# HG changeset patch # User haftmann # Date 1316036762 -7200 # Node ID afcbf23508af5435a873ba56c376299905649f0e # Parent 1886cddaf8a590df5923ed6e65901a40a21e3850 updated generated code diff -r 1886cddaf8a5 -r afcbf23508af src/HOL/Tools/Qelim/cooper_procedure.ML --- a/src/HOL/Tools/Qelim/cooper_procedure.ML Tue Sep 13 07:56:46 2011 +0200 +++ b/src/HOL/Tools/Qelim/cooper_procedure.ML Wed Sep 14 23:46:02 2011 +0200 @@ -1,69 +1,43 @@ (* Generated from Cooper.thy; DO NOT EDIT! *) structure Cooper_Procedure : 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 + type 'a equal + val equal : 'a equal -> 'a -> 'a -> bool + val eq : 'a equal -> 'a -> 'a -> bool + val suc : int -> int + datatype num = C of int | Bound of int | Cn of int * int * num | Neg of num | + Add of num * num | Sub of num * num | Mul of 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 + Eq of num | NEq of num | Dvd of int * num | NDvd of int * num | Not of fm | + And of fm * fm | Or of fm * fm | Imp of fm * fm | Iff of fm * fm | E of fm | + A of fm | Closed of int | NClosed of int 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 equal_numa : num -> num -> bool + val equal_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 disjuncts : fm -> fm list val dj : (fm -> fm) -> fm -> fm + val prep : fm -> fm + val conj : 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 + val nota : fm -> fm + val iffa : fm -> fm -> fm + val impa : fm -> fm -> fm + type 'a times + val times : 'a times -> 'a -> 'a -> 'a + type 'a dvd + val times_dvd : 'a dvd -> 'a times + 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 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 @@ -89,16 +63,40 @@ 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 one + val one : 'a one -> 'a 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 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 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 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 cancel_semigroup_add val semigroup_add_cancel_semigroup_add : 'a cancel_semigroup_add -> 'a semigroup_add @@ -120,27 +118,6 @@ 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 @@ -153,603 +130,532 @@ '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 dvd : 'a semiring_div * 'a equal -> 'a -> 'a -> bool + val abs_int : int -> int + val equal_int : int equal + val numadd : num * num -> num + val nummul : 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 one_inta : int + val zero_inta : int + val times_int : int times + val dvd_int : int dvd + val fst : 'a * 'b -> 'a + val sgn_int : int -> int + val apsnd : ('a -> 'b) -> 'c * 'a -> 'c * 'b + val divmod_int : int -> int -> int * int + val div_inta : int -> int -> int + val snd : 'a * 'b -> 'b + val mod_int : int -> int -> int + val div_int : int diva + val zero_int : int zero + val no_zero_divisors_int : int no_zero_divisors + val semigroup_mult_int : int semigroup_mult + val plus_int : int plus + val semigroup_add_int : int semigroup_add + val ab_semigroup_add_int : int ab_semigroup_add + val semiring_int : int semiring + val mult_zero_int : int mult_zero + val monoid_add_int : int monoid_add + val comm_monoid_add_int : int comm_monoid_add + val semiring_0_int : int semiring_0 + val one_int : int one + val power_int : int power + val monoid_mult_int : int monoid_mult + val zero_neq_one_int : int zero_neq_one + val semiring_1_int : int semiring_1 + val ab_semigroup_mult_int : int ab_semigroup_mult + val comm_semiring_int : int comm_semiring + val comm_semiring_0_int : int comm_semiring_0 + val comm_monoid_mult_int : int comm_monoid_mult + val comm_semiring_1_int : int comm_semiring_1 + val cancel_semigroup_add_int : int cancel_semigroup_add + val cancel_ab_semigroup_add_int : int cancel_ab_semigroup_add + val cancel_comm_monoid_add_int : int cancel_comm_monoid_add + val semiring_0_cancel_int : int semiring_0_cancel + val semiring_1_cancel_int : int semiring_1_cancel + val comm_semiring_0_cancel_int : int comm_semiring_0_cancel + val comm_semiring_1_cancel_int : int comm_semiring_1_cancel + val semiring_div_int : int semiring_div 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 qelim : fm -> (fm -> fm) -> fm + val maps : ('a -> 'b list) -> 'a list -> 'b list + val uptoa : int -> int -> int list + val minus_nat : int -> int -> int + val decrnum : num -> num + val decr : fm -> fm 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 gcd_int : int -> int -> int + val lcm_int : int -> int -> int + val zeta : fm -> int + val zsplit0 : num -> int * num val zlfm : fm -> fm - val unita : fm -> fm * (num list * IntInf.int) + val alpha : fm -> num list + val delta : fm -> int + val member : 'a equal -> 'a list -> 'a -> bool + val remdups : 'a equal -> 'a list -> 'a list + val a_beta : fm -> int -> fm + val mirror : fm -> fm + val size_list : 'a list -> int + val equal_num : num equal + val unita : fm -> fm * (num list * int) + val numsubst0 : num -> num -> num + val subst0 : num -> fm -> fm + val minusinf : fm -> fm 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}; -val eq = #eq : 'a eq -> 'a -> 'a -> bool; +type 'a equal = {equal : 'a -> 'a -> bool}; +val equal = #equal : 'a equal -> 'a -> 'a -> bool; -fun eqa A_ a b = eq A_ a b; +fun eq A_ a b = equal A_ a b; -fun leta s f = f s; - -fun suc n = IntInf.+ (n, (1 : IntInf.int)); +fun suc n = n + (1 : 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 num = C of int | Bound of int | Cn of int * int * num | Neg of num | + Add of num * num | Sub of num * num | Mul of 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; + | NEq of num | Dvd of int * num | NDvd of int * num | Not of fm | + And of fm * fm | Or of fm * fm | Imp of fm * fm | Iff of fm * fm | E of fm | + A of fm | Closed of int | NClosed of int; fun map f [] = [] | map f (x :: xs) = f x :: map f xs; -fun append [] ys = ys - | append (x :: xs) ys = x :: append xs ys; - -fun disjuncts (Or (p, q)) = append (disjuncts p) (disjuncts q) - | disjuncts F = [] - | disjuncts T = [T] - | disjuncts (Lt u) = [Lt u] - | disjuncts (Le v) = [Le v] - | disjuncts (Gt w) = [Gt w] - | disjuncts (Ge x) = [Ge x] - | disjuncts (Eq y) = [Eq y] - | disjuncts (NEq z) = [NEq z] - | disjuncts (Dvd (aa, ab)) = [Dvd (aa, ab)] - | disjuncts (NDvd (ac, ad)) = [NDvd (ac, ad)] - | disjuncts (Not ae) = [Not ae] - | disjuncts (And (af, ag)) = [And (af, ag)] - | disjuncts (Imp (aj, ak)) = [Imp (aj, ak)] - | disjuncts (Iff (al, am)) = [Iff (al, am)] - | disjuncts (E an) = [E an] - | disjuncts (A ao) = [A ao] - | disjuncts (Closed ap) = [Closed ap] - | disjuncts (NClosed aq) = [NClosed aq]; - -fun fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (NClosed nat) = f19 nat - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Closed nat) = f18 nat - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (A fm) = f17 fm - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (E fm) = f16 fm - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Iff (fm1, fm2)) = f15 fm1 fm2 - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Imp (fm1, fm2)) = f14 fm1 fm2 - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Or (fm1, fm2)) = f13 fm1 fm2 - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (And (fm1, fm2)) = f12 fm1 fm2 - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Not fm) = f11 fm - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (NDvd (inta, num)) = f10 inta num - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Dvd (inta, num)) = f9 inta num - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (NEq num) = f8 num - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Eq num) = f7 num - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Ge num) = f6 num - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Gt num) = f5 num - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Le num) = f4 num - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 - (Lt num) = f3 num - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 F - = f2 - | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 T - = f1; - -fun eq_num (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 equal_numa (Mul (inta, num)) (Sub (num1, num2)) = false + | equal_numa (Sub (num1, num2)) (Mul (inta, num)) = false + | equal_numa (Mul (inta, num)) (Add (num1, num2)) = false + | equal_numa (Add (num1, num2)) (Mul (inta, num)) = false + | equal_numa (Sub (num1a, num2a)) (Add (num1, num2)) = false + | equal_numa (Add (num1a, num2a)) (Sub (num1, num2)) = false + | equal_numa (Mul (inta, numa)) (Neg num) = false + | equal_numa (Neg numa) (Mul (inta, num)) = false + | equal_numa (Sub (num1, num2)) (Neg num) = false + | equal_numa (Neg num) (Sub (num1, num2)) = false + | equal_numa (Add (num1, num2)) (Neg num) = false + | equal_numa (Neg num) (Add (num1, num2)) = false + | equal_numa (Mul (intaa, numa)) (Cn (nat, inta, num)) = false + | equal_numa (Cn (nat, intaa, numa)) (Mul (inta, num)) = false + | equal_numa (Sub (num1, num2)) (Cn (nat, inta, num)) = false + | equal_numa (Cn (nat, inta, num)) (Sub (num1, num2)) = false + | equal_numa (Add (num1, num2)) (Cn (nat, inta, num)) = false + | equal_numa (Cn (nat, inta, num)) (Add (num1, num2)) = false + | equal_numa (Neg numa) (Cn (nat, inta, num)) = false + | equal_numa (Cn (nat, inta, numa)) (Neg num) = false + | equal_numa (Mul (inta, num)) (Bound nat) = false + | equal_numa (Bound nat) (Mul (inta, num)) = false + | equal_numa (Sub (num1, num2)) (Bound nat) = false + | equal_numa (Bound nat) (Sub (num1, num2)) = false + | equal_numa (Add (num1, num2)) (Bound nat) = false + | equal_numa (Bound nat) (Add (num1, num2)) = false + | equal_numa (Neg num) (Bound nat) = false + | equal_numa (Bound nat) (Neg num) = false + | equal_numa (Cn (nata, inta, num)) (Bound nat) = false + | equal_numa (Bound nata) (Cn (nat, inta, num)) = false + | equal_numa (Mul (intaa, num)) (C inta) = false + | equal_numa (C intaa) (Mul (inta, num)) = false + | equal_numa (Sub (num1, num2)) (C inta) = false + | equal_numa (C inta) (Sub (num1, num2)) = false + | equal_numa (Add (num1, num2)) (C inta) = false + | equal_numa (C inta) (Add (num1, num2)) = false + | equal_numa (Neg num) (C inta) = false + | equal_numa (C inta) (Neg num) = false + | equal_numa (Cn (nat, intaa, num)) (C inta) = false + | equal_numa (C intaa) (Cn (nat, inta, num)) = false + | equal_numa (Bound nat) (C inta) = false + | equal_numa (C inta) (Bound nat) = false + | equal_numa (Mul (intaa, numa)) (Mul (inta, num)) = + intaa = inta andalso equal_numa numa num + | equal_numa (Sub (num1a, num2a)) (Sub (num1, num2)) = + equal_numa num1a num1 andalso equal_numa num2a num2 + | equal_numa (Add (num1a, num2a)) (Add (num1, num2)) = + equal_numa num1a num1 andalso equal_numa num2a num2 + | equal_numa (Neg numa) (Neg num) = equal_numa numa num + | equal_numa (Cn (nata, intaa, numa)) (Cn (nat, inta, num)) = + nata = nat andalso (intaa = inta andalso equal_numa numa num) + | equal_numa (Bound nata) (Bound nat) = nata = nat + | equal_numa (C intaa) (C inta) = intaa = inta; -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 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 equal_fm (NClosed nata) (Closed nat) = false + | equal_fm (Closed nata) (NClosed nat) = false + | equal_fm (NClosed nat) (A fm) = false + | equal_fm (A fm) (NClosed nat) = false + | equal_fm (Closed nat) (A fm) = false + | equal_fm (A fm) (Closed nat) = false + | equal_fm (NClosed nat) (E fm) = false + | equal_fm (E fm) (NClosed nat) = false + | equal_fm (Closed nat) (E fm) = false + | equal_fm (E fm) (Closed nat) = false + | equal_fm (A fma) (E fm) = false + | equal_fm (E fma) (A fm) = false + | equal_fm (NClosed nat) (Iff (fm1, fm2)) = false + | equal_fm (Iff (fm1, fm2)) (NClosed nat) = false + | equal_fm (Closed nat) (Iff (fm1, fm2)) = false + | equal_fm (Iff (fm1, fm2)) (Closed nat) = false + | equal_fm (A fm) (Iff (fm1, fm2)) = false + | equal_fm (Iff (fm1, fm2)) (A fm) = false + | equal_fm (E fm) (Iff (fm1, fm2)) = false + | equal_fm (Iff (fm1, fm2)) (E fm) = false + | equal_fm (NClosed nat) (Imp (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (NClosed nat) = false + | equal_fm (Closed nat) (Imp (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (Closed nat) = false + | equal_fm (A fm) (Imp (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (A fm) = false + | equal_fm (E fm) (Imp (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (E fm) = false + | equal_fm (Iff (fm1a, fm2a)) (Imp (fm1, fm2)) = false + | equal_fm (Imp (fm1a, fm2a)) (Iff (fm1, fm2)) = false + | equal_fm (NClosed nat) (Or (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (NClosed nat) = false + | equal_fm (Closed nat) (Or (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (Closed nat) = false + | equal_fm (A fm) (Or (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (A fm) = false + | equal_fm (E fm) (Or (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (E fm) = false + | equal_fm (Iff (fm1a, fm2a)) (Or (fm1, fm2)) = false + | equal_fm (Or (fm1a, fm2a)) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1a, fm2a)) (Or (fm1, fm2)) = false + | equal_fm (Or (fm1a, fm2a)) (Imp (fm1, fm2)) = false + | equal_fm (NClosed nat) (And (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (NClosed nat) = false + | equal_fm (Closed nat) (And (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (Closed nat) = false + | equal_fm (A fm) (And (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (A fm) = false + | equal_fm (E fm) (And (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (E fm) = false + | equal_fm (Iff (fm1a, fm2a)) (And (fm1, fm2)) = false + | equal_fm (And (fm1a, fm2a)) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1a, fm2a)) (And (fm1, fm2)) = false + | equal_fm (And (fm1a, fm2a)) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1a, fm2a)) (And (fm1, fm2)) = false + | equal_fm (And (fm1a, fm2a)) (Or (fm1, fm2)) = false + | equal_fm (NClosed nat) (Not fm) = false + | equal_fm (Not fm) (NClosed nat) = false + | equal_fm (Closed nat) (Not fm) = false + | equal_fm (Not fm) (Closed nat) = false + | equal_fm (A fma) (Not fm) = false + | equal_fm (Not fma) (A fm) = false + | equal_fm (E fma) (Not fm) = false + | equal_fm (Not fma) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (Not fm) = false + | equal_fm (Not fm) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (Not fm) = false + | equal_fm (Not fm) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (Not fm) = false + | equal_fm (Not fm) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (Not fm) = false + | equal_fm (Not fm) (And (fm1, fm2)) = false + | equal_fm (NClosed nat) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (NClosed nat) = false + | equal_fm (Closed nat) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (Closed nat) = false + | equal_fm (A fm) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (A fm) = false + | equal_fm (E fm) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (And (fm1, fm2)) = false + | equal_fm (Not fm) (NDvd (inta, num)) = false + | equal_fm (NDvd (inta, num)) (Not fm) = false + | equal_fm (NClosed nat) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (NClosed nat) = false + | equal_fm (Closed nat) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (Closed nat) = false + | equal_fm (A fm) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (A fm) = false + | equal_fm (E fm) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (And (fm1, fm2)) = false + | equal_fm (Not fm) (Dvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) (Not fm) = false + | equal_fm (NDvd (intaa, numa)) (Dvd (inta, num)) = false + | equal_fm (Dvd (intaa, numa)) (NDvd (inta, num)) = false + | equal_fm (NClosed nat) (NEq num) = false + | equal_fm (NEq num) (NClosed nat) = false + | equal_fm (Closed nat) (NEq num) = false + | equal_fm (NEq num) (Closed nat) = false + | equal_fm (A fm) (NEq num) = false + | equal_fm (NEq num) (A fm) = false + | equal_fm (E fm) (NEq num) = false + | equal_fm (NEq num) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (NEq num) = false + | equal_fm (NEq num) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (NEq num) = false + | equal_fm (NEq num) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (NEq num) = false + | equal_fm (NEq num) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (NEq num) = false + | equal_fm (NEq num) (And (fm1, fm2)) = false + | equal_fm (Not fm) (NEq num) = false + | equal_fm (NEq num) (Not fm) = false + | equal_fm (NDvd (inta, numa)) (NEq num) = false + | equal_fm (NEq numa) (NDvd (inta, num)) = false + | equal_fm (Dvd (inta, numa)) (NEq num) = false + | equal_fm (NEq numa) (Dvd (inta, num)) = false + | equal_fm (NClosed nat) (Eq num) = false + | equal_fm (Eq num) (NClosed nat) = false + | equal_fm (Closed nat) (Eq num) = false + | equal_fm (Eq num) (Closed nat) = false + | equal_fm (A fm) (Eq num) = false + | equal_fm (Eq num) (A fm) = false + | equal_fm (E fm) (Eq num) = false + | equal_fm (Eq num) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (Eq num) = false + | equal_fm (Eq num) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (Eq num) = false + | equal_fm (Eq num) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (Eq num) = false + | equal_fm (Eq num) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (Eq num) = false + | equal_fm (Eq num) (And (fm1, fm2)) = false + | equal_fm (Not fm) (Eq num) = false + | equal_fm (Eq num) (Not fm) = false + | equal_fm (NDvd (inta, numa)) (Eq num) = false + | equal_fm (Eq numa) (NDvd (inta, num)) = false + | equal_fm (Dvd (inta, numa)) (Eq num) = false + | equal_fm (Eq numa) (Dvd (inta, num)) = false + | equal_fm (NEq numa) (Eq num) = false + | equal_fm (Eq numa) (NEq num) = false + | equal_fm (NClosed nat) (Ge num) = false + | equal_fm (Ge num) (NClosed nat) = false + | equal_fm (Closed nat) (Ge num) = false + | equal_fm (Ge num) (Closed nat) = false + | equal_fm (A fm) (Ge num) = false + | equal_fm (Ge num) (A fm) = false + | equal_fm (E fm) (Ge num) = false + | equal_fm (Ge num) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (Ge num) = false + | equal_fm (Ge num) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (Ge num) = false + | equal_fm (Ge num) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (Ge num) = false + | equal_fm (Ge num) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (Ge num) = false + | equal_fm (Ge num) (And (fm1, fm2)) = false + | equal_fm (Not fm) (Ge num) = false + | equal_fm (Ge num) (Not fm) = false + | equal_fm (NDvd (inta, numa)) (Ge num) = false + | equal_fm (Ge numa) (NDvd (inta, num)) = false + | equal_fm (Dvd (inta, numa)) (Ge num) = false + | equal_fm (Ge numa) (Dvd (inta, num)) = false + | equal_fm (NEq numa) (Ge num) = false + | equal_fm (Ge numa) (NEq num) = false + | equal_fm (Eq numa) (Ge num) = false + | equal_fm (Ge numa) (Eq num) = false + | equal_fm (NClosed nat) (Gt num) = false + | equal_fm (Gt num) (NClosed nat) = false + | equal_fm (Closed nat) (Gt num) = false + | equal_fm (Gt num) (Closed nat) = false + | equal_fm (A fm) (Gt num) = false + | equal_fm (Gt num) (A fm) = false + | equal_fm (E fm) (Gt num) = false + | equal_fm (Gt num) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (Gt num) = false + | equal_fm (Gt num) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (Gt num) = false + | equal_fm (Gt num) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (Gt num) = false + | equal_fm (Gt num) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (Gt num) = false + | equal_fm (Gt num) (And (fm1, fm2)) = false + | equal_fm (Not fm) (Gt num) = false + | equal_fm (Gt num) (Not fm) = false + | equal_fm (NDvd (inta, numa)) (Gt num) = false + | equal_fm (Gt numa) (NDvd (inta, num)) = false + | equal_fm (Dvd (inta, numa)) (Gt num) = false + | equal_fm (Gt numa) (Dvd (inta, num)) = false + | equal_fm (NEq numa) (Gt num) = false + | equal_fm (Gt numa) (NEq num) = false + | equal_fm (Eq numa) (Gt num) = false + | equal_fm (Gt numa) (Eq num) = false + | equal_fm (Ge numa) (Gt num) = false + | equal_fm (Gt numa) (Ge num) = false + | equal_fm (NClosed nat) (Le num) = false + | equal_fm (Le num) (NClosed nat) = false + | equal_fm (Closed nat) (Le num) = false + | equal_fm (Le num) (Closed nat) = false + | equal_fm (A fm) (Le num) = false + | equal_fm (Le num) (A fm) = false + | equal_fm (E fm) (Le num) = false + | equal_fm (Le num) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (Le num) = false + | equal_fm (Le num) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (Le num) = false + | equal_fm (Le num) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (Le num) = false + | equal_fm (Le num) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (Le num) = false + | equal_fm (Le num) (And (fm1, fm2)) = false + | equal_fm (Not fm) (Le num) = false + | equal_fm (Le num) (Not fm) = false + | equal_fm (NDvd (inta, numa)) (Le num) = false + | equal_fm (Le numa) (NDvd (inta, num)) = false + | equal_fm (Dvd (inta, numa)) (Le num) = false + | equal_fm (Le numa) (Dvd (inta, num)) = false + | equal_fm (NEq numa) (Le num) = false + | equal_fm (Le numa) (NEq num) = false + | equal_fm (Eq numa) (Le num) = false + | equal_fm (Le numa) (Eq num) = false + | equal_fm (Ge numa) (Le num) = false + | equal_fm (Le numa) (Ge num) = false + | equal_fm (Gt numa) (Le num) = false + | equal_fm (Le numa) (Gt num) = false + | equal_fm (NClosed nat) (Lt num) = false + | equal_fm (Lt num) (NClosed nat) = false + | equal_fm (Closed nat) (Lt num) = false + | equal_fm (Lt num) (Closed nat) = false + | equal_fm (A fm) (Lt num) = false + | equal_fm (Lt num) (A fm) = false + | equal_fm (E fm) (Lt num) = false + | equal_fm (Lt num) (E fm) = false + | equal_fm (Iff (fm1, fm2)) (Lt num) = false + | equal_fm (Lt num) (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) (Lt num) = false + | equal_fm (Lt num) (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) (Lt num) = false + | equal_fm (Lt num) (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) (Lt num) = false + | equal_fm (Lt num) (And (fm1, fm2)) = false + | equal_fm (Not fm) (Lt num) = false + | equal_fm (Lt num) (Not fm) = false + | equal_fm (NDvd (inta, numa)) (Lt num) = false + | equal_fm (Lt numa) (NDvd (inta, num)) = false + | equal_fm (Dvd (inta, numa)) (Lt num) = false + | equal_fm (Lt numa) (Dvd (inta, num)) = false + | equal_fm (NEq numa) (Lt num) = false + | equal_fm (Lt numa) (NEq num) = false + | equal_fm (Eq numa) (Lt num) = false + | equal_fm (Lt numa) (Eq num) = false + | equal_fm (Ge numa) (Lt num) = false + | equal_fm (Lt numa) (Ge num) = false + | equal_fm (Gt numa) (Lt num) = false + | equal_fm (Lt numa) (Gt num) = false + | equal_fm (Le numa) (Lt num) = false + | equal_fm (Lt numa) (Le num) = false + | equal_fm (NClosed nat) F = false + | equal_fm F (NClosed nat) = false + | equal_fm (Closed nat) F = false + | equal_fm F (Closed nat) = false + | equal_fm (A fm) F = false + | equal_fm F (A fm) = false + | equal_fm (E fm) F = false + | equal_fm F (E fm) = false + | equal_fm (Iff (fm1, fm2)) F = false + | equal_fm F (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) F = false + | equal_fm F (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) F = false + | equal_fm F (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) F = false + | equal_fm F (And (fm1, fm2)) = false + | equal_fm (Not fm) F = false + | equal_fm F (Not fm) = false + | equal_fm (NDvd (inta, num)) F = false + | equal_fm F (NDvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) F = false + | equal_fm F (Dvd (inta, num)) = false + | equal_fm (NEq num) F = false + | equal_fm F (NEq num) = false + | equal_fm (Eq num) F = false + | equal_fm F (Eq num) = false + | equal_fm (Ge num) F = false + | equal_fm F (Ge num) = false + | equal_fm (Gt num) F = false + | equal_fm F (Gt num) = false + | equal_fm (Le num) F = false + | equal_fm F (Le num) = false + | equal_fm (Lt num) F = false + | equal_fm F (Lt num) = false + | equal_fm (NClosed nat) T = false + | equal_fm T (NClosed nat) = false + | equal_fm (Closed nat) T = false + | equal_fm T (Closed nat) = false + | equal_fm (A fm) T = false + | equal_fm T (A fm) = false + | equal_fm (E fm) T = false + | equal_fm T (E fm) = false + | equal_fm (Iff (fm1, fm2)) T = false + | equal_fm T (Iff (fm1, fm2)) = false + | equal_fm (Imp (fm1, fm2)) T = false + | equal_fm T (Imp (fm1, fm2)) = false + | equal_fm (Or (fm1, fm2)) T = false + | equal_fm T (Or (fm1, fm2)) = false + | equal_fm (And (fm1, fm2)) T = false + | equal_fm T (And (fm1, fm2)) = false + | equal_fm (Not fm) T = false + | equal_fm T (Not fm) = false + | equal_fm (NDvd (inta, num)) T = false + | equal_fm T (NDvd (inta, num)) = false + | equal_fm (Dvd (inta, num)) T = false + | equal_fm T (Dvd (inta, num)) = false + | equal_fm (NEq num) T = false + | equal_fm T (NEq num) = false + | equal_fm (Eq num) T = false + | equal_fm T (Eq num) = false + | equal_fm (Ge num) T = false + | equal_fm T (Ge num) = false + | equal_fm (Gt num) T = false + | equal_fm T (Gt num) = false + | equal_fm (Le num) T = false + | equal_fm T (Le num) = false + | equal_fm (Lt num) T = false + | equal_fm T (Lt num) = false + | equal_fm F T = false + | equal_fm T F = false + | equal_fm (NClosed nata) (NClosed nat) = nata = nat + | equal_fm (Closed nata) (Closed nat) = nata = nat + | equal_fm (A fma) (A fm) = equal_fm fma fm + | equal_fm (E fma) (E fm) = equal_fm fma fm + | equal_fm (Iff (fm1a, fm2a)) (Iff (fm1, fm2)) = + equal_fm fm1a fm1 andalso equal_fm fm2a fm2 + | equal_fm (Imp (fm1a, fm2a)) (Imp (fm1, fm2)) = + equal_fm fm1a fm1 andalso equal_fm fm2a fm2 + | equal_fm (Or (fm1a, fm2a)) (Or (fm1, fm2)) = + equal_fm fm1a fm1 andalso equal_fm fm2a fm2 + | equal_fm (And (fm1a, fm2a)) (And (fm1, fm2)) = + equal_fm fm1a fm1 andalso equal_fm fm2a fm2 + | equal_fm (Not fma) (Not fm) = equal_fm fma fm + | equal_fm (NDvd (intaa, numa)) (NDvd (inta, num)) = + intaa = inta andalso equal_numa numa num + | equal_fm (Dvd (intaa, numa)) (Dvd (inta, num)) = + intaa = inta andalso equal_numa numa num + | equal_fm (NEq numa) (NEq num) = equal_numa numa num + | equal_fm (Eq numa) (Eq num) = equal_numa numa num + | equal_fm (Ge numa) (Ge num) = equal_numa numa num + | equal_fm (Gt numa) (Gt num) = equal_numa numa num + | equal_fm (Le numa) (Le num) = equal_numa numa num + | equal_fm (Lt numa) (Lt num) = equal_numa numa num + | equal_fm F F = true + | equal_fm T T = true; fun djf f p q = - (if eq_fm q T then T - else (if eq_fm q F then f p + (if equal_fm q T then T + else (if equal_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) @@ -765,1395 +671,27 @@ fun evaldjf f ps = foldr (djf f) ps F; -fun dj f p = evaldjf f (disjuncts p); - -fun disj p q = - (if eq_fm p T orelse eq_fm q T then T - else (if eq_fm p F then q else (if eq_fm q F then p else Or (p, q)))); - -fun minus_nat n m = IntInf.max (0, (IntInf.- (n, m))); - -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 : IntInf.int), i, decrnum a) - | decrnum (C u) = C u; - -fun decr (Lt a) = Lt (decrnum a) - | decr (Le a) = Le (decrnum a) - | decr (Gt a) = Gt (decrnum a) - | decr (Ge a) = Ge (decrnum a) - | decr (Eq a) = Eq (decrnum a) - | decr (NEq a) = NEq (decrnum a) - | decr (Dvd (i, a)) = Dvd (i, decrnum a) - | decr (NDvd (i, a)) = NDvd (i, decrnum a) - | decr (Not p) = Not (decr p) - | decr (And (p, q)) = And (decr p, decr q) - | decr (Or (p, q)) = Or (decr p, decr q) - | decr (Imp (p, q)) = Imp (decr p, decr q) - | decr (Iff (p, q)) = Iff (decr p, decr q) - | decr T = T - | decr F = F - | decr (E ao) = E ao - | decr (A ap) = A ap - | decr (Closed aq) = Closed aq - | decr (NClosed ar) = NClosed ar; - -fun concat_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 ((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 ((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 - | subst0 t (Lt a) = Lt (numsubst0 t a) - | subst0 t (Le a) = Le (numsubst0 t a) - | subst0 t (Gt a) = Gt (numsubst0 t a) - | subst0 t (Ge a) = Ge (numsubst0 t a) - | subst0 t (Eq a) = Eq (numsubst0 t a) - | subst0 t (NEq a) = NEq (numsubst0 t a) - | subst0 t (Dvd (i, a)) = Dvd (i, numsubst0 t a) - | subst0 t (NDvd (i, a)) = NDvd (i, numsubst0 t a) - | subst0 t (Not p) = Not (subst0 t p) - | subst0 t (And (p, q)) = And (subst0 t p, subst0 t q) - | subst0 t (Or (p, q)) = Or (subst0 t p, subst0 t q) - | subst0 t (Imp (p, q)) = Imp (subst0 t p, subst0 t q) - | subst0 t (Iff (p, q)) = Iff (subst0 t p, subst0 t q) - | subst0 t (Closed p) = Closed p - | subst0 t (NClosed p) = NClosed p; - -fun minusinf (And (p, q)) = And (minusinf p, minusinf q) - | minusinf (Or (p, q)) = Or (minusinf p, minusinf q) - | minusinf T = T - | minusinf F = F - | minusinf (Lt (C bo)) = Lt (C bo) - | minusinf (Lt (Bound bp)) = Lt (Bound bp) - | minusinf (Lt (Neg bt)) = Lt (Neg bt) - | minusinf (Lt (Add (bu, bv))) = Lt (Add (bu, bv)) - | minusinf (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx)) - | minusinf (Lt (Mul (by, bz))) = Lt (Mul (by, bz)) - | minusinf (Le (C co)) = Le (C co) - | minusinf (Le (Bound cp)) = Le (Bound cp) - | minusinf (Le (Neg ct)) = Le (Neg ct) - | minusinf (Le (Add (cu, cv))) = Le (Add (cu, cv)) - | minusinf (Le (Sub (cw, cx))) = Le (Sub (cw, cx)) - | minusinf (Le (Mul (cy, cz))) = Le (Mul (cy, cz)) - | minusinf (Gt (C doa)) = Gt (C doa) - | minusinf (Gt (Bound dp)) = Gt (Bound dp) - | minusinf (Gt (Neg dt)) = Gt (Neg dt) - | minusinf (Gt (Add (du, dv))) = Gt (Add (du, dv)) - | minusinf (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx)) - | minusinf (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz)) - | minusinf (Ge (C eo)) = Ge (C eo) - | minusinf (Ge (Bound ep)) = Ge (Bound ep) - | minusinf (Ge (Neg et)) = Ge (Neg et) - | minusinf (Ge (Add (eu, ev))) = Ge (Add (eu, ev)) - | minusinf (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex)) - | minusinf (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez)) - | minusinf (Eq (C fo)) = Eq (C fo) - | minusinf (Eq (Bound fp)) = Eq (Bound fp) - | minusinf (Eq (Neg ft)) = Eq (Neg ft) - | minusinf (Eq (Add (fu, fv))) = Eq (Add (fu, fv)) - | minusinf (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx)) - | minusinf (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz)) - | minusinf (NEq (C go)) = NEq (C go) - | minusinf (NEq (Bound gp)) = NEq (Bound gp) - | minusinf (NEq (Neg gt)) = NEq (Neg gt) - | minusinf (NEq (Add (gu, gv))) = NEq (Add (gu, gv)) - | minusinf (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx)) - | minusinf (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz)) - | minusinf (Dvd (aa, ab)) = Dvd (aa, ab) - | minusinf (NDvd (ac, ad)) = NDvd (ac, ad) - | minusinf (Not ae) = Not ae - | minusinf (Imp (aj, ak)) = Imp (aj, ak) - | minusinf (Iff (al, am)) = Iff (al, am) - | minusinf (E an) = E an - | minusinf (A ao) = A ao - | minusinf (Closed ap) = Closed ap - | minusinf (NClosed aq) = NClosed aq - | minusinf (Lt (Cn (cm, c, e))) = - (if ((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 ((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 ((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 ((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 ((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 ((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 ((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 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 (((sgn_int k) : IntInf.int) = (sgn_int l)) - then IntInf.divMod (IntInf.abs k, IntInf.abs l) - else let - val (r, s) = - IntInf.divMod (IntInf.abs k, IntInf.abs l); - in - (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 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 - | num_case f1 f2 f3 f4 f5 f6 f7 (Add (num1, num2)) = f5 num1 num2 - | num_case f1 f2 f3 f4 f5 f6 f7 (Neg num) = f4 num - | num_case f1 f2 f3 f4 f5 f6 f7 (Cn (nat, inta, num)) = f3 nat inta num - | num_case f1 f2 f3 f4 f5 f6 f7 (Bound nat) = f2 nat - | num_case f1 f2 f3 f4 f5 f6 f7 (C inta) = f1 inta; - -fun nummul i (C j) = C (IntInf.* (i, j)) - | nummul i (Cn (n, c, t)) = Cn (n, IntInf.* (c, i), nummul i t) - | nummul i (Bound v) = Mul (i, Bound v) - | nummul i (Neg v) = Mul (i, Neg v) - | nummul i (Add (v, va)) = Mul (i, Add (v, va)) - | nummul i (Sub (v, va)) = Mul (i, Sub (v, va)) - | nummul i (Mul (v, va)) = Mul (i, Mul (v, va)); - -fun numneg t = nummul (IntInf.~ (1 : IntInf.int)) t; - -fun numadd (Cn (n1, c1, r1), Cn (n2, c2, r2)) = - (if ((n1 : IntInf.int) = n2) - then let - val c = IntInf.+ (c1, c2); - in - (if ((c : IntInf.int) = (0 : IntInf.int)) then numadd (r1, r2) - else Cn (n1, c, numadd (r1, r2))) - end - else (if IntInf.<= (n1, n2) - then Cn (n1, c1, numadd (r1, Add (Mul (c2, Bound n2), r2))) - else Cn (n2, c2, numadd (Add (Mul (c1, Bound n1), r1), r2)))) - | numadd (Cn (n1, c1, r1), C dd) = Cn (n1, c1, numadd (r1, C dd)) - | numadd (Cn (n1, c1, r1), Bound de) = Cn (n1, c1, numadd (r1, Bound de)) - | numadd (Cn (n1, c1, r1), Neg di) = Cn (n1, c1, numadd (r1, Neg di)) - | numadd (Cn (n1, c1, r1), Add (dj, dk)) = - Cn (n1, c1, numadd (r1, Add (dj, dk))) - | numadd (Cn (n1, c1, r1), Sub (dl, dm)) = - Cn (n1, c1, numadd (r1, Sub (dl, dm))) - | numadd (Cn (n1, c1, r1), Mul (dn, doa)) = - Cn (n1, c1, numadd (r1, Mul (dn, doa))) - | numadd (C w, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (C w, r2)) - | numadd (Bound x, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Bound x, r2)) - | numadd (Neg ac, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Neg ac, r2)) - | numadd (Add (ad, ae), Cn (n2, c2, r2)) = - Cn (n2, c2, numadd (Add (ad, ae), r2)) - | numadd (Sub (af, ag), Cn (n2, c2, r2)) = - Cn (n2, c2, numadd (Sub (af, ag), r2)) - | numadd (Mul (ah, ai), Cn (n2, c2, r2)) = - Cn (n2, c2, numadd (Mul (ah, ai), r2)) - | numadd (C b1, C b2) = C (IntInf.