--- a/src/HOL/Tools/Qelim/cooper_procedure.ML Fri Feb 15 08:31:30 2013 +0100
+++ b/src/HOL/Tools/Qelim/cooper_procedure.ML Fri Feb 15 08:31:31 2013 +0100
@@ -1,57 +1,60 @@
(* Generated from Cooper.thy; DO NOT EDIT! *)
structure Cooper_Procedure : sig
+ val id : 'a -> 'a
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 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
+ datatype inta = Int_of_integer of int
+ datatype nat = Nat of int
+ datatype num = One | Bit0 of num | Bit1 of num
+ type 'a ord
+ val less_eq : 'a ord -> 'a -> 'a -> bool
+ val less : 'a ord -> 'a -> 'a -> bool
+ val ord_integer : int ord
+ val max : 'a ord -> 'a -> 'a -> 'a
+ val nat_of_integer : int -> nat
+ val integer_of_nat : nat -> int
+ val plus_nat : nat -> nat -> nat
+ val suc : nat -> nat
+ datatype numa = C of inta | Bound of nat | Cn of nat * inta * numa |
+ Neg of numa | Add of numa * numa | Sub of numa * numa | Mul of inta * numa
+ datatype fm = T | F | Lt of numa | Le of numa | Gt of numa | Ge of numa |
+ Eq of numa | NEq of numa | Dvd of inta * numa | NDvd of inta * numa |
+ 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 nat | NClosed of nat
val map : ('a -> 'b) -> 'a list -> 'b list
- val equal_numa : num -> num -> bool
+ val disjuncts : fm -> fm list
+ val foldr : ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b
+ val equal_nat : nat -> nat -> bool
+ val integer_of_int : inta -> int
+ val equal_inta : inta -> inta -> bool
+ val equal_numa : numa -> numa -> 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 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
- 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
- type 'a semigroup_mult
- val times_semigroup_mult : 'a semigroup_mult -> 'a times
+ val minus_nat : nat -> nat -> nat
+ val zero_nat : nat
+ val minusinf : fm -> fm
+ val numsubst0 : numa -> numa -> numa
+ val subst0 : numa -> fm -> fm
type 'a plus
val plus : 'a plus -> 'a -> 'a -> 'a
type 'a semigroup_add
val plus_semigroup_add : 'a semigroup_add -> 'a plus
+ type 'a cancel_semigroup_add
+ val semigroup_add_cancel_semigroup_add :
+ 'a cancel_semigroup_add -> 'a semigroup_add
type 'a ab_semigroup_add
val semigroup_add_ab_semigroup_add : 'a ab_semigroup_add -> 'a semigroup_add
- type 'a semiring
- val ab_semigroup_add_semiring : 'a semiring -> 'a ab_semigroup_add
- val semigroup_mult_semiring : 'a semiring -> 'a semigroup_mult
- type 'a mult_zero
- val times_mult_zero : 'a mult_zero -> 'a times
- val zero_mult_zero : 'a mult_zero -> 'a zero
+ type 'a cancel_ab_semigroup_add
+ val ab_semigroup_add_cancel_ab_semigroup_add :
+ 'a cancel_ab_semigroup_add -> 'a ab_semigroup_add
+ val cancel_semigroup_add_cancel_ab_semigroup_add :
+ 'a cancel_ab_semigroup_add -> 'a cancel_semigroup_add
+ type 'a zero
+ val zero : 'a zero -> 'a
type 'a monoid_add
val semigroup_add_monoid_add : 'a monoid_add -> 'a semigroup_add
val zero_monoid_add : 'a monoid_add -> 'a zero
@@ -59,25 +62,29 @@
val ab_semigroup_add_comm_monoid_add :
'a comm_monoid_add -> 'a ab_semigroup_add
val monoid_add_comm_monoid_add : 'a comm_monoid_add -> 'a monoid_add
+ type 'a cancel_comm_monoid_add
+ val cancel_ab_semigroup_add_cancel_comm_monoid_add :
+ 'a cancel_comm_monoid_add -> 'a cancel_ab_semigroup_add
+ val comm_monoid_add_cancel_comm_monoid_add :
+ 'a cancel_comm_monoid_add -> 'a comm_monoid_add
+ type 'a times
+ val times : 'a times -> 'a -> 'a -> 'a
+ type 'a mult_zero
+ val times_mult_zero : 'a mult_zero -> 'a times
+ val zero_mult_zero : 'a mult_zero -> 'a zero
+ type 'a semigroup_mult
+ val times_semigroup_mult : 'a semigroup_mult -> 'a times
+ type 'a semiring
+ val ab_semigroup_add_semiring : 'a semiring -> 'a ab_semigroup_add
+ val semigroup_mult_semiring : 'a semiring -> 'a semigroup_mult
type 'a semiring_0
val comm_monoid_add_semiring_0 : 'a semiring_0 -> 'a comm_monoid_add
val mult_zero_semiring_0 : 'a semiring_0 -> 'a mult_zero
val semiring_semiring_0 : 'a semiring_0 -> 'a semiring
- type 'a 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 semiring_0_cancel
+ val cancel_comm_monoid_add_semiring_0_cancel :
+ 'a semiring_0_cancel -> 'a cancel_comm_monoid_add
+ val semiring_0_semiring_0_cancel : 'a semiring_0_cancel -> 'a semiring_0
type 'a ab_semigroup_mult
val semigroup_mult_ab_semigroup_mult :
'a ab_semigroup_mult -> 'a semigroup_mult
@@ -87,42 +94,49 @@
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_semiring_0_cancel
+ val comm_semiring_0_comm_semiring_0_cancel :
+ 'a comm_semiring_0_cancel -> 'a comm_semiring_0
+ val semiring_0_cancel_comm_semiring_0_cancel :
+ 'a comm_semiring_0_cancel -> 'a semiring_0_cancel
+ type 'a 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 numeral
+ val one_numeral : 'a numeral -> 'a one
+ val semigroup_add_numeral : 'a numeral -> 'a semigroup_add
+ type 'a semiring_numeral
+ val monoid_mult_semiring_numeral : 'a semiring_numeral -> 'a monoid_mult
+ val numeral_semiring_numeral : 'a semiring_numeral -> 'a numeral
+ val semiring_semiring_numeral : 'a semiring_numeral -> 'a semiring
+ 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 semiring_numeral_semiring_1 : 'a semiring_1 -> 'a semiring_numeral
