# HG changeset patch # User haftmann # Date 1184052195 -7200 # Node ID 0410269099dcb3da3df9f10c0b82d32bbe4f5106 # Parent 7cd68def72b2c5f011bacb553dc1eef5efa6125d replaced code generator framework for reflected cooper diff -r 7cd68def72b2 -r 0410269099dc src/HOL/Tools/Qelim/cooper.ML --- a/src/HOL/Tools/Qelim/cooper.ML Tue Jul 10 09:23:14 2007 +0200 +++ b/src/HOL/Tools/Qelim/cooper.ML Tue Jul 10 09:23:15 2007 +0200 @@ -11,11 +11,13 @@ structure Cooper: COOPER = struct + open Conv; +open Normalizer; structure Integertab = TableFun(type key = Integer.int val ord = Integer.ord); + exception COOPER of string * exn; val simp_thms_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms); - val FWD = Drule.implies_elim_list; val true_tm = @{cterm "True"}; @@ -23,7 +25,7 @@ val zdvd1_eq = @{thm "zdvd1_eq"}; val presburger_ss = @{simpset} addsimps [zdvd1_eq]; val lin_ss = presburger_ss addsimps (@{thm "dvd_eq_mod_eq_0"}::zdvd1_eq::@{thms zadd_ac}); -(* Some types and constants *) + val iT = HOLogic.intT val bT = HOLogic.boolT; val dest_numeral = HOLogic.dest_number #> snd; @@ -59,7 +61,7 @@ val eval_ss = presburger_ss addsimps [simp_from_to] delsimps [insert_iff,bex_triv]; val eval_conv = Simplifier.rewrite eval_ss; -(* recongnising cterm without moving to terms *) +(* recognising cterm without moving to terms *) datatype fm = And of cterm*cterm| Or of cterm*cterm| Eq of cterm | NEq of cterm | Lt of cterm | Le of cterm | Gt of cterm | Ge of cterm @@ -78,9 +80,9 @@ | Const ("Orderings.ord_class.less_eq",_)$y$z => if term_of x aconv y then Le (Thm.dest_arg ct) else if term_of x aconv z then Ge (Thm.dest_arg1 ct) else Nox -| Const ("Divides.dvd",_)$_$(Const(@{const_name "HOL.plus"},_)$y$_) => +| Const (@{const_name Divides.dvd},_)$_$(Const(@{const_name "HOL.plus"},_)$y$_) => if term_of x aconv y then Dvd (Thm.dest_binop ct ||> Thm.dest_arg) else Nox -| Const("Not",_) $ (Const ("Divides.dvd",_)$_$(Const(@{const_name "HOL.plus"},_)$y$_)) => +| Const("Not",_) $ (Const (@{const_name Divides.dvd},_)$_$(Const(@{const_name "HOL.plus"},_)$y$_)) => if term_of x aconv y then NDvd (Thm.dest_binop (Thm.dest_arg ct) ||> Thm.dest_arg) else Nox | _ => Nox) @@ -117,10 +119,10 @@ val cadd = @{cterm "op + :: int => _"} val cmulC = @{cterm "op * :: int => _"} val cminus = @{cterm "op - :: int => _"} -val cone = @{cterm "1:: int"} +val cone = @{cterm "1 :: int"} val cneg = @{cterm "uminus :: int => _"} val [addC, mulC, subC, negC] = map term_of [cadd, cmulC, cminus, cneg] -val [zero, one] = [@{term "0::int"}, @{term "1::int"}]; +val [zero, one] = [@{term "0 :: int"}, @{term "1 :: int"}]; val is_numeral = can dest_numeral; @@ -225,8 +227,8 @@ | lin (vs as x::_) (Const("Not",_)$(Const("Orderings.ord_class.less_eq",T)$s$t)) = lin vs (Const("Orderings.ord_class.less",T)$t$s) | lin vs (Const ("Not",T)$t) = Const ("Not",T)$ (lin vs t) - | lin (vs as x::_) (Const("Divides.dvd",_)$d$t) = - HOLogic.mk_binrel "Divides.dvd" (numeral1 abs d, lint vs t) + | lin (vs as