) + else NEq (Cn (0, IntInf.~ c, Neg r)))) end - | zlfm (Iffa (p, q)) = - Or (And (zlfm p, zlfm q), And (zlfm (Nota p), zlfm (Nota q))) - | zlfm (Impa (p, q)) = Or (zlfm (Nota p), zlfm q) - | zlfm (Or (p, q)) = Or (zlfm p, zlfm q) - | zlfm (And (p, q)) = And (zlfm p, zlfm q); - -fun zeta (NClosed aq) = (1 : IntInf.int) - | zeta (Closed ap) = (1 : IntInf.int) - | zeta (A ao) = (1 : IntInf.int) - | zeta (E an) = (1 : IntInf.int) - | zeta (Iffa (al, am)) = (1 : IntInf.int) - | zeta (Impa (aj, ak)) = (1 : IntInf.int) - | zeta (Nota ae) = (1 : IntInf.int) - | zeta (NDvd (ac, Mul (hv, hw))) = (1 : IntInf.int) - | zeta (NDvd (ac, Sub (ht, hu))) = (1 : IntInf.int) - | zeta (NDvd (ac, Add (hr, hs))) = (1 : IntInf.int) - | zeta (NDvd (ac, Neg hq)) = (1 : IntInf.int) - | zeta (NDvd (ac, Bound hn)) = (1 : IntInf.int) - | zeta (NDvd (ac, C hm)) = (1 : IntInf.int) - | zeta (Dvd (aa, Mul (gz, ha))) = (1 : IntInf.int) - | zeta (Dvd (aa, Sub (gx, gy))) = (1 : IntInf.int) - | zeta (Dvd (aa, Add (gv, gw))) = (1 : IntInf.int) - | zeta (Dvd (aa, Neg gu)) = (1 : IntInf.int) - | zeta (Dvd (aa, Bound gr)) = (1 : IntInf.int) - | zeta (Dvd (aa, C gq)) = (1 : IntInf.int) - | zeta (NEq (Mul (gd, ge))) = (1 : IntInf.int) - | zeta (NEq (Sub (gb, gc))) = (1 : IntInf.int) - | zeta (NEq (Add (fz, ga))) = (1 : IntInf.int) - | zeta (NEq (Neg fy)) = (1 : IntInf.int) - | zeta (NEq (Bound fv)) = (1 : IntInf.int) - | zeta (NEq (C fu)) = (1 : IntInf.int) - | zeta (Eq (Mul (fh, fi))) = (1 : IntInf.int) - | zeta (Eq (Sub (ff, fg))) = (1 : IntInf.int) - | zeta (Eq (Add (fd, fe))) = (1 : IntInf.int) - | zeta (Eq (Neg fc)) = (1 : IntInf.int) - | zeta (Eq (Bound ez)) = (1 : IntInf.int) - | zeta (Eq (C ey)) = (1 : IntInf.int) - | zeta (Ge (Mul (el, em))) = (1 : IntInf.int) - | zeta (Ge (Sub (ej, ek))) = (1 : IntInf.int) - | zeta (Ge (Add (eh, ei))) = (1 : IntInf.int) - | zeta (Ge (Neg eg)) = (1 : IntInf.int) - | zeta (Ge (Bound ed)) = (1 : IntInf.int) - | zeta (Ge (C ec)) = (1 : IntInf.int) - | zeta (Gt (Mul (dp, dq))) = (1 : IntInf.int) - | zeta (Gt (Sub (dn, doa))) = (1 : IntInf.int) - | zeta (Gt (Add (dl, dm))) = (1 : IntInf.int) - | zeta (Gt (Neg dk)) = (1 : IntInf.int) - | zeta (Gt (Bound dh)) = (1 : IntInf.int) - | zeta (Gt (C dg)) = (1 : IntInf.int) - | zeta (Le (Mul (ct, cu))) = (1 : IntInf.int) - | zeta (Le (Sub (cr, cs))) = (1 : IntInf.int) - | zeta (Le (Add (cp, cq))) = (1 : IntInf.int) - | zeta (Le (Neg co)) = (1 : IntInf.int) - | zeta (Le (Bound cl)) = (1 : IntInf.int) - | zeta (Le (C ck)) = (1 : IntInf.int) - | zeta (Lt (Mul (bx, by))) = (1 : IntInf.int) - | zeta (Lt (Sub (bv, bw))) = (1 : IntInf.int) - | zeta (Lt (Add (bt, bu))) = (1 : IntInf.int) - | zeta (Lt (Neg bs)) = (1 : IntInf.int) - | zeta (Lt (Bound bp)) = (1 : IntInf.int) - | zeta (Lt (C bo)) = (1 : IntInf.int) - | zeta F = (1 : IntInf.int) - | zeta T = (1 : IntInf.int) - | zeta (NDvd (i, Cx (y, e))) = y - | zeta (Dvd (i, Cx (y, e))) = y - | zeta (Ge (Cx (y, e))) = y - | zeta (Gt (Cx (y, e))) = y - | zeta (Le (Cx (y, e))) = y - | zeta (Lt (Cx (y, e))) = y - | zeta (NEq (Cx (y, e))) = y - | zeta (Eq (Cx (y, e))) = y - | zeta (Or (p, q)) = ilcm (zeta p) (zeta q) - | zeta (And (p, q)) = ilcm (zeta p) (zeta q); - -fun a_beta (NClosed aq) = (fn k => NClosed aq) - | a_beta (Closed ap) = (fn k => Closed ap) - | a_beta (A ao) = (fn k => A ao) - | a_beta (E an) = (fn k => E an) - | a_beta (Iffa (al, am)) = (fn k => Iffa (al, am)) - | a_beta (Impa (aj, ak)) = (fn k => Impa (aj, ak)) - | a_beta (Nota ae) = (fn k => Nota ae) - | a_beta (NDvd (ac, Mul (hv, hw))) = (fn k => NDvd (ac, Mul (hv, hw))) - | a_beta (NDvd (ac, Sub (ht, hu))) = (fn k => NDvd (ac, Sub (ht, hu))) - | a_beta (NDvd (ac, Add (hr, hs))) = (fn k => NDvd (ac, Add (hr, hs))) - | a_beta (NDvd (ac, Neg hq)) = (fn k => NDvd (ac, Neg hq)) - | a_beta (NDvd (ac, Bound hn)) = (fn k => NDvd (ac, Bound hn)) - | a_beta (NDvd (ac, C hm)) = (fn k => NDvd (ac, C hm)) - | a_beta (Dvd (aa, Mul (gz, ha))) = (fn k => Dvd (aa, Mul (gz, ha))) - | a_beta (Dvd (aa, Sub (gx, gy))) = (fn k => Dvd (aa, Sub (gx, gy))) - | a_beta (Dvd (aa, Add (gv, gw))) = (fn k => Dvd (aa, Add (gv, gw))) - | a_beta (Dvd (aa, Neg gu)) = (fn k => Dvd (aa, Neg gu)) - | a_beta (Dvd (aa, Bound gr)) = (fn k => Dvd (aa, Bound gr)) - | a_beta (Dvd (aa, C gq)) = (fn k => Dvd (aa, C gq)) - | a_beta (NEq (Mul (gd, ge))) = (fn k => NEq (Mul (gd, ge))) - | a_beta (NEq (Sub (gb, gc))) = (fn k => NEq (Sub (gb, gc))) - | a_beta (NEq (Add (fz, ga))) = (fn k => NEq (Add (fz, ga))) - | a_beta (NEq (Neg fy)) = (fn k => NEq (Neg fy)) - | a_beta (NEq (Bound fv)) = (fn k => NEq (Bound fv)) - | a_beta (NEq (C fu)) = (fn k => NEq (C fu)) - | a_beta (Eq (Mul (fh, fi))) = (fn k => Eq (Mul (fh, fi))) - | a_beta (Eq (Sub (ff, fg))) = (fn k => Eq (Sub (ff, fg))) - | a_beta (Eq (Add (fd, fe))) = (fn k => Eq (Add (fd, fe))) - | a_beta (Eq (Neg fc)) = (fn k => Eq (Neg fc)) - | a_beta (Eq (Bound ez)) = (fn k => Eq (Bound ez)) - | a_beta (Eq (C ey)) = (fn k => Eq (C ey)) - | a_beta (Ge (Mul (el, em))) = (fn k => Ge (Mul (el, em))) - | a_beta (Ge (Sub (ej, ek))) = (fn k => Ge (Sub (ej, ek))) - | a_beta (Ge (Add (eh, ei))) = (fn k => Ge (Add (eh, ei))) - | a_beta (Ge (Neg eg)) = (fn k => Ge (Neg eg)) - | a_beta (Ge (Bound ed)) = (fn k => Ge (Bound ed)) - | a_beta (Ge (C ec)) = (fn k => Ge (C ec)) - | a_beta (Gt (Mul (dp, dq))) = (fn k => Gt (Mul (dp, dq))) - | a_beta (Gt (Sub (dn, doa))) = (fn k => Gt (Sub (dn, doa))) - | a_beta (Gt (Add (dl, dm))) = (fn k => Gt (Add (dl, dm))) - | a_beta (Gt (Neg dk)) = (fn k => Gt (Neg dk)) - | a_beta (Gt (Bound dh)) = (fn k => Gt (Bound dh)) - | a_beta (Gt (C dg)) = (fn k => Gt (C dg)) - | a_beta (Le (Mul (ct, cu))) = (fn k => Le (Mul (ct, cu))) - | a_beta (Le (Sub (cr, cs))) = (fn k => Le (Sub (cr, cs))) - | a_beta (Le (Add (cp, cq))) = (fn k => Le (Add (cp, cq))) - | a_beta (Le (Neg co)) = (fn k => Le (Neg co)) - | a_beta (Le (Bound cl)) = (fn k => Le (Bound cl)) - | a_beta (Le (C ck)) = (fn k => Le (C ck)) - | a_beta (Lt (Mul (bx, by))) = (fn k => Lt (Mul (bx, by))) - | a_beta (Lt (Sub (bv, bw))) = (fn k => Lt (Sub (bv, bw))) - | a_beta (Lt (Add (bt, bu))) = (fn k => Lt (Add (bt, bu))) - | a_beta (Lt (Neg bs)) = (fn k => Lt (Neg bs)) - | a_beta (Lt (Bound bp)) = (fn k => Lt (Bound bp)) - | a_beta (Lt (C bo)) = (fn k => Lt (C bo)) - | a_beta F = (fn k => F) - | a_beta T = (fn k => T) - | a_beta (NDvd (i, Cx (c, e))) = - (fn k => - NDvd (IntInf.* (div_int k c, i), - Cx ((1 : IntInf.int), Mul (div_int k c, e)))) - | a_beta (Dvd (i, Cx (c, e))) = - (fn k => - Dvd (IntInf.