+ 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 semiring_1_cancel
+ val semiring_0_cancel_semiring_1_cancel :
+ 'a semiring_1_cancel -> 'a semiring_0_cancel
+ val semiring_1_semiring_1_cancel : 'a semiring_1_cancel -> 'a semiring_1
type 'a 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 dvd
+ val times_dvd : 'a dvd -> 'a times
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
- type 'a cancel_ab_semigroup_add
- val ab_semigroup_add_cancel_ab_semigroup_add :
- 'a cancel_ab_semigroup_add -> 'a ab_semigroup_add
- val cancel_semigroup_add_cancel_ab_semigroup_add :
- 'a cancel_ab_semigroup_add -> 'a cancel_semigroup_add
- type 'a cancel_comm_monoid_add
- val cancel_ab_semigroup_add_cancel_comm_monoid_add :
- 'a cancel_comm_monoid_add -> 'a cancel_ab_semigroup_add
- val comm_monoid_add_cancel_comm_monoid_add :
- 'a cancel_comm_monoid_add -> 'a comm_monoid_add
- type 'a semiring_0_cancel
- val cancel_comm_monoid_add_semiring_0_cancel :
- 'a semiring_0_cancel -> 'a cancel_comm_monoid_add
- val semiring_0_semiring_0_cancel : 'a semiring_0_cancel -> 'a semiring_0
- type 'a semiring_1_cancel
- val semiring_0_cancel_semiring_1_cancel :
- 'a semiring_1_cancel -> 'a semiring_0_cancel
- val semiring_1_semiring_1_cancel : 'a semiring_1_cancel -> 'a semiring_1
- type 'a comm_semiring_0_cancel
- val comm_semiring_0_comm_semiring_0_cancel :
- 'a comm_semiring_0_cancel -> 'a comm_semiring_0
- val semiring_0_cancel_comm_semiring_0_cancel :
- 'a comm_semiring_0_cancel -> 'a semiring_0_cancel
type 'a comm_semiring_1_cancel
val comm_semiring_0_cancel_comm_semiring_1_cancel :
'a comm_semiring_1_cancel -> 'a comm_semiring_0_cancel
@@ -130,107 +144,182 @@
'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 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
+ 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 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 numsub : num -> num -> num
- val simpnum : num -> num
- 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 plus_inta : inta -> inta -> inta
+ val plus_int : inta plus
+ val semigroup_add_int : inta semigroup_add
+ val cancel_semigroup_add_int : inta cancel_semigroup_add
+ val ab_semigroup_add_int : inta ab_semigroup_add
+ val cancel_ab_semigroup_add_int : inta cancel_ab_semigroup_add
+ val zero_inta : inta
+ val zero_int : inta zero
+ val monoid_add_int : inta monoid_add
+ val comm_monoid_add_int : inta comm_monoid_add
+ val cancel_comm_monoid_add_int : inta cancel_comm_monoid_add
+ val times_inta : inta -> inta -> inta
+ val times_int : inta times
+ val mult_zero_int : inta mult_zero
+ val semigroup_mult_int : inta semigroup_mult
+ val semiring_int : inta semiring
+ val semiring_0_int : inta semiring_0
+ val semiring_0_cancel_int : inta semiring_0_cancel
+ val ab_semigroup_mult_int : inta ab_semigroup_mult
+ val comm_semiring_int : inta comm_semiring
+ val comm_semiring_0_int : inta comm_semiring_0
+ val comm_semiring_0_cancel_int : inta comm_semiring_0_cancel
+ val one_inta : inta
+ val one_int : inta one
+ val power_int : inta power
+ val monoid_mult_int : inta monoid_mult
+ val numeral_int : inta numeral
+ val semiring_numeral_int : inta semiring_numeral
+ val zero_neq_one_int : inta zero_neq_one
+ val semiring_1_int : inta semiring_1
+ val semiring_1_cancel_int : inta semiring_1_cancel
+ val comm_monoid_mult_int : inta comm_monoid_mult
+ val dvd_int : inta dvd
+ val comm_semiring_1_int : inta comm_semiring_1
+ val comm_semiring_1_cancel_int : inta comm_semiring_1_cancel
+ val no_zero_divisors_int : inta no_zero_divisors
+ val sgn_integer : int -> int
+ val abs_integer : int -> int
val apsnd : ('a -> 'b) -> 'c * 'a -> 'c * 'b
- val divmod_int : int -> int -> int * int
- val div_inta : int -> int -> int
+ val divmod_integer : int -> 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 mod_integer : int -> int -> int
+ val mod_int : inta -> inta -> inta
+ val fst : 'a * 'b -> 'a
+ val div_integer : int -> int -> int
+ val div_inta : inta -> inta -> inta
+ val div_int : inta diva
+ val semiring_div_int : inta semiring_div
+ val less_eq_int : inta -> inta -> bool
+ val uminus_int : inta -> inta
+ val nummul : inta -> numa -> numa
+ val numneg : numa -> numa
+ val less_eq_nat : nat -> nat -> bool
+ val numadd : numa * numa -> numa
+ val numsub : numa -> numa -> numa
+ val simpnum : numa -> numa
+ val less_int : inta -> inta -> bool
+ val equal_int : inta equal
+ val abs_int : inta -> inta
+ val nota : fm -> fm
+ val impa : fm -> fm -> fm
+ val iffa : fm -> fm -> fm
+ val disj : fm -> fm -> fm
+ val conj : fm -> fm -> fm
+ val dvd : 'a semiring_div * 'a equal -> 'a -> 'a -> bool
val simpfm : fm -> fm
- 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 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 alpha : fm -> num list
- val delta : fm -> int
+ val equal_num : numa equal
+ val gen_length : nat -> 'a list -> nat
+ val size_list : 'a list -> nat
+ val mirror : fm -> fm
+ val a_beta : fm -> inta -> fm
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 gcd_int : inta -> inta -> inta
+ val lcm_int : inta -> inta -> inta
+ val delta : fm -> inta