+ (b1, b2)) - | numadd (C aj, Bound bi) = Add (C aj, Bound bi) - | numadd (C aj, Neg bm) = Add (C aj, Neg bm) - | numadd (C aj, Add (bn, bo)) = Add (C aj, Add (bn, bo)) - | numadd (C aj, Sub (bp, bq)) = Add (C aj, Sub (bp, bq)) - | numadd (C aj, Mul (br, bs)) = Add (C aj, Mul (br, bs)) - | numadd (Bound ak, C cf) = Add (Bound ak, C cf) - | numadd (Bound ak, Bound cg) = Add (Bound ak, Bound cg) - | numadd (Bound ak, Neg ck) = Add (Bound ak, Neg ck) - | numadd (Bound ak, Add (cl, cm)) = Add (Bound ak, Add (cl, cm)) - | numadd (Bound ak, Sub (cn, co)) = Add (Bound ak, Sub (cn, co)) - | numadd (Bound ak, Mul (cp, cq)) = Add (Bound ak, Mul (cp, cq)) - | numadd (Neg ao, C en) = Add (Neg ao, C en) - | numadd (Neg ao, Bound eo) = Add (Neg ao, Bound eo) - | numadd (Neg ao, Neg es) = Add (Neg ao, Neg es) - | numadd (Neg ao, Add (et, eu)) = Add (Neg ao, Add (et, eu)) - | numadd (Neg ao, Sub (ev, ew)) = Add (Neg ao, Sub (ev, ew)) - | numadd (Neg ao, Mul (ex, ey)) = Add (Neg ao, Mul (ex, ey)) - | numadd (Add (ap, aq), C fl) = Add (Add (ap, aq), C fl) - | numadd (Add (ap, aq), Bound fm) = Add (Add (ap, aq), Bound fm) - | numadd (Add (ap, aq), Neg fq) = Add (Add (ap, aq), Neg fq) - | numadd (Add (ap, aq), Add (fr, fs)) = Add (Add (ap, aq), Add (fr, fs)) - | numadd (Add (ap, aq), Sub (ft, fu)) = Add (Add (ap, aq), Sub (ft, fu)) - | numadd (Add (ap, aq), Mul (fv, fw)) = Add (Add (ap, aq), Mul (fv, fw)) - | numadd (Sub (ar, asa), C gj) = Add (Sub (ar, asa), C gj) - | numadd (Sub (ar, asa), Bound gk) = Add (Sub (ar, asa), Bound gk) - | numadd (Sub (ar, asa), Neg go) = Add (Sub (ar, asa), Neg go) - | numadd (Sub (ar, asa), Add (gp, gq)) = Add (Sub (ar, asa), Add (gp, gq)) - | numadd (Sub (ar, asa), Sub (gr, gs)) = Add (Sub (ar, asa), Sub (gr, gs)) - | numadd (Sub (ar, asa), Mul (gt, gu)) = Add (Sub (ar, asa), Mul (gt, gu)) - | numadd (Mul (at, au), C hh) = Add (Mul (at, au), C hh) - | numadd (Mul (at, au), Bound hi) = Add (Mul (at, au), Bound hi) - | numadd (Mul (at, au), Neg hm) = Add (Mul (at, au), Neg hm) - | numadd (Mul (at, au), Add (hn, ho)) = Add (Mul (at, au), Add (hn, ho)) - | numadd (Mul (at, au), Sub (hp, hq)) = Add (Mul (at, au), Sub (hp, hq)) - | numadd (Mul (at, au), Mul (hr, hs)) = Add (Mul (at, au), Mul (hr, hs)); - -fun numsub s 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)) - | simpnum (Neg t) = numneg (simpnum t) - | simpnum (Add (t, s)) = numadd (simpnum t, simpnum s) - | simpnum (Sub (t, s)) = numsub (simpnum t) (simpnum s) - | simpnum (Mul (i, t)) = - (if ((i : IntInf.int) = (0 : IntInf.int)) then C (0 : IntInf.int) - else nummul i (simpnum t)) - | simpnum (Cn (v, va, vb)) = Cn (v, va, vb); - -fun nota (Not p) = p - | nota T = F - | nota F = T - | nota (Lt v) = Not (Lt v) - | nota (Le v) = Not (Le v) - | nota (Gt v) = Not (Gt v) - | nota (Ge v) = Not (Ge v) - | nota (Eq v) = Not (Eq v) - | nota (NEq v) = Not (NEq v) - | nota (Dvd (v, va)) = Not (Dvd (v, va)) - | nota (NDvd (v, va)) = Not (NDvd (v, va)) - | nota (And (v, va)) = Not (And (v, va)) - | nota (Or (v, va)) = Not (Or (v, va)) - | nota (Imp (v, va)) = Not (Imp (v, va)) - | nota (Iff (v, va)) = Not (Iff (v, va)) - | nota (E v) = Not (E v) - | nota (A v) = Not (A v) - | nota (Closed v) = Not (Closed v) - | nota (NClosed v) = Not (NClosed v); - -fun iffa p q = - (if 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 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 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 disjuncts (Or (p, q)) = disjuncts p @ disjuncts q + | disjuncts F = [] + | disjuncts T = [T] + | disjuncts (Lt v) = [Lt v] + | disjuncts (Le v) = [Le v] + | disjuncts (Gt v) = [Gt v] + | disjuncts (Ge v) = [Ge v] + | disjuncts (Eq v) = [Eq v] + | disjuncts (NEq v) = [NEq v] + | disjuncts (Dvd (v, va)) = [Dvd (v, va)] + | disjuncts (NDvd (v, va)) = [NDvd (v, va)] + | disjuncts (Not v) = [Not v] + | disjuncts (And (v, va)) = [And (v, va)] + | disjuncts (Imp (v, va)) = [Imp (v, va)] + | disjuncts (Iff (v, va)) = [Iff (v, va)] + | disjuncts (E v) = [E v] + | disjuncts (A v) = [A v] + | disjuncts (Closed v) = [Closed v] + | disjuncts (NClosed v) = [NClosed v]; -fun simpfm (And (p, q)) = conj (simpfm p) (simpfm q) - | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q) - | simpfm (Imp (p, q)) = impa (simpfm p) (simpfm q) - | simpfm (Iff (p, q)) = iffa (simpfm p) (simpfm q) - | simpfm (Not p) = nota (simpfm p) - | simpfm (Lt a) = - let - val aa = simpnum a; - in - (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 aa = simpnum a; - in - (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 aa = simpnum a; - in - (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 aa = simpnum a; - in - (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 aa = simpnum a; - in - (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 aa = simpnum a; - in - (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 ((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 aa = simpnum a; - in - (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 ((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 aa = simpnum a; - in - (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 - | simpfm (E v) = E v - | simpfm (A v) = A v - | simpfm (Closed v) = Closed v - | simpfm (NClosed v) = NClosed v; - -fun iupt i j = - (if IntInf.< (j, i) then [] - else i :: iupt (IntInf.+ (i, (1 : IntInf.int))) j); - -fun mirror (And (p, q)) = And (mirror p, mirror q) - | mirror (Or (p, q)) = Or (mirror p, mirror q) - | mirror T = T - | mirror F = F - | mirror (Lt (C bo)) = Lt (C bo) - | mirror (Lt (Bound bp)) = Lt (Bound bp) - | mirror (Lt (Neg bt)) = Lt (Neg bt) - | mirror (Lt (Add (bu, bv))) = Lt (Add (bu, bv)) - | mirror (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx)) - | mirror (Lt (Mul (by, bz))) = Lt (Mul (by, bz)) - | mirror (Le (C co)) = Le (C co) - | mirror (Le (Bound cp)) = Le (Bound cp) - | mirror (Le (Neg ct)) = Le (Neg ct) - | mirror (Le (Add (cu, cv))) = Le (Add (cu, cv)) - | mirror (Le (Sub (cw, cx))) = Le (Sub (cw, cx)) - | mirror (Le (Mul (cy, cz))) = Le (Mul (cy, cz)) - | mirror (Gt (C doa)) = Gt (C doa) - | mirror (Gt (Bound dp)) = Gt (Bound dp) - | mirror (Gt (Neg dt)) = Gt (Neg dt) - | mirror (Gt (Add (du, dv))) = Gt (Add (du, dv)) - | mirror (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx)) - | mirror (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz)) - | mirror (Ge (C eo)) = Ge (C eo) - | mirror (Ge (Bound ep)) = Ge (Bound ep) - | mirror (Ge (Neg et)) = Ge (Neg et) - | mirror (Ge (Add (eu, ev))) = Ge (Add (eu, ev)) - | mirror (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex)) - | mirror (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez)) - | mirror (Eq (C fo)) = Eq (C fo) - | mirror (Eq (Bound fp)) = Eq (Bound fp) - | mirror (Eq (Neg ft)) = Eq (Neg ft) - | mirror (Eq (Add (fu, fv))) = Eq (Add (fu, fv)) - | mirror (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx)) - | mirror (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz)) - | mirror (NEq (C go)) = NEq (C go) - | mirror (NEq (Bound gp)) = NEq (Bound gp) - | mirror (NEq (Neg gt)) = NEq (Neg gt) - | mirror (NEq (Add (gu, gv))) = NEq (Add (gu, gv)) - | mirror (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx)) - | mirror (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz)) - | mirror (Dvd (aa, C ho)) = Dvd (aa, C ho) - | mirror (Dvd (aa, Bound hp)) = Dvd (aa, Bound hp) - | mirror (Dvd (aa, Neg ht)) = Dvd (aa, Neg ht) - | mirror (Dvd (aa, Add (hu, hv))) = Dvd (aa, Add (hu, hv)) - | mirror (Dvd (aa, Sub (hw, hx))) = Dvd (aa, Sub (hw, hx)) - | mirror (Dvd (aa, Mul (hy, hz))) = Dvd (aa, Mul (hy, hz)) - | mirror (NDvd (ac, C io)) = NDvd (ac, C io) - | mirror (NDvd (ac, Bound ip)) = NDvd (ac, Bound ip) - | mirror (NDvd (ac, Neg it)) = NDvd (ac, Neg it) - | mirror (NDvd (ac, Add (iu, iv))) = NDvd (ac, Add (iu, iv)) - | mirror (NDvd (ac, Sub (iw, ix))) = NDvd (ac, Sub (iw, ix)) - | mirror (NDvd (ac, Mul (iy, iz))) = NDvd (ac, Mul (iy, iz)) - | mirror (Not ae) = Not ae - | mirror (Imp (aj, ak)) = Imp (aj, ak) - | mirror (Iff (al, am)) = Iff (al, am) - | mirror (E an) = E an - | mirror (A ao) = A ao - | mirror (Closed ap) = Closed ap - | mirror (NClosed aq) = NClosed aq - | mirror (Lt (Cn (cm, c, e))) = - (if ((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 ((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 ((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 ((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 ((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 ((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 ((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 ((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 : 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) - | alpha T = [] - | alpha F = [] - | alpha (Lt (C bo)) = [] - | alpha (Lt (Bound bp)) = [] - | alpha (Lt (Neg bt)) = [] - | alpha (Lt (Add (bu, bv))) = [] - | alpha (Lt (Sub (bw, bx))) = [] - | alpha (Lt (Mul (by, bz))) = [] - | alpha (Le (C co)) = [] - | alpha (Le (Bound cp)) = [] - | alpha (Le (Neg ct)) = [] - | alpha (Le (Add (cu, cv))) = [] - | alpha (Le (Sub (cw, cx))) = [] - | alpha (Le (Mul (cy, cz))) = [] - | alpha (Gt (C doa)) = [] - | alpha (Gt (Bound dp)) = [] - | alpha (Gt (Neg dt)) = [] - | alpha (Gt (Add (du, dv))) = [] - | alpha (Gt (Sub (dw, dx))) = [] - | alpha (Gt (Mul (dy, dz))) = [] - | alpha (Ge (C eo)) = [] - | alpha (Ge (Bound ep)) = [] - | alpha (Ge (Neg et)) = [] - | alpha (Ge (Add (eu, ev))) = [] - | alpha (Ge (Sub (ew, ex))) = [] - | alpha (Ge (Mul (ey, ez))) = [] - | alpha (Eq (C fo)) = [] - | alpha (Eq (Bound fp)) = [] - | alpha (Eq (Neg ft)) = [] - | alpha (Eq (Add (fu, fv))) = [] - | alpha (Eq (Sub (fw, fx))) = [] - | alpha (Eq (Mul (fy, fz))) = [] - | alpha (NEq (C go)) = [] - | alpha (NEq (Bound gp)) = [] - | alpha (NEq (Neg gt)) = [] - | alpha (NEq (Add (gu, gv))) = [] - | alpha (NEq (Sub (gw, gx))) = [] - | alpha (NEq (Mul (gy, gz))) = [] - | alpha (Dvd (aa, ab)) = [] - | alpha (NDvd (ac, ad)) = [] - | alpha (Not ae) = [] - | alpha (Imp (aj, ak)) = [] - | alpha (Iff (al, am)) = [] - | alpha (E an) = [] - | alpha (A ao) = [] - | alpha (Closed ap) = [] - | alpha (NClosed aq) = [] - | alpha (Lt (Cn (cm, c, e))) = - (if ((cm : IntInf.int) = (0 : IntInf.int)) then [e] else []) - | alpha (Le (Cn (dm, c, e))) = - (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 ((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) - | beta T = [] - | beta F = [] - | beta (Lt (C bo)) = [] - | beta (Lt (Bound bp)) = [] - | beta (Lt (Neg bt)) = [] - | beta (Lt (Add (bu, bv))) = [] - | beta (Lt (Sub (bw, bx))) = [] - | beta (Lt (Mul (by, bz))) = [] - | beta (Le (C co)) = [] - | beta (Le (Bound cp)) = [] - | beta (Le (Neg ct)) = [] - | beta (Le (Add (cu, cv))) = [] - | beta (Le (Sub (cw, cx))) = [] - | beta (Le (Mul (cy, cz))) = [] - | beta (Gt (C doa)) = [] - | beta (Gt (Bound dp)) = [] - | beta (Gt (Neg dt)) = [] - | beta (Gt (Add (du, dv))) = [] - | beta (Gt (Sub (dw, dx))) = [] - | beta (Gt (Mul (dy, dz))) = [] - | beta (Ge (C eo)) = [] - | beta (Ge (Bound ep)) = [] - | beta (Ge (Neg et)) = [] - | beta (Ge (Add (eu, ev))) = [] - | beta (Ge (Sub (ew, ex))) = [] - | beta (Ge (Mul (ey, ez))) = [] - | beta (Eq (C fo)) = [] - | beta (Eq (Bound fp)) = [] - | beta (Eq (Neg ft)) = [] - | beta (Eq (Add (fu, fv))) = [] - | beta (Eq (Sub (fw, fx))) = [] - | beta (Eq (Mul (fy, fz))) = [] - | beta (NEq (C go)) = [] - | beta (NEq (Bound gp)) = [] - | beta (NEq (Neg gt)) = [] - | beta (NEq (Add (gu, gv))) = [] - | beta (NEq (Sub (gw, gx))) = [] - | beta (NEq (Mul (gy, gz))) = [] - | beta (Dvd (aa, ab)) = [] - | beta (NDvd (ac, ad)) = [] - | beta (Not ae) = [] - | beta (Imp (aj, ak)) = [] - | beta (Iff (al, am)) = [] - | beta (E an) = [] - | beta (A ao) = [] - | beta (Closed ap) = [] - | beta (NClosed aq) = [] - | beta (Lt (Cn (cm, c, e))) = - (if ((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 ((fm : IntInf.int) = (0 : IntInf.int)) - then [Sub (C (~1 : IntInf.int), e)] else []) - | beta (Eq (Cn (gm, c, e))) = - (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) = 