x::_) (Const(@{const_name Divides.dvd},_)$d$t) = + HOLogic.mk_binrel @{const_name Divides.dvd} (numeral1 abs d, lint vs t) | lin (vs as x::_) ((b as Const("op =",_))$s$t) = (case lint vs (subC$t$s) of (t as a$(m$c$y)$r) => @@ -255,7 +257,7 @@ fun linearize_conv ctxt vs ct = case (term_of ct) of - Const("Divides.dvd",_)$d$t => + Const(@{const_name Divides.dvd},_)$d$t => let val th = binop_conv (lint_conv ctxt vs) ct val (d',t') = Thm.dest_binop (Thm.rhs_of th) @@ -280,7 +282,7 @@ | _ => dth end end -| Const("Not",_)$(Const("Divides.dvd",_)$_$_) => arg_conv (linearize_conv ctxt vs) ct +| Const("Not",_)$(Const(@{const_name Divides.dvd},_)$_$_) => arg_conv (linearize_conv ctxt vs) ct | t => if is_intrel t then (provelin ctxt ((HOLogic.eq_const bT)$t$(lin vs t) |> HOLogic.mk_Trueprop)) RS eq_reflection @@ -303,7 +305,7 @@ if x aconv y andalso s mem ["Orderings.ord_class.less", "Orderings.ord_class.less_eq"] then (ins (dest_numeral c) acc, dacc) else (acc,dacc) - | Const("Divides.dvd",_)$_$(Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_) => + | Const(@{const_name Divides.dvd},_)$_$(Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_) => if x aconv y then (acc,ins (dest_numeral c) dacc) else (acc,dacc) | Const("op &",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t) | Const("op |",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t) @@ -320,8 +322,8 @@ val th = Simplifier.rewrite lin_ss (Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} - (Thm.capply (Thm.capply @{cterm "op = :: int => _"} (Numeral.mk_cnumber @{ctyp "int"} x)) - @{cterm "0::int"}))) + (Thm.capply (Thm.capply @{cterm "op = :: int => _"} (Numeral.mk_cnumber @{ctyp "int"} x)) + @{cterm "0::int"}))) in equal_elim (Thm.symmetric th) TrueI end; val notz = let val tab = fold Integertab.update (ds ~~ (map (fn x => nzprop (Integer.div l x)) ds)) Integertab.empty @@ -341,7 +343,7 @@ | Const(s,_)$_$(Const("HOL.times_class.times",_)$c$y) => if x=y andalso s mem ["Orderings.ord_class.less", "Orderings.ord_class.less_eq"] then cv (Integer.div l (dest_numeral c)) t else Thm.reflexive t - | Const("Divides.dvd",_)$d$(r as (Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_)) => + | Const(@{const_name Divides.dvd},_)$d$(r as (Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_)) => if x=y then let val k = Integer.div l (dest_numeral c) @@ -357,7 +359,7 @@ else Thm.reflexive t | _ => Thm.reflexive t val uth = unit_conv p - val clt = Numeral.mk_cnumber @{ctyp "int"} l + val clt = Numeral.mk_cnumber @{ctyp "int"} l val ltx = Thm.capply (Thm.capply cmulC clt) cx val th = Drule.arg_cong_rule e (Thm.abstract_rule (fst (dest_Free x )) cx uth) val th' = inst' [Thm.cabs ltx (Thm.rhs_of uth), clt] unity_coeff_ex @@ -508,7 +510,7 @@ fun integer_nnf_conv ctxt env = nnf_conv then_conv literals_conv [HOLogic.conj, HOLogic.disj] [] env (linearize_conv ctxt); -(* val my_term = ref (@{cterm "NOTHING"}); *) +(* val my_term = ref (@{cterm "NotaHING"}); *) local val pcv = Simplifier.rewrite (HOL_basic_ss addsimps (simp_thms @ (List.take(ex_simps,4)) @@ -533,7 +535,8 @@ structure Coopereif = struct -open GeneratedCooper; +open GeneratedCooper.Reflected_Presburger; + fun cooper s = raise Cooper.COOPER ("Cooper Oracle Failed", ERROR s); fun i_of_term vs t = case t of @@ -560,22 +563,22 @@ | Const("False",_) => F | Const(@{const_name "Orderings.less"},_)$t1$t2 => Lt (Sub (i_of_term vs t1,i_of_term vs t2)) | Const(@{const_name "Orderings.less_eq"},_)$t1$t2 => Le (Sub(i_of_term vs t1,i_of_term vs t2)) - | Const(@{const_name "Divides.dvd"},_)$t1$t2 => + | Const(@{const_name Divides.dvd},_)$t1$t2 => (Dvd(HOLogic.dest_number t1 |> snd |> Integer.machine_int, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd") | @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2)) - | @{term "op = :: bool => _ "}$t1$t2 => Iff(qf_of_term ps vs t1,qf_of_term ps vs t2) + | @{term "op = :: bool => _ "}$t1$t2 => Iffa(qf_of_term ps vs t1,qf_of_term ps vs t2) | Const("op &",_)$t1$t2 => And(qf_of_term ps vs t1,qf_of_term ps vs t2) | Const("op |",_)$t1$t2 => Or(qf_of_term ps vs t1,qf_of_term ps vs t2) - | Const("op -->",_)$t1$t2 => Imp(qf_of_term ps vs t1,qf_of_term ps vs t2) - | Const("Not",_)$t' => NOT(qf_of_term ps vs t') + | Const("op -->",_)$t1$t2 => Impa(qf_of_term ps vs t1,qf_of_term ps vs t2) + | Const("Not",_)$t' => Nota(qf_of_term ps vs t') | Const("Ex",_)$Abs(xn,xT,p) => let val (xn',p') = variant_abs (xn,xT,p) - val vs' = (Free (xn',xT), nat 0) :: (map (fn(v,n) => (v,1+ n)) vs) + val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs) in E (qf_of_term ps vs' p') end | Const("All",_)$Abs(xn,xT,p) => let val (xn',p') = variant_abs (xn,xT,p) - val vs' = (Free (xn',xT), nat 0) :: (map (fn(v,n) => (v,1+ n)) vs) + val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs) in A (qf_of_term ps vs' p') end | _ =>(case AList.lookup (op aconv) ps t of @@ -612,7 +615,7 @@ fun myassoc2 l v = case l of [] => NONE - | (x,v': int)::xs => if v = v' then SOME x + | (x,v')::xs => if v = v' then SOME x else myassoc2 xs v; fun term_of_i vs t = @@ -625,7 +628,7 @@ (term_of_i vs t1)$(term_of_i vs t2) | Mul(i,t2) => Const(@{const_name "HOL.times"},[iT,iT] ---> iT)$ (HOLogic.mk_number HOLogic.intT (Integer.int i))$(term_of_i vs t2) - | CX(i,t')=> term_of_i vs (Add(Mul (i,Bound (nat 0)),t')); + | Cx(i,t')=> term_of_i vs (Add(Mul (i, Bound 0),t')); fun term_of_qf ps vs t = case t of @@ -636,17 +639,17 @@ | Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t' | Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t' | Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"} - | NEq t' => term_of_qf ps vs (NOT(Eq t')) + | NEq t' => term_of_qf ps vs (Nota(Eq t')) | Dvd(i,t') => @{term "op dvd :: int => _ "}$ (HOLogic.mk_number HOLogic.intT (Integer.int i))$(term_of_i vs t') - | NDvd(i,t')=> term_of_qf ps vs (NOT(Dvd(i,t'))) - | NOT t' => HOLogic.Not$(term_of_qf ps vs t') + | NDvd(i,t')=> term_of_qf ps vs (Nota(Dvd(i,t'))) + | Nota t' => HOLogic.Not$(term_of_qf