(* Title: HOL/Tools/Qelim/generated_cooper.ML
This file is generated from HOL/Decision_Procs/Cooper.thy. DO NOT EDIT.
*)
structure GeneratedCooper =
struct
type 'a eq = {eq : 'a -> 'a -> bool};
fun eq (A_:'a eq) = #eq A_;
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, b) = b;
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 (a, b) = a;
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 [] ys = ys
| append (x :: xs) ys = x :: append xs ys;
fun disjuncts (Or (p, q)) = append (disjuncts p) (disjuncts q)
| disjuncts F = []
| disjuncts T = [T]
| disjuncts (Lt u) = [Lt u]
| disjuncts (Le v) = [Le v]
| disjuncts (Gt w) = [Gt w]
| disjuncts (Ge x) = [Ge x]
| disjuncts (Eq y) = [Eq y]
| disjuncts (NEq z) = [NEq z]
| disjuncts (Dvd (aa, ab)) = [Dvd (aa, ab)]
| disjuncts (NDvd (ac, ad)) = [NDvd (ac, ad)]
| disjuncts (Not ae) = [Not ae]
| disjuncts (And (af, ag)) = [And (af, ag)]
| disjuncts (Imp (aj, ak)) = [Imp (aj, ak)]
| disjuncts (Iff (al, am)) = [Iff (al, am)]
| disjuncts (E an) = [E an]
| disjuncts (A ao) = [A ao]
| disjuncts (Closed ap) = [Closed ap]
| disjuncts (NClosed aq) = [NClosed aq];
fun 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 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 F
= f2
| fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 T
= f1;
fun eq_num (Mul (c, d)) (Sub (a, b)) = false
| eq_num (Mul (c, d)) (Add (a, b)) = false
| eq_num (Sub (c, d)) (Add (a, b)) = false
| eq_num (Mul (b, c)) (Neg a) = false
| eq_num (Sub (b, c)) (Neg a) = false
| eq_num (Add (b, c)) (Neg a) = false
| eq_num (Mul (d, e)) (Cn (a, b, c)) = false
| eq_num (Sub (d, e)) (Cn (a, b, c)) = false
| eq_num (Add (d, e)) (Cn (a, b, c)) = false
| eq_num (Neg d) (Cn (a, b, c)) = false
| eq_num (Mul (b, c)) (Bound a) = false
| eq_num (Sub (b, c)) (Bound a) = false
| eq_num (Add (b, c)) (Bound a) = false
| eq_num (Neg b) (Bound a) = false
| eq_num (Cn (b, c, d)) (Bound a) = false
| eq_num (Mul (b, c)) (C a) = false
| eq_num (Sub (b, c)) (C a) = false
| eq_num (Add (b, c)) (C a) = false
| eq_num (Neg b) (C a) = false
| eq_num (Cn (b, c, d)) (C a) = false
| eq_num (Bound b) (C a) = false
| eq_num (Sub (a, b)) (Mul (c, d)) = false
| eq_num (Add (a, b)) (Mul (c, d)) = false
| eq_num (Add (a, b)) (Sub (c, d)) = false
| eq_num (Neg a) (Mul (b, c)) = false
| eq_num (Neg a) (Sub (b, c)) = false
| eq_num (Neg a) (Add (b, c)) = false
| eq_num (Cn (a, b, c)) (Mul (d, e)) = false
| eq_num (Cn (a, b, c)) (Sub (d, e)) = false
| eq_num (Cn (a, b, c)) (Add (d, e)) = false
| eq_num (Cn (a, b, c)) (Neg d) = false
| eq_num (Bound a) (Mul (b, c)) = false
| eq_num (Bound a) (Sub (b, c)) = false
| eq_num (Bound a) (Add (b, c)) = false
| eq_num (Bound a) (Neg b) = false
| eq_num (Bound a) (Cn (b, c, d)) = false
| eq_num (C a) (Mul (b, c)) = false
| eq_num (C a) (Sub (b, c)) = false
| eq_num (C a) (Add (b, c)) = false
| eq_num (C a) (Neg b) = false
| eq_num (C a) (Cn (b, c, d)) = false
| eq_num (C a) (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 b) (Closed a) = false
| eq_fm (NClosed b) (A a) = false
| eq_fm (Closed b) (A a) = false
| eq_fm (NClosed b) (E a) = false