+ val alpha : fm -> numa list
+ val minus_int : inta -> inta -> inta
+ val zsplit0 : numa -> inta * numa
+ val zlfm : fm -> fm
+ val zeta : fm -> inta
+ val beta : fm -> numa list
+ val unita : fm -> fm * (numa list * inta)
+ val decrnum : numa -> numa
+ val decr : fm -> fm
+ val uptoa : inta -> inta -> inta list
+ val maps : ('a -> 'b list) -> 'a list -> 'b list
val cooper : fm -> fm
+ val qelim : fm -> (fm -> fm) -> fm
+ val prep : fm -> fm
val pa : fm -> fm
end = struct
+fun id x = (fn xa => xa) x;
+
type 'a equal = {equal : 'a -> 'a -> bool};
val equal = #equal : 'a equal -> 'a -> 'a -> bool;
fun eq A_ a b = equal A_ a b;
-fun suc n = n + (1 : IntInf.int);
+datatype inta = Int_of_integer of int;
+
+datatype nat = Nat of int;
+
+datatype num = One | Bit0 of num | Bit1 of 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;
+type 'a ord = {less_eq : 'a -> 'a -> bool, less : 'a -> 'a -> bool};
+val less_eq = #less_eq : 'a ord -> 'a -> 'a -> bool;
+val less = #less : 'a ord -> 'a -> 'a -> bool;
+
+val ord_integer =
+ {less_eq = (fn a => fn b => a <= b), less = (fn a => fn b => a < b)} :
+ int ord;
+
+fun max A_ a b = (if less_eq A_ a b then b else a);
-datatype fm = T | F | Lt of num | Le of num | Gt of num | Ge of num | 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;
+fun nat_of_integer k = Nat (max ord_integer 0 k);
+
+fun integer_of_nat (Nat x) = x;
+
+fun plus_nat m n = Nat (integer_of_nat m + integer_of_nat n);
+
+fun suc n = plus_nat n (nat_of_integer (1 : IntInf.int));
+
+datatype numa = C of inta | Bound of nat | Cn of nat * inta * numa | Neg of numa
+ | Add of numa * numa | Sub of numa * numa | Mul of inta * numa;
+
+datatype fm = T | F | Lt of numa | Le of numa | Gt of numa | Ge of numa |
+ Eq of numa | NEq of numa | Dvd of inta * numa | NDvd of inta * numa |
+ 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 nat | NClosed of nat;
fun map f [] = []
| map f (x :: xs) = f x :: map f xs;
+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 foldr f [] = id
+ | foldr f (x :: xs) = f x o foldr f xs;
+
+fun equal_nat m n = integer_of_nat m = integer_of_nat n;
+
+fun integer_of_int (Int_of_integer k) = k;
+
+fun equal_inta k l = integer_of_int k = integer_of_int l;
+
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
@@ -274,16 +363,17 @@
| 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_inta 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;
+ equal_nat nata nat andalso
+ (equal_inta intaa inta andalso equal_numa numa num)
+ | equal_numa (Bound nata) (Bound nat) = equal_nat nata nat
+ | equal_numa (C intaa) (C inta) = equal_inta intaa inta;
fun equal_fm (NClosed nata) (Closed nat) = false
| equal_fm (Closed nata) (NClosed nat) = false
@@ -627,8 +717,8 @@
| 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 (NClosed nata) (NClosed nat) = equal_nat nata nat
+ | equal_fm (Closed nata) (Closed nat) = equal_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)) =
@@ -641,9 +731,9 @@
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_inta intaa inta andalso equal_numa numa num
| equal_fm (Dvd (intaa, numa)) (Dvd (inta, num)) =
- intaa = inta andalso equal_numa numa num
+ equal_inta 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
@@ -666,32 +756,1457 @@
| E _ => Or (f p, q) | A _ => Or (f p, q)
| Closed _ => Or (f p, q) | NClosed _ => Or (f p, q))));
-fun foldr f [] a = a
- | foldr f (x :: xs) a = f x (foldr f xs a);
-
fun evaldjf f ps = foldr (djf f) ps F;
-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 dj f p = evaldjf f (disjuncts p);
+
+fun minus_nat m n =
+ Nat (max ord_integer 0 (integer_of_nat m - integer_of_nat n));
+
+val zero_nat : nat = Nat 0;
+
+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 equal_nat cm zero_nat then T
+ else Lt (Cn (suc (minus_nat cm (nat_of_integer (1 : IntInf.int))), c, e)))
+ | minusinf (Le (Cn (dm, c, e))) =
+ (if equal_nat dm zero_nat then T
+ else Le (Cn (suc (minus_nat dm (nat_of_integer (1 : IntInf.int))), c, e)))
+ | minusinf (Gt (Cn (em, c, e))) =
+ (if equal_nat em zero_nat then F
+ else Gt (Cn (suc (minus_nat em (nat_of_integer (1 : IntInf.int))), c, e)))
+ | minusinf (Ge (Cn (fm, c, e))) =
+ (if equal_nat fm zero_nat then F
+ else Ge (Cn (suc (minus_nat fm (nat_of_integer (1 : IntInf.int))), c, e)))
+ | minusinf (Eq (Cn (gm, c, e))) =
+ (if equal_nat gm zero_nat then F
+ else Eq (Cn (suc (minus_nat gm (nat_of_integer (1 : IntInf.int))), c, e)))
+ | minusinf (NEq (Cn (hm, c, e))) =
+ (if equal_nat hm zero_nat then T
+ else NEq (Cn (suc (minus_nat hm (nat_of_integer (1 : IntInf.int))), c,
+ e)));
+
+fun numsubst0 t (C c) = C c
+ | numsubst0 t (Bound n) = (if equal_nat n zero_nat 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 equal_nat v zero_nat then Add (Mul (i, t), numsubst0 t a)
+ else Cn (suc (minus_nat v (nat_of_integer (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;
+
+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 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 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 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 zero = {zero : 'a};
+val zero = #zero : 'a zero -> 'a;
