# HG changeset patch # User haftmann # Date 1272483666 -7200 # Node ID 48c35032d060998c887b752871d1f12646499921 # Parent 68a837d1a7542f4fe09cacb7211e049b00a272ef updated generated file diff -r 68a837d1a754 -r 48c35032d060 src/HOL/Tools/Qelim/generated_cooper.ML --- a/src/HOL/Tools/Qelim/generated_cooper.ML Wed Apr 28 21:41:05 2010 +0200 +++ b/src/HOL/Tools/Qelim/generated_cooper.ML Wed Apr 28 21:41:06 2010 +0200 @@ -1,49 +1,263 @@ -(* Title: HOL/Tools/Qelim/generated_cooper.ML +(* Generated from Cooper.thy; DO NOT EDIT! *) -This file is generated from HOL/Decision_Procs/Cooper.thy. DO NOT EDIT. -*) - -structure GeneratedCooper = -struct +structure Generated_Cooper : sig + type 'a eq + val eq : 'a eq -> 'a -> 'a -> bool + val eqa : 'a eq -> 'a -> 'a -> bool + val leta : 'a -> ('a -> 'b) -> 'b + val suc : IntInf.int -> IntInf.int + datatype num = C of IntInf.int | Bound of IntInf.int | + Cn of IntInf.int * 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 + | Not of fm | And of fm * fm | Or of fm * fm | Imp of fm * fm | + Iff of fm * fm | E of fm | A of fm | Closed of IntInf.int | + NClosed of IntInf.int + val map : ('a -> 'b) -> 'a list -> 'b list + val append : 'a list -> 'a list -> 'a list + val disjuncts : fm -> fm list + val fm_case : + 'a -> 'a -> (num -> 'a) -> + (num -> 'a) -> + (num -> 'a) -> + (num -> 'a) -> + (num -> 'a) -> + (num -> 'a) -> + (IntInf.int -> num -> 'a) -> + (IntInf.int -> num -> 'a) -> + (fm -> 'a) -> + (fm -> fm -> 'a) -> + (fm -> fm -> 'a) -> + (fm -> fm -> 'a) -> +(fm -> fm -> 'a) -> + (fm -> 'a) -> + (fm -> 'a) -> (IntInf.int -> 'a) -> (IntInf.int -> 'a) -> fm -> 'a + val eq_num : num -> num -> bool + val eq_fm : fm -> fm -> bool + val djf : ('a -> fm) -> 'a -> fm -> fm + val foldr : ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b + val evaldjf : ('a -> fm) -> 'a list -> fm + val dj : (fm -> fm) -> fm -> fm + val disj : fm -> fm -> fm + val minus_nat : IntInf.int -> IntInf.int -> IntInf.int + val decrnum : num -> num + val decr : fm -> fm + val concat_map : ('a -> 'b list) -> 'a list -> 'b list + val numsubst0 : num -> num -> num + val subst0 : num -> fm -> fm + val minusinf : fm -> fm + val eq_int : IntInf.int eq + val zero_int : IntInf.int + type 'a zero + val zero : 'a zero -> 'a + val zero_inta : IntInf.int zero + type 'a times + val times : 'a times -> 'a -> 'a -> 'a + type 'a no_zero_divisors + val times_no_zero_divisors : 'a no_zero_divisors -> 'a times + val zero_no_zero_divisors : 'a no_zero_divisors -> 'a zero + val times_int : IntInf.int times + val no_zero_divisors_int : IntInf.int no_zero_divisors + type 'a one + val one : 'a one -> 'a + type 'a zero_neq_one + val one_zero_neq_one : 'a zero_neq_one -> 'a one + val zero_zero_neq_one : 'a zero_neq_one -> 'a zero + type 'a semigroup_mult + val times_semigroup_mult : 'a semigroup_mult -> 'a times + type 'a plus + val plus : 'a plus -> 'a -> 'a -> 'a + type 'a semigroup_add + val plus_semigroup_add : 'a semigroup_add -> 'a plus + type 'a ab_semigroup_add + val semigroup_add_ab_semigroup_add : 'a ab_semigroup_add -> 'a semigroup_add + type 'a semiring + val ab_semigroup_add_semiring : 'a semiring -> 'a ab_semigroup_add + val semigroup_mult_semiring : 'a semiring -> 'a semigroup_mult + type 'a mult_zero + val times_mult_zero : 'a mult_zero -> 'a times + val zero_mult_zero : 'a mult_zero -> 'a zero + type 'a monoid_add + val semigroup_add_monoid_add : 'a monoid_add -> 'a semigroup_add + val zero_monoid_add : 'a monoid_add -> 'a zero + type 'a comm_monoid_add + val ab_semigroup_add_comm_monoid_add : + 'a comm_monoid_add -> 'a ab_semigroup_add + val monoid_add_comm_monoid_add : 'a comm_monoid_add -> 'a monoid_add + type 'a semiring_0 + val comm_monoid_add_semiring_0 : 'a semiring_0 -> 'a comm_monoid_add + val mult_zero_semiring_0 : 'a semiring_0 -> 'a mult_zero + val semiring_semiring_0 : 'a semiring_0 -> 'a semiring + type 'a power + val one_power : 'a power -> 'a one + val times_power : 'a power -> 'a times + type 'a monoid_mult + val semigroup_mult_monoid_mult : 'a monoid_mult -> 'a semigroup_mult + val power_monoid_mult : 'a monoid_mult -> 'a power + type 'a semiring_1 + val monoid_mult_semiring_1 : 'a semiring_1 -> 'a monoid_mult + val semiring_0_semiring_1 : 'a semiring_1 -> 'a semiring_0 + val zero_neq_one_semiring_1 : 'a semiring_1 -> 'a zero_neq_one + type 'a cancel_semigroup_add + val semigroup_add_cancel_semigroup_add : + 'a cancel_semigroup_add -> 'a semigroup_add + type 'a cancel_ab_semigroup_add + val ab_semigroup_add_cancel_ab_semigroup_add : + 'a cancel_ab_semigroup_add -> 'a ab_semigroup_add + val cancel_semigroup_add_cancel_ab_semigroup_add : + 'a cancel_ab_semigroup_add -> 'a cancel_semigroup_add + type 'a cancel_comm_monoid_add + val cancel_ab_semigroup_add_cancel_comm_monoid_add : + 'a cancel_comm_monoid_add -> 'a cancel_ab_semigroup_add + val comm_monoid_add_cancel_comm_monoid_add : + 'a cancel_comm_monoid_add -> 'a comm_monoid_add + type 'a semiring_0_cancel + val cancel_comm_monoid_add_semiring_0_cancel : + 'a semiring_0_cancel -> 'a cancel_comm_monoid_add + val semiring_0_semiring_0_cancel : 'a semiring_0_cancel -> 'a semiring_0 + type 'a semiring_1_cancel + val semiring_0_cancel_semiring_1_cancel : + 'a semiring_1_cancel -> 'a semiring_0_cancel + val semiring_1_semiring_1_cancel : 'a semiring_1_cancel -> 'a semiring_1 + type 'a dvd + val times_dvd : 'a dvd -> 'a times + type 'a ab_semigroup_mult + val semigroup_mult_ab_semigroup_mult : + 'a ab_semigroup_mult -> 'a semigroup_mult + type 'a comm_semiring + val ab_semigroup_mult_comm_semiring : 'a comm_semiring -> 'a ab_semigroup_mult + val semiring_comm_semiring : 'a comm_semiring -> 'a semiring + type 'a comm_semiring_0 + val comm_semiring_comm_semiring_0 : 'a comm_semiring_0 -> 'a comm_semiring + val semiring_0_comm_semiring_0 : 'a comm_semiring_0 -> 'a semiring_0 + type 'a comm_monoid_mult + val ab_semigroup_mult_comm_monoid_mult : + 'a comm_monoid_mult -> 'a ab_semigroup_mult + val monoid_mult_comm_monoid_mult : 'a comm_monoid_mult -> 'a monoid_mult + type 'a comm_semiring_1 + val comm_monoid_mult_comm_semiring_1 : + 'a comm_semiring_1 -> 'a comm_monoid_mult + val comm_semiring_0_comm_semiring_1 : 'a comm_semiring_1 -> 'a comm_semiring_0 + val dvd_comm_semiring_1 : 'a comm_semiring_1 -> 'a dvd + val semiring_1_comm_semiring_1 : 'a comm_semiring_1 -> 'a semiring_1 + type 'a comm_semiring_0_cancel + val comm_semiring_0_comm_semiring_0_cancel : + 'a comm_semiring_0_cancel -> 'a comm_semiring_0 + val semiring_0_cancel_comm_semiring_0_cancel : + 'a comm_semiring_0_cancel -> 'a semiring_0_cancel + type 'a comm_semiring_1_cancel + val comm_semiring_0_cancel_comm_semiring_1_cancel : + 'a comm_semiring_1_cancel -> 'a comm_semiring_0_cancel + val comm_semiring_1_comm_semiring_1_cancel : + 'a comm_semiring_1_cancel -> 'a comm_semiring_1 + val semiring_1_cancel_comm_semiring_1_cancel : + 'a comm_semiring_1_cancel -> 'a semiring_1_cancel + type 'a diva + val dvd_div : 'a diva -> 'a dvd + val diva : 'a diva -> 'a -> 'a -> 'a + val moda : 'a diva -> 'a -> 'a -> 'a + type 'a semiring_div + val div_semiring_div : 'a semiring_div -> 'a diva + val comm_semiring_1_cancel_semiring_div : + 'a semiring_div -> 'a comm_semiring_1_cancel + val no_zero_divisors_semiring_div : 'a semiring_div -> 'a no_zero_divisors + val one_int : IntInf.int + val one_inta : IntInf.int one + val zero_neq_one_int : IntInf.int zero_neq_one + val semigroup_mult_int : IntInf.int semigroup_mult + val plus_int : IntInf.int plus + val semigroup_add_int : IntInf.int semigroup_add + val ab_semigroup_add_int : IntInf.int ab_semigroup_add + val semiring_int : IntInf.int semiring + val mult_zero_int : IntInf.int mult_zero + val monoid_add_int : IntInf.int monoid_add + val comm_monoid_add_int : IntInf.int comm_monoid_add + val semiring_0_int : IntInf.int semiring_0 + val power_int : IntInf.int power + val monoid_mult_int : IntInf.int monoid_mult + val semiring_1_int : IntInf.int semiring_1 + val cancel_semigroup_add_int : IntInf.int cancel_semigroup_add + val cancel_ab_semigroup_add_int : IntInf.int cancel_ab_semigroup_add + val cancel_comm_monoid_add_int : IntInf.int cancel_comm_monoid_add + val semiring_0_cancel_int : IntInf.int semiring_0_cancel + val semiring_1_cancel_int : IntInf.int semiring_1_cancel + val dvd_int : IntInf.int dvd + val ab_semigroup_mult_int : IntInf.int ab_semigroup_mult + val comm_semiring_int : IntInf.int comm_semiring + val comm_semiring_0_int : IntInf.int comm_semiring_0 + val comm_monoid_mult_int : IntInf.int comm_monoid_mult + val comm_semiring_1_int : IntInf.int comm_semiring_1 + val comm_semiring_0_cancel_int : IntInf.int comm_semiring_0_cancel + val comm_semiring_1_cancel_int : IntInf.int comm_semiring_1_cancel + val abs_int : IntInf.int -> IntInf.int + val split : ('a -> 'b -> 'c) -> 'a * 'b -> 'c + val sgn_int : IntInf.int -> IntInf.int + val apsnd : ('a -> 'b) -> 'c * 'a -> 'c * 'b + val divmod_int : IntInf.int -> IntInf.int -> IntInf.int * IntInf.int + val snd : 'a * 'b -> 'b + val mod_int : IntInf.int -> IntInf.int -> IntInf.int + val fst : 'a * 'b -> 'a + val div_int : IntInf.int -> IntInf.int -> IntInf.int + val div_inta : IntInf.int diva + val semiring_div_int : IntInf.int semiring_div + val dvd : 'a semiring_div * 'a eq -> 'a -> 'a -> bool + val num_case : + (IntInf.int -> 'a) -> + (IntInf.int -> 'a) -> + (IntInf.int -> IntInf.int -> num -> 'a) -> + (num -> 'a) -> + (num -> num -> 'a) -> + (num -> num -> 'a) -> (IntInf.int -> num -> 'a) -> num -> 'a + val nummul : IntInf.int -> num -> num + val numneg : num -> num + val numadd : num * num -> num + val numsub : num -> num -> num + val simpnum : num -> num + val nota : fm -> fm + val iffa : fm -> fm -> fm + val impa : fm -> fm -> fm + val conj : fm -> fm -> fm + val simpfm : fm -> fm + val iupt : IntInf.int -> IntInf.int -> IntInf.int list + val mirror : fm -> fm + val size_list : 'a list -> IntInf.int + val alpha : fm -> num list + val beta : fm -> num list + val eq_numa : num eq + val member : 'a eq -> 'a -> 'a list -> bool + val remdups : 'a eq -> 'a list -> 'a list + val gcd_int : IntInf.int -> IntInf.int -> IntInf.int + val lcm_int : IntInf.int -> IntInf.int -> IntInf.int + val delta : fm -> IntInf.int + val a_beta : fm -> IntInf.int -> fm + val zeta : fm -> IntInf.int + val zsplit0 : num -> IntInf.int * num + val zlfm : fm -> fm + val unita : fm -> fm * (num list * IntInf.int) + val cooper : fm -> fm + val prep : fm -> fm + val qelim : fm -> (fm -> fm) -> fm + val pa : fm -> fm +end = 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; +val eq = #eq : 'a eq -> 'a -> 'a -> bool; -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 eqa A_ a b = eq A_ a b; 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; +fun suc n = IntInf.+ (n, (1 : 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; +datatype num = C of IntInf.int | Bound of IntInf.int | + Cn of IntInf.int * IntInf.int * num | Neg of num | Add of num * num | + Sub of num * num | Mul of IntInf.int * num; -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)))); +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 | Not of fm + | And of fm * fm | Or of fm * fm | Imp of fm * fm | Iff of fm * fm | E of fm | + A of fm | Closed of IntInf.int | NClosed of IntInf.int; fun map f [] = [] | map f (x :: xs) = f x :: map f xs; @@ -110,449 +324,441 @@ | 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_num (C intaa) (C inta) = ((intaa : IntInf.int) = inta) + | eq_num (Bound nata) (Bound nat) = ((nata : IntInf.int) = nat) + | eq_num (Cn (nata, intaa, numa)) (Cn (nat, inta, num)) = + ((nata : IntInf.int) = nat) andalso + (((intaa : IntInf.int) = inta) andalso eq_num numa num) + | eq_num (Neg numa) (Neg num) = eq_num numa num + | eq_num (Add (num1a, num2a)) (Add (num1, num2)) = + eq_num num1a num1 andalso eq_num num2a num2 + | eq_num (Sub (num1a, num2a)) (Sub (num1, num2)) = + eq_num num1a num1 andalso eq_num num2a num2 + | eq_num (Mul (intaa, numa)) (Mul (inta, num)) = + ((intaa : IntInf.int) = inta) andalso eq_num numa num + | eq_num (C inta) (Bound nat) = false + | eq_num (Bound nat) (C inta) = false + | eq_num (C intaa) (Cn (nat, inta, num)) = false + | eq_num (Cn (nat, intaa, num)) (C inta) = false + | eq_num (C inta) (Neg num) = false + | eq_num (Neg num) (C inta) = false + | eq_num (C inta) (Add (num1, num2)) = false + | eq_num (Add (num1, num2)) (C inta) = false + | eq_num (C inta) (Sub (num1, num2)) = false + | eq_num (Sub (num1, num2)) (C inta) = false + | eq_num (C intaa) (Mul (inta, num)) = false + | eq_num (Mul (intaa, num)) (C inta) = false + | eq_num (Bound nata) (Cn (nat, inta, num)) = false + | eq_num (Cn (nata, inta, num)) (Bound nat) = false + | eq_num (Bound nat) (Neg num) = false + | eq_num (Neg num) (Bound nat) = false + | eq_num (Bound nat) (Add (num1, num2)) = false + | eq_num (Add (num1, num2)) (Bound nat) = false + | eq_num (Bound nat) (Sub (num1, num2)) = false + | eq_num (Sub (num1, num2)) (Bound nat) = false + | eq_num (Bound nat) (Mul (inta, num)) = false + | eq_num (Mul (inta, num)) (Bound nat) = false + | eq_num (Cn (nat, inta, numa)) (Neg num) = false + | eq_num (Neg numa) (Cn (nat, inta, num)) = false + | eq_num (Cn (nat, inta, num)) (Add (num1, num2)) = false + | eq_num (Add (num1, num2)) (Cn (nat, inta, num)) = false + | eq_num (Cn (nat, inta, num)) (Sub (num1, num2)) = false + | eq_num (Sub (num1, num2)) (Cn (nat, inta, num)) = false + | eq_num (Cn (nat, intaa, numa)) (Mul (inta, num)) = false + | eq_num (Mul (intaa, numa)) (Cn (nat, inta, num)) = false + | eq_num (Neg num) (Add (num1, num2)) = false + | eq_num (Add (num1, num2)) (Neg num) = false + | eq_num (Neg num) (Sub (num1, num2)) = false + | eq_num (Sub (num1, num2)) (Neg num) = false + | eq_num (Neg numa) (Mul (inta, num)) = false + | eq_num (Mul (inta, numa)) (Neg num) = false + | eq_num (Add (num1a, num2a)) (Sub (num1, num2)) = false + | eq_num (Sub (num1a, num2a)) (Add (num1, num2)) = false + | eq_num (Add (num1, num2)) (Mul (inta, num)) = false + | eq_num (Mul (inta, num)) (Add (num1, num2)) = false + | eq_num (Sub (num1, num2)) (Mul (inta, num)) = false + | eq_num (Mul (inta, num)) (Sub (num1, num2)) = false; -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 +fun eq_fm T T = true + | eq_fm F F = true + | eq_fm (Lt numa) (Lt num) = eq_num numa num + | eq_fm (Le numa) (Le num) = eq_num numa num + | eq_fm (Gt numa) (Gt num) = eq_num numa num + | eq_fm (Ge numa) (Ge num) = eq_num numa num + | eq_fm (Eq numa) (Eq num) = eq_num numa num + | eq_fm (NEq numa) (NEq num) = eq_num numa num + | eq_fm (Dvd (intaa, numa)) (Dvd (inta, num)) = + ((intaa : IntInf.int) = inta) andalso eq_num numa num + | eq_fm (NDvd (intaa, numa)) (NDvd (inta, num)) = + ((intaa : IntInf.int) = inta) andalso eq_num numa num + | eq_fm (Not fma) (Not fm) = eq_fm fma fm + | eq_fm (And (fm1a, fm2a)) (And (fm1, fm2)) = + eq_fm fm1a fm1 andalso eq_fm fm2a fm2 + | eq_fm (Or (fm1a, fm2a)) (Or (fm1, fm2)) = + eq_fm fm1a fm1 andalso eq_fm fm2a fm2 + | eq_fm (Imp (fm1a, fm2a)) (Imp (fm1, fm2)) = + eq_fm fm1a fm1 andalso eq_fm fm2a fm2 + | eq_fm (Iff (fm1a, fm2a)) (Iff (fm1, fm2)) = + eq_fm fm1a fm1 andalso eq_fm fm2a fm2 + | eq_fm (E fma) (E fm) = eq_fm fma fm + | eq_fm (A fma) (A fm) = eq_fm fma fm + | eq_fm (Closed nata) (Closed nat) = ((nata : IntInf.int) = nat) + | eq_fm (NClosed nata) (NClosed nat) = ((nata : IntInf.int) = nat) + | eq_fm T F = 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; + | eq_fm T (Lt num) = false + | eq_fm (Lt num) T = false + | eq_fm T (Le num) = false + | eq_fm (Le num) T = false + | eq_fm T (Gt num) = false + | eq_fm (Gt num) T = false + | eq_fm T (Ge num) = false + | eq_fm (Ge num) T = false + | eq_fm T (Eq num) = false + | eq_fm (Eq num) T = false + | eq_fm T (NEq num) = false + | eq_fm (NEq num) T = false + | eq_fm T (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) T = false + | eq_fm T (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) T = false + | eq_fm T (Not fm) = false + | eq_fm (Not fm) T = false + | eq_fm T (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) T = false + | eq_fm T (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) T = false + | eq_fm T (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) T = false + | eq_fm T (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) T = false + | eq_fm T (E fm) = false + | eq_fm (E fm) T = false + | eq_fm T (A fm) = false + | eq_fm (A fm) T = false + | eq_fm T (Closed nat) = false + | eq_fm (Closed nat) T = false + | eq_fm T (NClosed nat) = false + | eq_fm (NClosed nat) T = false + | eq_fm F (Lt num) = false + | eq_fm (Lt num) F = false + | eq_fm F (Le num) = false + | eq_fm (Le num) F = false + | eq_fm F (Gt num) = false + | eq_fm (Gt num) F = false + | eq_fm F (Ge num) = false + | eq_fm (Ge num) F = false + | eq_fm F (Eq num) = false + | eq_fm (Eq num) F = false + | eq_fm F (NEq num) = false + | eq_fm (NEq num) F = false + | eq_fm F (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) F = false + | eq_fm F (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) F = false + | eq_fm F (Not fm) = false + | eq_fm (Not fm) F = false + | eq_fm F (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) F = false + | eq_fm F (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) F = false + | eq_fm F (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) F = false + | eq_fm F (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) F = false + | eq_fm F (E fm) = false + | eq_fm (E fm) F = false + | eq_fm F (A fm) = false + | eq_fm (A fm) F = false + | eq_fm F (Closed nat) = false + | eq_fm (Closed nat) F = false + | eq_fm F (NClosed nat) = false + | eq_fm (NClosed nat) F = false + | eq_fm (Lt numa) (Le num) = false + | eq_fm (Le numa) (Lt num) = false + | eq_fm (Lt numa) (Gt num) = false + | eq_fm (Gt numa) (Lt num) = false + | eq_fm (Lt numa) (Ge num) = false + | eq_fm (Ge numa) (Lt num) = false + | eq_fm (Lt numa) (Eq num) = false + | eq_fm (Eq numa) (Lt num) = false + | eq_fm (Lt numa) (NEq num) = false + | eq_fm (NEq numa) (Lt num) = false + | eq_fm (Lt numa) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, numa)) (Lt num) = false + | eq_fm (Lt numa) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, numa)) (Lt num) = false + | eq_fm (Lt num) (Not fm) = false + | eq_fm (Not fm) (Lt num) = false + | eq_fm (Lt num) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (Lt num) = false + | eq_fm (Lt num) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (Lt num) = false + | eq_fm (Lt num) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (Lt num) = false + | eq_fm (Lt num) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (Lt num) = false + | eq_fm (Lt num) (E fm) = false + | eq_fm (E fm) (Lt num) = false + | eq_fm (Lt num) (A fm) = false + | eq_fm (A fm) (Lt num) = false + | eq_fm (Lt num) (Closed nat) = false + | eq_fm (Closed nat) (Lt num) = false + | eq_fm (Lt num) (NClosed nat) = false + | eq_fm (NClosed nat) (Lt num) = false + | eq_fm (Le numa) (Gt num) = false + | eq_fm (Gt numa) (Le num) = false + | eq_fm (Le numa) (Ge num) = false + | eq_fm (Ge numa) (Le num) = false + | eq_fm (Le numa) (Eq num) = false + | eq_fm (Eq numa) (Le num) = false + | eq_fm (Le numa) (NEq num) = false + | eq_fm (NEq numa) (Le num) = false + | eq_fm (Le numa) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, numa)) (Le num) = false + | eq_fm (Le numa) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, numa)) (Le num) = false + | eq_fm (Le num) (Not fm) = false + | eq_fm (Not fm) (Le num) = false + | eq_fm (Le num) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (Le num) = false + | eq_fm (Le num) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (Le num) = false + | eq_fm (Le num) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (Le num) = false + | eq_fm (Le num) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (Le num) = false + | eq_fm (Le num) (E fm) = false + | eq_fm (E fm) (Le num) = false + | eq_fm (Le num) (A fm) = false + | eq_fm (A fm) (Le num) = false + | eq_fm (Le num) (Closed nat) = false + | eq_fm (Closed nat) (Le num) = false + | eq_fm (Le num) (NClosed nat) = false + | eq_fm (NClosed nat) (Le num) = false + | eq_fm (Gt numa) (Ge num) = false + | eq_fm (Ge numa) (Gt num) = false + | eq_fm (Gt numa) (Eq num) = false + | eq_fm (Eq numa) (Gt num) = false + | eq_fm (Gt numa) (NEq num) = false + | eq_fm (NEq numa) (Gt num) = false + | eq_fm (Gt numa) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, numa)) (Gt num) = false + | eq_fm (Gt numa) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, numa)) (Gt num) = false + | eq_fm (Gt num) (Not fm) = false + | eq_fm (Not fm) (Gt num) = false + | eq_fm (Gt num) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (Gt num) = false + | eq_fm (Gt num) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (Gt num) = false + | eq_fm (Gt num) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (Gt num) = false + | eq_fm (Gt num) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (Gt num) = false + | eq_fm (Gt num) (E fm) = false + | eq_fm (E fm) (Gt num) = false + | eq_fm (Gt num) (A fm) = false + | eq_fm (A fm) (Gt num) = false + | eq_fm (Gt num) (Closed nat) = false + | eq_fm (Closed nat) (Gt num) = false + | eq_fm (Gt num) (NClosed nat) = false + | eq_fm (NClosed nat) (Gt num) = false + | eq_fm (Ge numa) (Eq num) = false + | eq_fm (Eq numa) (Ge num) = false + | eq_fm (Ge numa) (NEq num) = false + | eq_fm (NEq numa) (Ge num) = false + | eq_fm (Ge numa) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, numa)) (Ge num) = false + | eq_fm (Ge numa) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, numa)) (Ge num) = false + | eq_fm (Ge num) (Not fm) = false + | eq_fm (Not fm) (Ge num) = false + | eq_fm (Ge num) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (Ge num) = false + | eq_fm (Ge num) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (Ge num) = false + | eq_fm (Ge num) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (Ge num) = false + | eq_fm (Ge num) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (Ge num) = false + | eq_fm (Ge num) (E fm) = false + | eq_fm (E fm) (Ge num) = false + | eq_fm (Ge num) (A fm) = false + | eq_fm (A fm) (Ge num) = false + | eq_fm (Ge num) (Closed nat) = false + | eq_fm (Closed nat) (Ge num) = false + | eq_fm (Ge num) (NClosed nat) = false + | eq_fm (NClosed nat) (Ge num) = false + | eq_fm (Eq numa) (NEq num) = false + | eq_fm (NEq numa) (Eq num) = false + | eq_fm (Eq numa) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, numa)) (Eq num) = false + | eq_fm (Eq numa) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, numa)) (Eq num) = false + | eq_fm (Eq num) (Not fm) = false + | eq_fm (Not fm) (Eq num) = false + | eq_fm (Eq num) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (Eq num) = false + | eq_fm (Eq num) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (Eq num) = false + | eq_fm (Eq num) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (Eq num) = false + | eq_fm (Eq num) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (Eq num) = false + | eq_fm (Eq num) (E fm) = false + | eq_fm (E fm) (Eq num) = false + | eq_fm (Eq num) (A fm) = false + | eq_fm (A fm) (Eq num) = false + | eq_fm (Eq num) (Closed nat) = false + | eq_fm (Closed nat) (Eq num) = false + | eq_fm (Eq num) (NClosed nat) = false + | eq_fm (NClosed nat) (Eq num) = false + | eq_fm (NEq numa) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, numa)) (NEq num) = false + | eq_fm (NEq numa) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, numa)) (NEq num) = false + | eq_fm (NEq num) (Not fm) = false + | eq_fm (Not fm) (NEq num) = false + | eq_fm (NEq num) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (NEq num) = false + | eq_fm (NEq num) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (NEq num) = false + | eq_fm (NEq num) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (NEq num) = false + | eq_fm (NEq num) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (NEq num) = false + | eq_fm (NEq num) (E fm) = false + | eq_fm (E fm) (NEq num) = false + | eq_fm (NEq num) (A fm) = false + | eq_fm (A fm) (NEq num) = false + | eq_fm (NEq num) (Closed nat) = false + | eq_fm (Closed nat) (NEq num) = false + | eq_fm (NEq num) (NClosed nat) = false + | eq_fm (NClosed nat) (NEq num) = false + | eq_fm (Dvd (intaa, numa)) (NDvd (inta, num)) = false + | eq_fm (NDvd (intaa, numa)) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (Not fm) = false + | eq_fm (Not fm) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (E fm) = false + | eq_fm (E fm) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (A fm) = false + | eq_fm (A fm) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (Closed nat) = false + | eq_fm (Closed nat) (Dvd (inta, num)) = false + | eq_fm (Dvd (inta, num)) (NClosed nat) = false + | eq_fm (NClosed nat) (Dvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (Not fm) = false + | eq_fm (Not fm) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (E fm) = false + | eq_fm (E fm) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (A fm) = false + | eq_fm (A fm) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (Closed nat) = false + | eq_fm (Closed nat) (NDvd (inta, num)) = false + | eq_fm (NDvd (inta, num)) (NClosed nat) = false + | eq_fm (NClosed nat) (NDvd (inta, num)) = false + | eq_fm (Not fm) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (Not fm) = false + | eq_fm (Not fm) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (Not