| eq_fm (Closed b) (E a) = false
| eq_fm (A b) (E a) = false
| eq_fm (NClosed c) (Iff (a, b)) = false
| eq_fm (Closed c) (Iff (a, b)) = false
| eq_fm (A c) (Iff (a, b)) = false
| eq_fm (E c) (Iff (a, b)) = false
| eq_fm (NClosed c) (Imp (a, b)) = false
| eq_fm (Closed c) (Imp (a, b)) = false
| eq_fm (A c) (Imp (a, b)) = false
| eq_fm (E c) (Imp (a, b)) = false
| eq_fm (Iff (c, d)) (Imp (a, b)) = false
| eq_fm (NClosed c) (Or (a, b)) = false
| eq_fm (Closed c) (Or (a, b)) = false
| eq_fm (A c) (Or (a, b)) = false
| eq_fm (E c) (Or (a, b)) = false
| eq_fm (Iff (c, d)) (Or (a, b)) = false
| eq_fm (Imp (c, d)) (Or (a, b)) = false
| eq_fm (NClosed c) (And (a, b)) = false
| eq_fm (Closed c) (And (a, b)) = false
| eq_fm (A c) (And (a, b)) = false
| eq_fm (E c) (And (a, b)) = false
| eq_fm (Iff (c, d)) (And (a, b)) = false
| eq_fm (Imp (c, d)) (And (a, b)) = false
| eq_fm (Or (c, d)) (And (a, b)) = false
| eq_fm (NClosed b) (Not a) = false
| eq_fm (Closed b) (Not a) = false
| eq_fm (A b) (Not a) = false
| eq_fm (E b) (Not a) = false
| eq_fm (Iff (b, c)) (Not a) = false
| eq_fm (Imp (b, c)) (Not a) = false
| eq_fm (Or (b, c)) (Not a) = false
| eq_fm (And (b, c)) (Not a) = false
| eq_fm (NClosed c) (NDvd (a, b)) = false
| eq_fm (Closed c) (NDvd (a, b)) = false
| eq_fm (A c) (NDvd (a, b)) = false
| eq_fm (E c) (NDvd (a, b)) = false
| eq_fm (Iff (c, d)) (NDvd (a, b)) = false
| eq_fm (Imp (c, d)) (NDvd (a, b)) = false
| eq_fm (Or (c, d)) (NDvd (a, b)) = false
| eq_fm (And (c, d)) (NDvd (a, b)) = false
| eq_fm (Not c) (NDvd (a, b)) = false
| eq_fm (NClosed c) (Dvd (a, b)) = false
| eq_fm (Closed c) (Dvd (a, b)) = false
| eq_fm (A c) (Dvd (a, b)) = false
| eq_fm (E c) (Dvd (a, b)) = false
| eq_fm (Iff (c, d)) (Dvd (a, b)) = false
| eq_fm (Imp (c, d)) (Dvd (a, b)) = false
| eq_fm (Or (c, d)) (Dvd (a, b)) = false
| eq_fm (And (c, d)) (Dvd (a, b)) = false
| eq_fm (Not c) (Dvd (a, b)) = false
| eq_fm (NDvd (c, d)) (Dvd (a, b)) = false
| eq_fm (NClosed b) (NEq a) = false
| eq_fm (Closed b) (NEq a) = false
| eq_fm (A b) (NEq a) = false
| eq_fm (E b) (NEq a) = false
| eq_fm (Iff (b, c)) (NEq a) = false
| eq_fm (Imp (b, c)) (NEq a) = false
| eq_fm (Or (b, c)) (NEq a) = false
| eq_fm (And (b, c)) (NEq a) = false
| eq_fm (Not b) (NEq a) = false
| eq_fm (NDvd (b, c)) (NEq a) = false
| eq_fm (Dvd (b, c)) (NEq a) = false
| eq_fm (NClosed b) (Eq a) = false
| eq_fm (Closed b) (Eq a) = false
| eq_fm (A b) (Eq a) = false
| eq_fm (E b) (Eq a) = false
| eq_fm (Iff (b, c)) (Eq a) = false
| eq_fm (Imp (b, c)) (Eq a) = false
| eq_fm (Or (b, c)) (Eq a) = false
| eq_fm (And (b, c)) (Eq a) = false
| eq_fm (Not b) (Eq a) = false
| eq_fm (NDvd (b, c)) (Eq a) = false
| eq_fm (Dvd (b, c)) (Eq a) = false
| eq_fm (NEq b) (Eq a) = false
| eq_fm (NClosed b) (Ge a) = false
| eq_fm (Closed b) (Ge a) = false
| eq_fm (A b) (Ge a) = false
| eq_fm (E b) (Ge a) = false
| eq_fm (Iff (b, c)) (Ge a) = false
| eq_fm (Imp (b, c)) (Ge a) = false
| eq_fm (Or (b, c)) (Ge a) = false
| eq_fm (And (b, c)) (Ge a) = false
| eq_fm (Not b) (Ge a) = false
| eq_fm (NDvd (b, c)) (Ge a) = false
| eq_fm (Dvd (b, c)) (Ge a) = false