ps vs t') | And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2) | Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2) - | Imp(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2) - | Iff(t1,t2) => (HOLogic.eq_const bT)$(term_of_qf ps vs t1)$ (term_of_qf ps vs t2) + | Impa(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2) + | Iffa(t1,t2) => (HOLogic.eq_const bT)$(term_of_qf ps vs t1)$ (term_of_qf ps vs t2) | Closed n => valOf (myassoc2 ps n) - | NClosed n => term_of_qf ps vs (NOT (Closed n)) + | NClosed n => term_of_qf ps vs (Nota (Closed n)) | _ => cooper "If this is raised, Isabelle/HOL or generate_code is inconsistent!"; (* The oracle *) diff -r 7cd68def72b2 -r 0410269099dc src/HOL/Tools/Qelim/generated_cooper.ML --- a/src/HOL/Tools/Qelim/generated_cooper.ML Tue Jul 10 09:23:14 2007 +0200 +++ b/src/HOL/Tools/Qelim/generated_cooper.ML Tue Jul 10 09:23:15 2007 +0200 @@ -1,1693 +1,2244 @@ -structure GeneratedCooper = +(* Title: HOL/Tools/Presburger/generated_cooper.ML + ID: $Id$ + +This file is generated from HOL/ex/Reflected_Presburger.thy. DO NOT EDIT. +*) + +structure GeneratedCooper = struct -nonfix oo; -fun nat i = if i < 0 then 0 else i; -val one_def0 : int = (0 + 1); +structure Product_Type = +struct + +fun fst (y, b) = y; -datatype num = C of int | Bound of int | CX of int * num | Neg of num - | Add of num * num | Sub of num * num | Mul of int * num; +fun snd (a, y) = y; + +end; (*struct Product_Type*) + +structure Integer = +struct -fun snd (a, b) = b; +datatype bit = B0 | B1; + +fun suc n = (IntInf.+ (n, (1 : IntInf.int))); -fun negateSnd x = (fn (q, r) => (q, ~ r)) x; +val zero_nat : IntInf.int = (0 : IntInf.int); -fun minus_def2 z w = (z + ~ w); +fun nat k = (if IntInf.< (k, (0 : IntInf.int)) then zero_nat else k); fun adjust b = - (fn (q, r) => - (if (0 <= minus_def2 r b) then (((2 * q) + 1), minus_def2 r b) - else ((2 * q), r))); + (fn a as (q, r) => + (if IntInf.<= ((0 : IntInf.int), IntInf.- (r, b)) + then (IntInf.+ (IntInf.* ((2 : IntInf.int), q), (1 : IntInf.int)), + IntInf.- (r, b)) + else (IntInf.* ((2 : IntInf.int), q), r))); fun negDivAlg a b = - (if ((0 <= (a + b)) orelse (b <= 0)) then (~1, (a + b)) - else adjust b (negDivAlg a (2 * b))); + (if IntInf.<= ((0 : IntInf.int), IntInf.+ (a, b)) orelse + IntInf.<= (b, (0 : IntInf.int)) + then ((~1 : IntInf.int), IntInf.+ (a, b)) + else adjust b (negDivAlg a (IntInf.* ((2 : IntInf.int), b)))); + +val negateSnd : IntInf.int * IntInf.int -> IntInf.int * IntInf.int = + (fn a as (q, r) => (q, IntInf.~ r)); fun posDivAlg a b = - (if ((a < b) orelse (b <= 0)) then (0, a) - else adjust b (posDivAlg a (2 * b))); + (if IntInf.< (a, b) orelse IntInf.<= (b, (0 : IntInf.int)) + then ((0 : IntInf.int), a) + else adjust b (posDivAlg a (IntInf.* ((2 : IntInf.int), b)))); + +val divAlg : IntInf.int * IntInf.int -> IntInf.int * IntInf.int = + (fn a as (aa, b) => + (if IntInf.<= ((0 : IntInf.int), aa) + then (if IntInf.