fm) = false + | eq_fm (Not fm) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (Not fm) = false + | eq_fm (Not fm) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (Not fm) = false + | eq_fm (Not fma) (E fm) = false + | eq_fm (E fma) (Not fm) = false + | eq_fm (Not fma) (A fm) = false + | eq_fm (A fma) (Not fm) = false + | eq_fm (Not fm) (Closed nat) = false + | eq_fm (Closed nat) (Not fm) = false + | eq_fm (Not fm) (NClosed nat) = false + | eq_fm (NClosed nat) (Not fm) = false + | eq_fm (And (fm1a, fm2a)) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1a, fm2a)) (And (fm1, fm2)) = false + | eq_fm (And (fm1a, fm2a)) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1a, fm2a)) (And (fm1, fm2)) = false + | eq_fm (And (fm1a, fm2a)) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1a, fm2a)) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (E fm) = false + | eq_fm (E fm) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (A fm) = false + | eq_fm (A fm) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (Closed nat) = false + | eq_fm (Closed nat) (And (fm1, fm2)) = false + | eq_fm (And (fm1, fm2)) (NClosed nat) = false + | eq_fm (NClosed nat) (And (fm1, fm2)) = false + | eq_fm (Or (fm1a, fm2a)) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1a, fm2a)) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1a, fm2a)) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1a, fm2a)) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (E fm) = false + | eq_fm (E fm) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (A fm) = false + | eq_fm (A fm) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (Closed nat) = false + | eq_fm (Closed nat) (Or (fm1, fm2)) = false + | eq_fm (Or (fm1, fm2)) (NClosed nat) = false + | eq_fm (NClosed nat) (Or (fm1, fm2)) = false + | eq_fm (Imp (fm1a, fm2a)) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1a, fm2a)) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (E fm) = false + | eq_fm (E fm) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (A fm) = false + | eq_fm (A fm) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (Closed nat) = false + | eq_fm (Closed nat) (Imp (fm1, fm2)) = false + | eq_fm (Imp (fm1, fm2)) (NClosed nat) = false + | eq_fm (NClosed nat) (Imp (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (E fm) = false + | eq_fm (E fm) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (A fm) = false + | eq_fm (A fm) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (Closed nat) = false + | eq_fm (Closed nat) (Iff (fm1, fm2)) = false + | eq_fm (Iff (fm1, fm2)) (NClosed nat) = false + | eq_fm (NClosed nat) (Iff (fm1, fm2)) = false + | eq_fm (E fma) (A fm) = false + | eq_fm (A fma) (E fm) = false + | eq_fm (E fm) (Closed nat) = false + | eq_fm (Closed nat) (E fm) = false + | eq_fm (E fm) (NClosed nat) = false + | eq_fm (NClosed nat) (E fm) = false + | eq_fm (A fm) (Closed nat) = false + | eq_fm (Closed nat) (A fm) = false + | eq_fm (A fm) (NClosed nat) = false + | eq_fm (NClosed nat) (A fm) = false + | eq_fm (Closed nata) (NClosed nat) = false + | eq_fm (NClosed nata) (Closed nat) = false; 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)); + (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 _ => Or (f p, q) + | Le _ => Or (f p, q) | Gt _ => Or (f p, q) + | Ge _ => Or (f p, q) | Eq _ => Or (f p, q) + | NEq _ => Or (f p, q) | Dvd (_, _) => Or (f p, q) + | NDvd (_, _) => Or (f p, q) | Not _ => Or (f p, q) + | And (_, _) => Or (f p, q) | Or (_, _) => Or (f p, q) + | Imp (_, _) => Or (f p, q) | Iff (_, _) => Or (f p, q) + | E _ => Or (f p, q) | A _ => Or (f p, q) + | Closed _ => Or (f p, q) | NClosed _ => Or (f p, q)))); fun foldr f [] a = a | foldr f (x :: xs) a = f x (foldr f xs a); @@ -562,18 +768,17 @@ 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)))); + (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 minus_nat n m = IntInf.max (0, (IntInf.- (n, m))); -fun decrnum (Bound n) = Bound (minus_nat n 1) +fun decrnum (Bound n) = Bound (minus_nat n (1 : IntInf.int)) | 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 (Cn (n, i, a)) = Cn (minus_nat n (1 : IntInf.int), i, decrnum a) | decrnum (C u) = C u; fun decr (Lt a) = Lt (decrnum a) @@ -596,20 +801,20 @@ | 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 concat_map f [] = [] + | concat_map f (x :: xs) = append (f x) (concat_map f xs); fun numsubst0 t (C c) = C c - | numsubst0 t (Bound n) = (if eqop eq_nat n 0 then t else Bound n) + | numsubst0 t (Bound n) = + (if ((n : IntInf.int) = (0 : IntInf.int)) 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)); + (if ((v : IntInf.int) = (0 : IntInf.int)) + then Add (Mul (i, t), numsubst0 t a) + else Cn (suc (minus_nat v (1 : IntInf.int)), i, numsubst0 t a)); fun subst0 t T = T | subst0 t F = F @@ -679,49 +884,417 @@ | 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))) + (if ((cm : IntInf.int) = (0 : IntInf.int)) then T + else Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), 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))) + (if ((dm : IntInf.int) = (0 : IntInf.int)) then T + else Le (Cn (suc (minus_nat dm (1 : IntInf.int)), 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))) + (if ((em : IntInf.int) = (0 : IntInf.int)) then F + else Gt (Cn (suc (minus_nat em (1 : IntInf.int)), 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))) + (if ((fm : IntInf.int) = (0 : IntInf.int)) then F + else Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), 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))) + (if ((gm : IntInf.int) = (0 : IntInf.int)) then F + else Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), 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))); + (if ((hm : IntInf.int) = (0 : IntInf.int)) then T + else NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e))); val eq_int = {eq = (fn a => fn b => ((a : IntInf.int) = b))} : IntInf.int eq; +val zero_int : IntInf.int = (0 : IntInf.int); + +type 'a zero = {zero : 'a}; +val zero = #zero : 'a zero -> 'a; + +val zero_inta = {zero = zero_int} : IntInf.int zero; + +type 'a times = {times : 'a -> 'a -> 'a}; +val times = #times : 'a times -> 'a -> 'a -> 'a; + +type 'a no_zero_divisors = + {times_no_zero_divisors : 'a times, zero_no_zero_divisors : 'a zero}; +val times_no_zero_divisors = #times_no_zero_divisors : + 'a no_zero_divisors -> 'a times; +val zero_no_zero_divisors = #zero_no_zero_divisors : + 'a no_zero_divisors -> 'a zero; + +val times_int = {times = (fn a => fn b => IntInf.* (a, b))} : IntInf.int times; + +val no_zero_divisors_int = + {times_no_zero_divisors = times_int, zero_no_zero_divisors = zero_inta} : + IntInf.int no_zero_divisors; + +type 'a one = {one : 'a}; +val one = #one : 'a one -> 'a; + +type 'a zero_neq_one = {one_zero_neq_one : 'a one, zero_zero_neq_one : 'a zero}; +val one_zero_neq_one = #one_zero_neq_one : 'a zero_neq_one -> 'a one; +val zero_zero_neq_one = #zero_zero_neq_one : 'a zero_neq_one -> 'a zero; + +type 'a semigroup_mult = {times_semigroup_mult : 'a times}; +val times_semigroup_mult = #times_semigroup_mult : + 'a semigroup_mult -> 'a times; + +type 'a plus = {plus : 'a -> 'a -> 'a}; +val plus = #plus : 'a plus -> 'a -> 'a -> 'a; + +type 'a semigroup_add = {plus_semigroup_add : 'a plus}; +val plus_semigroup_add = #plus_semigroup_add : 'a semigroup_add -> 'a plus; + +type 'a ab_semigroup_add = {semigroup_add_ab_semigroup_add : 'a semigroup_add}; +val semigroup_add_ab_semigroup_add = #semigroup_add_ab_semigroup_add : + 'a ab_semigroup_add -> 'a semigroup_add; + +type 'a semiring = + {ab_semigroup_add_semiring : 'a ab_semigroup_add, + semigroup_mult_semiring : 'a semigroup_mult}; +val ab_semigroup_add_semiring = #ab_semigroup_add_semiring : + 'a semiring -> 'a ab_semigroup_add; +val semigroup_mult_semiring = #semigroup_mult_semiring : + 'a semiring -> 'a semigroup_mult; + +type 'a mult_zero = {times_mult_zero : 'a times, zero_mult_zero : 'a zero}; +val times_mult_zero = #times_mult_zero : 'a mult_zero -> 'a times; +val zero_mult_zero = #zero_mult_zero : 'a mult_zero -> 'a zero; + +type 'a monoid_add = + {semigroup_add_monoid_add : 'a semigroup_add, zero_monoid_add : 'a zero}; +val semigroup_add_monoid_add = #semigroup_add_monoid_add : + 'a monoid_add -> 'a semigroup_add; +val zero_monoid_add = #zero_monoid_add : 'a monoid_add -> 'a zero; + +type 'a comm_monoid_add = + {ab_semigroup_add_comm_monoid_add : 'a ab_semigroup_add, + monoid_add_comm_monoid_add : 'a monoid_add}; +val ab_semigroup_add_comm_monoid_add = #ab_semigroup_add_comm_monoid_add : + 'a comm_monoid_add -> 'a ab_semigroup_add; +val monoid_add_comm_monoid_add = #monoid_add_comm_monoid_add : + 'a comm_monoid_add -> 'a monoid_add; + +type 'a semiring_0 = + {comm_monoid_add_semiring_0 : 'a comm_monoid_add, + mult_zero_semiring_0 : 'a mult_zero, semiring_semiring_0 : 'a semiring}; +val comm_monoid_add_semiring_0 = #comm_monoid_add_semiring_0 : + 'a semiring_0 -> 'a comm_monoid_add; +val mult_zero_semiring_0 = #mult_zero_semiring_0 : + 'a semiring_0 -> 'a mult_zero; +val semiring_semiring_0 = #semiring_semiring_0 : 'a semiring_0 -> 'a semiring; + +type 'a power = {one_power : 'a one, times_power : 'a times}; +val one_power = #one_power : 'a power -> 'a one; +val times_power = #times_power : 'a power -> 'a times; + +type 'a monoid_mult = + {semigroup_mult_monoid_mult : 'a semigroup_mult, + power_monoid_mult : 'a power}; +val semigroup_mult_monoid_mult = #semigroup_mult_monoid_mult : + 'a monoid_mult -> 'a semigroup_mult; +val power_monoid_mult = #power_monoid_mult : 'a monoid_mult -> 'a power; + +type 'a semiring_1 = + {monoid_mult_semiring_1 : 'a monoid_mult, + semiring_0_semiring_1 : 'a semiring_0, + zero_neq_one_semiring_1 : 'a zero_neq_one}; +val monoid_mult_semiring_1 = #monoid_mult_semiring_1 : + 'a semiring_1 -> 'a monoid_mult; +val semiring_0_semiring_1 = #semiring_0_semiring_1 : + 'a semiring_1 -> 'a semiring_0; +val zero_neq_one_semiring_1 = #zero_neq_one_semiring_1 : + 'a semiring_1 -> 'a zero_neq_one; + +type 'a cancel_semigroup_add = + {semigroup_add_cancel_semigroup_add : 'a semigroup_add}; +val semigroup_add_cancel_semigroup_add = #semigroup_add_cancel_semigroup_add : + 'a cancel_semigroup_add -> 'a semigroup_add; + +type 'a cancel_ab_semigroup_add = + {ab_semigroup_add_cancel_ab_semigroup_add : 'a ab_semigroup_add, + cancel_semigroup_add_cancel_ab_semigroup_add : 'a cancel_semigroup_add}; +val ab_semigroup_add_cancel_ab_semigroup_add = + #ab_semigroup_add_cancel_ab_semigroup_add : + 'a cancel_ab_semigroup_add -> 'a ab_semigroup_add; +val cancel_semigroup_add_cancel_ab_semigroup_add = + #cancel_semigroup_add_cancel_ab_semigroup_add : + 'a cancel_ab_semigroup_add -> 'a cancel_semigroup_add; + +type 'a cancel_comm_monoid_add = + {cancel_ab_semigroup_add_cancel_comm_monoid_add : 'a cancel_ab_semigroup_add, + comm_monoid_add_cancel_comm_monoid_add : 'a comm_monoid_add}; +val cancel_ab_semigroup_add_cancel_comm_monoid_add = + #cancel_ab_semigroup_add_cancel_comm_monoid_add : + 'a cancel_comm_monoid_add -> 'a cancel_ab_semigroup_add; +val comm_monoid_add_cancel_comm_monoid_add = + #comm_monoid_add_cancel_comm_monoid_add : + 'a cancel_comm_monoid_add -> 'a comm_monoid_add; + +type 'a semiring_0_cancel = + {cancel_comm_monoid_add_semiring_0_cancel : 'a cancel_comm_monoid_add, + semiring_0_semiring_0_cancel : 'a semiring_0}; +val cancel_comm_monoid_add_semiring_0_cancel = + #cancel_comm_monoid_add_semiring_0_cancel : + 'a semiring_0_cancel -> 'a cancel_comm_monoid_add; +val semiring_0_semiring_0_cancel = #semiring_0_semiring_0_cancel : + 'a semiring_0_cancel -> 'a semiring_0; + +type 'a semiring_1_cancel = + {semiring_0_cancel_semiring_1_cancel : 'a semiring_0_cancel, + semiring_1_semiring_1_cancel : 'a semiring_1}; +val semiring_0_cancel_semiring_1_cancel = #semiring_0_cancel_semiring_1_cancel : + 'a semiring_1_cancel -> 'a semiring_0_cancel; +val semiring_1_semiring_1_cancel = #semiring_1_semiring_1_cancel : + 'a semiring_1_cancel -> 'a semiring_1; + +type 'a dvd = {times_dvd : 'a times}; +val times_dvd = #times_dvd : 'a dvd -> 'a times; + +type 'a ab_semigroup_mult = + {semigroup_mult_ab_semigroup_mult : 'a semigroup_mult}; +val semigroup_mult_ab_semigroup_mult = #semigroup_mult_ab_semigroup_mult : + 'a ab_semigroup_mult -> 'a semigroup_mult; + +type 'a comm_semiring = + {ab_semigroup_mult_comm_semiring : 'a ab_semigroup_mult, + semiring_comm_semiring : 'a semiring}; +val ab_semigroup_mult_comm_semiring = #ab_semigroup_mult_comm_semiring : + 'a comm_semiring -> 'a ab_semigroup_mult; +val semiring_comm_semiring = #semiring_comm_semiring : + 'a comm_semiring -> 'a semiring; + +type 'a comm_semiring_0 = + {comm_semiring_comm_semiring_0 : 'a comm_semiring, + semiring_0_comm_semiring_0 : 'a semiring_0}; +val comm_semiring_comm_semiring_0 = #comm_semiring_comm_semiring_0 : + 'a comm_semiring_0 -> 'a comm_semiring; +val semiring_0_comm_semiring_0 = #semiring_0_comm_semiring_0 : + 'a comm_semiring_0 -> 'a semiring_0; + +type 'a comm_monoid_mult = + {ab_semigroup_mult_comm_monoid_mult : 'a ab_semigroup_mult, + monoid_mult_comm_monoid_mult : 'a monoid_mult}; +val ab_semigroup_mult_comm_monoid_mult = #ab_semigroup_mult_comm_monoid_mult : + 'a comm_monoid_mult -> 'a ab_semigroup_mult; +val monoid_mult_comm_monoid_mult = #monoid_mult_comm_monoid_mult : + 'a comm_monoid_mult -> 'a monoid_mult; + +type 'a comm_semiring_1 = + {comm_monoid_mult_comm_semiring_1 : 'a comm_monoid_mult, + comm_semiring_0_comm_semiring_1 : 'a comm_semiring_0, + dvd_comm_semiring_1 : 'a dvd, semiring_1_comm_semiring_1 : 'a semiring_1}; +val comm_monoid_mult_comm_semiring_1 = #comm_monoid_mult_comm_semiring_1 : + 'a comm_semiring_1 -> 'a comm_monoid_mult; +val comm_semiring_0_comm_semiring_1 = #comm_semiring_0_comm_semiring_1 : + 'a comm_semiring_1 -> 'a comm_semiring_0; +val dvd_comm_semiring_1 = #dvd_comm_semiring_1 : 'a comm_semiring_1 -> 'a dvd; +val semiring_1_comm_semiring_1 = #semiring_1_comm_semiring_1 : + 'a comm_semiring_1 -> 'a semiring_1; + +type 'a comm_semiring_0_cancel = + {comm_semiring_0_comm_semiring_0_cancel : 'a comm_semiring_0, + semiring_0_cancel_comm_semiring_0_cancel : 'a semiring_0_cancel}; +val comm_semiring_0_comm_semiring_0_cancel = + #comm_semiring_0_comm_semiring_0_cancel : + 'a comm_semiring_0_cancel -> 'a comm_semiring_0; +val semiring_0_cancel_comm_semiring_0_cancel = + #semiring_0_cancel_comm_semiring_0_cancel : + 'a comm_semiring_0_cancel -> 'a semiring_0_cancel; + +type 'a comm_semiring_1_cancel = + {comm_semiring_0_cancel_comm_semiring_1_cancel : 'a comm_semiring_0_cancel, + comm_semiring_1_comm_semiring_1_cancel : 'a comm_semiring_1, + semiring_1_cancel_comm_semiring_1_cancel : 'a semiring_1_cancel}; +val comm_semiring_0_cancel_comm_semiring_1_cancel = + #comm_semiring_0_cancel_comm_semiring_1_cancel : + 'a comm_semiring_1_cancel -> 'a comm_semiring_0_cancel; +val comm_semiring_1_comm_semiring_1_cancel = + #comm_semiring_1_comm_semiring_1_cancel : + 'a comm_semiring_1_cancel -> 'a comm_semiring_1; +val semiring_1_cancel_comm_semiring_1_cancel = + #semiring_1_cancel_comm_semiring_1_cancel : + 'a comm_semiring_1_cancel -> 'a semiring_1_cancel; + +type 'a diva = {dvd_div : 'a dvd, diva : 'a -> 'a -> 'a, moda : 'a -> 'a -> 'a}; +val dvd_div = #dvd_div : 'a diva -> 'a dvd; +val diva = #diva : 'a diva -> 'a -> 'a -> 'a; +val moda = #moda : 'a diva -> 'a -> 'a -> 'a; + +type 'a semiring_div = + {div_semiring_div : 'a diva, + comm_semiring_1_cancel_semiring_div : 'a comm_semiring_1_cancel, + no_zero_divisors_semiring_div : 'a no_zero_divisors}; +val div_semiring_div = #div_semiring_div : 'a semiring_div -> 'a diva; +val comm_semiring_1_cancel_semiring_div = #comm_semiring_1_cancel_semiring_div : + 'a semiring_div -> 'a comm_semiring_1_cancel; +val no_zero_divisors_semiring_div = #no_zero_divisors_semiring_div : + 'a semiring_div -> 'a no_zero_divisors; + +val one_int : IntInf.int = (1 : IntInf.int); + +val one_inta = {one = one_int} : IntInf.int one; + +val zero_neq_one_int = + {one_zero_neq_one = one_inta, zero_zero_neq_one = zero_inta} : + IntInf.int zero_neq_one; + +val semigroup_mult_int = {times_semigroup_mult = times_int} : + IntInf.int semigroup_mult; + +val plus_int = {plus = (fn a => fn b => IntInf.