| eq_fm (NEq b) (Ge a) = false
| eq_fm (Eq b) (Ge a) = false
| eq_fm (NClosed b) (Gt a) = false
| eq_fm (Closed b) (Gt a) = false
| eq_fm (A b) (Gt a) = false
| eq_fm (E b) (Gt a) = false
| eq_fm (Iff (b, c)) (Gt a) = false
| eq_fm (Imp (b, c)) (Gt a) = false
| eq_fm (Or (b, c)) (Gt a) = false
| eq_fm (And (b, c)) (Gt a) = false
| eq_fm (Not b) (Gt a) = false
| eq_fm (NDvd (b, c)) (Gt a) = false
| eq_fm (Dvd (b, c)) (Gt a) = false
| eq_fm (NEq b) (Gt a) = false
| eq_fm (Eq b) (Gt a) = false
| eq_fm (Ge b) (Gt a) = false
| eq_fm (NClosed b) (Le a) = false
| eq_fm (Closed b) (Le a) = false
| eq_fm (A b) (Le a) = false
| eq_fm (E b) (Le a) = false
| eq_fm (Iff (b, c)) (Le a) = false
| eq_fm (Imp (b, c)) (Le a) = false
| eq_fm (Or (b, c)) (Le a) = false
| eq_fm (And (b, c)) (Le a) = false
| eq_fm (Not b) (Le a) = false
| eq_fm (NDvd (b, c)) (Le a) = false
| eq_fm (Dvd (b, c)) (Le a) = false
| eq_fm (NEq b) (Le a) = false
| eq_fm (Eq b) (Le a) = false
| eq_fm (Ge b) (Le a) = false
| eq_fm (Gt b) (Le a) = false
| eq_fm (NClosed b) (Lt a) = false
| eq_fm (Closed b) (Lt a) = false
| eq_fm (A b) (Lt a) = false
| eq_fm (E b) (Lt a) = false
| eq_fm (Iff (b, c)) (Lt a) = false
| eq_fm (Imp (b, c)) (Lt a) = false
| eq_fm (Or (b, c)) (Lt a) = false
| eq_fm (And (b, c)) (Lt a) = false
| eq_fm (Not b) (Lt a) = false
| eq_fm (NDvd (b, c)) (Lt a) = false
| eq_fm (Dvd (b, c)) (Lt a) = false
| eq_fm (NEq b) (Lt a) = false
| eq_fm (Eq b) (Lt a) = false
| eq_fm (Ge b) (Lt a) = false
| eq_fm (Gt b) (Lt a) = false
| eq_fm (Le b) (Lt a) = false
| eq_fm (NClosed a) F = false
| eq_fm (Closed a) F = false
| eq_fm (A a) F = false
| eq_fm (E a) F = false
| eq_fm (Iff (a, b)) F = false
| eq_fm (Imp (a, b)) F = false
| eq_fm (Or (a, b)) F = false
| eq_fm (And (a, b)) F = false
| eq_fm (Not a) F = false
| eq_fm (NDvd (a, b)) F = false
| eq_fm (Dvd (a, b)) F = false
| eq_fm (NEq a) F = false
| eq_fm (Eq a) F = false
| eq_fm (Ge a) F = false
| eq_fm (Gt a) F = false
| eq_fm (Le a) F = false
| eq_fm (Lt a) F = false
| eq_fm (NClosed a) T = false
| eq_fm (Closed a) T = false
| eq_fm (A a) T = false
| eq_fm (E a) T = false
| eq_fm (Iff (a, b)) T = false
| eq_fm (Imp (a, b)) T = false
| eq_fm (Or (a, b)) T = false
| eq_fm (And (a, b)) T = false
| eq_fm (Not a) T = false
| eq_fm (NDvd (a, b)) T = false
| eq_fm (Dvd (a, b)) T = false
| eq_fm (NEq a) T = false
| eq_fm (Eq a) T = false
| eq_fm (Ge a) T = false
| eq_fm (Gt a) T = false
| eq_fm (Le a) T = false
| eq_fm (Lt a) T = false
| eq_fm F T = false
| eq_fm (Closed a) (NClosed b) = false
| eq_fm (A a) (NClosed b) = false
| eq_fm (A a) (Closed b) = false
| eq_fm (E a) (NClosed b) = false
| eq_fm (E a) (Closed b) = false
| eq_fm (E a) (A b) = false
| eq_fm (Iff (a, b)) (NClosed c) = false
| eq_fm (Iff (a, b)) (Closed c) = false
| eq_fm (Iff (a, b)) (A c) = false
| eq_fm (Iff (a, b)) (E c) = false
| eq_fm (Imp (a, b)) (NClosed c) = false
| eq_fm (Imp (a, b)) (Closed c) = false
| eq_fm (Imp (a, b)) (A c) = false
| eq_fm (Imp (a, b)) (E c) = false
| eq_fm (Imp (a, b)) (Iff (c, d)) = false
| eq_fm (Or (a, b)) (NClosed c) = false