* (div_int k c, i), - Cx ((1 : IntInf.int), Mul (div_int k c, e)))) - | a_beta (Ge (Cx (c, e))) = - (fn k => Ge (Cx ((1 : IntInf.int), Mul (div_int k c, e)))) - | a_beta (Gt (Cx (c, e))) = - (fn k => Gt (Cx ((1 : IntInf.int), Mul (div_int k c, e)))) - | a_beta (Le (Cx (c, e))) = - (fn k => Le (Cx ((1 : IntInf.int), Mul (div_int k c, e)))) - | a_beta (Lt (Cx (c, e))) = - (fn k => Lt (Cx ((1 : IntInf.int), Mul (div_int k c, e)))) - | a_beta (NEq (Cx (c, e))) = - (fn k => NEq (Cx ((1 : IntInf.int), Mul (div_int k c, e)))) - | a_beta (Eq (Cx (c, e))) = - (fn k => Eq (Cx ((1 : IntInf.int), Mul (div_int k c, e)))) - | a_beta (Or (p, q)) = (fn k => Or (a_beta p k, a_beta q k)) - | a_beta (And (p, q)) = (fn k => And (a_beta p k, a_beta q k)); - -fun delta (NClosed aq) = (1 : IntInf.int) - | delta (Closed ap) = (1 : IntInf.int) - | delta (A ao) = (1 : IntInf.int) - | delta (E an) = (1 : IntInf.int) - | delta (Iffa (al, am)) = (1 : IntInf.int) - | delta (Impa (aj, ak)) = (1 : IntInf.int) - | delta (Nota ae) = (1 : IntInf.int) - | delta (NDvd (ac, Mul (ct, cu))) = (1 : IntInf.int) - | delta (NDvd (ac, Sub (cr, cs))) = (1 : IntInf.int) - | delta (NDvd (ac, Add (cp, cq))) = (1 : IntInf.int) - | delta (NDvd (ac, Neg co)) = (1 : IntInf.int) - | delta (NDvd (ac, Bound cl)) = (1 : IntInf.int) - | delta (NDvd (ac, C ck)) = (1 : IntInf.int) - | delta (Dvd (aa, Mul (bx, by))) = (1 : IntInf.int) - | delta (Dvd (aa, Sub (bv, bw))) = (1 : IntInf.int) - | delta (Dvd (aa, Add (bt, bu))) = (1 : IntInf.int) - | delta (Dvd (aa, Neg bs)) = (1 : IntInf.int) - | delta (Dvd (aa, Bound bp)) = (1 : IntInf.int) - | delta (Dvd (aa, C bo)) = (1 : IntInf.int) - | delta (NEq z) = (1 : IntInf.int) - | delta (Eq y) = (1 : IntInf.int) - | delta (Ge x) = (1 : IntInf.int) - | delta (Gt w) = (1 : IntInf.int) - | delta (Le v) = (1 : IntInf.int) - | delta (Lt u) = (1 : IntInf.int) - | delta F = (1 : IntInf.int) - | delta T = (1 : IntInf.int) - | delta (NDvd (y, Cx (c, e))) = y - | delta (Dvd (y, Cx (c, e))) = y - | delta (Or (p, q)) = ilcm (delta p) (delta q) - | delta (And (p, q)) = ilcm (delta p) (delta q); - -fun beta (NClosed aq) = [] - | beta (Closed ap) = [] - | beta (A ao) = [] - | beta (E an) = [] - | beta (Iffa (al, am)) = [] - | beta (Impa (aj, ak)) = [] - | beta (Nota ae) = [] - | beta (NDvd (ac, ad)) = [] - | beta (Dvd (aa, ab)) = [] - | beta (NEq (Mul (gd, ge))) = [] - | beta (NEq (Sub (gb, gc))) = [] - | beta (NEq (Add (fz, ga))) = [] - | beta (NEq (Neg fy)) = [] - | beta (NEq (Bound fv)) = [] - | beta (NEq (C fu)) = [] - | beta (Eq (Mul (fh, fi))) = [] - | beta (Eq (Sub (ff, fg))) = [] - | beta (Eq (Add (fd, fe))) = [] - | beta (Eq (Neg fc)) = [] - | beta (Eq (Bound ez)) = [] - | beta (Eq (C ey)) = [] - | beta (Ge (Mul (el, em))) = [] - | beta (Ge (Sub (ej, ek))) = [] - | beta (Ge (Add (eh, ei))) = [] - | beta (Ge (Neg eg)) = [] - | beta (Ge (Bound ed)) = [] - | beta (Ge (C ec)) = [] - | beta (Gt (Mul (dp, dq))) = [] - | beta (Gt (Sub (dn, doa))) = [] - | beta (Gt (Add (dl, dm))) = [] - | beta (Gt (Neg dk)) = [] - | beta (Gt (Bound dh)) = [] - | beta (Gt (C dg)) = [] - | beta (Le (Mul (ct, cu))) = [] - | beta (Le (Sub (cr, cs))) = [] - | beta (Le (Add (cp, cq))) = [] - | beta (Le (Neg co)) = [] - | beta (Le (Bound cl)) = [] - | beta (Le (C ck)) = [] - | beta (Lt (Mul (bx, by))) = [] - | beta (Lt (Sub (bv, bw))) = [] - | beta (Lt (Add (bt, bu))) = [] - | beta (Lt (Neg bs)) = [] - | beta (Lt (Bound bp)) = [] - | beta (Lt (C bo)) = [] - | beta F = [] - | beta T = [] - | beta (Ge (Cx (c, e))) = [Sub (C (~1 : IntInf.int), e)] - | beta (Gt (Cx (c, e))) = [Neg e] - | beta (Le (Cx (c, e))) = [] - | beta (Lt (Cx (c, e))) = [] - | beta (NEq (Cx (c, e))) = [Neg e] - | beta (Eq (Cx (c, e))) = [Sub (C (~1 : IntInf.int), e)] - | beta (Or (p, q)) = append (beta p) (beta q) - | beta (And (p, q)) = append (beta p) (beta q); - -fun alpha (NClosed aq) = [] - | alpha (Closed ap) = [] - | alpha (A ao) = [] - | alpha (E an) = [] - | alpha (Iffa (al, am)) = [] - | alpha (Impa (aj, ak)) = [] - | alpha (Nota ae) = [] - | alpha (NDvd (ac, ad)) = [] - | alpha (Dvd (aa, ab)) = [] - | alpha (NEq (Mul (gd, ge))) = [] - | alpha (NEq (Sub (gb, gc))) = [] - | alpha (NEq (Add (fz, ga))) = [] - | alpha (NEq (Neg fy)) = [] - | alpha (NEq (Bound fv)) = [] - | alpha (NEq (C fu)) = [] - | alpha (Eq (Mul (fh, fi))) = [] - | alpha (Eq (Sub (ff, fg))) = [] - | alpha (Eq (Add (fd, fe))) = [] - | alpha (Eq (Neg fc)) = [] - | alpha (Eq (Bound ez)) = [] - | alpha (Eq (C ey)) = [] - | alpha (Ge (Mul (el, em))) = [] - | alpha (Ge (Sub (ej, ek))) = [] - | alpha (Ge (Add (eh, ei))) = [] - | alpha (Ge (Neg eg)) = [] - | alpha (Ge (Bound ed)) = [] - | alpha (Ge (C ec)) = [] - | alpha (Gt (Mul (dp, dq))) = [] - | alpha (Gt (Sub (dn, doa))) = [] - | alpha (Gt (Add (dl, dm))) = [] - | alpha (Gt (Neg dk)) = [] - | alpha (Gt (Bound dh)) = [] - | alpha (Gt (C dg)) = [] - | alpha (Le (Mul (ct, cu))) = [] - | alpha (Le (Sub (cr, cs))) = [] - | alpha (Le (Add (cp, cq))) = [] - | alpha (Le (Neg co)) = [] - | alpha (Le (Bound cl)) = [] - | alpha (Le (C ck)) = [] - | alpha (Lt (Mul (bx, by))) = [] - | alpha (Lt (Sub (bv, bw))) = [] - | alpha (Lt (Add (bt, bu))) = [] - | alpha (Lt (Neg bs)) = [] - | alpha (Lt (Bound bp)) = [] - | alpha (Lt (C bo)) = [] - | alpha F = [] - | alpha T = [] - | alpha (Ge (Cx (c, e))) = [] - | alpha (Gt (Cx (c, e))) = [] - | alpha (Le (Cx (c, e))) = [Add (C (~1 : IntInf.int), e)] - | alpha (Lt (Cx (c, e))) = [e] - | alpha (NEq (Cx (c, e))) = [e] - | alpha (Eq (Cx (c, e))) = [Add (C (~1 : IntInf.int), e)] - | alpha (Or (p, q)) = append (alpha p) (alpha q) - | alpha (And (p, q)) = append (alpha p) (alpha q); - -fun numadd (Mul (ar, asa), Mul (aza, azb)) = Add (Mul (ar, asa), Mul (aza, azb)) - | numadd (Mul (ar, asa), Sub (ayy, ayz)) = Add (Mul (ar, asa), Sub (ayy, ayz)) - | numadd (Mul (ar, asa), Add (Mul (azw, Mul (bas, bat)), ayx)) = - Add (Mul (ar, asa), Add (Mul (azw, Mul (bas, bat)), ayx)) - | numadd (Mul (ar, asa), Add (Mul (azw, Sub (baq, bar)), ayx)) = - Add (Mul (ar, asa), Add (Mul (azw, Sub (baq, bar)), ayx)) - | numadd (Mul (ar, asa), Add (Mul (azw, Add (bao, bap)), ayx)) = - Add (Mul (ar, asa), Add (Mul (azw, Add (bao, bap)), ayx)) - | numadd (Mul (ar, asa), Add (Mul (azw, Neg ban), ayx)) = - Add (Mul (ar, asa), Add (Mul (azw, Neg ban), ayx)) - | numadd (Mul (ar, asa), Add (Mul (azw, Cx (bal, bam)), ayx)) = - Add (Mul (ar, asa), Add (Mul (azw, Cx (bal, bam)), ayx)) - | numadd (Mul (ar, asa), Add (Mul (azw, C baj), ayx)) = - Add (Mul (ar, asa), Add (Mul (azw, C baj), ayx)) - | numadd (Mul (ar, asa), Add (Sub (azu, azv), ayx)) = - Add (Mul (ar, asa), Add (Sub (azu, azv), ayx)) - | numadd (Mul (ar, asa), Add (Add (azs, azt), ayx)) = - Add (Mul (ar, asa), Add (Add (azs, azt), ayx)) - | numadd (Mul (ar, asa), Add (Neg azr, ayx)) = - Add (Mul (ar, asa), Add (Neg azr, ayx)) - | numadd (Mul (ar, asa), Add (Cx (azp, azq), ayx)) = - Add (Mul (ar, asa), Add (Cx (azp, azq), ayx)) - | numadd (Mul (ar, asa), Add (Bound azo, ayx)) = - Add (Mul (ar, asa), Add (Bound azo, ayx)) - | numadd (Mul (ar, asa), Add (C azn, ayx)) = - Add (Mul (ar, asa), Add (C azn, ayx)) - | numadd (Mul (ar, asa), Neg ayv) = Add (Mul (ar, asa), Neg ayv) - | numadd (Mul (ar, asa), Cx (ayt, ayu)) = Add (Mul (ar, asa), Cx (ayt, ayu)) - | numadd (Mul (ar, asa), Bound ays) = Add (Mul (ar, asa), Bound ays) - | numadd (Mul (ar, asa), C ayr) = Add (Mul (ar, asa), C ayr) - | numadd (Sub (ap, aq), Mul (awm, awn)) = Add (Sub (ap, aq), Mul (awm, awn)) - | numadd (Sub (ap, aq), Sub (awk, awl)) = Add (Sub (ap, aq), Sub (awk, awl)) - | 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), 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, Add (aya, ayb)), awj)) = - Add (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), 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, Cx (axx, axy)), awj)) = - Add (Sub (ap, aq), Add (Mul (axi, Cx (axx, axy)), 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 (Sub (axg, axh), awj)) = - Add (Sub (ap, aq), Add (Sub (axg, axh), awj)) - | numadd (Sub (ap, aq), Add (Add (axe, axf), awj)) = - Add (Sub (ap, aq), Add (Add (axe, axf), awj)) - | numadd (Sub (ap, aq), Add (Neg axd, awj)) = - Add (Sub (ap, aq), Add (Neg axd, awj)) - | numadd (Sub (ap, aq), Add (Cx (axb, axc), awj)) = - Add (Sub (ap, aq), Add (Cx (axb, axc), awj)) - | numadd (Sub (ap, aq), Add (Bound axa, awj)) = - Add (Sub (ap, aq), Add (Bound axa, awj)) - | numadd (Sub (ap, aq), Add (C awz, awj)) = - Add (Sub (ap, aq), Add (C awz, awj)) - | numadd (Sub (ap, aq), Neg awh) = Add (Sub (ap, aq), Neg awh) - | numadd (Sub (ap, aq), Cx (awf, awg)) = Add (Sub (ap, aq), Cx (awf, awg)) - | numadd (Sub (ap, aq), Bound awe) = Add (Sub (ap, aq), Bound awe) - | numadd (Sub (ap, aq), C awd) = Add (Sub (ap, aq), C awd) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) - | 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), 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), 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, 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, 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, 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, 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 (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 (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 (Neg aup, atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, 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 (Bound aum, atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) - | 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), Neg att) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) - | 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), Bound atq) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), C atp) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), C atp) - | 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, 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), 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), 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, 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, 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, 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, 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 (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 (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 (Neg asb, arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, 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 (Bound ary, arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) - | 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), Neg arf) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) - | 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), Bound arc) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), C arb) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), C arb) - | 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, 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), 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), 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, 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, 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, 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, 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 (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 (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 (Neg apn, aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, 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 (Bound apk, aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) - | 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), Neg aor) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) - | 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), Bound aoo) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), C aon) = - Add (Add (Mul (mc, Add (acl, acm)), ao), C aon) - | numadd (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) = - Add (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) - | 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), 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), 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, 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, 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, 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, C anr), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, anda), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, anda), 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 (Neg amz, amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Neg amz, 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 (Bound amw, amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) - | 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), Neg amd) = - Add (Add (Mul (mc, Neg ack), ao), Neg amd) - | 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), Bound ama) = - Add (Add (Mul (mc, Neg ack), ao), Bound ama) - | numadd (Add (Mul (mc, Neg ack), ao), C alz) = - Add (Add (Mul (mc, Neg ack), ao), C alz) - | 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, 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), 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), 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, 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, 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, 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, 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 (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 (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 (Neg akl, ajr)) = - Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Neg akl, 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 (Bound aki, ajr)) = - Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Bound aki, ajr)) - | 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), Neg ajp) = - Add (Add (Mul (mc, Cx (aci, acj)), ao), Neg ajp) - | 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), Bound ajm) = - Add (Add (Mul (mc, Cx (aci, acj)), ao), Bound ajm) - | numadd (Add (Mul (mc, Cx (aci, acj)), ao), C ajl) = - Add (Add (Mul (mc, Cx (aci, acj)), ao), C ajl) - | numadd (Add (Mul (mc, C acg), ao), Mul (adl, adm)) = - Add (Add (Mul (mc, C acg), ao), Mul (adl, adm)) - | 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), 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), 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, 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, 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, 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, C aeu), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), 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 (Add (aed, aee), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), 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 (Cx (aea, aeb), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Cx (aea, aeb), 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 (C ady, adi)) = - Add (Add (Mul (mc, C acg), ao), Add (C ady, adi)) - | 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), Cx (ade, adf)) = - Add (Add (Mul (mc, C acg), ao), Cx (ade, adf)) - | 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), C adc) = - Add (Add (Mul (mc, C acg), ao), C adc) - | numadd (Add (Sub (ma, mb), ao), Mul (zq, zr)) = - Add (Add (Sub (ma, mb), ao), Mul (zq, zr)) - | numadd (Add (Sub (ma, mb), ao), Sub (zo, zp)) = - Add (Add (Sub (ma, mb), ao), Sub (zo, zp)) - | 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), 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, 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, Neg abd), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), 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, C aaz), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), 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 (Add (aai, aaj), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), 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 (Cx (aaf, aag), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Cx (aaf, aag), 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 (C aad, zn)) = - Add (Add (Sub (ma, mb), ao), Add (C aad, zn)) - | numadd (Add (Sub (ma, mb), ao), Neg zl) = - Add (Add (Sub (ma, mb), ao), Neg zl) - | numadd (Add (Sub (ma, mb), ao), Cx (zj, zk)) = - Add (Add (Sub (ma, mb), ao), Cx (zj, zk)) - | numadd (Add (Sub (ma, mb), ao), Bound zi) = - Add (Add (Sub (ma, mb), ao), Bound zi) - | numadd (Add (Sub (ma, mb), ao), C zh) = Add (Add (Sub (ma, mb), ao), C zh) - | numadd (Add (Add (ly, lz), ao), Mul (xc, xd)) = - Add (Add (Add (ly, lz), ao), Mul (xc, xd)) - | numadd (Add (Add (ly, lz), ao), Sub (xa, xb)) = - Add (Add (Add (ly, lz), ao), Sub (xa, xb)) - | 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), 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, 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, Neg yp), wz)) = - Add (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), 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, C yl), wz)) = - Add (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), 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 (Add (xu, xv), wz)) = - Add (Add (Add (ly, lz), ao), Add (Add (xu, xv), 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 (Cx (xr, xs), wz)) = - Add (Add (Add (ly, lz), ao), Add (Cx (xr, xs), 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 (C xp, wz)) = - Add (Add (Add (ly, lz), ao), Add (C xp, wz)) - | numadd (Add (Add (ly, lz), ao), Neg wx) = - Add (Add (Add (ly, lz), ao), Neg wx) - | numadd (Add (Add (ly, lz), ao), Cx (wv, ww)) = - Add (Add (Add (ly, lz), ao), Cx (wv, ww)) - | numadd (Add (Add (ly, lz), ao), Bound wu) = - Add (Add (Add (ly, lz), ao), Bound wu) - | numadd (Add (Add (ly, lz), ao), C wt) = Add (Add (Add (ly, lz), ao), C wt) - | numadd (Add (Neg lx, ao), Mul (uo, up)) = - Add (Add (Neg lx, ao), Mul (uo, up)) - | numadd (Add (Neg lx, ao), Sub (um, un)) = - Add (Add (Neg lx, ao), Sub (um, un)) - | 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), 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, Add (wc, wd)), ul)) = - Add (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), 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, Cx (vz, wa)), ul)) = - Add (Add (Neg lx, ao), Add (Mul (vk, Cx (vz, wa)), 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 (Sub (vi, vj), ul)) = - Add (Add (Neg lx, ao), Add (Sub (vi, vj), 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 (Neg vf, ul)) = - Add (Add (Neg lx, ao), Add (Neg vf, 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 (Bound vc, ul)) = - Add (Add (Neg lx, ao), Add (Bound vc, ul)) - | numadd (Add (Neg lx, ao), Add (C vb, ul)) = - Add (Add (Neg lx, ao), Add (C vb, ul)) - | numadd (Add (Neg lx, ao), Neg uj) = Add (Add (Neg lx, ao), Neg uj) - | numadd (Add (Neg lx, ao), Cx (uh, ui)) = Add (Add (Neg lx, ao), Cx (uh, ui)) - | numadd (Add (Neg lx, ao), Bound ug) = Add (Add (Neg lx, ao), Bound ug) - | numadd (Add (Neg lx, ao), C uf) = Add (Add (Neg lx, ao), C uf) - | numadd (Add (Cx (lv, lw), ao), Mul (sa, sb)) = - Add (Add (Cx (lv, lw), ao), Mul (sa, sb)) - | numadd (Add (Cx (lv, lw), ao), Sub (ry, rz)) = - Add (Add (Cx (lv, lw), ao), Sub (ry, rz)) - | 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), 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, 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, Neg tn), rx)) = - Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Neg tn), 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, C tj), rx)) = - Add (Add (Cx (lv, lw), ao), Add (Mul (sw, C tj), 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 (Add (ss, st), rx)) = - Add (Add (Cx (lv, lw), ao), Add (Add (ss, st), 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 (Cx (sp, sq), rx)) = - Add (Add (Cx (lv, lw), ao), Add (Cx (sp, sq), 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 (C sn, rx)) = - Add (Add (Cx (lv, lw), ao), Add (C sn, rx)) - | numadd (Add (Cx (lv, lw), ao), Neg rv) = Add (Add (Cx (lv, lw), ao), Neg rv) - | numadd (Add (Cx (lv, lw), ao), Cx (rt, ru)) = - Add (Add (Cx (lv, lw), ao), Cx (rt, ru)) - | numadd (Add (Cx (lv, lw), ao), Bound rs) = - Add (Add (Cx (lv, lw), ao), Bound rs) - | numadd (Add (Cx (lv, lw), ao), C rr) = Add (Add (Cx (lv, lw), ao), C rr) - | numadd (Add (Bound lu, ao), Mul (pm, pn)) = - Add (Add (Bound lu, ao), Mul (pm, pn)) - | numadd (Add (Bound lu, ao), Sub (pk, pl)) = - Add (Add (Bound lu, ao), Sub (pk, pl)) - | 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), 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, Add (ra, rb)), pj)) = - Add (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), 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, Cx (qx, qy)), pj)) = - Add (Add (Bound lu, ao), Add (Mul (qi, Cx (qx, qy)), 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 (Sub (qg, qh), pj)) = - Add (Add (Bound lu, ao), Add (Sub (qg, qh), 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 (Neg qd, pj)) = - Add (Add (Bound lu, ao), Add (Neg qd, 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 (Bound qa, pj)) = - Add (Add (Bound lu, ao), Add (Bound qa, pj)) - | numadd (Add (Bound lu, ao), Add (C pz, pj)) = - Add (Add (Bound lu, ao), Add (C pz, pj)) - | numadd (Add (Bound lu, ao), Neg ph) = Add (Add (Bound lu, ao), Neg ph) - | numadd (Add (Bound lu, ao), Cx (pf, pg)) = - Add (Add (Bound lu, ao), Cx (pf, pg)) - | numadd (Add (Bound lu, ao), Bound pe) = Add (Add (Bound lu, ao), Bound pe) - | numadd (Add (Bound lu, ao), C pd) = Add (Add (Bound lu, ao), C pd) - | numadd (Add (C lt, ao), Mul (my, mz)) = Add (Add (C lt, ao), Mul (my, mz)) - | numadd (Add (C lt, ao), Sub (mw, mx)) = Add (Add (C lt, ao), Sub (mw, mx)) - | 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), Add (Mul (nu, Sub (ooa, opa)), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, Sub (ooa, opa)), 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, Neg ol), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, Neg ol), 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, C oh), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, C oh), 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 (Add (nq, nr), mv)) = - Add (Add (C lt, ao), Add (Add (nq, nr), 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 (Cx (nn, no), mv)) = - Add (Add (C lt, ao), Add (Cx (nn, no), 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 (C nl, mv)) = - Add (Add (C lt, ao), Add (C nl, mv)) - | numadd (Add (C lt, ao), Neg mt) = Add (Add (C lt, ao), Neg mt) - | numadd (Add (C lt, ao), Cx (mr, ms)) = Add (Add (C lt, ao), Cx (mr, ms)) - | numadd (Add (C lt, ao), Bound mq) = Add (Add (C lt, ao), Bound mq) - | numadd (Add (C lt, ao), C mp) = Add (Add (C lt, ao), C mp) - | numadd (Neg am, Mul (jd, je)) = Add (Neg am, Mul (jd, je)) - | numadd (Neg am, Sub (jb, jc)) = Add (Neg am, Sub (jb, jc)) - | numadd (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) = - Add (Neg am, Add (Mul (jz, Mul (kv, kw)), 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, Add (kr, ks)), ja)) = - Add (Neg am, Add (Mul (jz, Add (kr, ks)), 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, Cx (ko, kp)), ja)) = - Add (Neg am, Add (Mul (jz, Cx (ko, kp)), ja)) - | numadd (Neg am, Add (Mul (jz, C km), ja)) = - Add (Neg am, Add (Mul (jz, C km), ja)) - | numadd (Neg am, Add (Sub (jx, jy), ja)) = - Add (Neg am, Add (Sub (jx, jy), ja)) - | numadd (Neg am, Add (Add (jv, jw), ja)) = - Add (Neg am, Add (Add (jv, jw), ja)) - | numadd (Neg am, Add (Neg ju, ja)) = Add (Neg am, Add (Neg ju, ja)) - | numadd (Neg am, Add (Cx (js, jt), ja)) = Add (Neg am, Add (Cx (js, jt), ja)) - | numadd (Neg am, Add (Bound jr, ja)) = Add (Neg am, Add (Bound jr, ja)) - | numadd (Neg am, Add (C jq, ja)) = Add (Neg am, Add (C jq, ja)) - | numadd (Neg am, Neg iy) = Add (Neg am, Neg iy) - | numadd (Neg am, Cx (iw, ix)) = Add (Neg am, Cx (iw, ix)) - | numadd (Neg am, Bound iv) = Add (Neg am, Bound iv) - | numadd (Neg am, C iu) = Add (Neg am, C iu) - | numadd (Cx (ak, al), Mul (gp, gq)) = Add (Cx (ak, al), Mul (gp, gq)) - | numadd (Cx (ak, al), Sub (gn, go)) = Add (Cx (ak, al), Sub (gn, go)) - | 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), Add (Mul (hl, Sub (ifa, ig)), gm)) = - Add (Cx (ak, al), Add (Mul (hl, Sub (ifa, ig)), 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, Neg ic), gm)) = - Add (Cx (ak, al), Add (Mul (hl, Neg ic), 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, C hy), gm)) = - Add (Cx (ak, al), Add (Mul (hl, C hy), gm)) - | numadd (Cx (ak, al), Add (Sub (hj, hk), gm)) = - Add (Cx (ak, al), Add (Sub (hj, hk), gm)) - | numadd (Cx (ak, al), Add (Add (hh, hi), gm)) = - Add (Cx (ak, al), Add (Add (hh, hi), gm)) - | numadd (Cx (ak, al), Add (Neg hg, gm)) = Add (Cx (ak, al), Add (Neg hg, gm)) - | numadd (Cx (ak, al), Add (Cx (he, hf), gm)) = - Add (Cx (ak, al), Add (Cx (he, hf), gm)) - | numadd (Cx (ak, al), Add (Bound hd, gm)) = - Add (Cx (ak, al), Add (Bound hd, gm)) - | numadd (Cx (ak, al), Add (C hc, gm)) = Add (Cx (ak, al), Add (C hc, gm)) - | numadd (Cx (ak, al), Neg gk) = Add (Cx (ak, al), Neg gk) - | numadd (Cx (ak, al), Cx (gi, gj)) = Add (Cx (ak, al), Cx (gi, gj)) - | numadd (Cx (ak, al), Bound gh) = Add (Cx (ak, al), Bound gh) - | numadd (Cx (ak, al), C gg) = Add (Cx (ak, al), C gg) - | numadd (Bound aj, Mul (eb, ec)) = Add (Bound aj, Mul (eb, ec)) - | numadd (Bound aj, Sub (dz, ea)) = Add (Bound aj, Sub (dz, ea)) - | numadd (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) = - Add (Bound aj, Add (Mul (ex, Mul (ft, fu)), 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, Add (fp, fq)), dy)) = - Add (Bound aj, Add (Mul (ex, Add (fp, fq)), 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, Cx (fm, fna)), dy)) = - Add (Bound aj, Add (Mul (ex, Cx (fm, fna)), dy)) - | numadd (Bound aj, Add (Mul (ex, C fk), dy)) = - Add (Bound aj, Add (Mul (ex, C fk), dy)) - | numadd (Bound aj, Add (Sub (ev, ew), dy)) = - Add (Bound aj, Add (Sub (ev, ew), dy)) - | numadd (Bound aj, Add (Add (et, eu), dy)) = - Add (Bound aj, Add (Add (et, eu), dy)) - | numadd (Bound aj, Add (Neg es, dy)) = Add (Bound aj, Add (Neg es, dy)) - | numadd (Bound aj, Add (Cx (eq, er), dy)) = - Add (Bound aj, Add (Cx (eq, er), dy)) - | numadd (Bound aj, Add (Bound ep, dy)) = Add (Bound aj, Add (Bound ep, dy)) - | numadd (Bound aj, Add (C eo, dy)) = Add (Bound aj, Add (C eo, dy)) - | numadd (Bound aj, Neg dw) = Add (Bound aj, Neg dw) - | numadd (Bound aj, Cx (du, dv)) = Add (Bound aj, Cx (du, dv)) - | numadd (Bound aj, Bound dt) = Add (Bound aj, Bound dt) - | numadd (Bound aj, C ds) = Add (Bound aj, C ds) - | numadd (C ai, Mul (bn, bo)) = Add (C ai, Mul (bn, bo)) - | numadd (C ai, Sub (bl, bm)) = Add (C ai, Sub (bl, bm)) - | numadd (C ai, Add (Mul (cj, Mul (df, dg)), bk)) = - Add (C ai, Add (Mul (cj, Mul (df, dg)), 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, Add (db, dc)), bk)) = - Add (C ai, Add (Mul (cj, Add (db, dc)), 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, Cx (cy, cz)), bk)) = - Add (C ai, Add (Mul (cj, Cx (cy, cz)), bk)) - | numadd (C ai, Add (Mul (cj, C cw), bk)) = - Add (C ai, Add (Mul (cj, C cw), bk)) - | numadd (C ai, Add (Sub (ch, ci), bk)) = Add (C ai, Add (Sub (ch, ci), bk)) - | numadd (C ai, Add (Add (cf, cg), bk)) = Add (C ai, Add (Add (cf, cg), bk)) - | numadd (C ai, Add (Neg ce, bk)) = Add (C ai, Add (Neg ce, bk)) - | numadd (C ai, Add (Cx (cc, cd), bk)) = Add (C ai, Add (Cx (cc, cd), bk)) - | numadd (C ai, Add (Bound cb, bk)) = Add (C ai, Add (Bound cb, bk)) - | numadd (C ai, Add (C ca, bk)) = Add (C ai, Add (C ca, bk)) - | numadd (C ai, Neg bi) = Add (C ai, Neg bi) - | numadd (C ai, Cx (bg, bh)) = Add (C ai, Cx (bg, bh)) - | numadd (C ai, Bound bf) = Add (C ai, Bound bf) - | numadd (C b1, C b2) = C (IntInf.+ (b1, b2)) - | numadd (Mul (ag, ah), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Mul (ag, ah), r2)) - | numadd (Sub (ae, af), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Sub (ae, af), 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 (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, 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, 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, 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, C abv), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, C abv), 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 (Add (ln, lo), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Add (ln, lo), 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 (Cx (lk, ll), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Cx (lk, ll), 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 (C li, ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (C li, ad), r2)) - | numadd (Neg ab, Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Neg ab, r2)) - | numadd (Cx (y, z), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Cx (y, z), r2)) - | numadd (Bound x, Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Bound x, r2)) - | numadd (C w, Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (C w, r2)) - | numadd (Add (Mul (c1, Bound n1), r1), Mul (afz, aga)) = - Add (Mul (c1, Bound n1), numadd (r1, Mul (afz, aga))) - | 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), 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), 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, 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, 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, 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, C aie), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, C aie), 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 (Add (ahc, ahd), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Add (ahc, ahd), 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 (Cx (agz, aha), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Cx (agz, aha), 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 (C agx, afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (C agx, afw))) - | 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), Cx (afs, aft)) = - Add (Mul (c1, Bound n1), numadd (r1, Cx (afs, aft))) - | 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), C afq) = - Add (Mul (c1, Bound n1), numadd (r1, C afq)) - | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (c2, Bound n2), r2)) = - (if ((n1 : IntInf.int) = n2) - then let - val c = IntInf.