+
+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 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 times = {times : 'a -> 'a -> 'a};
+val times = #times : 'a times -> 'a -> 'a -> 'a;
+
+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 semigroup_mult = {times_semigroup_mult : 'a times};
+val times_semigroup_mult = #times_semigroup_mult :
+ 'a semigroup_mult -> 'a times;
+
+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 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 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 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_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 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 numeral =
+ {one_numeral : 'a one, semigroup_add_numeral : 'a semigroup_add};
+val one_numeral = #one_numeral : 'a numeral -> 'a one;
+val semigroup_add_numeral = #semigroup_add_numeral :
+ 'a numeral -> 'a semigroup_add;
+
+type 'a semiring_numeral =
+ {monoid_mult_semiring_numeral : 'a monoid_mult,
+ numeral_semiring_numeral : 'a numeral,
+ semiring_semiring_numeral : 'a semiring};
+val monoid_mult_semiring_numeral = #monoid_mult_semiring_numeral :
+ 'a semiring_numeral -> 'a monoid_mult;
+val numeral_semiring_numeral = #numeral_semiring_numeral :
+ 'a semiring_numeral -> 'a numeral;
+val semiring_semiring_numeral = #semiring_semiring_numeral :
+ 'a semiring_numeral -> 'a semiring;
+
+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 =
+ {semiring_numeral_semiring_1 : 'a semiring_numeral,
+ semiring_0_semiring_1 : 'a semiring_0,
+ zero_neq_one_semiring_1 : 'a zero_neq_one};
+val semiring_numeral_semiring_1 = #semiring_numeral_semiring_1 :
+ 'a semiring_1 -> 'a semiring_numeral;
+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 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_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 dvd = {times_dvd : 'a times};
+val times_dvd = #times_dvd : 'a dvd -> 'a times;
+
+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_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 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 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;
+
+fun plus_inta k l = Int_of_integer (integer_of_int k + integer_of_int l);
+
+val plus_int = {plus = plus_inta} : inta plus;
+
+val semigroup_add_int = {plus_semigroup_add = plus_int} : inta semigroup_add;
+
+val cancel_semigroup_add_int =
+ {semigroup_add_cancel_semigroup_add = semigroup_add_int} :
+ inta cancel_semigroup_add;
+
+val ab_semigroup_add_int = {semigroup_add_ab_semigroup_add = semigroup_add_int}
+ : inta ab_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}
+ : inta cancel_ab_semigroup_add;
+
+val zero_inta : inta = Int_of_integer 0;
+
+val zero_int = {zero = zero_inta} : inta zero;
+
+val monoid_add_int =
+ {semigroup_add_monoid_add = semigroup_add_int, zero_monoid_add = zero_int} :
+ inta 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}
+ : inta comm_monoid_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}
+ : inta cancel_comm_monoid_add;
+
+fun times_inta k l = Int_of_integer (integer_of_int k * integer_of_int l);
+
+val times_int = {times = times_inta} : inta times;
+
+val mult_zero_int = {times_mult_zero = times_int, zero_mult_zero = zero_int} :
+ inta mult_zero;
+
+val semigroup_mult_int = {times_semigroup_mult = times_int} :
+ inta semigroup_mult;
+
+val semiring_int =
+ {ab_semigroup_add_semiring = ab_semigroup_add_int,
+ semigroup_mult_semiring = semigroup_mult_int}
+ : inta semiring;
+
+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}
+ : inta semiring_0;
+
+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}
+ : inta semiring_0_cancel;
+
+val ab_semigroup_mult_int =
+ {semigroup_mult_ab_semigroup_mult = semigroup_mult_int} :
+ inta ab_semigroup_mult;
+
+val comm_semiring_int =
+ {ab_semigroup_mult_comm_semiring = ab_semigroup_mult_int,
+ semiring_comm_semiring = semiring_int}
+ : inta comm_semiring;
+
+val comm_semiring_0_int =
+ {comm_semiring_comm_semiring_0 = comm_semiring_int,
+ semiring_0_comm_semiring_0 = semiring_0_int}
+ : inta comm_semiring_0;
+
+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}
+ : inta comm_semiring_0_cancel;
+
+val one_inta : inta = Int_of_integer (1 : IntInf.int);
+
+val one_int = {one = one_inta} : inta one;
+
+val power_int = {one_power = one_int, times_power = times_int} : inta power;
+
+val monoid_mult_int =
+ {semigroup_mult_monoid_mult = semigroup_mult_int,
+ power_monoid_mult = power_int}
+ : inta monoid_mult;
+
+val numeral_int =
+ {one_numeral = one_int, semigroup_add_numeral = semigroup_add_int} :
+ inta numeral;
+
+val semiring_numeral_int =
+ {monoid_mult_semiring_numeral = monoid_mult_int,
+ numeral_semiring_numeral = numeral_int,
+ semiring_semiring_numeral = semiring_int}
+ : inta semiring_numeral;
+
+val zero_neq_one_int =
+ {one_zero_neq_one = one_int, zero_zero_neq_one = zero_int} :
+ inta zero_neq_one;
+
+val semiring_1_int =
+ {semiring_numeral_semiring_1 = semiring_numeral_int,
+ semiring_0_semiring_1 = semiring_0_int,
+ zero_neq_one_semiring_1 = zero_neq_one_int}
+ : inta semiring_1;