+ (a, b))} : IntInf.int plus; + +val semigroup_add_int = {plus_semigroup_add = plus_int} : + IntInf.int semigroup_add; + +val ab_semigroup_add_int = {semigroup_add_ab_semigroup_add = semigroup_add_int} + : IntInf.int ab_semigroup_add; + +val semiring_int = + {ab_semigroup_add_semiring = ab_semigroup_add_int, + semigroup_mult_semiring = semigroup_mult_int} + : IntInf.int semiring; + +val mult_zero_int = {times_mult_zero = times_int, zero_mult_zero = zero_inta} : + IntInf.int mult_zero; + +val monoid_add_int = + {semigroup_add_monoid_add = semigroup_add_int, zero_monoid_add = zero_inta} : + IntInf.int monoid_add; + +val comm_monoid_add_int = + {ab_semigroup_add_comm_monoid_add = ab_semigroup_add_int, + monoid_add_comm_monoid_add = monoid_add_int} + : IntInf.int comm_monoid_add; + +val semiring_0_int = + {comm_monoid_add_semiring_0 = comm_monoid_add_int, + mult_zero_semiring_0 = mult_zero_int, semiring_semiring_0 = semiring_int} + : IntInf.int semiring_0; + +val power_int = {one_power = one_inta, times_power = times_int} : + IntInf.int power; + +val monoid_mult_int = + {semigroup_mult_monoid_mult = semigroup_mult_int, + power_monoid_mult = power_int} + : IntInf.int monoid_mult; + +val semiring_1_int = + {monoid_mult_semiring_1 = monoid_mult_int, + semiring_0_semiring_1 = semiring_0_int, + zero_neq_one_semiring_1 = zero_neq_one_int} + : IntInf.int semiring_1; + +val cancel_semigroup_add_int = + {semigroup_add_cancel_semigroup_add = semigroup_add_int} : + IntInf.int cancel_semigroup_add; + +val cancel_ab_semigroup_add_int = + {ab_semigroup_add_cancel_ab_semigroup_add = ab_semigroup_add_int, + cancel_semigroup_add_cancel_ab_semigroup_add = cancel_semigroup_add_int} + : IntInf.int cancel_ab_semigroup_add; + +val cancel_comm_monoid_add_int = + {cancel_ab_semigroup_add_cancel_comm_monoid_add = cancel_ab_semigroup_add_int, + comm_monoid_add_cancel_comm_monoid_add = comm_monoid_add_int} + : IntInf.int cancel_comm_monoid_add; + +val semiring_0_cancel_int = + {cancel_comm_monoid_add_semiring_0_cancel = cancel_comm_monoid_add_int, + semiring_0_semiring_0_cancel = semiring_0_int} + : IntInf.int semiring_0_cancel; + +val semiring_1_cancel_int = + {semiring_0_cancel_semiring_1_cancel = semiring_0_cancel_int, + semiring_1_semiring_1_cancel = semiring_1_int} + : IntInf.int semiring_1_cancel; + +val dvd_int = {times_dvd = times_int} : IntInf.int dvd; + +val ab_semigroup_mult_int = + {semigroup_mult_ab_semigroup_mult = semigroup_mult_int} : + IntInf.int ab_semigroup_mult; + +val comm_semiring_int = + {ab_semigroup_mult_comm_semiring = ab_semigroup_mult_int, + semiring_comm_semiring = semiring_int} + : IntInf.int comm_semiring; + +val comm_semiring_0_int = + {comm_semiring_comm_semiring_0 = comm_semiring_int, + semiring_0_comm_semiring_0 = semiring_0_int} + : IntInf.int comm_semiring_0; + +val comm_monoid_mult_int = + {ab_semigroup_mult_comm_monoid_mult = ab_semigroup_mult_int, + monoid_mult_comm_monoid_mult = monoid_mult_int} + : IntInf.int comm_monoid_mult; + +val comm_semiring_1_int = + {comm_monoid_mult_comm_semiring_1 = comm_monoid_mult_int, + comm_semiring_0_comm_semiring_1 = comm_semiring_0_int, + dvd_comm_semiring_1 = dvd_int, semiring_1_comm_semiring_1 = semiring_1_int} + : IntInf.int comm_semiring_1; + +val comm_semiring_0_cancel_int = + {comm_semiring_0_comm_semiring_0_cancel = comm_semiring_0_int, + semiring_0_cancel_comm_semiring_0_cancel = semiring_0_cancel_int} + : IntInf.int comm_semiring_0_cancel; + +val comm_semiring_1_cancel_int = + {comm_semiring_0_cancel_comm_semiring_1_cancel = comm_semiring_0_cancel_int, + comm_semiring_1_comm_semiring_1_cancel = comm_semiring_1_int, + semiring_1_cancel_comm_semiring_1_cancel = semiring_1_cancel_int} + : IntInf.int comm_semiring_1_cancel; + +fun abs_int i = (if IntInf.< (i, (0 : IntInf.int)) then IntInf.~ i else i); + +fun split f (a, b) = f a b; + fun sgn_int i = - (if eqop eq_int i (0 : IntInf.int) then (0 : IntInf.int) + (if ((i : IntInf.int) = (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) +fun divmod_int k l = + (if ((k : IntInf.int) = (0 : IntInf.int)) + then ((0 : IntInf.int), (0 : IntInf.int)) + else (if ((l : IntInf.int) = (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 + (if (((sgn_int k) : IntInf.int) = (sgn_int l)) + then IntInf.divMod (IntInf.abs k, IntInf.abs l) else let - val a = - (fn k => fn l => IntInf.divMod (IntInf.abs k, - IntInf.abs l)) - k l; - val (r, s) = a; + val (r, s) = + IntInf.divMod (IntInf.abs k, IntInf.abs l); in - (if eqop eq_int s (0 : IntInf.int) + (if ((s : IntInf.int) = (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 snd (a, b) = b; + +fun mod_int a b = snd (divmod_int a b); + +fun fst (a, b) = a; + +fun div_int a b = fst (divmod_int a b); + +val div_inta = {dvd_div = dvd_int, diva = div_int, moda = mod_int} : + IntInf.int diva; + +val semiring_div_int = + {div_semiring_div = div_inta, + comm_semiring_1_cancel_semiring_div = comm_semiring_1_cancel_int, + no_zero_divisors_semiring_div = no_zero_divisors_int} + : IntInf.int semiring_div; + +fun dvd (A1_, A2_) a b = + eqa A2_ (moda (div_semiring_div A1_) b a) + (zero ((zero_no_zero_divisors o no_zero_divisors_semiring_div) A1_)); 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 @@ -742,11 +1315,11 @@ 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 + (if ((n1 : IntInf.int) = n2) then let val c = IntInf.+ (c1, c2); in - (if eqop eq_int c (0 : IntInf.int) then numadd (r1, r2) + (if ((c : IntInf.int) = (0 : IntInf.int)) then numadd (r1, r2) else Cn (n1, c, numadd (r1, r2))) end else (if IntInf.<= (n1, n2) @@ -807,10 +1380,8 @@ | 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)); + (if eq_num 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)) @@ -818,7 +1389,7 @@ | 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) + (if ((i : IntInf.int) = (0 : IntInf.int)) then C (0 : IntInf.int) else nummul i (simpnum t)) | simpnum (Cn (v, va, vb)) = Cn (v, va, vb); @@ -843,23 +1414,20 @@ | 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))))))); + (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 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)))); + (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 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)))); + (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 simpfm (And (p, q)) = conj (simpfm p) (simpfm q) | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q) @@ -868,91 +1436,80 @@ | simpfm (Not p) = nota (simpfm p) | simpfm (Lt a) = let - val a' = simpnum a; + val aa = 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') + (case aa of C v => (if IntInf.< (v, (0 : IntInf.int)) then T else F) + | Bound _ => Lt aa | Cn (_, _, _) => Lt aa | Neg _ => Lt aa + | Add (_, _) => Lt aa | Sub (_, _) => Lt aa | Mul (_, _) => Lt aa) end | simpfm (Le a) = let - val a' = simpnum a; + val aa = 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') + (case aa of C v => (if IntInf.<= (v, (0 : IntInf.int)) then T else F) + | Bound _ => Le aa | Cn (_, _, _) => Le aa | Neg _ => Le aa + | Add (_, _) => Le aa | Sub (_, _) => Le aa | Mul (_, _) => Le aa) end | simpfm (Gt a) = let - val a' = simpnum a; + val aa = 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') + (case aa of C v => (if IntInf.< ((0 : IntInf.int), v) then T else F) + | Bound _ => Gt aa | Cn (_, _, _) => Gt aa | Neg _ => Gt aa + | Add (_, _) => Gt aa | Sub (_, _) => Gt aa | Mul (_, _) => Gt aa) end | simpfm (Ge a) = let - val a' = simpnum a; + val aa = 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') + (case aa of C v => (if IntInf.<= ((0 : IntInf.int), v) then T else F) + | Bound _ => Ge aa | Cn (_, _, _) => Ge aa | Neg _ => Ge aa + | Add (_, _) => Ge aa | Sub (_, _) => Ge aa | Mul (_, _) => Ge aa) end | simpfm (Eq a) = let - val a' = simpnum a; + val aa = 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') + (case aa + of C v => (if ((v : IntInf.int) = (0 : IntInf.int)) then T else F) + | Bound _ => Eq aa | Cn (_, _, _) => Eq aa | Neg _ => Eq aa + | Add (_, _) => Eq aa | Sub (_, _) => Eq aa | Mul (_, _) => Eq aa) end | simpfm (NEq a) = let - val a' = simpnum a; + val aa = 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') + (case aa + of C v => (if not ((v : IntInf.int) = (0 : IntInf.int)) then T else F) + | Bound _ => NEq aa | Cn (_, _, _) => NEq aa | Neg _ => NEq aa + | Add (_, _) => NEq aa | Sub (_, _) => NEq aa | Mul (_, _) => NEq aa) 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 + (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; + val aa = 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')) + (case aa + of C v => + (if dvd (semiring_div_int, eq_int) i v then T else F) + | Bound _ => Dvd (i, aa) | Cn (_, _, _) => Dvd (i, aa) + | Neg _ => Dvd (i, aa) | Add (_, _) => Dvd (i, aa) + | Sub (_, _) => Dvd (i, aa) | Mul (_, _) => Dvd (i, aa)) 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 + (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; + val aa = 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')) + (case aa + of C v => + (if not (dvd (semiring_div_int, eq_int) i v) then T + else F) + | Bound _ => NDvd (i, aa) | Cn (_, _, _) => NDvd (i, aa) + | Neg _ => NDvd (i, aa) | Add (_, _) => NDvd (i, aa) + | Sub (_, _) => NDvd (i, aa) | Mul (_, _) => NDvd (i, aa)) end)) | simpfm