+ (c1, c2); + | zlfm (Dvd (i, a)) = + (if eqop eq_int i (0 : IntInf.int) then zlfm (Eq a) + else let + val aa = zsplit0 a; + val (c, r) = aa; + in + (if eqop eq_int c (0 : IntInf.int) then Dvd (abs_int i, r) + else (if IntInf.< ((0 : IntInf.int), c) + then Dvd (abs_int i, Cn (0, c, r)) + else Dvd (abs_int i, Cn (0, IntInf.~ c, Neg r)))) + end) + | zlfm (NDvd (i, a)) = + (if eqop eq_int i (0 : IntInf.int) then zlfm (NEq a) + else let + val aa = zsplit0 a; + val (c, r) = aa; in - (if ((c : IntInf.int) = (0 : IntInf.int)) then numadd (r1, r2) - else Add (Mul (c, Bound n1), numadd (r1, r2))) - end - else (if IntInf.<= (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)))); - -fun nummul i (Sub (v, va)) = Mul (i, Sub (v, va)) - | nummul i (Neg v) = Mul (i, Neg v) - | nummul i (Cx (v, va)) = Mul (i, Cx (v, va)) - | nummul i (Bound v) = Mul (i, Bound v) - | nummul i (Mul (c, t)) = nummul (IntInf.* (i, c)) t - | nummul i (Add (a, b)) = numadd (nummul i a, nummul i b) - | nummul i (C j) = C (IntInf.* (i, j)); - -fun numneg t = nummul (IntInf.~ (1 : IntInf.int)) t; - -fun numsub s t = - (if eq_num s t then C (0 : IntInf.int) else numadd (s, numneg t)); - -fun simpnum (Cx (v, va)) = Cx (v, va) - | simpnum (Mul (i, t)) = - (if ((i : IntInf.int) = (0 : IntInf.int)) then C (0 : IntInf.int) - else nummul i (simpnum t)) - | simpnum (Sub (t, s)) = numsub (simpnum t) (simpnum s) - | simpnum (Add (t, s)) = numadd (simpnum t, simpnum s) - | simpnum (Neg t) = numneg (simpnum t) - | simpnum (Bound n) = - Add (Mul ((1 : IntInf.int), Bound n), C (0 : IntInf.int)) - | simpnum (C j) = C j; - -val eq_numa = {eq = eq_num} : num eq; - -fun mirror (NClosed aq) = NClosed aq - | mirror (Closed ap) = Closed ap - | mirror (A ao) = A ao - | mirror (E an) = E an - | mirror (Iffa (al, am)) = Iffa (al, am) - | mirror (Impa (aj, ak)) = Impa (aj, ak) - | mirror (Nota ae) = Nota ae - | mirror (NDvd (ac, Mul (hv, hw))) = NDvd (ac, Mul (hv, hw)) - | mirror (NDvd (ac, Sub (ht, hu))) = NDvd (ac, Sub (ht, hu)) - | mirror (NDvd (ac, Add (hr, hs))) = NDvd (ac, Add (hr, hs)) - | mirror (NDvd (ac, Neg hq)) = NDvd (ac, Neg hq) - | mirror (NDvd (ac, Bound hn)) = NDvd (ac, Bound hn) - | mirror (NDvd (ac, C hm)) = NDvd (ac, C hm) - | mirror (Dvd (aa, Mul (gz, ha))) = Dvd (aa, Mul (gz, ha)) - | mirror (Dvd (aa, Sub (gx, gy))) = Dvd (aa, Sub (gx, gy)) - | mirror (Dvd (aa, Add (gv, gw))) = Dvd (aa, Add (gv, gw)) - | mirror (Dvd (aa, Neg gu)) = Dvd (aa, Neg gu) - | mirror (Dvd (aa, Bound gr)) = Dvd (aa, Bound gr) - | mirror (Dvd (aa, C gq)) = Dvd (aa, C gq) - | mirror (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge)) - | mirror (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc)) - | mirror (NEq (Add (fz, ga))) = NEq (Add (fz, ga)) - | mirror (NEq (Neg fy)) = NEq (Neg fy) - | mirror (NEq (Bound fv)) = NEq (Bound fv) - | mirror (NEq (C fu)) = NEq (C fu) - | mirror (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi)) - | mirror (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg)) - | mirror (Eq (Add (fd, fe))) = Eq (Add (fd, fe)) - | mirror (Eq (Neg fc)) = Eq (Neg fc) - | mirror (Eq (Bound ez)) = Eq (Bound ez) - | mirror (Eq (C ey)) = Eq (C ey) - | mirror (Ge (Mul (el, em))) = Ge (Mul (el, em)) - | mirror (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek)) - | mirror (Ge (Add (eh, ei))) = Ge (Add (eh, ei)) - | mirror (Ge (Neg eg)) = Ge (Neg eg) - | mirror (Ge (Bound ed)) = Ge (Bound ed) - | mirror (Ge (C ec)) = Ge (C ec) - | mirror (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq)) - | mirror (Gt (Sub (dn, doa))) = Gt (Sub (dn, doa)) - | mirror (Gt (Add (dl, dm))) = Gt (Add (dl, dm)) - | mirror (Gt (Neg dk)) = Gt (Neg dk) - | mirror (Gt (Bound dh)) = Gt (Bound dh) - | mirror (Gt (C dg)) = Gt (C dg) - | mirror (Le (Mul (ct, cu))) = Le (Mul (ct, cu)) - | mirror (Le (Sub (cr, cs))) = Le (Sub (cr, cs)) - | mirror (Le (Add (cp, cq))) = Le (Add (cp, cq)) - | mirror (Le (Neg co)) = Le (Neg co) - | mirror (Le (Bound cl)) = Le (Bound cl) - | mirror (Le (C ck)) = Le (C ck) - | mirror (Lt (Mul (bx, by))) = Lt (Mul (bx, by)) - | mirror (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw)) - | mirror (Lt (Add (bt, bu))) = Lt (Add (bt, bu)) - | mirror (Lt (Neg bs)) = Lt (Neg bs) - | mirror (Lt (Bound bp)) = Lt (Bound bp) - | mirror (Lt (C bo)) = Lt (C bo) - | mirror F = F - | mirror T = T - | mirror (NDvd (i, Cx (c, e))) = NDvd (i, Cx (c, Neg e)) - | mirror (Dvd (i, Cx (c, e))) = Dvd (i, Cx (c, Neg e)) - | mirror (Ge (Cx (c, e))) = Le (Cx (c, Neg e)) - | mirror (Gt (Cx (c, e))) = Lt (Cx (c, Neg e)) - | mirror (Le (Cx (c, e))) = Ge (Cx (c, Neg e)) - | mirror (Lt (Cx (c, e))) = Gt (Cx (c, Neg e)) - | mirror (NEq (Cx (c, e))) = NEq (Cx (c, Neg e)) - | mirror (Eq (Cx (c, e))) = Eq (Cx (c, Neg e)) - | mirror (Or (p, q)) = Or (mirror p, mirror q) - | mirror (And (p, q)) = And (mirror p, mirror q); + (if eqop eq_int c (0 : IntInf.int) then NDvd (abs_int i, r) + else (if IntInf.< ((0 : IntInf.int), c) + then NDvd (abs_int i, Cn (0, c, r)) + else NDvd (abs_int i, Cn (0, IntInf.~ c, Neg r)))) + 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 unita p = let val p' = zlfm p; val l = zeta p'; val q = - And (Dvd (l, Cx ((1 : IntInf.int), C (0 : IntInf.int))), a_beta p' l); + And (Dvd (l, Cn (0, (1 : IntInf.int), C (0 : IntInf.int))), a_beta p' l); val d = delta q; val b = remdups eq_numa (map simpnum (beta q)); val a = remdups eq_numa (map simpnum (alpha q)); in - (if IntInf.<= ((size_list b), (size_list a)) then (q, (b, d)) + (if IntInf.<= (size_list b, size_list a) then (q, (b, d)) else (mirror q, (a, d))) end; -fun iupt i j = - (if IntInf.< (j, i) then [] - else i :: iupt (IntInf.+ (i, (1 : IntInf.int))) j); - -fun minusinf (NClosed aq) = NClosed aq - | minusinf (Closed ap) = Closed ap - | minusinf (A ao) = A ao - | minusinf (E an) = E an - | minusinf (Iffa (al, am)) = Iffa (al, am) - | minusinf (Impa (aj, ak)) = Impa (aj, ak) - | minusinf (Nota ae) = Nota ae - | minusinf (NDvd (ac, ad)) = NDvd (ac, ad) - | minusinf (Dvd (aa, ab)) = Dvd (aa, ab) - | minusinf (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge)) - | minusinf (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc)) - | minusinf (NEq (Add (fz, ga))) = NEq (Add (fz, ga)) - | minusinf (NEq (Neg fy)) = NEq (Neg fy) - | minusinf (NEq (Bound fv)) = NEq (Bound fv) - | minusinf (NEq (C fu)) = NEq (C fu) - | minusinf (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi)) - | minusinf (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg)) - | minusinf (Eq (Add (fd, fe))) = Eq (Add (fd, fe)) - | minusinf (Eq (Neg fc)) = Eq (Neg fc) - | minusinf (Eq (Bound ez)) = Eq (Bound ez) - | minusinf (Eq (C ey)) = Eq (C ey) - | minusinf (Ge (Mul (el, em))) = Ge (Mul (el, em)) - | minusinf (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek)) - | minusinf (Ge (Add (eh, ei))) = Ge (Add (eh, ei)) - | minusinf (Ge (Neg eg)) = Ge (Neg eg) - | minusinf (Ge (Bound ed)) = Ge (Bound