+
+val semiring_1_cancel_int =
+ {semiring_0_cancel_semiring_1_cancel = semiring_0_cancel_int,
+ semiring_1_semiring_1_cancel = semiring_1_int}
+ : inta semiring_1_cancel;
+
+val comm_monoid_mult_int =
+ {ab_semigroup_mult_comm_monoid_mult = ab_semigroup_mult_int,
+ monoid_mult_comm_monoid_mult = monoid_mult_int}
+ : inta comm_monoid_mult;
+
+val dvd_int = {times_dvd = times_int} : inta dvd;
+
+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}
+ : inta comm_semiring_1;
+
+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}
+ : inta comm_semiring_1_cancel;
+
+val no_zero_divisors_int =
+ {times_no_zero_divisors = times_int, zero_no_zero_divisors = zero_int} :
+ inta no_zero_divisors;
+
+fun sgn_integer k =
+ (if k = 0 then 0
+ else (if k < 0 then ~ (1 : IntInf.int) else (1 : IntInf.int)));
+
+fun abs_integer k = (if k < 0 then ~ k else k);
+
+fun apsnd f (x, y) = (x, f y);
+
+fun divmod_integer k l =
+ (if k = 0 then (0, 0)
+ else (if l = 0 then (0, k)
+ else (apsnd o (fn a => fn b => a * b) o sgn_integer) l
+ (if sgn_integer k = sgn_integer 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 then (~ r, 0)
+ else (~ r - (1 : IntInf.int), abs_integer l - s))
+ end)));
+
+fun snd (a, b) = b;
+
+fun mod_integer k l = snd (divmod_integer k l);
+
+fun mod_int k l =
+ Int_of_integer (mod_integer (integer_of_int k) (integer_of_int l));
+
+fun fst (a, b) = a;
+
+fun div_integer k l = fst (divmod_integer k l);
+
+fun div_inta k l =
+ Int_of_integer (div_integer (integer_of_int k) (integer_of_int l));
+
+val div_int = {dvd_div = dvd_int, diva = div_inta, moda = mod_int} : inta diva;
+
+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}
+ : inta semiring_div;
+
+fun less_eq_int k l = integer_of_int k <= integer_of_int l;
+
+fun uminus_int k = Int_of_integer (~ (integer_of_int k));
+
+fun nummul i (C j) = C (times_inta i j)
+ | nummul i (Cn (n, c, t)) = Cn (n, times_inta 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 (uminus_int (Int_of_integer (1 : IntInf.int))) t;
+
+fun less_eq_nat m n = integer_of_nat m <= integer_of_nat n;
+
+fun numadd (Cn (n1, c1, r1), Cn (n2, c2, r2)) =
+ (if equal_nat n1 n2
+ then let
+ val c = plus_inta c1 c2;
+ in
+ (if equal_inta c zero_inta then numadd (r1, r2)
+ else Cn (n1, c, numadd (r1, r2)))
+ end
+ else (if less_eq_nat 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 (plus_inta 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 numsub s t = (if equal_numa s t then C zero_inta else numadd (s, numneg t));
+
+fun simpnum (C j) = C j
+ | simpnum (Bound n) = Cn (n, Int_of_integer (1 : IntInf.int), C zero_inta)
+ | 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 equal_inta i zero_inta then C zero_inta else nummul i (simpnum t))
+ | simpnum (Cn (v, va, vb)) = Cn (v, va, vb);
+
+fun less_int k l = integer_of_int k < integer_of_int l;
+
+val equal_int = {equal = equal_inta} : inta equal;
+
+fun abs_int i = (if less_int i zero_inta then uminus_int i else i);
+
+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 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))));
+
+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 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 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 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 dj f p = evaldjf f (disjuncts p);
+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 less_int v zero_inta 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 less_eq_int v zero_inta 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 less_int zero_inta 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 less_eq_int zero_inta 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 equal_inta v zero_inta 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 (equal_inta v zero_inta) 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 equal_inta i zero_inta then simpfm (Eq a)
+ else (if equal_inta (abs_int i) (Int_of_integer (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 equal_inta i zero_inta then simpfm (NEq a)
+ else (if equal_inta (abs_int i) (Int_of_integer (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;
+
+val equal_num = {equal = equal_numa} : numa equal;
+
+fun gen_length n (x :: xs) = gen_length (suc n) xs
+ | gen_length n [] = n;
+
+fun size_list x = gen_length zero_nat x;
+
+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 equal_nat cm zero_nat then Gt (Cn (zero_nat, c, Neg e))
+ else Lt (Cn (suc (minus_nat cm (nat_of_integer (1 : IntInf.int))), c, e)))
+ | mirror (Le (Cn (dm, c, e))) =
+ (if equal_nat dm zero_nat then Ge (Cn (zero_nat, c, Neg e))
+ else Le (Cn (suc (minus_nat dm (nat_of_integer (1 : IntInf.int))), c, e)))
+ | mirror (Gt (Cn (em, c, e))) =
+ (if equal_nat em zero_nat then Lt (Cn (zero_nat, c, Neg e))
+ else Gt (Cn (suc (minus_nat em (nat_of_integer (1 : IntInf.int))), c, e)))
+ | mirror (Ge (Cn (fm, c, e))) =
+ (if equal_nat fm zero_nat then Le (Cn (zero_nat, c, Neg e))
+ else Ge (Cn (suc (minus_nat fm (nat_of_integer (1 : IntInf.int))), c, e)))
+ | mirror (Eq (Cn (gm, c, e))) =
+ (if equal_nat gm zero_nat then Eq (Cn (zero_nat, c, Neg e))