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 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) - | delta (Le v) = (1 : IntInf.int) - | delta (Gt w) = (1 : IntInf.int) - | delta (Ge x) = (1 : IntInf.int) - | delta (Eq y) = (1 : IntInf.int) - | delta (NEq z) = (1 : IntInf.int) - | delta (Dvd (aa, C bo)) = (1 : IntInf.int) - | delta (Dvd (aa, Bound bp)) = (1 : IntInf.int) - | delta (Dvd (aa, Neg bt)) = (1 : IntInf.int) - | delta (Dvd (aa, Add (bu, bv))) = (1 : IntInf.int) - | delta (Dvd (aa, Sub (bw, bx))) = (1 : IntInf.int) - | delta (Dvd (aa, Mul (by, bz))) = (1 : IntInf.int) - | delta (NDvd (ac, C co)) = (1 : IntInf.int) - | delta (NDvd (ac, Bound cp)) = (1 : IntInf.int) - | delta (NDvd (ac, Neg ct)) = (1 : IntInf.int) - | delta (NDvd (ac, Add (cu, cv))) = (1 : IntInf.int) - | delta (NDvd (ac, Sub (cw, cx))) = (1 : IntInf.int) - | delta (NDvd (ac, Mul (cy, cz))) = (1 : IntInf.int) - | delta (Not ae) = (1 : IntInf.int) - | delta (Imp (aj, ak)) = (1 : IntInf.int) - | delta (Iff (al, am)) = (1 : IntInf.int) - | delta (E an) = (1 : IntInf.int) - | delta (A ao) = (1 : IntInf.int) - | delta (Closed ap) = (1 : IntInf.int) - | delta (NClosed aq) = (1 : IntInf.int) - | delta (Dvd (i, Cn (cm, c, e))) = - (if ((cm : IntInf.int) = (0 : IntInf.int)) then i else (1 : IntInf.int)) - | delta (NDvd (i, Cn (dm, c, e))) = - (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 _ => 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 ((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 ((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 ((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 ((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 ((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 ((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 ((im : IntInf.int) = (0 : IntInf.int)) - then (fn k => - Dvd (IntInf.* (div_int k c, i), - 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 ((jm : IntInf.int) = (0 : IntInf.int)) - then (fn k => - NDvd (IntInf.* (div_int k c, i), - 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)) = 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) - | zeta (Lt (Bound bp)) = (1 : IntInf.int) - | zeta (Lt (Neg bt)) = (1 : IntInf.int) - | zeta (Lt (Add (bu, bv))) = (1 : IntInf.int) - | zeta (Lt (Sub (bw, bx))) = (1 : IntInf.int) - | zeta (Lt (Mul (by, bz))) = (1 : IntInf.int) - | zeta (Le (C co)) = (1 : IntInf.int) - | zeta (Le (Bound cp)) = (1 : IntInf.int) - | zeta (Le (Neg ct)) = (1 : IntInf.int) - | zeta (Le (Add (cu, cv))) = (1 : IntInf.int) - | zeta (Le (Sub (cw, cx))) = (1 : IntInf.int) - | zeta (Le (Mul (cy, cz))) = (1 : IntInf.int) - | zeta (Gt (C doa)) = (1 : IntInf.int) - | zeta (Gt (Bound dp)) = (1 : IntInf.int) - | zeta (Gt (Neg dt)) = (1 : IntInf.int) - | zeta (Gt (Add (du, dv))) = (1 : IntInf.int) - | zeta (Gt (Sub (dw, dx))) = (1 : IntInf.int) - | zeta (Gt (Mul (dy, dz))) = (1 : IntInf.int) - | zeta (Ge (C eo)) = (1 : IntInf.int) - | zeta (Ge (Bound ep)) = (1 : IntInf.int) - | zeta (Ge (Neg et)) = (1 : IntInf.int) - | zeta (Ge (Add (eu, ev))) = (1 : IntInf.int) - | zeta (Ge (Sub (ew, ex))) = (1 : IntInf.int) - | zeta (Ge (Mul (ey, ez))) = (1 : IntInf.int) - | zeta (Eq (C fo)) = (1 : IntInf.int) - | zeta (Eq (Bound fp)) = (1 : IntInf.int) - | zeta (Eq (Neg ft)) = (1 : IntInf.int) - | zeta (Eq (Add (fu, fv))) = (1 : IntInf.int) - | zeta (Eq (Sub (fw, fx))) = (1 : IntInf.int) - | zeta (Eq (Mul (fy, fz))) = (1 : IntInf.int) - | zeta (NEq (C go)) = (1 : IntInf.int) - | zeta (NEq (Bound gp)) = (1 : IntInf.int) - | zeta (NEq (Neg gt)) = (1 : IntInf.int) - | zeta (NEq (Add (gu, gv))) = (1 : IntInf.int) - | zeta (NEq (Sub (gw, gx))) = (1 : IntInf.int) - | zeta (NEq (Mul (gy, gz))) = (1 : IntInf.int) - | zeta (Dvd (aa, C ho)) = (1 : IntInf.int) - | zeta (Dvd (aa, Bound hp)) = (1 : IntInf.int) - | zeta (Dvd (aa, Neg ht)) = (1 : IntInf.int) - | zeta (Dvd (aa, Add (hu, hv))) = (1 : IntInf.int) - | zeta (Dvd (aa, Sub (hw, hx))) = (1 : IntInf.int) - | zeta (Dvd (aa, Mul (hy, hz))) = (1 : IntInf.int) - | zeta (NDvd (ac, C io)) = (1 : IntInf.int) - | zeta (NDvd (ac, Bound ip)) = (1 : IntInf.int) - | zeta (NDvd (ac, Neg it)) = (1 : IntInf.int) - | zeta (NDvd (ac, Add (iu, iv))) = (1 : IntInf.int) - | zeta (NDvd (ac, Sub (iw, ix))) = (1 : IntInf.int) - | zeta (NDvd (ac, Mul (iy, iz))) = (1 : IntInf.int) - | zeta (Not ae) = (1 : IntInf.int) - | zeta (Imp (aj, ak)) = (1 : IntInf.int) - | zeta (Iff (al, am)) = (1 : IntInf.int) - | zeta (E an) = (1 : IntInf.int) - | zeta (A ao) = (1 : IntInf.int) - | zeta (Closed ap) = (1 : IntInf.int) - | zeta (NClosed aq) = (1 : IntInf.int) - | zeta (Lt (Cn (cm, c, e))) = - (if ((cm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) - | zeta (Le (Cn (dm, c, e))) = - (if ((dm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) - | zeta (Gt (Cn (em, c, e))) = - (if ((em : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) - | zeta (Ge (Cn (fm, c, e))) = - (if ((fm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) - | zeta (Eq (Cn (gm, c, e))) = - (if ((gm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) - | zeta (NEq (Cn (hm, c, e))) = - (if ((hm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) - | zeta (Dvd (i, Cn (im, c, e))) = - (if ((im : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int)) - | zeta (NDvd (i, Cn (jm, c, e))) = - (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 ((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 (ia, aa) = zsplit0 a; - in - (if ((n : IntInf.int) = (0 : IntInf.int)) then (IntInf.+ (i, ia), aa) - else (ia, Cn (n, i, aa))) - end - | zsplit0 (Neg a) = - let - val (i, aa) = zsplit0 a; - in - (IntInf.~ i, Neg aa) - end - | zsplit0 (Add (a, b)) = - let - val (ia, aa) = zsplit0 a; - val (ib, ba) = zsplit0 b; - in - (IntInf.+ (ia, ib), Add (aa, ba)) - end - | zsplit0 (Sub (a, b)) = - let - val (ia, aa) = zsplit0 a; - val (ib, ba) = zsplit0 b; - in - (IntInf.- (ia, ib), Sub (aa, ba)) - end - | zsplit0 (Mul (i, a)) = - let - val (ia, aa) = zsplit0 a; - in - (IntInf.* (i, ia), Mul (i, aa)) - end; - -fun zlfm (And (p, q)) = And (zlfm p, zlfm q) - | zlfm (Or (p, q)) = Or (zlfm p, zlfm q) - | zlfm (Imp (p, q)) = Or (zlfm (Not p), zlfm q) - | zlfm (Iff (p, q)) = - Or (And (zlfm p, zlfm q), And (zlfm (Not p), zlfm (Not q))) - | zlfm (Lt a) = - let - val (c, r) = zsplit0 a; - in - (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 (c, r) = zsplit0 a; - in - (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 (c, r) = zsplit0 a; - in - (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 (c, r) = zsplit0 a; - in - (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 (c, r) = zsplit0 a; - in - (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 (c, r) = zsplit0 a; - in - (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 ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (Eq a) - else let - val (c, r) = zsplit0 a; - in - (if ((c : IntInf.int) = (0 : IntInf.int)) then Dvd (abs_int i, r) - else (if IntInf.< ((0 : IntInf.int), c) - then Dvd (abs_int i, 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 ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (NEq a) - else let - val (c, r) = zsplit0 a; - in - (if ((c : IntInf.int) = (0 : IntInf.int)) then NDvd (abs_int i, r) - else (if IntInf.< ((0 : IntInf.int), c) - then NDvd (abs_int i, 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)) - | zlfm (Not (Imp (p, q))) = And (zlfm p, zlfm (Not q)) - | zlfm (Not (Iff (p, q))) = - Or (And (zlfm p, zlfm (Not q)), And (zlfm (Not p), zlfm q)) - | zlfm (Not (Not p)) = zlfm p - | zlfm (Not T) = F - | zlfm (Not F) = T - | zlfm (Not (Lt a)) = zlfm (Ge a) - | zlfm (Not (Le a)) = zlfm (Gt a) - | zlfm (Not (Gt a)) = zlfm (Le a) - | zlfm (Not (Ge a)) = zlfm (Lt a) - | zlfm (Not (Eq a)) = zlfm (NEq a) - | zlfm (Not (NEq a)) = zlfm (Eq a) - | zlfm (Not (Dvd (i, a))) = zlfm (NDvd (i, a)) - | zlfm (Not (NDvd (i, a))) = zlfm (Dvd (i, a)) - | zlfm (Not (Closed p)) = NClosed p - | zlfm (Not (NClosed p)) = Closed p - | zlfm T = T - | zlfm F = F - | zlfm (Not (E ci)) = Not (E ci) - | zlfm (Not (A cj)) = Not (A cj) - | zlfm (E ao) = E ao - | zlfm (A ap) = A ap - | zlfm (Closed aq) = Closed aq - | zlfm (NClosed ar) = NClosed ar; - -fun unita p = - let - val pa = zlfm p; - val l = zeta pa; - val q = - 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)); - in - (if IntInf.<= (size_list b, size_list a) then (q, (b, d)) - else (mirror q, (a, d))) - end; - -fun cooper p = - let - 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 eq_fm md T then T - else let - val qd = - 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) - end; +fun dj f p = evaldjf f (disjuncts p); fun prep (E T) = T | prep (E F) = F @@ -2249,6 +787,611 @@ | prep (Closed ap) = Closed ap | prep (NClosed aq) = NClosed aq; +fun conj p q = + (if equal_fm p F orelse equal_fm q F then F + else (if equal_fm p T then q + else (if equal_fm q T then p else And (p, q)))); + +fun disj p q = + (if equal_fm p T orelse equal_fm q T then T + else (if equal_fm p F then q else (if equal_fm q F then p else Or (p, q)))); + +fun nota (Not p) = p + | nota T = F + | nota F = T + | nota (Lt v) = Not (Lt v) + | nota (Le v) = Not (Le v) + | nota (Gt v) = Not (Gt v) + | nota (Ge v) = Not (Ge v) + | nota (Eq v) = Not (Eq v) + | nota (NEq v) = Not (NEq v) + | nota (Dvd (v, va)) = Not (Dvd (v, va)) + | nota (NDvd (v, va)) = Not (NDvd (v, va)) + | nota (And (v, va)) = Not (And (v, va)) + | nota (Or (v, va)) = Not (Or (v, va)) + | nota (Imp (v, va)) = Not (Imp (v, va)) + | nota (Iff (v, va)) = Not (Iff (v, va)) + | nota (E v) = Not (E v) + | nota (A v) = Not (A v) + | nota (Closed v) = Not (Closed v) + | nota (NClosed v) = Not (NClosed v); + +fun iffa p q = + (if equal_fm p q then T + else (if equal_fm p (nota q) orelse equal_fm (nota p) q then F + else (if equal_fm p F then nota q + else (if equal_fm q F then nota p + else (if equal_fm p T then q + else (if equal_fm q T then p + else Iff (p, q))))))); + +fun impa p q = + (if equal_fm p F orelse equal_fm q T then T + else (if equal_fm p T then q + else (if equal_fm q F then nota p else Imp (p, q)))); + +type 'a times = {times : 'a -> 'a -> 'a}; +val times = #times : 'a times -> 'a -> 'a -> 'a; + +type 'a dvd = {times_dvd : 'a times}; +val times_dvd = #times_dvd : 'a dvd -> 'a times; + +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 zero = {zero : 'a}; +val zero = #zero : 'a zero -> '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; + +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 one = {one : 'a}; +val one = #one : 'a one -> 'a; + +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 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 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 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 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 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 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; + +fun dvd (A1_, A2_) a b = + eq A2_ (moda (div_semiring_div A1_) b a) + (zero ((zero_mult_zero o mult_zero_semiring_0 o semiring_0_semiring_1 o + semiring_1_comm_semiring_1 o + comm_semiring_1_comm_semiring_1_cancel o + comm_semiring_1_cancel_semiring_div) + A1_)); + +fun abs_int i = (if i < (0 : IntInf.int) then ~ i else i); + +val equal_int = {equal = (fn a => fn b => a = b)} : int equal; + +fun numadd (Cn (n1, c1, r1), Cn (n2, c2, r2)) = + (if n1 = n2 + then let + val