| eq_fm (Or (a, b)) (Closed c) = false
| eq_fm (Or (a, b)) (A c) = false
| eq_fm (Or (a, b)) (E c) = false
| eq_fm (Or (a, b)) (Iff (c, d)) = false
| eq_fm (Or (a, b)) (Imp (c, d)) = false
| eq_fm (And (a, b)) (NClosed c) = false
| eq_fm (And (a, b)) (Closed c) = false
| eq_fm (And (a, b)) (A c) = false
| eq_fm (And (a, b)) (E c) = false
| eq_fm (And (a, b)) (Iff (c, d)) = false
| eq_fm (And (a, b)) (Imp (c, d)) = false
| eq_fm (And (a, b)) (Or (c, d)) = false
| eq_fm (Not a) (NClosed b) = false
| eq_fm (Not a) (Closed b) = false
| eq_fm (Not a) (A b) = false
| eq_fm (Not a) (E b) = false
| eq_fm (Not a) (Iff (b, c)) = false
| eq_fm (Not a) (Imp (b, c)) = false
| eq_fm (Not a) (Or (b, c)) = false
| eq_fm (Not a) (And (b, c)) = false
| eq_fm (NDvd (a, b)) (NClosed c) = false
| eq_fm (NDvd (a, b)) (Closed c) = false
| eq_fm (NDvd (a, b)) (A c) = false
| eq_fm (NDvd (a, b)) (E c) = false
| eq_fm (NDvd (a, b)) (Iff (c, d)) = false
| eq_fm (NDvd (a, b)) (Imp (c, d)) = false
| eq_fm (NDvd (a, b)) (Or (c, d)) = false
| eq_fm (NDvd (a, b)) (And (c, d)) = false
| eq_fm (NDvd (a, b)) (Not c) = false
| eq_fm (Dvd (a, b)) (NClosed c) = false
| eq_fm (Dvd (a, b)) (Closed c) = false
| eq_fm (Dvd (a, b)) (A c) = false
| eq_fm (Dvd (a, b)) (E c) = false
| eq_fm (Dvd (a, b)) (Iff (c, d)) = false
| eq_fm (Dvd (a, b)) (Imp (c, d)) = false
| eq_fm (Dvd (a, b)) (Or (c, d)) = false
| eq_fm (Dvd (a, b)) (And (c, d)) = false
| eq_fm (Dvd (a, b)) (Not c) = false
| eq_fm (Dvd (a, b)) (NDvd (c, d)) = false
| eq_fm (NEq a) (NClosed b) = false
| eq_fm (NEq a) (Closed b) = false
| eq_fm (NEq a) (A b) = false
| eq_fm (NEq a) (E b) = false
| eq_fm (NEq a) (Iff (b, c)) = false
| eq_fm (NEq a) (Imp (b, c)) = false
| eq_fm (NEq a) (Or (b, c)) = false
| eq_fm (NEq a) (And (b, c)) = false
| eq_fm (NEq a) (Not b) = false
| eq_fm (NEq a) (NDvd (b, c)) = false
| eq_fm (NEq a) (Dvd (b, c)) = false
| eq_fm (Eq a) (NClosed b) = false
| eq_fm (Eq a) (Closed b) = false
| eq_fm (Eq a) (A b) = false
| eq_fm (Eq a) (E b) = false
| eq_fm (Eq a) (Iff (b, c)) = false
| eq_fm (Eq a) (Imp (b, c)) = false
| eq_fm (Eq a) (Or (b, c)) = false
| eq_fm (Eq a) (And (b, c)) = false
| eq_fm (Eq a) (Not b) = false
| eq_fm (Eq a) (NDvd (b, c)) = false
| eq_fm (Eq a) (Dvd (b, c)) = false
| eq_fm (Eq a) (NEq b) = false
| eq_fm (Ge a) (NClosed b) = false
| eq_fm (Ge a) (Closed b) = false
| eq_fm (Ge a) (A b) = false
| eq_fm (Ge a) (E b) = false
| eq_fm (Ge a) (Iff (b, c)) = false
| eq_fm (Ge a) (Imp (b, c)) = false
| eq_fm (Ge a) (Or (b, c)) = false
| eq_fm (Ge a) (And (b, c)) = false
| eq_fm (Ge a) (Not b) = false
| eq_fm (Ge a) (NDvd (b, c)) = false
| eq_fm (Ge a) (Dvd (b, c)) = false
| eq_fm (Ge a) (NEq b) = false
| eq_fm (Ge a) (Eq b) = false
| eq_fm (Gt a) (NClosed b) = false
| eq_fm (Gt a) (Closed b) = false
| eq_fm (Gt a) (A b) = false
| eq_fm (Gt a) (E b) = false
| eq_fm (Gt a) (Iff (b, c)) = false
| eq_fm (Gt a) (Imp (b, c)) = false
| eq_fm (Gt a) (Or (b, c)) = false
| eq_fm (Gt a) (And (b, c)) = false
| eq_fm (Gt a) (Not b) = false
| eq_fm (Gt a) (NDvd (b, c)) = false
| eq_fm (Gt a) (Dvd (b, c)) = false
| eq_fm (Gt a) (NEq b) = false