+ else Eq (Cn (suc (minus_nat gm (nat_of_integer (1 : IntInf.int))), c, e)))
+ | mirror (NEq (Cn (hm, c, e))) =
+ (if equal_nat hm zero_nat then NEq (Cn (zero_nat, c, Neg e))
+ else NEq (Cn (suc (minus_nat hm (nat_of_integer (1 : IntInf.int))), c,
+ e)))
+ | mirror (Dvd (i, Cn (im, c, e))) =
+ (if equal_nat im zero_nat then Dvd (i, Cn (zero_nat, c, Neg e))
+ else Dvd (i, Cn (suc (minus_nat im (nat_of_integer (1 : IntInf.int))), c,
+ e)))
+ | mirror (NDvd (i, Cn (jm, c, e))) =
+ (if equal_nat jm zero_nat then NDvd (i, Cn (zero_nat, c, Neg e))
+ else NDvd (i, Cn (suc (minus_nat jm (nat_of_integer (1 : IntInf.int))), c,
+ e)));
+
+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 equal_nat cm zero_nat
+ then (fn k =>
+ Lt (Cn (zero_nat, Int_of_integer (1 : IntInf.int),
+ Mul (div_inta k c, e))))
+ else (fn _ =>
+ Lt (Cn (suc (minus_nat cm (nat_of_integer (1 : IntInf.int))), c,
+ e))))
+ | a_beta (Le (Cn (dm, c, e))) =
+ (if equal_nat dm zero_nat
+ then (fn k =>
+ Le (Cn (zero_nat, Int_of_integer (1 : IntInf.int),
+ Mul (div_inta k c, e))))
+ else (fn _ =>
+ Le (Cn (suc (minus_nat dm (nat_of_integer (1 : IntInf.int))), c,
+ e))))
+ | a_beta (Gt (Cn (em, c, e))) =
+ (if equal_nat em zero_nat
+ then (fn k =>
+ Gt (Cn (zero_nat, Int_of_integer (1 : IntInf.int),
+ Mul (div_inta k c, e))))
+ else (fn _ =>
+ Gt (Cn (suc (minus_nat em (nat_of_integer (1 : IntInf.int))), c,
+ e))))
+ | a_beta (Ge (Cn (fm, c, e))) =
+ (if equal_nat fm zero_nat
+ then (fn k =>
+ Ge (Cn (zero_nat, Int_of_integer (1 : IntInf.int),
+ Mul (div_inta k c, e))))
+ else (fn _ =>
+ Ge (Cn (suc (minus_nat fm (nat_of_integer (1 : IntInf.int))), c,
+ e))))
+ | a_beta (Eq (Cn (gm, c, e))) =
+ (if equal_nat gm zero_nat
+ then (fn k =>
+ Eq (Cn (zero_nat, Int_of_integer (1 : IntInf.int),
+ Mul (div_inta k c, e))))
+ else (fn _ =>
+ Eq (Cn (suc (minus_nat gm (nat_of_integer (1 : IntInf.int))), c,
+ e))))
+ | a_beta (NEq (Cn (hm, c, e))) =
+ (if equal_nat hm zero_nat
+ then (fn k =>
+ NEq (Cn (zero_nat, Int_of_integer (1 : IntInf.int),
+ Mul (div_inta k c, e))))
+ else (fn _ =>
+ NEq (Cn (suc (minus_nat hm (nat_of_integer (1 : IntInf.int))), c,
+ e))))
+ | a_beta (Dvd (i, Cn (im, c, e))) =
+ (if equal_nat im zero_nat
+ then (fn k =>
+ Dvd (times_inta (div_inta k c) i,
+ Cn (zero_nat, Int_of_integer (1 : IntInf.int),
+ Mul (div_inta k c, e))))
+ else (fn _ =>
+ Dvd (i, Cn (suc (minus_nat im (nat_of_integer (1 : IntInf.int))),
+ c, e))))
+ | a_beta (NDvd (i, Cn (jm, c, e))) =
+ (if equal_nat jm zero_nat
+ then (fn k =>
+ NDvd (times_inta (div_inta k c) i,
+ Cn (zero_nat, Int_of_integer (1 : IntInf.int),
+ Mul (div_inta k c, e))))
+ else (fn _ =>
+ NDvd (i, Cn (suc (minus_nat jm (nat_of_integer (1 : IntInf.int))),
+ c, e))));
+
+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 gcd_int k l =
+ abs_int
+ (if equal_inta l zero_inta then k
+ else gcd_int l (mod_int (abs_int k) (abs_int l)));
+
+fun lcm_int a b = div_inta (times_inta (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 = Int_of_integer (1 : IntInf.int)
+ | delta F = Int_of_integer (1 : IntInf.int)
+ | delta (Lt u) = Int_of_integer (1 : IntInf.int)
+ | delta (Le v) = Int_of_integer (1 : IntInf.int)
+ | delta (Gt w) = Int_of_integer (1 : IntInf.int)
+ | delta (Ge x) = Int_of_integer (1 : IntInf.int)
+ | delta (Eq y) = Int_of_integer (1 : IntInf.int)
+ | delta (NEq z) = Int_of_integer (1 : IntInf.int)
+ | delta (Dvd (aa, C bo)) = Int_of_integer (1 : IntInf.int)
+ | delta (Dvd (aa, Bound bp)) = Int_of_integer (1 : IntInf.int)
+ | delta (Dvd (aa, Neg bt)) = Int_of_integer (1 : IntInf.int)
+ | delta (Dvd (aa, Add (bu, bv))) = Int_of_integer (1 : IntInf.int)
+ | delta (Dvd (aa, Sub (bw, bx))) = Int_of_integer (1 : IntInf.int)
+ | delta (Dvd (aa, Mul (by, bz))) = Int_of_integer (1 : IntInf.int)
+ | delta (NDvd (ac, C co)) = Int_of_integer (1 : IntInf.int)
+ | delta (NDvd (ac, Bound cp)) = Int_of_integer (1 : IntInf.int)
+ | delta (NDvd (ac, Neg ct)) = Int_of_integer (1 : IntInf.int)
+ | delta (NDvd (ac, Add (cu, cv))) = Int_of_integer (1 : IntInf.int)
+ | delta (NDvd (ac, Sub (cw, cx))) = Int_of_integer (1 : IntInf.int)
+ | delta (NDvd (ac, Mul (cy, cz))) = Int_of_integer (1 : IntInf.int)
+ | delta (Not ae) = Int_of_integer (1 : IntInf.int)
+ | delta (Imp (aj, ak)) = Int_of_integer (1 : IntInf.int)
+ | delta (Iff (al, am)) = Int_of_integer (1 : IntInf.int)
+ | delta (E an) = Int_of_integer (1 : IntInf.int)
+ | delta (A ao) = Int_of_integer (1 : IntInf.int)
+ | delta (Closed ap) = Int_of_integer (1 : IntInf.int)
+ | delta (NClosed aq) = Int_of_integer (1 : IntInf.int)
+ | delta (Dvd (i, Cn (cm, c, e))) =
+ (if equal_nat cm zero_nat then i else Int_of_integer (1 : IntInf.int))
+ | delta (NDvd (i, Cn (dm, c, e))) =
+ (if equal_nat dm zero_nat then i else Int_of_integer (1 : IntInf.int));
+
+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 equal_nat cm zero_nat then [e] else [])
+ | alpha (Le (Cn (dm, c, e))) =
+ (if equal_nat dm zero_nat
+ then [Add (C (Int_of_integer (~1 : IntInf.int)), e)] else [])
+ | alpha (Gt (Cn (em, c, e))) = (if equal_nat em zero_nat then [] else [])