c = c1 + c2; + in + (if c = (0 : IntInf.int) then numadd (r1, r2) + else Cn (n1, c, numadd (r1, r2))) + end + else (if n1 <= n2 + then Cn (n1, c1, numadd (r1, Add (Mul (c2, Bound n2), r2))) + else Cn (n2, c2, numadd (Add (Mul (c1, Bound n1), r1), r2)))) + | numadd (Cn (n1, c1, r1), C dd) = Cn (n1, c1, numadd (r1, C dd)) + | numadd (Cn (n1, c1, r1), Bound de) = Cn (n1, c1, numadd (r1, Bound de)) + | numadd (Cn (n1, c1, r1), Neg di) = Cn (n1, c1, numadd (r1, Neg di)) + | numadd (Cn (n1, c1, r1), Add (dj, dk)) = + Cn (n1, c1, numadd (r1, Add (dj, dk))) + | numadd (Cn (n1, c1, r1), Sub (dl, dm)) = + Cn (n1, c1, numadd (r1, Sub (dl, dm))) + | numadd (Cn (n1, c1, r1), Mul (dn, doa)) = + Cn (n1, c1, numadd (r1, Mul (dn, doa))) + | numadd (C w, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (C w, r2)) + | numadd (Bound x, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Bound x, r2)) + | numadd (Neg ac, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Neg ac, r2)) + | numadd (Add (ad, ae), Cn (n2, c2, r2)) = + Cn (n2, c2, numadd (Add (ad, ae), r2)) + | numadd (Sub (af, ag), Cn (n2, c2, r2)) = + Cn (n2, c2, numadd (Sub (af, ag), r2)) + | numadd (Mul (ah, ai), Cn (n2, c2, r2)) = + Cn (n2, c2, numadd (Mul (ah, ai), r2)) + | numadd (C b1, C b2) = C (b1 + b2) + | numadd (C aj, Bound bi) = Add (C aj, Bound bi) + | numadd (C aj, Neg bm) = Add (C aj, Neg bm) + | numadd (C aj, Add (bn, bo)) = Add (C aj, Add (bn, bo)) + | numadd (C aj, Sub (bp, bq)) = Add (C aj, Sub (bp, bq)) + | numadd (C aj, Mul (br, bs)) = Add (C aj, Mul (br, bs)) + | numadd (Bound ak, C cf) = Add (Bound ak, C cf) + | numadd (Bound ak, Bound cg) = Add (Bound ak, Bound cg) + | numadd (Bound ak, Neg ck) = Add (Bound ak, Neg ck) + | numadd (Bound ak, Add (cl, cm)) = Add (Bound ak, Add (cl, cm)) + | numadd (Bound ak, Sub (cn, co)) = Add (Bound ak, Sub (cn, co)) + | numadd (Bound ak, Mul (cp, cq)) = Add (Bound ak, Mul (cp, cq)) + | numadd (Neg ao, C en) = Add (Neg ao, C en) + | numadd (Neg ao, Bound eo) = Add (Neg ao, Bound eo) + | numadd (Neg ao, Neg et) = Add (Neg ao, Neg et) + | numadd (Neg ao, Add (eu, ev)) = Add (Neg ao, Add (eu, ev)) + | numadd (Neg ao, Sub (ew, ex)) = Add (Neg ao, Sub (ew, ex)) + | numadd (Neg ao, Mul (ey, ez)) = Add (Neg ao, Mul (ey, ez)) + | numadd (Add (ap, aq), C fm) = Add (Add (ap, aq), C fm) + | numadd (Add (ap, aq), Bound fna) = Add (Add (ap, aq), Bound fna) + | numadd (Add (ap, aq), Neg fr) = Add (Add (ap, aq), Neg fr) + | numadd (Add (ap, aq), Add (fs, ft)) = Add (Add (ap, aq), Add (fs, ft)) + | numadd (Add (ap, aq), Sub (fu, fv)) = Add (Add (ap, aq), Sub (fu, fv)) + | numadd (Add (ap, aq), Mul (fw, fx)) = Add (Add (ap, aq), Mul (fw, fx)) + | numadd (Sub (ar, asa), C gk) = Add (Sub (ar, asa), C gk) + | numadd (Sub (ar, asa), Bound gl) = Add (Sub (ar, asa), Bound gl) + | numadd (Sub (ar, asa), Neg gp) = Add (Sub (ar, asa), Neg gp) + | numadd (Sub (ar, asa), Add (gq, gr)) = Add (Sub (ar, asa), Add (gq, gr)) + | numadd (Sub (ar, asa), Sub (gs, gt)) = Add (Sub (ar, asa), Sub (gs, gt)) + | numadd (Sub (ar, asa), Mul (gu, gv)) = Add (Sub (ar, asa), Mul (gu, gv)) + | numadd (Mul (at, au), C hi) = Add (Mul (at, au), C hi) + | numadd (Mul (at, au), Bound hj) = Add (Mul (at, au), Bound hj) + | numadd (Mul (at, au), Neg hn) = Add (Mul (at, au), Neg hn) + | numadd (Mul (at, au), Add (ho, hp)) = Add (Mul (at, au), Add (ho, hp)) + | numadd (Mul (at, au), Sub (hq, hr)) = Add (Mul (at, au), Sub (hq, hr)) + | numadd (Mul (at, au), Mul (hs, ht)) = Add (Mul (at, au), Mul (hs, ht)); + +fun nummul i (C j) = C (i * j) + | nummul i (Cn (n, c, t)) = Cn (n, c * i, nummul i t) + | nummul i (Bound v) = Mul (i, Bound v) + | nummul i (Neg v) = Mul (i, Neg v) + | nummul i (Add (v, va)) = Mul (i, Add (v, va)) + | nummul i (Sub (v, va)) = Mul (i, Sub (v, va)) + | nummul i (Mul (v, va)) = Mul (i, Mul (v, va)); + +fun numneg t = nummul (~ (1 : IntInf.int)) t; + +fun numsub s t = + (if equal_numa s t then C (0 : IntInf.int) else numadd (s, numneg t)); + +fun simpnum (C j) = C j + | simpnum (Bound n) = Cn (n, (1 : IntInf.int), C (0 : IntInf.int)) + | simpnum (Neg t) = numneg (simpnum t) + | simpnum (Add (t, s)) = numadd (simpnum t, simpnum s) + | simpnum (Sub (t, s)) = numsub (simpnum t) (simpnum s) + | simpnum (Mul (i, t)) = + (if i = (0 : IntInf.int) then C (0 : IntInf.int) else nummul i (simpnum t)) + | simpnum (Cn (v, va, vb)) = Cn (v, va, vb); + +val one_inta : int = (1 : IntInf.int); + +val zero_inta : int = (0 : IntInf.int); + +val times_int = {times = (fn a => fn b => a * b)} : int times; + +val dvd_int = {times_dvd = times_int} : int dvd; + +fun fst (a, b) = a; + +fun sgn_int i = + (if i = (0 : IntInf.int) then (0 : IntInf.int) + else (if (0 : IntInf.int) < i then (1 : IntInf.int) + else ~ (1 : IntInf.int))); + +fun apsnd f (x, y) = (x, f y); + +fun divmod_int k l = + (if k = (0 : IntInf.int) then ((0 : IntInf.int), (0 : IntInf.int)) + else (if l = (0 : IntInf.int) then ((0 : IntInf.int), k) + else apsnd (fn a => sgn_int l * a) + (if sgn_int k = sgn_int l then Integer.div_mod (abs k) (abs l) + else let + val (r, s) = Integer.div_mod (abs k) (abs l); + in + (if s = (0 : IntInf.int) then (~ r, (0 : IntInf.int)) + else (~ r - (1 : IntInf.int), abs_int l - s)) + end))); + +fun div_inta a b = fst (divmod_int a b); + +fun snd (a, b) = b; + +fun mod_int a b = snd (divmod_int a b); + +val div_int = {dvd_div = dvd_int, diva = div_inta, moda = mod_int} : int diva; + +val zero_int = {zero = zero_inta} : int zero; + +val no_zero_divisors_int = + {times_no_zero_divisors = times_int, zero_no_zero_divisors = zero_int} : + int no_zero_divisors; + +val semigroup_mult_int = {times_semigroup_mult = times_int} : + int semigroup_mult; + +val plus_int = {plus = (fn a => fn b => a + b)} : int plus; + +val semigroup_add_int = {plus_semigroup_add = plus_int} : int semigroup_add; + +val ab_semigroup_add_int = {semigroup_add_ab_semigroup_add = semigroup_add_int} + : int ab_semigroup_add; + +val semiring_int = + {ab_semigroup_add_semiring = ab_semigroup_add_int, + semigroup_mult_semiring = semigroup_mult_int} + : int semiring; + +val mult_zero_int = {times_mult_zero = times_int, zero_mult_zero = zero_int} : + int mult_zero; + +val monoid_add_int = + {semigroup_add_monoid_add = semigroup_add_int, zero_monoid_add = zero_int} : + 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} + : 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} + : int semiring_0; + +val one_int = {one = one_inta} : int one; + +val power_int = {one_power = one_int, times_power = times_int} : int power; + +val monoid_mult_int = + {semigroup_mult_monoid_mult = semigroup_mult_int, + power_monoid_mult = power_int} + : int monoid_mult; + +val zero_neq_one_int = + {one_zero_neq_one = one_int, zero_zero_neq_one = zero_int} : int zero_neq_one; + +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} + : int semiring_1; + +val ab_semigroup_mult_int = + {semigroup_mult_ab_semigroup_mult = semigroup_mult_int} : + int ab_semigroup_mult; + +val comm_semiring_int = + {ab_semigroup_mult_comm_semiring = ab_semigroup_mult_int, + semiring_comm_semiring = semiring_int} + : int comm_semiring; + +val comm_semiring_0_int = + {comm_semiring_comm_semiring_0 = comm_semiring_int, + semiring_0_comm_semiring_0 = semiring_0_int} + : 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} + : 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} + : int comm_semiring_1; + +val cancel_semigroup_add_int = + {semigroup_add_cancel_semigroup_add = semigroup_add_int} : + 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} + : 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} + : 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} + : 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} + : int semiring_1_cancel; + +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} + : 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} + : int comm_semiring_1_cancel; + +val semiring_div_int = + {div_semiring_div = div_int, + comm_semiring_1_cancel_semiring_div = comm_semiring_1_cancel_int, + no_zero_divisors_semiring_div = no_zero_divisors_int} + : int semiring_div; + +fun simpfm (And (p, q)) = conj (simpfm p) (simpfm q) + | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q) + | simpfm (Imp (p, q)) = impa (simpfm p) (simpfm q) + | simpfm (Iff (p, q)) = iffa (simpfm p) (simpfm q) + | simpfm (Not p) = nota (simpfm p) + | simpfm (Lt a) = + let + val aa = simpnum a; + in + (case aa of C v => (if 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 aa = simpnum a; + in + (case aa of C v => (if 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 aa = simpnum a; + in + (case aa of C v => (if (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 aa = simpnum a; + in + (case aa of C v => (if (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 aa = simpnum a; + in + (case aa of C v => (if v = (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 aa = simpnum a; + in + (case aa of C v => (if not (v = (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 i = (0 : IntInf.int) then simpfm (Eq a) + else (if abs_int i = (1 : IntInf.int) then T + else let + val aa = simpnum a; + in + (case aa + of C v => + (if dvd (semiring_div_int, equal_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 i = (0 : IntInf.int) then simpfm (NEq a) + else (if abs_int i = (1 : IntInf.int) then F + else let + val aa = simpnum a; + in + (case aa + of C v => + (if not (dvd (semiring_div_int, equal_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 + | simpfm (E v) = E v + | simpfm (A v) = A v + | simpfm (Closed v) = Closed v + | simpfm (NClosed v) = NClosed v; + fun qelim (E p) = (fn qe => dj qe (qelim p qe)) | qelim (A p) = (fn qe => nota (qe (qelim (Not p) qe))) | qelim (Not p) = (fn qe => nota (qelim p qe)) @@ -2258,16 +1401,754 @@ | qelim (Iff (p, q)) = (fn qe => iffa (qelim p qe) (qelim q qe)) | qelim T = (fn _ => simpfm T) | qelim F = (fn _ => simpfm F) - | qelim (Lt u) = (fn _ => simpfm (Lt u)) + | qelim (Lt v) = (fn _ => simpfm (Lt v)) | 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)); + | qelim (Gt v) = (fn _ => simpfm (Gt v)) + | qelim (Ge v) = (fn _ => simpfm (Ge v)) + | qelim (Eq v) = (fn _ => simpfm (Eq v)) + | qelim (NEq v) = (fn _ => simpfm (NEq v)) + | qelim (Dvd (v, va)) = (fn _ => simpfm (Dvd (v, va))) + | qelim (NDvd (v, va)) = (fn _ => simpfm (NDvd (v, va))) + | qelim (Closed v) = (fn _ => simpfm (Closed v)) + | qelim (NClosed v) = (fn _ => simpfm (NClosed v)); + +fun maps f [] = [] + | maps f (x :: xs) = f x @ maps f xs; + +fun uptoa i j = (if i <= j then i :: uptoa (i + (1 : IntInf.int)) j else []); + +fun minus_nat n m = Integer.max (n - m) 0; + +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 : IntInf.int), i, decrnum a) + | decrnum (C v) = C v; + +fun decr (Lt a) = Lt (decrnum a) + | decr (Le a) = Le (decrnum a) + | decr (Gt a) = Gt (decrnum a) + | decr (Ge a) = Ge (decrnum a) + | decr (Eq a) = Eq (decrnum a) + | decr (NEq a) = NEq (decrnum a) + | decr (Dvd (i, a)) = Dvd (i, decrnum a) + | decr (NDvd (i, a)) = NDvd (i, decrnum a) + | decr (Not p) = Not (decr p) + | decr (And (p, q)) = And (decr p, decr q) + | decr (Or (p, q)) = Or (decr p, decr q) + | decr (Imp (p, q)) = Imp (decr p, decr q) + | decr (Iff (p, q)) = Iff (decr p, decr q) + | decr T = T + | decr F = F + | decr (E v) = E v + | decr (A v) = A v + | decr (Closed v) = Closed v + | decr (NClosed v) = NClosed v; + +fun beta (And (p, q)) = beta p @ beta q + | beta (Or (p, q)) = beta p @ beta q + | beta T = [] + | beta F = [] + | beta (Lt (C bo)) = [] + | beta (Lt (Bound bp)) = [] + | beta (Lt (Neg bt)) = [] + | beta (Lt (Add (bu, bv))) = [] + | beta (Lt (Sub (bw, bx))) = [] + | beta (Lt (Mul (by, bz))) = [] + | beta (Le (C co)) = [] + | beta (Le (Bound cp)) = [] + | beta (Le (Neg ct)) = [] + | beta (Le (Add (cu, cv))) = [] + | beta (Le (Sub (cw, cx))) = [] + | beta (Le (Mul (cy, cz))) = [] + | beta (Gt (C doa)) = [] + | beta (Gt (Bound dp)) = [] + | beta (Gt (Neg dt)) = [] + | beta (Gt (Add (du, dv))) = [] + | beta (Gt (Sub (dw, dx))) = [] + | beta (Gt (Mul (dy, dz))) = [] + | beta (Ge (C eo)) = [] + | beta (Ge (Bound ep)) = [] + | beta (Ge (Neg et)) = [] + | beta (Ge (Add (eu, ev))) = [] + | beta (Ge (Sub (ew, ex))) = [] + | beta (Ge (Mul (ey, ez))) = [] + | beta (Eq (C fo)) = [] + | beta (Eq (Bound fp)) = [] + | beta (Eq (Neg ft)) = [] + | beta (Eq (Add (fu, fv))) = [] + | beta (Eq (Sub (fw, fx))) = [] + | beta (Eq (Mul (fy, fz))) = [] + | beta (NEq (C go)) = [] + | beta (NEq (Bound gp)) = [] + | beta (NEq (Neg gt)) = [] + | beta (NEq (Add (gu, gv))) = [] + | beta (NEq (Sub (gw, gx))) = [] + | beta (NEq (Mul (gy, gz))) = [] + | beta (Dvd (aa, ab)) = [] + | beta (NDvd (ac, ad)) = [] + | beta (Not ae) = [] + | beta (Imp (aj, ak)) = [] + | beta (Iff (al, am)) = [] + | beta (E an) = [] + | beta (A ao) = [] + | beta (Closed ap) = [] + | beta (NClosed aq) = [] + | beta (Lt (Cn (cm, c, e))) = (if cm = (0 : IntInf.int) then [] else []) + | beta (Le (Cn (dm, c, e))) = (if dm = (0 : IntInf.int) then [] else []) + | beta (Gt (Cn (em, c, e))) = (if em = (0 : IntInf.int) then [Neg e] else []) + | beta (Ge (Cn (fm, c, e))) = + (if fm = (0 : IntInf.int) then [Sub (C (~1 : IntInf.int), e)] else []) + | beta (Eq (Cn (gm, c, e))) = + (if gm = (0 : IntInf.int) then [Sub (C (~1 : IntInf.int), e)] else []) + | beta (NEq (Cn (hm, c, e))) = + (if hm = (0 : IntInf.int) then [Neg e] else []); + +fun gcd_int k l = + abs_int + (if l = (0 : IntInf.int) then k + else gcd_int l (mod_int (abs_int k) (abs_int l))); + +fun lcm_int a b = div_inta (abs_int a * abs_int b) (gcd_int a b); + +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) + | zeta (Lt (Bound bp)) = (1 : IntInf.int) + | zeta (Lt (Neg bt)) = (1 : IntInf.int) + | zeta (Lt (Add (bu, bv))) = (1 : IntInf.int) + | zeta (Lt (Sub (bw, bx))) = (1 : IntInf.int) + | zeta (Lt (Mul (by, bz))) = (1 : IntInf.int) + | zeta (Le (C co)) = (1 : IntInf.int) + | zeta (Le (Bound cp)) = (1 : IntInf.int) + | zeta (Le (Neg ct)) = (1 : IntInf.int) + | zeta (Le (Add (cu, cv))) = (1 : IntInf.int) + | zeta (Le (Sub (cw, cx))) = (1 : IntInf.int) + | zeta (Le (Mul (cy, cz))) = (1 : IntInf.int) + | zeta (Gt (C doa)) = (1 : IntInf.int) + | zeta (Gt (Bound dp)) = (1 : IntInf.int) + | zeta (Gt (Neg dt)) = (1 : IntInf.int) + | zeta (Gt (Add (du, dv))) = (1 : IntInf.int) + | zeta (Gt (Sub (dw, dx))) = (1 : IntInf.int) + | zeta (Gt (Mul (dy, dz))) = (1 : IntInf.int) + | zeta (Ge (C eo)) = (1 : IntInf.int) + | zeta (Ge (Bound ep)) = (1 : IntInf.int) + | zeta (Ge (Neg et)) = (1 : IntInf.int) + | zeta (Ge (Add (eu, ev))) = (1 : IntInf.int) + | zeta (Ge (Sub (ew, ex))) = (1 : IntInf.int) + | zeta (Ge (Mul (ey, ez))) = (1 : IntInf.int) + | zeta (Eq (C fo)) = (1 : IntInf.int) + | zeta (Eq (Bound fp)) = (1 : IntInf.int) + | zeta (Eq (Neg ft)) = (1 : IntInf.int) + | zeta (Eq (Add (fu, fv))) = (1 : IntInf.int) + | zeta (Eq (Sub (fw, fx))) = (1 : IntInf.int) + | zeta (Eq (Mul (fy, fz))) = (1 : IntInf.int) + | zeta (NEq (C go)) = (1 : IntInf.int) + | zeta (NEq (Bound gp)) = (1 : IntInf.int) + | zeta (NEq (Neg gt)) = (1 : IntInf.int) + | zeta (NEq (Add (gu, gv))) = (1 : IntInf.int) + | zeta (NEq (Sub (gw, gx))) = (1 : IntInf.int) + | zeta (NEq (Mul (gy, gz))) = (1 : IntInf.int) + | zeta (Dvd (aa, C ho)) = (1 : IntInf.int) + | zeta (Dvd (aa, Bound hp)) = (1 : IntInf.int) + | zeta (Dvd (aa, Neg ht)) = (1 : IntInf.int) + | zeta (Dvd (aa, Add (hu, hv))) = (1 : IntInf.int) + | zeta (Dvd (aa, Sub (hw, hx))) = (1 : IntInf.int) + | zeta (Dvd (aa, Mul (hy, hz))) = (1 : IntInf.int) + | zeta (NDvd (ac, C io)) = (1 : IntInf.int) + | zeta (NDvd (ac, Bound ip)) = (1 : IntInf.int) + | zeta (NDvd (ac, Neg it)) = (1 : IntInf.int) + | zeta (NDvd (ac, Add (iu, iv))) = (1 : IntInf.int) + | zeta (NDvd (ac, Sub (iw, ix))) = (1 : IntInf.int) + | zeta (NDvd (ac, Mul (iy, iz))) = (1 : IntInf.int) + | zeta (Not ae) = (1 : IntInf.int) + | zeta (Imp (aj, ak)) = (1 : IntInf.int) + | zeta (Iff (al, am)) = (1 : IntInf.int) + | zeta (E an) = (1 : IntInf.int) + | zeta (A ao) = (1 : IntInf.int) + | zeta (Closed ap) = (1 : IntInf.int) + | zeta (NClosed aq) = (1 : IntInf.int) + | zeta (Lt (Cn (cm, c, e))) = + (if cm = (0 : IntInf.int) then c else (1 : IntInf.int)) + | zeta (Le (Cn (dm, c, e))) = + (if dm = (0 : IntInf.int) then c else (1 : IntInf.int)) + | zeta (Gt (Cn (em, c, e))) = + (if em = (0 : IntInf.int) then c else (1 : IntInf.int)) + | zeta (Ge (Cn (fm, c, e))) = + (if fm = (0 : IntInf.int) then c else (1 : IntInf.int)) + | zeta (Eq (Cn (gm, c, e))) = + (if gm = (0 : IntInf.int) then c else (1 : IntInf.int)) + | zeta (NEq (Cn (hm, c, e))) = + (if hm = (0 : IntInf.int) then c else (1 : IntInf.int)) + | zeta (Dvd (i, Cn (im, c, e))) = + (if im = (0 : IntInf.int) then c else (1 : IntInf.int)) + | zeta (NDvd (i, Cn (jm, c, e))) = + (if jm = (0 : IntInf.int) then c else (1 : IntInf.int)); + +fun zsplit0 (C c) = ((0 : IntInf.int), C c) + | zsplit0 (Bound n) = + (if n = (0 : IntInf.int) then ((1 : IntInf.int), C (0 : IntInf.int)) + else ((0 : IntInf.int), Bound n)) + | zsplit0 (Cn (n, i, a)) = + let + val (ia, aa) = zsplit0 a; + in + (if n = (0 : IntInf.int) then (i + ia, aa) else (ia, Cn (n, i, aa))) + end + | zsplit0 (Neg a) = let + val (i, aa) = zsplit0 a; + in + (~ i, Neg aa) + end + | zsplit0 (Add (a, b)) = + let + val (ia, aa) = zsplit0 a; + val (ib, ba) = zsplit0 b; + in + (ia + ib, Add (aa, ba)) + end + | zsplit0 (Sub (a, b)) = + let + val (ia, aa) = zsplit0 a; + val (ib, ba) = zsplit0 b; + in + (ia - ib, Sub (aa, ba)) + end + | zsplit0 (Mul (i, a)) = + let + val (ia, aa) = zsplit0 a; + in + (i * ia, Mul (i, aa)) + end; + +fun zlfm (And (p, q)) = And (zlfm p, zlfm q) + | zlfm (Or (p, q)) = Or (zlfm p, zlfm q) + | zlfm (Imp (p, q)) = Or (zlfm (Not p), zlfm q) + | zlfm (Iff (p, q)) = + Or (And (zlfm p, zlfm q), And (zlfm (Not p), zlfm (Not q))) + | zlfm (Lt a) = + let + val (c, r) = zsplit0 a; + in + (if c = (0 : IntInf.int) then Lt r + else (if (0 : IntInf.int) < c then Lt (Cn ((0 : IntInf.int), c, r)) + else Gt (Cn ((0 : IntInf.int), ~ c, Neg r)))) + end + | zlfm (Le a) = + let + val (c, r) = zsplit0 a; + in + (if c = (0 : IntInf.int) then Le r + else (if (0 : IntInf.int) < c then Le (Cn ((0 : IntInf.int), c, r)) + else Ge (Cn ((0 : IntInf.int), ~ c, Neg r)))) + end + | zlfm (Gt a) = + let + val (c, r) = zsplit0 a; + in + (if c = (0 : IntInf.int) then Gt r + else (if (0 : IntInf.int) < c then Gt (Cn ((0 : IntInf.int), c, r)) + else Lt (Cn ((0 : IntInf.int), ~ c, Neg r)))) + end + | zlfm (Ge a) = + let + val (c, r) = zsplit0 a; + in + (if c = (0 : IntInf.int) then Ge r + else (if (0 : IntInf.int) < c then Ge (Cn ((0 : IntInf.int), c, r)) + else Le (Cn ((0 : IntInf.int), ~ c, Neg r)))) + end + | zlfm (Eq a) = + let + val (c, r) = zsplit0 a; + in + (if c = (0 : IntInf.int) then Eq r + else (if (0 : IntInf.int) < c then Eq (Cn ((0 : IntInf.int), c, r)) + else Eq (Cn ((0 : IntInf.int), ~ c, Neg r)))) + end + | zlfm (NEq a) = + let + val (c, r) = zsplit0 a; + in + (if c = (0 : IntInf.int) then NEq r + else (if (0 : IntInf.int) < c then NEq (Cn ((0 : IntInf.int), c, r)) + else NEq (Cn ((0 : IntInf.int), ~ c, Neg r)))) + end + | zlfm (Dvd (i, a)) = + (if i = (0 : IntInf.int) then zlfm (Eq a) + else let + val (c, r) = zsplit0 a; + in + (if c = (0 : IntInf.int) then Dvd (abs_int i, r) + else (if (0 : IntInf.int) < c + then Dvd (abs_int i, Cn ((0 : IntInf.int), c, r)) + else Dvd (abs_int i, Cn ((0 : IntInf.int), ~ c, Neg r)))) + end) + | zlfm (NDvd (i, a)) = + (if i = (0 : IntInf.int) then zlfm (NEq a) + else let + val (c, r) = zsplit0 a; + in + (if c = (0 : IntInf.int) then NDvd (abs_int i, r) + else (if (0 : IntInf.int) < c + then NDvd (abs_int i, Cn ((0 : IntInf.int), c, r)) + else NDvd (abs_int i, Cn ((0 : IntInf.int), ~ c, Neg r)))) + end) + | zlfm (Not (And (p, q))) = Or (zlfm (Not p), zlfm (Not q)) + | zlfm (Not (Or (p, q))) = And (zlfm (Not p), zlfm (Not q)) + | zlfm (Not (Imp (p, q))) = And (zlfm p, zlfm (Not q)) + | zlfm (Not (Iff (p, q))) = + Or (And (zlfm p, zlfm (Not q)), And (zlfm (Not p), zlfm q)) + | zlfm (Not (Not p)) = zlfm p + | zlfm (Not T) = F + | zlfm (Not F) = T + | zlfm (Not (Lt a)) = zlfm (Ge a) + | zlfm (Not (Le a)) = zlfm (Gt a) + | zlfm (Not (Gt a)) = zlfm (Le a) + | zlfm (Not (Ge a)) = zlfm (Lt a) + | zlfm (Not (Eq a)) = zlfm (NEq a) + | zlfm (Not (NEq a)) = zlfm (Eq a) + | zlfm (Not (Dvd (i, a))) = zlfm (NDvd (i, a)) + | zlfm (Not (NDvd (i, a))) = zlfm (Dvd (i, a)) + | zlfm (Not (Closed p)) = NClosed p + | zlfm (Not (NClosed p)) = Closed p + | zlfm T = T + | zlfm F = F + | zlfm (Not (E ci)) = Not (E ci) + | zlfm (Not (A cj)) = Not (A cj) + | zlfm (E ao) = E ao + | zlfm (A ap) = A ap + | zlfm (Closed aq) = Closed aq + | zlfm (NClosed ar) = NClosed ar; + +fun alpha (And (p, q)) = alpha p @ alpha q + | alpha (Or (p, q)) = alpha p @ alpha q + | alpha T = [] + | alpha F = [] + | alpha (Lt (C bo)) = [] + | alpha (Lt (Bound bp)) = [] + | alpha (Lt (Neg bt)) = [] + | alpha (Lt (Add (bu, bv))) = [] + | alpha (Lt (Sub (bw, bx))) = [] + | alpha (Lt (Mul (by, bz))) = [] + | alpha (Le (C co)) = [] + | alpha (Le (Bound cp)) = [] + | alpha (Le (Neg ct)) = [] + | alpha (Le (Add (cu, cv))) = [] + | alpha (Le (Sub (cw, cx))) = [] + | alpha (Le (Mul (cy, cz))) = [] + | alpha (Gt (C doa)) = [] + | alpha (Gt (Bound dp)) = [] + | alpha (Gt (Neg dt)) = [] + | alpha (Gt (Add (du, dv))) = [] + | alpha (Gt (Sub (dw, dx))) = [] + | alpha (Gt (Mul (dy, dz))) = [] + | alpha (Ge (C eo)) = [] + | alpha (Ge (Bound ep)) = [] + | alpha (Ge (Neg et)) = [] + | alpha (Ge (Add (eu, ev))) = [] + | alpha (Ge (Sub (ew, ex))) = [] + | alpha (Ge (Mul (ey, ez))) = [] + | alpha (Eq (C fo)) = [] + | alpha (Eq (Bound fp)) = [] + | alpha (Eq (Neg ft)) = [] + | alpha (Eq (Add (fu, fv))) = [] + | alpha (Eq (Sub (fw, fx))) = [] + | alpha (Eq (Mul (fy, fz))) = [] + | alpha (NEq (C go)) = [] + | alpha (NEq (Bound gp)) = [] + | alpha (NEq (Neg gt)) = [] + | alpha (NEq (Add (gu, gv))) = [] + | alpha (NEq (Sub (gw, gx))) = [] + | alpha (NEq (Mul (gy, gz))) = [] + | alpha (Dvd (aa, ab)) = [] + | alpha (NDvd (ac, ad)) = [] + | alpha (Not ae) = [] + | alpha (Imp (aj, ak)) = [] + | alpha (Iff (al, am)) = [] + | alpha (E an) = [] + | alpha (A ao) = [] + | alpha (Closed ap) = [] + | alpha (NClosed aq) = [] + | alpha (Lt (Cn (cm, c, e))) = (if cm = (0 : IntInf.int) then [e] else []) + | alpha (Le (Cn (dm, c, e))) = + (if dm = (0 : IntInf.int) then [Add (C (~1 : IntInf.int), e)] else []) + | alpha (Gt (Cn (em, c, e))) = (if em = (0 : IntInf.int) then [] else []) + | alpha (Ge (Cn (fm, c, e))) = (if fm = (0 : IntInf.int) then [] else []) + | alpha (Eq (Cn (gm, c, e))) = + (if gm = (0 : IntInf.int) then [Add (C (~1 : IntInf.int), e)] else []) + | alpha (NEq (Cn (hm, c, e))) = (if hm = (0 : IntInf.int) then [e] else []); + +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) + | delta (Le v) = (1 : IntInf.int) + | delta (Gt w) = (1 : IntInf.int) + | delta (Ge x) = (1 : IntInf.int) + | delta (Eq y) = (1 : IntInf.int) + | delta (NEq z) = (1 : IntInf.int) + | delta (Dvd (aa, C bo)) = (1 : IntInf.int) + | delta (Dvd (aa, Bound bp)) = (1 : IntInf.int) + | delta (Dvd (aa, Neg bt)) = (1 : IntInf.int) + | delta (Dvd (aa, Add (bu, bv))) = (1 : IntInf.int) + | delta (Dvd (aa, Sub (bw, bx))) = (1 : IntInf.int) + | delta (Dvd (aa, Mul (by, bz))) = (1 : IntInf.int) + | delta (NDvd (ac, C co)) = (1 : IntInf.int) + | delta (NDvd (ac, Bound cp)) = (1 : IntInf.int) + | delta (NDvd (ac, Neg ct)) = (1 : IntInf.int) + | delta (NDvd (ac, Add (cu, cv))) = (1 : IntInf.int) + | delta (NDvd (ac, Sub (cw, cx))) = (1 : IntInf.int) + | delta (NDvd (ac, Mul (cy, cz))) = (1 : IntInf.int) + | delta (Not ae) = (1 : IntInf.int) + | delta (Imp (aj, ak)) = (1 : IntInf.int) + | delta (Iff (al, am)) = (1 : IntInf.int) + | delta (E an) = (1 : IntInf.int) + | delta (A ao) = (1 : IntInf.int) + | delta (Closed ap) = (1 : IntInf.int) + | delta (NClosed aq) = (1 : IntInf.int) + | delta (Dvd (i, Cn (cm, c, e))) = + (if cm = (0 : IntInf.int) then i else (1 : IntInf.int)) + | delta (NDvd (i, Cn (dm, c, e))) = + (if dm = (0 : IntInf.int) then i else (1 : IntInf.int)); + +fun member A_ [] y = false + | member A_ (x :: xs) y = eq A_ x y orelse member A_ xs y; + +fun remdups A_ [] = [] + | remdups A_ (x :: xs) = + (if member A_ xs x then remdups A_ xs else x :: remdups A_ xs); + +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 _ => 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 cm = (0 : IntInf.int) + then (fn k => + Lt (Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_inta k c, e)))) + else (fn _ => Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), c, e)))) + | a_beta (Le (Cn (dm, c, e))) = + (if dm = (0 : IntInf.int) + then (fn k => + Le (Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_inta k c, e)))) + else (fn _ => Le (Cn (suc (minus_nat dm (1 : IntInf.int)), c, e)))) + | a_beta (Gt (Cn (em, c, e))) = + (if em = (0 : IntInf.int) + then (fn k => + Gt (Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_inta k c, e)))) + else (fn _ => Gt (Cn (suc (minus_nat em (1 : IntInf.int)), c, e)))) + | a_beta (Ge (Cn (fm, c, e))) = + (if fm = (0 : IntInf.int) + then (fn k => + Ge (Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_inta k c, e)))) + else (fn _ => Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), c, e)))) + | a_beta (Eq (Cn (gm, c, e))) = + (if gm = (0 : IntInf.int) + then (fn k => + Eq (Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_inta k c, e)))) + else (fn _ => Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), c, e)))) + | a_beta (NEq (Cn (hm, c, e))) = + (if hm = (0 : IntInf.int) + then (fn k => + NEq (Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_inta 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 im = (0 : IntInf.int) + then (fn k => + Dvd (div_inta k c * i, + Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_inta 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 jm = (0 : IntInf.int) + then (fn k => + NDvd (div_inta k c * i, + Cn ((0 : IntInf.int), (1 : IntInf.int), + Mul (div_inta k c, e)))) + else (fn _ => NDvd (i, Cn (suc (minus_nat jm (1 : IntInf.int)), c, e)))); + +fun mirror (And (p, q)) = And (mirror p, mirror q) + | mirror (Or (p, q)) = Or (mirror p, mirror q) + | mirror T = T + | mirror F = F + | mirror (Lt (C bo)) = Lt (C bo) + | mirror (Lt (Bound bp)) = Lt (Bound bp) + | mirror (Lt (Neg bt)) = Lt (Neg bt) + | mirror (Lt (Add (bu, bv))) = Lt (Add (bu, bv)) + | mirror (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx)) + | mirror (Lt (Mul (by, bz))) = Lt (Mul (by, bz)) + | mirror (Le (C co)) = Le (C co) + | mirror (Le (Bound cp)) = Le (Bound cp) + | mirror (Le (Neg ct)) = Le (Neg ct) + | mirror (Le (Add (cu, cv))) = Le (Add (cu, cv)) + | mirror (Le (Sub (cw, cx))) = Le (Sub (cw, cx)) + | mirror (Le (Mul (cy, cz))) = Le (Mul (cy, cz)) + | mirror (Gt (C doa)) = Gt (C doa) + | mirror (Gt (Bound dp)) = Gt (Bound dp) + | mirror (Gt (Neg dt)) = Gt (Neg dt) + | mirror (Gt (Add (du, dv))) = Gt (Add (du, dv)) + | mirror (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx)) + | mirror (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz)) + | mirror (Ge (C eo)) = Ge (C eo) + | mirror (Ge (Bound ep)) = Ge (Bound ep) + | mirror (Ge (Neg et)) = Ge (Neg et) + | mirror (Ge (Add (eu, ev))) = Ge (Add (eu, ev)) + | mirror (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex)) + | mirror (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez)) + | mirror (Eq (C fo)) = Eq (C fo) + | mirror (Eq (Bound fp)) = Eq (Bound fp) + | mirror (Eq (Neg ft)) = Eq (Neg ft) + | mirror (Eq (Add (fu, fv))) = Eq (Add (fu, fv)) + | mirror (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx)) + | mirror (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz)) + | mirror (NEq (C go)) = NEq (C go) + | mirror (NEq (Bound gp)) = NEq (Bound gp) + | mirror (NEq (Neg gt)) = NEq (Neg gt) + | mirror (NEq (Add (gu, gv))) = NEq (Add (gu, gv)) + | mirror (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx)) + | mirror (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz)) + | mirror (Dvd (aa, C ho)) = Dvd (aa, C ho) + | mirror (Dvd (aa, Bound hp)) = Dvd (aa, Bound hp) + | mirror (Dvd (aa, Neg ht)) = Dvd (aa, Neg ht) + | mirror (Dvd (aa, Add (hu, hv))) = Dvd (aa, Add (hu, hv)) + | mirror (Dvd (aa, Sub (hw, hx))) = Dvd (aa, Sub (hw, hx)) + | mirror (Dvd (aa, Mul (hy, hz))) = Dvd (aa, Mul (hy, hz)) + | mirror (NDvd (ac, C io)) = NDvd (ac, C io) + | mirror (NDvd (ac, Bound ip)) = NDvd (ac, Bound ip) + | mirror (NDvd (ac, Neg it)) = NDvd (ac, Neg it) + | mirror (NDvd (ac, Add (iu, iv))) = NDvd (ac, Add (iu, iv)) + | mirror (NDvd (ac, Sub (iw, ix))) = NDvd (ac, Sub (iw, ix)) + | mirror (NDvd (ac, Mul (iy, iz))) = NDvd (ac, Mul (iy, iz)) + | mirror (Not ae) = Not ae + | mirror (Imp (aj, ak)) = Imp (aj, ak) + | mirror (Iff (al, am)) = Iff (al, am) + | mirror (E an) = E an + | mirror (A ao) = A ao + | mirror (Closed ap) = Closed ap + | mirror (NClosed aq) = NClosed aq + | mirror (Lt (Cn (cm, c, e))) = + (if cm = (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 dm = (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 em = (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 fm = (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 gm = (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 hm = (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 im = (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 jm = (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 : IntInf.int) + | size_list (a :: lista) = size_list lista + suc (0 : IntInf.int); + +val equal_num = {equal = equal_numa} : num equal; + +fun unita p = + let + val pa = zlfm p; + val l = zeta pa; + val q = + 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 equal_num (map simpnum (beta q)); + val a = remdups equal_num (map simpnum (alpha q)); + in + (if size_list b <= size_list a then (q, (b, d)) else (mirror q, (a, d))) + end; + +fun numsubst0 t (C c) = C c + | numsubst0 t (Bound n) = (if n = (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 v = (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 + | subst0 t (Lt a) = Lt (numsubst0 t a) + | subst0 t (Le a) = Le (numsubst0 t a) + | subst0 t (Gt a) = Gt (numsubst0 t a) + | subst0 t (Ge a) = Ge (numsubst0 t a) + | subst0 t (Eq a) = Eq (numsubst0 t a) + | subst0 t (NEq a) = NEq (numsubst0 t a) + | subst0 t (Dvd (i, a)) = Dvd (i, numsubst0 t a) + | subst0 t (NDvd (i, a)) = NDvd (i, numsubst0 t a) + | subst0 t (Not p) = Not (subst0 t p) + | subst0 t (And (p, q)) = And (subst0 t p, subst0 t q) + | subst0 t (Or (p, q)) = Or (subst0 t p, subst0 t q) + | subst0 t (Imp (p, q)) = Imp (subst0 t p, subst0 t q) + | subst0 t (Iff (p, q)) = Iff (subst0 t p, subst0 t q) + | subst0 t (Closed p) = Closed p + | subst0 t (NClosed p) = NClosed p; + +fun minusinf (And (p, q)) = And (minusinf p, minusinf q) + | minusinf (Or (p, q)) = Or (minusinf p, minusinf q) + | minusinf T = T + | minusinf F = F + | minusinf (Lt (C bo)) = Lt (C bo) + | minusinf (Lt (Bound bp)) = Lt (Bound bp) + | minusinf (Lt (Neg bt)) = Lt (Neg bt) + | minusinf (Lt (Add (bu, bv))) = Lt (Add (bu, bv)) + | minusinf (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx)) + | minusinf (Lt (Mul (by, bz))) = Lt (Mul (by, bz)) + | minusinf (Le (C co)) = Le (C co) + | minusinf (Le (Bound cp)) = Le (Bound cp) + | minusinf (Le (Neg ct)) = Le (Neg ct) + | minusinf (Le (Add (cu, cv))) = Le (Add (cu, cv)) + | minusinf (Le (Sub (cw, cx))) = Le (Sub (cw, cx)) + | minusinf (Le (Mul (cy, cz))) = Le (Mul (cy, cz)) + | minusinf (Gt (C doa)) = Gt (C doa) + | minusinf (Gt (Bound dp)) = Gt (Bound dp) + | minusinf (Gt (Neg dt)) = Gt (Neg dt) + | minusinf (Gt (Add (du, dv))) = Gt (Add (du, dv)) + | minusinf (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx)) + | minusinf (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz)) + | minusinf (Ge (C eo)) = Ge (C eo) + | minusinf (Ge (Bound ep)) = Ge (Bound ep) + | minusinf (Ge (Neg et)) = Ge (Neg et) + | minusinf (Ge (Add (eu, ev))) = Ge (Add (eu, ev)) + | minusinf (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex)) + | minusinf (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez)) + | minusinf (Eq (C fo)) = Eq (C fo) + | minusinf (Eq (Bound fp)) = Eq (Bound fp) + | minusinf (Eq (Neg ft)) = Eq (Neg ft) + | minusinf (Eq (Add (fu, fv))) = Eq (Add (fu, fv)) + | minusinf (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx)) + | minusinf (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz)) + | minusinf (NEq (C go)) = NEq (C go) + | minusinf (NEq (Bound gp)) = NEq (Bound gp) + | minusinf (NEq (Neg gt)) = NEq (Neg gt) + | minusinf (NEq (Add (gu, gv))) = NEq (Add (gu, gv)) + | minusinf (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx)) + | minusinf (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz)) + | minusinf (Dvd (aa, ab)) = Dvd (aa, ab) + | minusinf (NDvd (ac, ad)) = NDvd (ac, ad) + | minusinf (Not ae) = Not ae + | minusinf (Imp (aj, ak)) = Imp (aj, ak) + | minusinf (Iff (al, am)) = Iff (al, am) + | minusinf (E an) = E an + | minusinf (A ao) = A ao + | minusinf (Closed ap) = Closed ap + | minusinf (NClosed aq) = NClosed aq + | minusinf (Lt (Cn (cm, c, e))) = + (if cm = (0 : IntInf.int) then T + else Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), c, e))) + | minusinf (Le (Cn (dm, c, e))) = + (if dm = (0 : IntInf.int) then T + else Le (Cn (suc (minus_nat dm (1 : IntInf.int)), c, e))) + | minusinf (Gt (Cn (em, c, e))) = + (if em = (0 : IntInf.int) then F + else Gt (Cn (suc (minus_nat em (1 : IntInf.int)), c, e))) + | minusinf (Ge (Cn (fm, c, e))) = + (if fm = (0 : IntInf.int) then F + else Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), c, e))) + | minusinf (Eq (Cn (gm, c, e))) = + (if gm = (0 : IntInf.int) then F + else Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), c, e))) + | minusinf (NEq (Cn (hm, c, e))) = + (if hm = (0 : IntInf.int) then T + else NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e))); + +fun cooper p = + let + val (q, (b, d)) = unita p; + val js = uptoa (1 : IntInf.int) d; + val mq = simpfm (minusinf q); + val md = evaldjf (fn j => simpfm (subst0 (C j) mq)) js; + in + (if equal_fm md T then T + else let + val qd = + evaldjf (fn (ba, j) => simpfm (subst0 (Add (ba, C j)) q)) + (maps (fn ba => map (fn a => (ba, a)) js) b); + in + decr (disj md qd) + end) + end; fun pa p = qelim (prep p) cooper;