| eq_fm (Gt a) (Eq b) = false
| eq_fm (Gt a) (Ge b) = false
| eq_fm (Le a) (NClosed b) = false
| eq_fm (Le a) (Closed b) = false
| eq_fm (Le a) (A b) = false
| eq_fm (Le a) (E b) = false
| eq_fm (Le a) (Iff (b, c)) = false
| eq_fm (Le a) (Imp (b, c)) = false
| eq_fm (Le a) (Or (b, c)) = false
| eq_fm (Le a) (And (b, c)) = false
| eq_fm (Le a) (Not b) = false
| eq_fm (Le a) (NDvd (b, c)) = false
| eq_fm (Le a) (Dvd (b, c)) = false
| eq_fm (Le a) (NEq b) = false
| eq_fm (Le a) (Eq b) = false
| eq_fm (Le a) (Ge b) = false
| eq_fm (Le a) (Gt b) = false
| eq_fm (Lt a) (NClosed b) = false
| eq_fm (Lt a) (Closed b) = false
| eq_fm (Lt a) (A b) = false
| eq_fm (Lt a) (E b) = false
| eq_fm (Lt a) (Iff (b, c)) = false
| eq_fm (Lt a) (Imp (b, c)) = false
| eq_fm (Lt a) (Or (b, c)) = false
| eq_fm (Lt a) (And (b, c)) = false
| eq_fm (Lt a) (Not b) = false
| eq_fm (Lt a) (NDvd (b, c)) = false
| eq_fm (Lt a) (Dvd (b, c)) = false
| eq_fm (Lt a) (NEq b) = false
| eq_fm (Lt a) (Eq b) = false
| eq_fm (Lt a) (Ge b) = false
| eq_fm (Lt a) (Gt b) = false
| eq_fm (Lt a) (Le b) = false
| eq_fm F (NClosed a) = false
| eq_fm F (Closed a) = false
| eq_fm F (A a) = false
| eq_fm F (E a) = false
| eq_fm F (Iff (a, b)) = false
| eq_fm F (Imp (a, b)) = false
| eq_fm F (Or (a, b)) = false
| eq_fm F (And (a, b)) = false
| eq_fm F (Not a) = false
| eq_fm F (NDvd (a, b)) = false
| eq_fm F (Dvd (a, b)) = false
| eq_fm F (NEq a) = false
| eq_fm F (Eq a) = false
| eq_fm F (Ge a) = false
| eq_fm F (Gt a) = false
| eq_fm F (Le a) = false
| eq_fm F (Lt a) = false
| eq_fm T (NClosed a) = false
| eq_fm T (Closed a) = false
| eq_fm T (A a) = false
| eq_fm T (E a) = false
| eq_fm T (Iff (a, b)) = false
| eq_fm T (Imp (a, b)) = false
| eq_fm T (Or (a, b)) = false
| eq_fm T (And (a, b)) = false
| eq_fm T (Not a) = false
| eq_fm T (NDvd (a, b)) = false
| eq_fm T (Dvd (a, b)) = false
| eq_fm T (NEq a) = false
| eq_fm T (Eq a) = false
| eq_fm T (Ge a) = false
| eq_fm T (Gt a) = false
| eq_fm T (Le a) = false
| eq_fm T (Lt a) = 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 [] a = a
| 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 t (Cn (v, i, a)) =
(if eqop eq_nat v 0 then Add (Mul (i, t), numsubst0 t a)
else Cn (suc (minus_nat v 1), i, numsubst0 t a));
fun subst0 t T = T
| subst0 t F = F
| subst0 t (Lt a) = Lt (numsubst0 t a)
| subst0 t (Le a) = Le (numsubst0 t a)
| subst0 t (Gt a) = Gt (numsubst0 t a)
| subst0 t (Ge a) = Ge (numsubst0 t a)
| subst0 t (Eq a) = Eq (numsubst0 t a)
| subst0 t (NEq a) = NEq (numsubst0 t a)
| subst0 t (Dvd (i, a)) = Dvd (i, numsubst0 t a)
| subst0 t (NDvd (i, a)) = NDvd (i, numsubst0 t a)
| subst0 t (Not p) = Not (subst0 t p)
| subst0 t (And (p, q)) = And (subst0 t p, subst0 t q)
| subst0 t (Or (p, q)) = Or (subst0 t p, subst0 t q)
| subst0 t (Imp (p, q)) = Imp (subst0 t p, subst0 t q)
| subst0 t (Iff (p, q)) = Iff (subst0 t p, subst0 t q)
| subst0 t (Closed p) = Closed p
| subst0 t (NClosed p) = NClosed p;
fun minusinf (And (p, q)) = And (minusinf p, minusinf q)
| minusinf (Or (p, q)) = Or (minusinf p, minusinf q)
| minusinf 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)));