+ | alpha (Ge (Cn (fm, c, e))) = (if equal_nat fm zero_nat then [] else [])
+ | alpha (Eq (Cn (gm, c, e))) =
+ (if equal_nat gm zero_nat
+ then [Add (C (Int_of_integer (~1 : IntInf.int)), e)] else [])
+ | alpha (NEq (Cn (hm, c, e))) = (if equal_nat hm zero_nat then [e] else []);
+
+fun minus_int k l = Int_of_integer (integer_of_int k - integer_of_int l);
+
+fun zsplit0 (C c) = (zero_inta, C c)
+ | zsplit0 (Bound n) =
+ (if equal_nat n zero_nat then (Int_of_integer (1 : IntInf.int), C zero_inta)
+ else (zero_inta, Bound n))
+ | zsplit0 (Cn (n, i, a)) =
+ let
+ val (ia, aa) = zsplit0 a;
+ in
+ (if equal_nat n zero_nat then (plus_inta i ia, aa)
+ else (ia, Cn (n, i, aa)))
+ end
+ | zsplit0 (Neg a) = let
+ val (i, aa) = zsplit0 a;
+ in
+ (uminus_int i, Neg aa)
+ end
+ | zsplit0 (Add (a, b)) =
+ let
+ val (ia, aa) = zsplit0 a;
+ val (ib, ba) = zsplit0 b;
+ in
+ (plus_inta ia ib, Add (aa, ba))
+ end
+ | zsplit0 (Sub (a, b)) =
+ let
+ val (ia, aa) = zsplit0 a;
+ val (ib, ba) = zsplit0 b;
+ in
+ (minus_int ia ib, Sub (aa, ba))
+ end
+ | zsplit0 (Mul (i, a)) =
+ let
+ val (ia, aa) = zsplit0 a;
+ in
+ (times_inta 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 equal_inta c zero_inta then Lt r
+ else (if less_int zero_inta c then Lt (Cn (zero_nat, c, r))
+ else Gt (Cn (zero_nat, uminus_int c, Neg r))))
+ end
+ | zlfm (Le a) =
+ let
+ val (c, r) = zsplit0 a;
+ in
+ (if equal_inta c zero_inta then Le r
+ else (if less_int zero_inta c then Le (Cn (zero_nat, c, r))
+ else Ge (Cn (zero_nat, uminus_int c, Neg r))))
+ end
+ | zlfm (Gt a) =
+ let
+ val (c, r) = zsplit0 a;
+ in
+ (if equal_inta c zero_inta then Gt r
+ else (if less_int zero_inta c then Gt (Cn (zero_nat, c, r))
+ else Lt (Cn (zero_nat, uminus_int c, Neg r))))
+ end
+ | zlfm (Ge a) =
+ let
+ val (c, r) = zsplit0 a;
+ in
+ (if equal_inta c zero_inta then Ge r
+ else (if less_int zero_inta c then Ge (Cn (zero_nat, c, r))
+ else Le (Cn (zero_nat, uminus_int c, Neg r))))
+ end
+ | zlfm (Eq a) =
+ let
+ val (c, r) = zsplit0 a;
+ in
+ (if equal_inta c zero_inta then Eq r
+ else (if less_int zero_inta c then Eq (Cn (zero_nat, c, r))
+ else Eq (Cn (zero_nat, uminus_int c, Neg r))))
+ end
+ | zlfm (NEq a) =
+ let
+ val (c, r) = zsplit0 a;
+ in
+ (if equal_inta c zero_inta then NEq r
+ else (if less_int zero_inta c then NEq (Cn (zero_nat, c, r))
+ else NEq (Cn (zero_nat, uminus_int c, Neg r))))
+ end
+ | zlfm (Dvd (i, a)) =
+ (if equal_inta i zero_inta then zlfm (Eq a)
+ else let
+ val (c, r) = zsplit0 a;
+ in
+ (if equal_inta c zero_inta then Dvd (abs_int i, r)
+ else (if less_int zero_inta c
+ then Dvd (abs_int i, Cn (zero_nat, c, r))
+ else Dvd (abs_int i, Cn (zero_nat, uminus_int c, Neg r))))
+ end)
+ | zlfm (NDvd (i, a)) =
+ (if equal_inta i zero_inta then zlfm (NEq a)
+ else let
+ val (c, r) = zsplit0 a;
+ in
+ (if equal_inta c zero_inta then NDvd (abs_int i, r)
+ else (if less_int zero_inta c
+ then NDvd (abs_int i, Cn (zero_nat, c, r))
+ else NDvd (abs_int i,
+ Cn (zero_nat, uminus_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 zeta (And (p, q)) = lcm_int (zeta p) (zeta q)
+ | zeta (Or (p, q)) = lcm_int (zeta p) (zeta q)
+ | zeta T = Int_of_integer (1 : IntInf.int)
+ | zeta F = Int_of_integer (1 : IntInf.int)
+ | zeta (Lt (C bo)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Lt (Bound bp)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Lt (Neg bt)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Lt (Add (bu, bv))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Lt (Sub (bw, bx))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Lt (Mul (by, bz))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Le (C co)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Le (Bound cp)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Le (Neg ct)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Le (Add (cu, cv))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Le (Sub (cw, cx))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Le (Mul (cy, cz))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Gt (C doa)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Gt (Bound dp)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Gt (Neg dt)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Gt (Add (du, dv))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Gt (Sub (dw, dx))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Gt (Mul (dy, dz))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Ge (C eo)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Ge (Bound ep)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Ge (Neg et)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Ge (Add (eu, ev))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Ge (Sub (ew, ex))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Ge (Mul (ey, ez))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Eq (C fo)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Eq (Bound fp)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Eq (Neg ft)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Eq (Add (fu, fv))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Eq (Sub (fw, fx))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Eq (Mul (fy, fz))) = Int_of_integer (1 : IntInf.int)