ed) - | minusinf (Ge (C ec)) = Ge (C ec) - | minusinf (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq)) - | minusinf (Gt (Sub (dn, doa))) = Gt (Sub (dn, doa)) - | minusinf (Gt (Add (dl, dm))) = Gt (Add (dl, dm)) - | minusinf (Gt (Neg dk)) = Gt (Neg dk) - | minusinf (Gt (Bound dh)) = Gt (Bound dh) - | minusinf (Gt (C dg)) = Gt (C dg) - | minusinf (Le (Mul (ct, cu))) = Le (Mul (ct, cu)) - | minusinf (Le (Sub (cr, cs))) = Le (Sub (cr, cs)) - | minusinf (Le (Add (cp, cq))) = Le (Add (cp, cq)) - | minusinf (Le (Neg co)) = Le (Neg co) - | minusinf (Le (Bound cl)) = Le (Bound cl) - | minusinf (Le (C ck)) = Le (C ck) - | minusinf (Lt (Mul (bx, by))) = Lt (Mul (bx, by)) - | minusinf (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw)) - | minusinf (Lt (Add (bt, bu))) = Lt (Add (bt, bu)) - | minusinf (Lt (Neg bs)) = Lt (Neg bs) - | minusinf (Lt (Bound bp)) = Lt (Bound bp) - | minusinf (Lt (C bo)) = Lt (C bo) - | minusinf F = F - | minusinf T = T - | minusinf (Ge (Cx (c, e))) = F - | minusinf (Gt (Cx (c, e))) = F - | minusinf (Le (Cx (c, e))) = T - | minusinf (Lt (Cx (c, e))) = T - | minusinf (NEq (Cx (c, e))) = T - | minusinf (Eq (Cx (c, e))) = F - | minusinf (Or (p, q)) = Or (minusinf p, minusinf q) - | minusinf (And (p, q)) = And (minusinf p, minusinf q); - -fun numsubst0 t (Mul (i, a)) = Mul (i, numsubst0 t a) - | numsubst0 t (Sub (a, b)) = Sub (numsubst0 t a, numsubst0 t b) - | numsubst0 t (Add (a, b)) = Add (numsubst0 t a, numsubst0 t b) - | numsubst0 t (Neg a) = Neg (numsubst0 t a) - | numsubst0 t (Cx (i, a)) = Add (Mul (i, t), numsubst0 t a) - | numsubst0 t (Bound n) = - (if ((n : IntInf.int) = zero_nat) then t else Bound n) - | numsubst0 t (C c) = C c; - -fun subst0 t (NClosed p) = NClosed p - | subst0 t (Closed p) = Closed p - | subst0 t (Iffa (p, q)) = Iffa (subst0 t p, subst0 t q) - | subst0 t (Impa (p, q)) = Impa (subst0 t p, subst0 t q) - | subst0 t (Or (p, q)) = Or (subst0 t p, subst0 t q) - | subst0 t (And (p, q)) = And (subst0 t p, subst0 t q) - | subst0 t (Nota p) = Nota (subst0 t p) - | subst0 t (NDvd (i, a)) = NDvd (i, numsubst0 t a) - | subst0 t (Dvd (i, a)) = Dvd (i, numsubst0 t a) - | subst0 t (NEq a) = NEq (numsubst0 t a) - | subst0 t (Eq a) = Eq (numsubst0 t a) - | subst0 t (Ge a) = Ge (numsubst0 t a) - | subst0 t (Gt a) = Gt (numsubst0 t a) - | subst0 t (Le a) = Le (numsubst0 t a) - | subst0 t (Lt a) = Lt (numsubst0 t a) - | subst0 t F = F - | subst0 t T = T; - -fun conj p q = - (if eq_fm p F orelse eq_fm q F then F - else (if eq_fm p T then q else (if eq_fm q T then p else And (p, q)))); - -fun disj p q = - (if eq_fm p T orelse eq_fm q T then T - else (if eq_fm p F then q else (if eq_fm q F then p else Or (p, q)))); - -fun nota (NClosed v) = Nota (NClosed v) - | nota (Closed v) = Nota (Closed v) - | nota (A v) = Nota (A v) - | nota (E v) = Nota (E v) - | nota (Iffa (v, va)) = Nota (Iffa (v, va)) - | nota (Impa (v, va)) = Nota (Impa (v, va)) - | nota (Or (v, va)) = Nota (Or (v, va)) - | nota (And (v, va)) = Nota (And (v, va)) - | nota (NDvd (v, va)) = Nota (NDvd (v, va)) - | nota (Dvd (v, va)) = Nota (Dvd (v, va)) - | nota (NEq v) = Nota (NEq v) - | nota (Eq v) = Nota (Eq v) - | nota (Ge v) = Nota (Ge v) - | nota (Gt v) = Nota (Gt v) - | nota (Le v) = Nota (Le v) - | nota (Lt v) = Nota (Lt v) - | nota F = T - | nota T = F - | nota (Nota y) = y; - -fun imp p q = - (if eq_fm p F orelse eq_fm q T then T - else (if eq_fm p T then q - else (if eq_fm q F then nota p else Impa (p, q)))); - -fun iff p q = - (if eq_fm p q then T - else (if eq_fm p (nota q) orelse eq_fm (nota p) q then F - else (if eq_fm p F then nota q - else (if eq_fm q F then nota p - else (if eq_fm p T then q - else (if eq_fm q T then p - else Iffa (p, q))))))); - -fun simpfm (NClosed v) = NClosed v - | simpfm (Closed v) = Closed v - | simpfm (A v) = A v - | simpfm (E v) = E v - | simpfm F = F - | simpfm T = T - | simpfm (NDvd (i, a)) = - (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (NEq a) - else (if (((abs_int i) : IntInf.int) = (1 : IntInf.int)) then F - else let - val a' = simpnum a; - in - (case a' - of C v => - (if not (dvd (dvd_mod_int, eq_int) i v) then T else F) - | Bound nat => NDvd (i, a') - | Cx (int, num) => NDvd (i, a') | Neg num => NDvd (i, a') - | Add (num1, num2) => NDvd (i, a') - | Sub (num1, num2) => NDvd (i, a') - | Mul (int, num) => NDvd (i, a')) - end)) - | simpfm (Dvd (i, a)) = - (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (Eq a) - else (if (((abs_int i) : IntInf.int) = (1 : IntInf.int)) then T - else let - val a' = simpnum a; - in - (case a' - of C v => - (if dvd (dvd_mod_int, eq_int) i v then T else F) - | Bound nat => Dvd (i, a') | Cx (int, num) => Dvd (i, a') - | Neg num => Dvd (i, a') - | Add (num1, num2) => Dvd (i, a') - | Sub (num1, num2) => Dvd (i, a') - | Mul (int, num) => Dvd (i, a')) - end)) - | simpfm (NEq a) = - let - val a' = simpnum a; - in - (case a' - of C v => (if not ((v : IntInf.int) = (0 : IntInf.int)) then T else F) - | Bound nat => NEq a' | Cx (int, num) => NEq a' | Neg num => NEq a' - | Add (num1, num2) => NEq a' | Sub (num1, num2) => NEq a' - | Mul (int, num) => NEq a') - end - | simpfm (Eq a) = - let - val a' = simpnum a; - in - (case a' - of C v => (if ((v : IntInf.int) = (0 : IntInf.int)) then T else F) - | Bound nat => Eq a' | Cx (int, num) => Eq a' | Neg num => Eq a' - | Add (num1, num2) => Eq a' | Sub (num1, num2) => Eq a' - | Mul (int, num) => Eq a') - end - | simpfm (Ge a) = - let - val a' = simpnum a; - in - (case a' of C v => (if IntInf.<= ((0 : IntInf.int), v) then T else F) - | Bound nat => Ge a' | Cx (int, num) => Ge a' | Neg num => Ge a' - | Add (num1, num2) => Ge a' | Sub (num1, num2) => Ge a' - | Mul (int, num) => Ge a') - end - | simpfm (Gt a) = - let - val a' = simpnum a; - in - (case a' of C v => (if IntInf.< ((0 : IntInf.int), v) then T else F) - | Bound nat => Gt a' | Cx (int, num) => Gt a' | Neg num => Gt a' - | Add (num1, num2) => Gt a' | Sub (num1, num2) => Gt a' - | Mul (int, num) => Gt a') - end - | simpfm (Le a) = - let - val a' = simpnum a; - in - (case a' of C v => (if IntInf.<= (v, (0 : IntInf.int)) then T else F) - | Bound nat => Le a' | Cx (int, num) => Le a' | Neg num => Le a' - | Add (num1, num2) => Le a' | Sub (num1, num2) => Le a' - | Mul (int, num) => Le a') - end - | simpfm (Lt a) = - let - val a' = simpnum a; - in - (case a' of C v => (if IntInf.< (v, (0 : IntInf.int)) then T else F) - | Bound nat => Lt a' | Cx (int, num) => Lt a' | Neg num => Lt a' - | Add (num1, num2) => Lt a' | Sub (num1, num2) => Lt a' - | Mul (int, num) => Lt a') - end - | simpfm (Nota p) = nota (simpfm p) - | simpfm (Iffa (p, q)) = iff (simpfm p) (simpfm q) - | simpfm (Impa (p, q)) = imp (simpfm p) (simpfm q) - | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q) - | simpfm (And (p, q)) = conj (simpfm p) (simpfm q); - -fun decrnum (Cx (w, x)) = Cx (w, x) - | decrnum (C u) = C u - | decrnum (Mul (c, a)) = Mul (c, decrnum a) - | decrnum (Sub (a, b)) = Sub (decrnum a, decrnum b) - | decrnum (Add (a, b)) = Add (decrnum a, decrnum b) - | decrnum (Neg a) = Neg (decrnum a) - | decrnum (Bound n) = Bound (nat (IntInf.