T = T | simpfm F = F @@ -1025,32 +1582,40 @@ | 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))) + (if ((cm : IntInf.int) = (0 : IntInf.int)) + then Gt (Cn ((0 : IntInf.int), c, Neg e)) + else Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), 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))) + (if ((dm : IntInf.int) = (0 : IntInf.int)) + then Ge (Cn ((0 : IntInf.int), c, Neg e)) + else Le (Cn (suc (minus_nat dm (1 : IntInf.int)), 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))) + (if ((em : IntInf.int) = (0 : IntInf.int)) + then Lt (Cn ((0 : IntInf.int), c, Neg e)) + else Gt (Cn (suc (minus_nat em (1 : IntInf.int)), 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))) + (if ((fm : IntInf.int) = (0 : IntInf.int)) + then Le (Cn ((0 : IntInf.int), c, Neg e)) + else Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), 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))) + (if ((gm : IntInf.int) = (0 : IntInf.int)) + then Eq (Cn ((0 : IntInf.int), c, Neg e)) + else Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), 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))) + (if ((hm : IntInf.int) = (0 : IntInf.int)) + then NEq (Cn ((0 : IntInf.int), c, Neg e)) + else NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), 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))) + (if ((im : IntInf.int) = (0 : IntInf.int)) + then Dvd (i, Cn ((0 : IntInf.int), c, Neg e)) + else Dvd (i, Cn (suc (minus_nat im (1 : IntInf.int)), 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))); + (if ((jm : IntInf.int) = (0 : IntInf.int)) + then NDvd (i, Cn ((0 : IntInf.int), c, Neg e)) + else NDvd (i, Cn (suc (minus_nat jm (1 : IntInf.int)), c, e))); -fun size_list [] = 0 - | size_list (a :: lista) = IntInf.+ (size_list lista, suc 0); +fun size_list [] = (0 : IntInf.int) + | size_list (a :: lista) = IntInf.+ (size_list lista, suc (0 : IntInf.int)); fun alpha (And (p, q)) = append (alpha p) (alpha q) | alpha (Or (p, q)) = append (alpha p) (alpha q) @@ -1101,14 +1666,20 @@ | 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 (Lt (Cn (cm, c, e))) = + (if ((cm : IntInf.int) = (0 : IntInf.int)) 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 []) + (if ((dm : IntInf.int) = (0 : IntInf.int)) + then [Add (C (~1 : IntInf.int), e)] else []) + | alpha (Gt (Cn (em, c, e))) = + (if ((em : IntInf.int) = (0 : IntInf.int)) then [] else []) + | alpha (Ge (Cn (fm, c, e))) = + (if ((fm : IntInf.int) = (0 : IntInf.int)) 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 []); + (if ((gm : IntInf.int) = (0 : IntInf.int)) + then [Add (C (~1 : IntInf.int), e)] else []) + | alpha (NEq (Cn (hm, c, e))) = + (if ((hm : IntInf.int) = (0 : IntInf.int)) then [e] else []); fun beta (And (p, q)) = append (beta p) (beta q) | beta (Or (p, q)) = append (beta p) (beta q) @@ -1159,24 +1730,39 @@ | 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 (Lt (Cn (cm, c, e))) = + (if ((cm : IntInf.int) = (0 : IntInf.int)) then [] else []) + | beta (Le (Cn (dm, c, e))) = + (if ((dm : IntInf.int) = (0 : IntInf.int)) then [] else []) + | beta (Gt (Cn (em, c, e))) = + (if ((em : IntInf.int) = (0 : IntInf.int)) then [Neg e] else []) | beta (Ge (Cn (fm, c, e))) = - (if eqop eq_nat fm 0 then [Sub (C (~1 : IntInf.int), e)] else []) + (if ((fm : IntInf.int) = (0 : IntInf.int)) + 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 []); + (if ((gm : IntInf.int) = (0 : IntInf.int)) + then [Sub (C (~1 : IntInf.int), e)] else []) + | beta (NEq (Cn (hm, c, e))) = + (if ((hm : IntInf.int) = (0 : IntInf.int)) then [Neg e] else []); + +val eq_numa = {eq = eq_num} : num eq; fun member A_ x [] = false - | member A_ x (y :: ys) = eqop A_ x y orelse member A_ x ys; + | member A_ x (y :: ys) = eqa 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) +fun gcd_int k l = + abs_int + (if ((l : IntInf.int) = (0 : IntInf.int)) then k + else gcd_int l (mod_int (abs_int k) (abs_int l))); + +fun lcm_int a b = div_int (IntInf.* (abs_int a, abs_int b)) (gcd_int a b); + +fun delta (And (p, q)) = lcm_int (delta p) (delta q) + | delta (Or (p, q)) = lcm_int (delta p) (delta q) | delta T = (1 : IntInf.int) | delta F = (1 : IntInf.int) | delta (Lt u) = (1 : IntInf.int) @@ -1205,110 +1791,117 @@ | 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)) + (if ((cm : IntInf.int) = (0 : IntInf.int)) 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); + (if ((dm : IntInf.int) = (0 : IntInf.int)) then i else (1 : IntInf.int)); 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 T = (fn _ => T) + | a_beta F = (fn _ => F) + | a_beta (Lt (C bo)) = (fn _ => Lt (C bo)) + | a_beta (Lt (Bound bp)) = (fn _ => Lt (Bound bp)) + | a_beta (Lt (Neg bt)) = (fn _ => Lt (Neg bt)) + | a_beta (Lt (Add (bu, bv))) = (fn _ => Lt (Add (bu, bv))) + | a_beta (Lt (Sub (bw, bx))) = (fn _ => Lt (Sub (bw, bx))) + | a_beta (Lt (Mul (by, bz))) = (fn _ => Lt (Mul (by, bz))) + | a_beta (Le (C co)) = (fn _ => Le (C co)) + | a_beta (Le (Bound cp)) = (fn _ => Le (Bound cp)) + | a_beta (Le (Neg ct)) = (fn _ => Le (Neg ct)) + | a_beta (Le (Add (cu, cv))) = (fn _ => Le (Add (cu, cv))) + | a_beta (Le (Sub (cw, cx))) = (fn _ => Le (Sub (cw, cx))) + | a_beta (Le (Mul (cy, cz))) = (fn _ => Le (Mul (cy, cz))) + | a_beta (Gt (C doa)) = (fn _ => Gt (C doa)) + | a_beta (Gt (Bound dp)) = (fn _ => Gt (Bound dp)) + | a_beta (Gt (Neg dt)) = (fn _ => Gt (Neg dt)) + | a_beta (Gt (Add (du, dv))) = (fn _ => Gt (Add (du, dv))) + | a_beta (Gt (Sub (dw, dx))) = (fn _ => Gt (Sub (dw, dx))) + | a_beta (Gt (Mul (dy, dz))) = (fn _ => Gt (Mul (dy, dz))) + | a_beta (Ge (C eo)) = (fn _ => Ge (C eo)) + | a_beta (Ge (Bound ep)) = (fn _ => Ge (Bound ep)) + | a_beta (Ge (Neg et)) = (fn _ => Ge (Neg et)) + | a_beta (Ge (Add (eu, ev))) = (fn _ => Ge (Add (eu, ev))) + | a_beta (Ge (Sub (ew, ex))) = (fn _ => Ge (Sub (ew, ex))) + | a_beta (Ge (Mul (ey, ez))) = (fn _ => Ge (Mul (ey, ez))) + | a_beta (Eq (C fo)) = (fn _ => Eq (C fo)) + | a_beta (Eq (Bound fp)) = (fn _ => Eq (Bound fp)) + | a_beta (Eq (Neg ft)) = (fn _ => Eq (Neg ft)) + | a_beta (Eq (Add (fu, fv))) = (fn _ => Eq (Add (fu, fv))) + | a_beta (Eq (Sub (fw, fx))) = (fn _ => Eq (Sub (fw, fx))) + | a_beta (Eq (Mul (fy, fz))) = (fn _ => Eq (Mul (fy, fz))) + | a_beta (NEq (C go)) = (fn _ => NEq (C go)) + | a_beta (NEq (Bound gp)) = (fn _ => NEq (Bound gp)) + | a_beta (NEq (Neg gt)) = (fn _ => NEq (Neg gt)) + | a_beta (NEq (Add (gu, gv))) = (fn _ => NEq (Add (gu, gv))) + | a_beta (NEq (Sub (gw, gx))) = (fn _ => NEq (Sub (gw, gx))) + | a_beta (NEq (Mul (gy, gz))) = (fn _ => NEq (Mul (gy, gz))) + | a_beta (Dvd (aa, C ho)) = (fn _ => Dvd (aa, C ho)) + | a_beta (Dvd (aa, Bound hp)) = (fn _ => Dvd (aa, Bound hp)) + | a_beta (Dvd (aa, Neg ht)) = (fn _ => Dvd (aa, Neg ht)) + | a_beta (Dvd (aa, Add (hu, hv))) = (fn _ => Dvd (aa, Add (hu, hv))) + | a_beta (Dvd (aa, Sub (hw, hx))) = (fn _ => Dvd (aa, Sub (hw, hx))) + | a_beta (Dvd (aa, Mul (hy, hz))) = (fn _ => Dvd (aa, Mul (hy, hz))) + | a_beta (NDvd (ac, C io)) = (fn _ => NDvd (ac, C io)) + | a_beta (NDvd (ac, Bound ip)) = (fn _ => NDvd (ac, Bound ip)) + | a_beta (NDvd (ac, Neg it)) = (fn _ => NDvd (ac, Neg it)) + | a_beta (NDvd (ac, Add (iu, iv))) = (fn _ => NDvd (ac, Add (iu, iv))) + | a_beta (NDvd (ac, Sub (iw, ix))) = (fn _ => NDvd (ac, Sub (iw, ix))) + | a_beta (NDvd (ac, Mul (iy, iz))) = (fn _ => NDvd (ac, Mul (iy, iz))) + | a_beta (Not ae) = (fn _ => Not ae) + | a_beta (Imp (aj, ak)) = (fn _ => Imp (aj, ak)) + | a_beta (Iff (al, am)) = (fn _ => Iff (al, am)) + | a_beta (E an) = (fn _ => E an) + | a_beta (A ao) = (fn _ => A ao) + | a_beta (Closed ap) = (fn _ => Closed ap) + | a_beta (NClosed aq) = (fn _ => 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)))) + (if ((cm : IntInf.int) = (0 : IntInf.int)) + then (fn k => + Lt (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn _ => Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), 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)))) + (if ((dm : IntInf.int) = (0 : IntInf.int)) + then (fn k => + Le (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn _ => Le (Cn (suc (minus_nat dm (1 : IntInf.int)), 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)))) + (if ((em : IntInf.int) = (0 : IntInf.int)) + then (fn k => + Gt (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn _ => Gt (Cn (suc (minus_nat em (1 : IntInf.int)), 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)))) + (if ((fm : IntInf.int) = (0 : IntInf.int)) + then (fn k => + Ge (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn _ => Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), 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)))) + (if ((gm : IntInf.int) = (0 : IntInf.int)) + then (fn k => + Eq (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e)))) + else (fn _ => Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), 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)))) + (if ((hm : IntInf.int) = (0 : IntInf.int)) + then (fn k => + NEq (Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_int k c, e)))) + else (fn _ => NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e)))) | a_beta (Dvd (i, Cn (im, c, e))) = - (if eqop eq_nat im 0 + (if ((im : IntInf.int) = (0 : IntInf.int)) 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)))) + Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_int k c, e)))) + else (fn _ => Dvd (i, Cn (suc (minus_nat im (1 : IntInf.int)), c, e)))) | a_beta (NDvd (i, Cn (jm, c, e))) = - (if eqop eq_nat jm 0 + (if ((jm : IntInf.int) = (0 : IntInf.int)) 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)))); + Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_int k c, e)))) + else (fn _ => NDvd (i, Cn (suc (minus_nat jm (1 : IntInf.int)), c, e)))); -fun zeta (And (p, q)) = zlcm (zeta p) (zeta q) - | zeta (Or (p, q)) = zlcm (zeta p) (zeta q) +fun zeta (And (p, q)) = lcm_int (zeta p) (zeta q) + | zeta (Or (p, q)) = lcm_int (zeta p) (zeta q) | zeta T = (1 : IntInf.int) | zeta F = (1 : IntInf.int) | zeta (Lt (C bo)) = (1 : IntInf.int) @@ -1367,64 +1960,59 @@ | 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)) + (if ((cm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) | zeta (Le (Cn (dm, c, e))) = - (if eqop eq_nat dm 0 then c else (1 : IntInf.int)) + (if ((dm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) | zeta (Gt (Cn (em, c, e))) = - (if eqop eq_nat em 0 then c else (1 : IntInf.int)) + (if ((em : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) | zeta (Ge (Cn (fm, c, e))) = - (if eqop eq_nat fm 0 then c else (1 : IntInf.int)) + (if ((fm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) | zeta (Eq (Cn (gm, c, e))) = - (if eqop eq_nat gm 0 then c else (1 : IntInf.int)) + (if ((gm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) | zeta (NEq (Cn (hm, c, e))) = - (if eqop eq_nat hm 0 then c else (1 : IntInf.int)) + (if ((hm : IntInf.int) = (0 : IntInf.int)) 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)) + (if ((im : IntInf.int) = (0 : IntInf.int)) 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)); + (if ((jm : IntInf.int) = (0 : IntInf.int)) 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)) + (if ((n : IntInf.int) = (0 : IntInf.int)) + 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; + val (ia, aa) = zsplit0 a; in - (if eqop eq_nat n 0 then (IntInf.+ (i, i'), a') else (i', Cn (n, i, a'))) + (if ((n : IntInf.int) = (0 : IntInf.int)) then (IntInf.+ (i, ia), aa) + else (ia, Cn (n, i, aa))) end | zsplit0 (Neg a) = let - val aa = zsplit0 a; - val (i', a') = aa; + val (i, aa) = zsplit0 a; in - (IntInf.~ i', Neg a') + (IntInf.~ i, Neg aa) end | zsplit0 (Add (a, b)) = let - val aa = zsplit0 a; - val (ia, a') = aa; - val ab = zsplit0 b; - val (ib, b') = ab; + val (ia, aa) = zsplit0 a; + val (ib, ba) = zsplit0 b; in - (IntInf.+ (ia, ib), Add (a', b')) + (IntInf.+ (ia, ib), Add (aa, ba)) end | zsplit0 (Sub (a, b)) = let - val aa = zsplit0 a; - val (ia, a') = aa; - val ab = zsplit0 b; - val (ib, b') = ab; + val (ia, aa) = zsplit0 a; + val (ib, ba) = zsplit0 b; in - (IntInf.- (ia, ib), Sub (a', b')) + (IntInf.- (ia, ib), Sub (aa, ba)) end | zsplit0 (Mul (i, a)) = let - val aa = zsplit0 a; - val (i', a') = aa; + val (ia, aa) = zsplit0 a; in - (IntInf.* (i, i'), Mul (i, a')) + (IntInf.* (i, ia), Mul (i, aa)) end; fun zlfm (And (p, q)) = And (zlfm p, zlfm q) @@ -1434,79 +2022,79 @@ 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; + val (c, r) = zsplit0 a; 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)))) + (if ((c : IntInf.int) = (0 : IntInf.int)) then Lt r + else (if IntInf.< ((0 : IntInf.int), c) + then Lt (Cn ((0 : IntInf.int), c, r)) + else Gt (Cn ((0 : IntInf.int), IntInf.~ c, Neg r)))) end | zlfm (Le a) = let - val aa = zsplit0 a; - val (c, r) = aa; + val (c, r) = zsplit0 a; 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)))) + (if ((c : IntInf.int) = (0 : IntInf.int)) then Le r + else (if IntInf.< ((0 : IntInf.int), c) + then Le (Cn ((0 : IntInf.int), c, r)) + else Ge (Cn ((0 : IntInf.int), IntInf.~ c, Neg r)))) end | zlfm (Gt a) = let - val aa = zsplit0 a; - val (c, r) = aa; + val (c, r) = zsplit0 a; 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)))) + (if ((c : IntInf.int) = (0 : IntInf.int)) then Gt r + else (if IntInf.< ((0 : IntInf.int), c) + then Gt (Cn ((0 : IntInf.int), c, r)) + else Lt (Cn ((0 : IntInf.int), IntInf.~ c, Neg r)))) end | zlfm (Ge a) = let - val aa = zsplit0 a; - val (c, r) = aa; + val (c, r) = zsplit0 a; 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)))) + (if ((c : IntInf.int) = (0 : IntInf.int)) then Ge r + else (if IntInf.< ((0 : IntInf.int), c) + then Ge (Cn ((0 : IntInf.int), c, r)) + else Le (Cn ((0 : IntInf.int), IntInf.~ c, Neg r)))) end | zlfm (Eq a) = let - val aa = zsplit0 a; - val (c, r) = aa; + val (c, r) = zsplit0 a; 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)))) + (if ((c : IntInf.int) = (0 : IntInf.int)) then Eq r + else (if IntInf.< ((0 : IntInf.int), c) + then Eq (Cn ((0 : IntInf.int), c, r)) + else Eq (Cn ((0 : IntInf.int), IntInf.~ c, Neg r)))) end | zlfm (NEq a) = let - val aa = zsplit0 a; - val (c, r) = aa; + val (c, r) = zsplit0 a; 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)))) + (if ((c : IntInf.int) = (0 : IntInf.int)) then NEq r + else (if IntInf.< ((0 : IntInf.int), c) + then NEq (Cn ((0 : IntInf.int), c, r)) + else NEq (Cn ((0 : IntInf.int), IntInf.~ c, Neg r)))) end | zlfm (Dvd (i, a)) = - (if eqop eq_int i (0 : IntInf.int) then zlfm (Eq a) + (if ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (Eq a) else let - val aa = zsplit0 a; - val (c, r) = aa; + val (c, r) = zsplit0 a; in - (if eqop eq_int c (0 : IntInf.int) then Dvd (abs_int i, r) + (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, Cn (0, c, r)) - else Dvd (abs_int i, Cn (0, IntInf.~ c, Neg r)))) + then Dvd (abs_int i, Cn ((0 : IntInf.int), c, r)) + else Dvd (abs_int i, + Cn ((0 : IntInf.int), IntInf.~ c, Neg r)))) end) | zlfm (NDvd (i, a)) = - (if eqop eq_int i (0 : IntInf.int) then zlfm (NEq a) + (if ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (NEq a) else let - val aa = zsplit0 a; - val (c, r) = aa; + val (c, r) = zsplit0 a; in - (if eqop eq_int c (0 : IntInf.int) then NDvd (abs_int i, r) + (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, Cn (0, c, r)) - else NDvd (abs_int i, Cn (0, IntInf.~ c, Neg r)))) + then NDvd (abs_int i, Cn ((0 : IntInf.int), c, r)) + else NDvd (abs_int i, + Cn ((0 : IntInf.int), 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)) @@ -1537,10 +2125,11 @@ fun unita p = let - val p' = zlfm p; - val l = zeta p'; + val pa = zlfm p; + val l = zeta pa; val q = - And (Dvd (l, Cn (0, (1 : IntInf.int), C (0 : IntInf.int))), a_beta p' l); + And (Dvd (l, Cn ((0 : IntInf.int), (1 : IntInf.int), C (0 : IntInf.int))), + a_beta pa l); val d = delta q; val b = remdups eq_numa (map simpnum (beta q)); val a = remdups eq_numa (map simpnum (alpha q)); @@ -1551,18 +2140,16 @@ fun cooper p = let - val a = unita p; - val (q, aa) = a; - val (b, d) = aa; + val (q, (b, d)) = unita p; 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 + (if eq_fm 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)); + evaldjf (fn (ba, j) => simpfm (subst0 (Add (ba, C j)) q)) + (concat_map (fn ba => map (fn a => (ba, a)) js) b); in decr (disj md qd) end) @@ -1669,37 +2256,19 @@ | 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)); + | qelim T = (fn _ => simpfm T) + | qelim F = (fn _ => simpfm F) + | qelim (Lt u) = (fn _ => simpfm (Lt u)) + | qelim (Le v) = (fn _ => simpfm (Le v)) + | qelim (Gt w) = (fn _ => simpfm (Gt w)) + | qelim (Ge x) = (fn _ => simpfm (Ge x)) + | qelim (Eq y) = (fn _ => simpfm (Eq y)) + | qelim (NEq z) = (fn _ => simpfm (NEq z)) + | qelim (Dvd (aa, ab)) = (fn _ => simpfm (Dvd (aa, ab))) + | qelim (NDvd (ac, ad)) = (fn _ => simpfm (NDvd (ac, ad))) + | qelim (Closed ap) = (fn _ => simpfm (Closed ap)) + | qelim (NClosed aq) = (fn _ => 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*) +end; (*struct Generated_Cooper*)