val eq_int = {eq = (fn a => fn b => ((a : IntInf.int) = b))} : IntInf.int eq;
fun sgn_int i =
(if eqop eq_int i (0 : IntInf.int) then (0 : IntInf.int)
else (if IntInf.< ((0 : IntInf.int), i) then (1 : IntInf.int)
else IntInf.~ (1 : IntInf.int)));
fun apsnd f (x, y) = (x, f y);
fun divmoda k l =
(if eqop eq_int k (0 : IntInf.int) then ((0 : IntInf.int), (0 : IntInf.int))
else (if eqop eq_int l (0 : IntInf.int) then ((0 : IntInf.int), k)
else apsnd (fn a => IntInf.* (sgn_int l, a))
(if eqop eq_int (sgn_int k) (sgn_int l)
then (fn k => fn l => IntInf.divMod (IntInf.abs k,
IntInf.abs l))
k l
else let
val a =
(fn k => fn l => IntInf.divMod (IntInf.abs k,
IntInf.abs l))
k l;
val (r, s) = a;
in
(if eqop eq_int s (0 : IntInf.int)
then (IntInf.~ r, (0 : IntInf.int))
else (IntInf.- (IntInf.~ r, (1 : IntInf.int)),
IntInf.- (abs_int l, s)))
end)));
fun mod_int a b = snd (divmoda a b);
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 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));
fun numneg t = nummul (IntInf.~ (1 : IntInf.int)) t;
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));
val eq_numa = {eq = eq_num} : num eq;
fun numsub s t =
(if eqop eq_numa s t then C (0 : IntInf.int) else numadd (s, numneg t));
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 nota (Not p) = p
| nota T = F
| nota F = T
| nota (Lt v) = Not (Lt v)
| nota (Le v) = Not (Le v)
| nota (Gt v) = Not (Gt v)
| nota (Ge v) = Not (Ge v)
| nota (Eq v) = Not (Eq v)
| nota (NEq v) = Not (NEq v)
| nota (Dvd (v, va)) = Not (Dvd (v, va))
| nota (NDvd (v, va)) = Not (NDvd (v, va))
| nota (And (v, va)) = Not (And (v, va))
| nota (Or (v, va)) = Not (Or (v, va))
| nota (Imp (v, va)) = Not (Imp (v, va))
| nota (Iff (v, va)) = Not (Iff (v, va))
| nota (E v) = Not (E v)
| nota (A v) = Not (A v)
| nota (Closed v) = Not (Closed v)
| nota (NClosed v) = Not (NClosed v);
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)))))));
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))));
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 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 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 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 y) = (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 (i, Cn (cm, c, e))) =
(if eqop eq_nat cm 0 then i else (1 : IntInf.int))
| delta (NDvd (i, Cn (dm, c, e))) =
(if eqop eq_nat dm 0 then i else (1 : IntInf.int));
fun div_int a b = fst (divmoda a b);
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 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, c, e))) =
(if eqop eq_nat cm 0 then c else (1 : IntInf.int))
| zeta (Le (Cn (dm, c, e))) =
(if eqop eq_nat dm 0 then c else (1 : IntInf.int))
| zeta (Gt (Cn (em, c, e))) =
(if eqop eq_nat em 0 then c else (1 : IntInf.int))
| zeta (Ge (Cn (fm, c, e))) =
(if eqop eq_nat fm 0 then c else (1 : IntInf.int))
| zeta (Eq (Cn (gm, c, e))) =
(if eqop eq_nat gm 0 then c else (1 : IntInf.int))
| zeta (NEq (Cn (hm, c, e))) =
(if eqop eq_nat hm 0 then c else (1 : IntInf.int))
| zeta (Dvd (i, Cn (im, c, e))) =