+ | zeta (NEq (C go)) = Int_of_integer (1 : IntInf.int)
+ | zeta (NEq (Bound gp)) = Int_of_integer (1 : IntInf.int)
+ | zeta (NEq (Neg gt)) = Int_of_integer (1 : IntInf.int)
+ | zeta (NEq (Add (gu, gv))) = Int_of_integer (1 : IntInf.int)
+ | zeta (NEq (Sub (gw, gx))) = Int_of_integer (1 : IntInf.int)
+ | zeta (NEq (Mul (gy, gz))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Dvd (aa, C ho)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Dvd (aa, Bound hp)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Dvd (aa, Neg ht)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Dvd (aa, Add (hu, hv))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Dvd (aa, Sub (hw, hx))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Dvd (aa, Mul (hy, hz))) = Int_of_integer (1 : IntInf.int)
+ | zeta (NDvd (ac, C io)) = Int_of_integer (1 : IntInf.int)
+ | zeta (NDvd (ac, Bound ip)) = Int_of_integer (1 : IntInf.int)
+ | zeta (NDvd (ac, Neg it)) = Int_of_integer (1 : IntInf.int)
+ | zeta (NDvd (ac, Add (iu, iv))) = Int_of_integer (1 : IntInf.int)
+ | zeta (NDvd (ac, Sub (iw, ix))) = Int_of_integer (1 : IntInf.int)
+ | zeta (NDvd (ac, Mul (iy, iz))) = Int_of_integer (1 : IntInf.int)
+ | zeta (Not ae) = Int_of_integer (1 : IntInf.int)
+ | zeta (Imp (aj, ak)) = Int_of_integer (1 : IntInf.int)
+ | zeta (Iff (al, am)) = Int_of_integer (1 : IntInf.int)
+ | zeta (E an) = Int_of_integer (1 : IntInf.int)
+ | zeta (A ao) = Int_of_integer (1 : IntInf.int)
+ | zeta (Closed ap) = Int_of_integer (1 : IntInf.int)
+ | zeta (NClosed aq) = Int_of_integer (1 : IntInf.int)
+ | zeta (Lt (Cn (cm, c, e))) =
+ (if equal_nat cm zero_nat then c else Int_of_integer (1 : IntInf.int))
+ | zeta (Le (Cn (dm, c, e))) =
+ (if equal_nat dm zero_nat then c else Int_of_integer (1 : IntInf.int))
+ | zeta (Gt (Cn (em, c, e))) =
+ (if equal_nat em zero_nat then c else Int_of_integer (1 : IntInf.int))
+ | zeta (Ge (Cn (fm, c, e))) =
+ (if equal_nat fm zero_nat then c else Int_of_integer (1 : IntInf.int))
+ | zeta (Eq (Cn (gm, c, e))) =
+ (if equal_nat gm zero_nat then c else Int_of_integer (1 : IntInf.int))
+ | zeta (NEq (Cn (hm, c, e))) =
+ (if equal_nat hm zero_nat then c else Int_of_integer (1 : IntInf.int))
+ | zeta (Dvd (i, Cn (im, c, e))) =
+ (if equal_nat im zero_nat then c else Int_of_integer (1 : IntInf.int))
+ | zeta (NDvd (i, Cn (jm, c, e))) =
+ (if equal_nat jm zero_nat then c else Int_of_integer (1 : IntInf.int));
+
+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 equal_nat cm zero_nat then [] else [])
+ | beta (Le (Cn (dm, c, e))) = (if equal_nat dm zero_nat then [] else [])
+ | beta (Gt (Cn (em, c, e))) = (if equal_nat em zero_nat then [Neg e] else [])
+ | beta (Ge (Cn (fm, c, e))) =
+ (if equal_nat fm zero_nat
+ then [Sub (C (Int_of_integer (~1 : IntInf.int)), e)] else [])
+ | beta (Eq (Cn (gm, c, e))) =
+ (if equal_nat gm zero_nat
+ then [Sub (C (Int_of_integer (~1 : IntInf.int)), e)] else [])
+ | beta (NEq (Cn (hm, c, e))) =
+ (if equal_nat hm zero_nat then [Neg e] else []);
+
+fun unita p =
+ let
+ val pa = zlfm p;
+ val l = zeta pa;
+ val q =
+ And (Dvd (l, Cn (zero_nat, Int_of_integer (1 : IntInf.int), C zero_inta)),
+ 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 less_eq_nat (size_list b) (size_list a) then (q, (b, d))
+ else (mirror q, (a, d)))
+ end;
+
+fun decrnum (Bound n) = Bound (minus_nat n (nat_of_integer (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 (nat_of_integer (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 uptoa i j =
+ (if less_eq_int i j
+ then i :: uptoa (plus_inta i (Int_of_integer (1 : IntInf.int))) j else []);
+
+fun maps f [] = []
+ | maps f (x :: xs) = f x @ maps f xs;
+
+fun cooper p =
+ let
+ val (q, (b, d)) = unita p;
+ val js = uptoa (Int_of_integer (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 qelim (E p) = (fn qe => dj qe (qelim p qe))
+ | qelim (A p) = (fn qe => nota (qe (qelim (Not p) qe)))
+ | qelim (Not p) = (fn qe => nota (qelim p qe))
+ | qelim (And (p, q)) = (fn qe => conj (qelim p qe) (qelim q qe))
+ | qelim (Or (p, q)) = (fn qe => disj (qelim p qe) (qelim q qe))
+ | qelim (Imp (p, q)) = (fn qe => impa (qelim p qe) (qelim q qe))
+ | qelim (Iff (p, q)) = (fn qe => iffa (qelim p qe) (qelim q qe))
+ | qelim T = (fn _ => simpfm T)
+ | qelim F = (fn _ => simpfm F)
+ | qelim (Lt v) = (fn _ => simpfm (Lt v))
+ | qelim (Le v) = (fn _ => simpfm (Le v))
+ | 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 prep (E T) = T
| prep (E F) = F
@@ -787,1369 +2302,6 @@
| 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))
- | qelim (And (p, q)) = (fn qe => conj (qelim p qe) (qelim q qe))
- | qelim (Or (p, q)) = (fn qe => disj (qelim p qe) (qelim q qe))
- | qelim (Imp (p, q)) = (fn qe => impa (qelim p qe) (qelim q qe))
- | qelim (Iff (p, q)) = (fn qe => iffa (qelim p qe) (qelim q qe))
- | qelim T = (fn _ => simpfm T)
- | qelim F = (fn _ => simpfm F)
- | qelim (Lt v) = (fn _ => simpfm (Lt v))
- | qelim (Le v) = (fn _ => simpfm (Le v))
- | 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;
end; (*struct Cooper_Procedure*)