- (n, (1 : IntInf.int)))); - -fun decr (NClosed ar) = NClosed ar - | decr (Closed aq) = Closed aq - | decr (A ap) = A ap - | decr (E ao) = E ao - | decr F = F - | decr T = T - | decr (Iffa (p, q)) = Iffa (decr p, decr q) - | decr (Impa (p, q)) = Impa (decr p, decr q) - | decr (Or (p, q)) = Or (decr p, decr q) - | decr (And (p, q)) = And (decr p, decr q) - | decr (Nota p) = Nota (decr p) - | decr (NDvd (i, a)) = NDvd (i, decrnum a) - | decr (Dvd (i, a)) = Dvd (i, decrnum a) - | decr (NEq a) = NEq (decrnum a) - | decr (Eq a) = Eq (decrnum a) - | decr (Ge a) = Ge (decrnum a) - | decr (Gt a) = Gt (decrnum a) - | decr (Le a) = Le (decrnum a) - | decr (Lt a) = Lt (decrnum a); - fun cooper p = let - val (q, a) = unita p; - val (b, d) = a; + val a = unita p; + val (q, aa) = a; + val (b, d) = aa; val js = iupt (1 : IntInf.int) d; val mq = simpfm (minusinf q); val md = evaldjf (fn j => simpfm (subst0 (C j) mq)) js; in - (if eq_fm md T then T + (if eqop eq_fma md T then T else let val qd = - evaldjf (fn aa as (ba, j) => simpfm (subst0 (Add (ba, C j)) q)) - (allpairs (fn aa => fn ba => (aa, ba)) b js); + evaldjf (fn ab as (ba, j) => simpfm (subst0 (Add (ba, C j)) q)) + (concat (map (fn ba => map (fn ab => (ba, ab)) js) b)); in decr (disj md qd) end) end; -fun prep (NClosed aq) = NClosed aq - | prep (Closed ap) = Closed ap - | prep (NDvd (ac, ad)) = NDvd (ac, ad) - | prep (Dvd (aa, ab)) = Dvd (aa, ab) - | prep (NEq z) = NEq z - | prep (Eq y) = Eq y - | prep (Ge x) = Ge x - | prep (Gt w) = Gt w - | prep (Le v) = Le v - | prep (Lt u) = Lt u - | prep F = F - | prep T = T - | prep (Iffa (p, q)) = Or (prep (And (p, q)), prep (And (Nota p, Nota q))) - | prep (Impa (p, q)) = prep (Or (Nota p, q)) - | prep (And (p, q)) = And (prep p, prep q) - | prep (Or (p, q)) = Or (prep p, prep q) - | prep (Nota (NClosed ck)) = Nota (prep (NClosed ck)) - | prep (Nota (Closed cj)) = Nota (prep (Closed cj)) - | prep (Nota (E ch)) = Nota (prep (E ch)) - | prep (Nota (NDvd (bw, bx))) = Nota (prep (NDvd (bw, bx))) - | prep (Nota (Dvd (bu, bv))) = Nota (prep (Dvd (bu, bv))) - | prep (Nota (NEq bt)) = Nota (prep (NEq bt)) - | prep (Nota (Eq bs)) = Nota (prep (Eq bs)) - | prep (Nota (Ge br)) = Nota (prep (Ge br)) - | prep (Nota (Gt bq)) = Nota (prep (Gt bq)) - | prep (Nota (Le bp)) = Nota (prep (Le bp)) - | prep (Nota (Lt bo)) = Nota (prep (Lt bo)) - | prep (Nota F) = Nota (prep F) - | prep (Nota T) = Nota (prep T) - | prep (Nota (Iffa (p, q))) = - Or (prep (And (p, Nota q)), prep (And (Nota p, q))) - | prep (Nota (Impa (p, q))) = And (prep p, prep (Nota q)) - | prep (Nota (Or (p, q))) = And (prep (Nota p), prep (Nota q)) - | prep (Nota (A p)) = prep (E (Nota p)) - | prep (Nota (And (p, q))) = Or (prep (Nota p), prep (Nota q)) - | prep (Nota (Nota p)) = prep p - | prep (A (NClosed kj)) = prep (Nota (E (Nota (NClosed kj)))) - | prep (A (Closed ki)) = prep (Nota (E (Nota (Closed ki)))) - | prep (A (A kh)) = prep (Nota (E (Nota (A kh)))) - | prep (A (E kg)) = prep (Nota (E (Nota (E kg)))) - | prep (A (Iffa (ke, kf))) = prep (Nota (E (Nota (Iffa (ke, kf))))) - | prep (A (Impa (kc, kd))) = prep (Nota (E (Nota (Impa (kc, kd))))) - | prep (A (Or (ka, kb))) = prep (Nota (E (Nota (Or (ka, kb))))) - | prep (A (Nota jx)) = prep (Nota (E (Nota (Nota jx)))) - | prep (A (NDvd (jv, jw))) = prep (Nota (E (Nota (NDvd (jv, jw))))) - | prep (A (Dvd (jt, ju))) = prep (Nota (E (Nota (Dvd (jt, ju))))) - | prep (A (NEq js)) = prep (Nota (E (Nota (NEq js)))) - | prep (A (Eq jr)) = prep (Nota (E (Nota (Eq jr)))) - | prep (A (Ge jq)) = prep (Nota (E (Nota (Ge jq)))) - | prep (A (Gt jp)) = prep (Nota (E (Nota (Gt jp)))) - | prep (A (Le jo)) = prep (Nota (E (Nota (Le jo)))) - | prep (A (Lt jn)) = prep (Nota (E (Nota (Lt jn)))) - | prep (A F) = prep (Nota (E (Nota F))) - | prep (A T) = prep (Nota (E (Nota T))) - | prep (A (And (p, q))) = And (prep (A p), prep (A q)) - | prep (E (NClosed fb)) = E (prep (NClosed fb)) - | prep (E (Closed fa)) = E (prep (Closed fa)) +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 (E ey)) = E (prep (E ey)) - | prep (E (And (eq, er))) = E (prep (And (eq, er))) - | prep (E (Nota (NClosed hs))) = E (prep (Nota (NClosed hs))) - | prep (E (Nota (Closed hr))) = E (prep (Nota (Closed hr))) - | prep (E (Nota (A hq))) = E (prep (Nota (A hq))) - | prep (E (Nota (E hp))) = E (prep (Nota (E hp))) - | prep (E (Nota (Or (hj, hk)))) = E (prep (Nota (Or (hj, hk)))) - | prep (E (Nota (Nota hg))) = E (prep (Nota (Nota hg))) - | prep (E (Nota (NDvd (he, hf)))) = E (prep (Nota (NDvd (he, hf)))) - | prep (E (Nota (Dvd (hc, hd)))) = E (prep (Nota (Dvd (hc, hd)))) - | prep (E (Nota (NEq hb))) = E (prep (Nota (NEq hb))) - | prep (E (Nota (Eq ha))) = E (prep (Nota (Eq ha))) - | prep (E (Nota (Ge gz))) = E (prep (Nota (Ge gz))) - | prep (E (Nota (Gt gy))) = E (prep (Nota (Gt gy))) - | prep (E (Nota (Le gx))) = E (prep (Nota (Le gx))) - | prep (E (Nota (Lt gw))) = E (prep (Nota (Lt gw))) - | prep (E (Nota F)) = E (prep (Nota F)) - | prep (E (Nota T)) = E (prep (Nota T)) - | prep (E (NDvd (en, eo))) = E (prep (NDvd (en, eo))) - | prep (E (Dvd (el, em))) = E (prep (Dvd (el, em))) - | prep (E (NEq ek)) = E (prep (NEq ek)) - | prep (E (Eq ej)) = E (prep (Eq ej)) - | prep (E (Ge ei)) = E (prep (Ge ei)) - | prep (E (Gt eh)) = E (prep (Gt eh)) - | prep (E (Le eg)) = E (prep (Le eg)) - | prep (E (Lt ef)) = E (prep (Lt ef)) - | prep (E (Nota (Iffa (p, q)))) = - Or (prep (E (And (p, Nota q))), prep (E (And (Nota p, q)))) - | prep (E (Nota (Impa (p, q)))) = prep (E (And (p, Nota q))) - | prep (E (Nota (And (p, q)))) = Or (prep (E (Nota p)), prep (E (Nota q))) - | prep (E (Iffa (p, q))) = - Or (prep (E (And (p, q))), prep (E (And (Nota p, Nota q)))) - | prep (E (Impa (p, q))) = Or (prep (E (Nota p)), prep (E q)) - | prep (E (Or (p, q))) = Or (prep (E p), prep (E q)) - | prep (E F) = F - | prep (E T) = T; + | 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 qelim (NClosed aq) = (fn y => simpfm (NClosed aq)) - | qelim (Closed ap) = (fn y => simpfm (Closed ap)) - | qelim (NDvd (ac, ad)) = (fn y => simpfm (NDvd (ac, ad))) - | qelim (Dvd (aa, ab)) = (fn y => simpfm (Dvd (aa, ab))) - | qelim (NEq z) = (fn y => simpfm (NEq z)) - | qelim (Eq y) = (fn ya => simpfm (Eq y)) - | qelim (Ge x) = (fn y => simpfm (Ge x)) - | qelim (Gt w) = (fn y => simpfm (Gt w)) - | qelim (Le v) = (fn y => simpfm (Le v)) - | qelim (Lt u) = (fn y => simpfm (Lt u)) +fun qelim (E p) = (fn qe => dj qe (qelim p qe)) + | qelim (A p) = (fn qe => nota (qe (qelim (Not p) qe))) + | qelim (Not p) = (fn qe => nota (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 => impa (qelim p qe) (qelim q qe)) + | qelim (Iff (p, q)) = (fn qe => iffa (qelim p qe) (qelim q qe)) + | qelim T = (fn y => simpfm T) | qelim F = (fn y => simpfm F) - | qelim T = (fn y => simpfm T) - | qelim (Iffa (p, q)) = (fn qe => iff (qelim p qe) (qelim q qe)) - | qelim (Impa (p, q)) = (fn qe => imp (qelim p qe) (qelim q qe)) - | qelim (Or (p, q)) = (fn qe => disj (qelim p qe) (qelim q qe)) - | qelim (And (p, q)) = (fn qe => conj (qelim p qe) (qelim q qe)) - | qelim (Nota p) = (fn qe => nota (qelim p qe)) - | qelim (A p) = (fn qe => nota (qe (qelim (Nota p) qe))) - | qelim (E p) = (fn qe => dj qe (qelim p qe)); + | 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)); -val pa : fm -> fm = (fn p => qelim (prep p) cooper); +fun pa p = qelim (prep p) cooper; + +fun neg z = IntInf.< (z, (0 : IntInf.int)); + +fun nat_aux i n = + (if IntInf.<= (i, (0 : IntInf.int)) then n + else nat_aux (IntInf.- (i, (1 : IntInf.int))) (suc n)); end; (*struct GeneratedCooper*)