(if eqop eq_nat im 0 then c else (1 : IntInf.int))
| zeta (NDvd (i, Cn (jm, c, e))) =
(if eqop eq_nat jm 0 then c else (1 : IntInf.int));
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 aa = zsplit0 a;
val (ia, a') = aa;
val ab = zsplit0 b;
val (ib, b') = ab;
in
(IntInf.- (ia, ib), Sub (a', b'))
end
| zsplit0 (Mul (i, a)) =
let
val aa = zsplit0 a;
val (i', a') = aa;
in
(IntInf.* (i, i'), Mul (i, a'))
end;
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 aa = zsplit0 a;
val (c, r) = aa;
in
(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 aa = zsplit0 a;
val (c, r) = aa;
in
(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 (Ge a) =
let
val aa = zsplit0 a;
val (c, r) = aa;
in
(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 (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 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, 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))
else (mirror q, (a, d)))
end;
fun cooper p =
let
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 eqop eq_fma md T then T
else let
val qd =
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 (E T) = T
| prep (E F) = F
| prep (E (Or (p, q))) = Or (prep (E p), prep (E q))
| prep (E (Imp (p, q))) = Or (prep (E (Not p)), prep (E q))
| prep (E (Iff (p, q))) =
Or (prep (E (And (p, q))), prep (E (And (Not p, Not q))))
| prep (E (Not (And (p, q)))) = Or (prep (E (Not p)), prep (E (Not q)))
| prep (E (Not (Imp (p, q)))) = prep (E (And (p, Not q)))
| prep (E (Not (Iff (p, q)))) =
Or (prep (E (And (p, Not q))), prep (E (And (Not p, q))))
| prep (E (Lt ef)) = E (prep (Lt ef))
| prep (E (Le eg)) = E (prep (Le eg))
| prep (E (Gt eh)) = E (prep (Gt eh))
| prep (E (Ge ei)) = E (prep (Ge ei))
| prep (E (Eq ej)) = E (prep (Eq ej))
| prep (E (NEq ek)) = E (prep (NEq ek))
| prep (E (Dvd (el, em))) = E (prep (Dvd (el, em)))
| prep (E (NDvd (en, eo))) = E (prep (NDvd (en, eo)))
| prep (E (Not T)) = E (prep (Not T))
| prep (E (Not F)) = E (prep (Not F))
| prep (E (Not (Lt gw))) = E (prep (Not (Lt gw)))
| prep (E (Not (Le gx))) = E (prep (Not (Le gx)))
| prep (E (Not (Gt gy))) = E (prep (Not (Gt gy)))
| prep (E (Not (Ge gz))) = E (prep (Not (Ge gz)))
| prep (E (Not (Eq ha))) = E (prep (Not (Eq ha)))
| prep (E (Not (NEq hb))) = E (prep (Not (NEq hb)))
| prep (E (Not (Dvd (hc, hd)))) = E (prep (Not (Dvd (hc, hd))))
| prep (E (Not (NDvd (he, hf)))) = E (prep (Not (NDvd (he, hf))))
| prep (E (Not (Not hg))) = E (prep (Not (Not hg)))
| prep (E (Not (Or (hj, hk)))) = E (prep (Not (Or (hj, hk))))
| prep (E (Not (E hp))) = E (prep (Not (E hp)))
| prep (E (Not (A hq))) = E (prep (Not (A hq)))
| prep (E (Not (Closed hr))) = E (prep (Not (Closed hr)))
| prep (E (Not (NClosed hs))) = E (prep (Not (NClosed hs)))
| prep (E (And (eq, er))) = E (prep (And (eq, er)))
| prep (E (E ey)) = E (prep (E ey))
| prep (E (A ez)) = E (prep (A ez))
| prep (E (Closed fa)) = E (prep (Closed fa))
| prep (E (NClosed fb)) = E (prep (NClosed fb))
| prep (A (And (p, q))) = And (prep (A p), prep (A q))
| prep (A T) = prep (Not (E (Not T)))
| prep (A F) = prep (Not (E (Not F)))
| prep (A (Lt jn)) = prep (Not (E (Not (Lt jn))))
