--- 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;