| prep (A (Le jo)) = prep (Not (E (Not (Le jo))))
| prep (A (Gt jp)) = prep (Not (E (Not (Gt jp))))
| prep (A (Ge jq)) = prep (Not (E (Not (Ge jq))))
| prep (A (Eq jr)) = prep (Not (E (Not (Eq jr))))
| prep (A (NEq js)) = prep (Not (E (Not (NEq js))))
| prep (A (Dvd (jt, ju))) = prep (Not (E (Not (Dvd (jt, ju)))))
| prep (A (NDvd (jv, jw))) = prep (Not (E (Not (NDvd (jv, jw)))))
| prep (A (Not jx)) = prep (Not (E (Not (Not jx))))
| prep (A (Or (ka, kb))) = prep (Not (E (Not (Or (ka, kb)))))
| prep (A (Imp (kc, kd))) = prep (Not (E (Not (Imp (kc, kd)))))
| prep (A (Iff (ke, kf))) = prep (Not (E (Not (Iff (ke, kf)))))
| prep (A (E kg)) = prep (Not (E (Not (E kg))))
| prep (A (A kh)) = prep (Not (E (Not (A kh))))
| prep (A (Closed ki)) = prep (Not (E (Not (Closed ki))))
| prep (A (NClosed kj)) = prep (Not (E (Not (NClosed kj))))
| prep (Not (Not p)) = prep p
| prep (Not (And (p, q))) = Or (prep (Not p), prep (Not q))
| prep (Not (A p)) = prep (E (Not p))
| prep (Not (Or (p, q))) = And (prep (Not p), prep (Not q))
| prep (Not (Imp (p, q))) = And (prep p, prep (Not q))
| prep (Not (Iff (p, q))) = Or (prep (And (p, Not q)), prep (And (Not p, q)))
| prep (Not T) = Not (prep T)
| prep (Not F) = Not (prep F)
| prep (Not (Lt bo)) = Not (prep (Lt bo))
| prep (Not (Le bp)) = Not (prep (Le bp))
| prep (Not (Gt bq)) = Not (prep (Gt bq))
| prep (Not (Ge br)) = Not (prep (Ge br))
| prep (Not (Eq bs)) = Not (prep (Eq bs))
| prep (Not (NEq bt)) = Not (prep (NEq bt))
| prep (Not (Dvd (bu, bv))) = Not (prep (Dvd (bu, bv)))
| prep (Not (NDvd (bw, bx))) = Not (prep (NDvd (bw, bx)))
| prep (Not (E ch)) = Not (prep (E ch))
| prep (Not (Closed cj)) = Not (prep (Closed cj))
| prep (Not (NClosed ck)) = Not (prep (NClosed ck))
| prep (Or (p, q)) = Or (prep p, prep q)
| prep (And (p, q)) = And (prep p, prep q)
| prep (Imp (p, q)) = prep (Or (Not p, q))
| prep (Iff (p, q)) = Or (prep (And (p, q)), prep (And (Not p, Not q)))
| prep T = T
| prep F = F
| prep (Lt u) = Lt u
| prep (Le v) = Le v
| prep (Gt w) = Gt w
| prep (Ge x) = Ge x
| prep (Eq y) = Eq y
| prep (NEq z) = NEq z
| prep (Dvd (aa, ab)) = Dvd (aa, ab)
| prep (NDvd (ac, ad)) = NDvd (ac, ad)
| prep (Closed ap) = Closed ap
| prep (NClosed aq) = NClosed aq;
fun 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 (Lt u) = (fn y => simpfm (Lt u))
| qelim (Le v) = (fn y => simpfm (Le v))
| qelim (Gt w) = (fn y => simpfm (Gt w))
| qelim (Ge x) = (fn y => simpfm (Ge x))
| qelim (Eq y) = (fn ya => simpfm (Eq y))
| qelim (NEq z) = (fn y => simpfm (NEq z))
| qelim (Dvd (aa, ab)) = (fn y => simpfm (Dvd (aa, ab)))
| qelim (NDvd (ac, ad)) = (fn y => simpfm (NDvd (ac, ad)))
| qelim (Closed ap) = (fn y => simpfm (Closed ap))
| qelim (NClosed aq) = (fn y => simpfm (NClosed aq));
fun 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));
fun adjust b =
(fn a as (q, r) =>
(if IntInf.<= ((0 : IntInf.int), IntInf.- (r, b))
then (IntInf.+ (IntInf.* ((2 : IntInf.int), q), (1 : IntInf.int)),
IntInf.- (r, b))
else (IntInf.* ((2 : IntInf.int), q), r)));
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))));
end; (*struct GeneratedCooper*)