<= ((0 : IntInf.int), b) then posDivAlg aa b + else (if ((aa : IntInf.int) = (0 : IntInf.int)) + then ((0 : IntInf.int), (0 : IntInf.int)) + else negateSnd (negDivAlg (IntInf.~ aa) (IntInf.~ b)))) + else (if IntInf.< ((0 : IntInf.int), b) then negDivAlg aa b + else negateSnd (posDivAlg (IntInf.~ aa) (IntInf.~ b))))); + +fun abs_int i = (if IntInf.< (i, (0 : IntInf.int)) then IntInf.~ i else i); + +fun div_int a b = Product_Type.fst (divAlg (a, b)); + +fun mod_int a b = Product_Type.snd (divAlg (a, b)); + +fun dvd_int m n = (((mod_int n m) : IntInf.int) = (0 : IntInf.int)); + +fun eq_bit B0 B0 = true + | eq_bit B1 B1 = true + | eq_bit B0 B1 = false + | eq_bit B1 B0 = false; + +fun int_aux i n = + (if ((n : IntInf.int) = (0 : IntInf.int)) then i + else int_aux (IntInf.+ (i, (1 : IntInf.int))) + (IntInf.- (n, (1 : IntInf.int)))); + +end; (*struct Integer*) + +structure Nat = +struct + +fun div_nat m k = (Product_Type.fst (Integer.divAlg (m, k))); + +fun mod_nat m k = (Product_Type.snd (Integer.divAlg (m, k))); + +end; (*struct Nat*) + +structure GCD = +struct + +fun gcd (m, n) = + (if ((n : IntInf.int) = Integer.zero_nat) then m + else gcd (n, Nat.mod_nat m n)); + +val lcm : IntInf.int * IntInf.int -> IntInf.int = + (fn a as (m, n) => Nat.div_nat (IntInf.* (m, n)) (gcd (m, n))); + +val ilcm : IntInf.int -> IntInf.int -> IntInf.int = + (fn i => fn j => + Integer.int_aux (0 : IntInf.int) + (lcm (Integer.nat (Integer.abs_int i), Integer.nat (Integer.abs_int j)))); + +end; (*struct GCD*) + +structure HOL = +struct + +type 'a eq = {eq : 'a -> 'a -> bool}; +fun eq (A_:'a eq) = #eq A_; + +end; (*struct HOL*) + +structure List = +struct + +fun map f (x :: xs) = f x :: map f xs + | map f [] = []; + +fun foldr f (x :: xs) a = f x (foldr f xs a) + | foldr f [] y = y; + +fun append (x :: xs) ys = x :: append xs ys + | append [] y = y; + +fun memberl A_ x (y :: ys) = HOL.eq A_ x y orelse memberl A_ x ys + | memberl A_ x [] = false; + +fun remdups A_ (x :: xs) = + (if memberl A_ x xs then remdups A_ xs else x :: remdups A_ xs) + | remdups A_ [] = []; + +fun allpairs f (x :: xs) ys = append (map (f x) ys) (allpairs f xs ys) + | allpairs f [] ys = []; -fun divAlg x = - (fn (a, b) => - (if (0 <= a) - then (if (0 <= b) then posDivAlg a b - else (if (a = 0) then (0, 0) - else negateSnd (negDivAlg (~ a) (~ b)))) - else (if (0 < b) then negDivAlg a b - else negateSnd (posDivAlg (~ a) (~ b))))) - x; +fun size_list (a :: lista) = + (IntInf.+ ((size_list lista), (Integer.suc Integer.zero_nat))) + | size_list [] = Integer.zero_nat; + +end; (*struct List*) + +structure Reflected_Presburger = +struct + +datatype num = C of IntInf.int | Bound of IntInf.int | Cx of IntInf.int * num | + Neg of num | Add of num * num | Sub of num * num | Mul of IntInf.int * num; + +datatype fm = T | F | Lt of num | Le of num | Gt of num | Ge of num | Eq of num + | NEq of num | Dvd of IntInf.int * num | NDvd of IntInf.int * num | Nota of fm + | And of fm * fm | Or of fm * fm | Impa of fm * fm | Iffa of fm * fm | E of fm + | A of fm | Closed of IntInf.int | NClosed of IntInf.int; + +fun disjuncts (NClosed aq) = [NClosed aq] + | disjuncts (Closed ap) = [Closed ap] + | disjuncts (A ao) = [A ao] + | disjuncts (E an) = [E an] + | disjuncts (Iffa (al, am)) = [Iffa (al, am)] + | disjuncts (Impa (aj, ak)) = [Impa (aj, ak)] + | disjuncts (And (af, ag)) = [And (af, ag)] + | disjuncts (Nota ae) = [Nota ae] + | disjuncts (NDvd (ac, ad)) = [NDvd (ac, ad)] + | disjuncts (Dvd (aa, ab)) = [Dvd (aa, ab)] + | disjuncts (NEq z) = [NEq z] + | disjuncts (Eq y) = [Eq y] + | disjuncts (Ge x) = [Ge x] + | disjuncts (Gt w) = [Gt w] + | disjuncts (Le v) = [Le v] + | disjuncts (Lt u) = [Lt u] + | disjuncts T = [T] + | disjuncts F = [] + | disjuncts (Or (p, q)) = List.append (disjuncts p) (disjuncts q); -fun mod_def1 a b = snd (divAlg (a, b)); - -fun dvd m n = (mod_def1 n m = 0); - -fun abs i = (if (i < 0) then ~ i else i); - -fun less_def3 m n = ((m) < (n)); - -fun less_eq_def3 m n = Bool.not (less_def3 n m); +fun eq_num (C int) (C int') = ((int : IntInf.int) = int') + | eq_num (Bound nat) (Bound nat') = ((nat : IntInf.int) = nat') + | eq_num (Cx (int, num)) (Cx (int', num')) = + ((int : IntInf.int) = int') andalso eq_num num num' + | eq_num (Neg num) (Neg num') = eq_num num num' + | eq_num (Add (num1, num2)) (Add (num1', num2')) = + eq_num num1 num1' andalso eq_num num2 num2' + | eq_num (Sub (num1, num2)) (Sub (num1', num2')) = + eq_num num1 num1' andalso eq_num num2 num2' + | eq_num (Mul (int, num)) (Mul (int', num')) = + ((int : IntInf.int) = int') andalso eq_num num num' + | eq_num (C a) (Bound b) = false + | eq_num (C a) (Cx (b, c)) = false + | eq_num (C a) (Neg b) = false + | eq_num (C a) (Add (b, c)) = false + | eq_num (C a) (Sub (b, c)) = false + | eq_num (C a) (Mul (b, c)) = false + | eq_num (Bound a) (Cx (b, c)) = false + | eq_num (Bound a) (Neg b) = false + | eq_num (Bound a) (Add (b, c)) = false + | eq_num (Bound a) (Sub (b, c)) = false + | eq_num (Bound a) (Mul (b, c)) = false + | eq_num (Cx (a, b)) (Neg c) = false + | eq_num (Cx (a, b)) (Add (c, d)) = false + | eq_num (Cx (a, b)) (Sub (c, d)) = false + | eq_num (Cx (a, b)) (Mul (c, d)) = false + | eq_num (Neg a) (Add (b, c)) = false + | eq_num (Neg a) (Sub (b, c)) = false + | eq_num (Neg a) (Mul (b, c)) = false + | eq_num (Add (a, b)) (Sub (c, d)) = false + | eq_num (Add (a, b)) (Mul (c, d)) = false + | eq_num (Sub (a, b)) (Mul (c, d)) = false + | eq_num (Bound b) (C a) = false + | eq_num (Cx (b, c)) (C a) = false + | eq_num (Neg b) (C a) = false + | eq_num (Add (b, c)) (C a) = false + | eq_num (Sub (b, c)) (C a) = false + | eq_num (Mul (b, c)) (C a) = false + | eq_num (Cx (b, c)) (Bound a) = false + | eq_num (Neg b) (Bound a) = false + | eq_num (Add (b, c)) (Bound a) = false + | eq_num (Sub (b, c)) (Bound a) = false + | eq_num (Mul (b, c)) (Bound a) = false + | eq_num (Neg c) (Cx (a, b)) = false + | eq_num (Add (c, d)) (Cx (a, b)) = false + | eq_num (Sub (c, d)) (Cx (a, b)) = false + | eq_num (Mul (c, d)) (Cx (a, b)) = false + | eq_num (Add (b, c)) (Neg a) = false + | eq_num (Sub (b, c)) (Neg a) = false + | eq_num (Mul (b, c)) (Neg a) = false + | eq_num (Sub (c, d)) (Add (a, b)) = false + | eq_num (Mul (c, d)) (Add (a, b)) = false + | eq_num (Mul (c, d)) (Sub (a, b)) = false; -fun numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (c2, Bound n2), r2)) = - (if (n1 = n2) - then let val c = (c1 + c2) - in (if (c = 0) then numadd (r1, r2) - else Add (Mul (c, Bound n1), numadd (r1, r2))) - end - else (if less_eq_def3 n1 n2 - then Add (Mul (c1, Bound n1), - numadd (r1, Add (Mul (c2, Bound n2), r2))) - else Add (Mul (c2, Bound n2), - numadd (Add (Mul (c1, Bound n1), r1), r2)))) - | numadd (Add (Mul (c1, Bound n1), r1), C afq) = - Add (Mul (c1, Bound n1), numadd (r1, C afq)) - | numadd (Add (Mul (c1, Bound n1), r1), Bound afr) = - Add (Mul (c1, Bound n1), numadd (r1, Bound afr)) - | numadd (Add (Mul (c1, Bound n1), r1), CX (afs, aft)) = - Add (Mul (c1, Bound n1), numadd (r1, CX (afs, aft))) - | numadd (Add (Mul (c1, Bound n1), r1), Neg afu) = - Add (Mul (c1, Bound n1), numadd (r1, Neg afu)) - | numadd (Add (Mul (c1, Bound n1), r1), Add (C agx, afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (C agx, afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Add (Bound agy, afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Bound agy, afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Add (CX (agz, aha), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (CX (agz, aha), afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Add (Neg ahb, afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Neg ahb, afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Add (Add (ahc, ahd), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Add (ahc, ahd), afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Add (Sub (ahe, ahf), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Sub (ahe, ahf), afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, C aie), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, C aie), afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, CX (aig, aih)), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, CX (aig, aih)), afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Neg aii), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Neg aii), afw))) - | numadd - (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Add (aij, aik)), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Add (aij, aik)), afw))) - | numadd - (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Sub (ail, aim)), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Sub (ail, aim)), afw))) +fun eq_fm T T = true + | eq_fm F F = true + | eq_fm (Lt num) (Lt num') = eq_num num num' + | eq_fm (Le num) (Le num') = eq_num num num' + | eq_fm (Gt num) (Gt num') = eq_num num num' + | eq_fm (Ge num) (Ge num') = eq_num num num' + | eq_fm (Eq num) (Eq num') = eq_num num num' + | eq_fm (NEq num) (NEq num') = eq_num num num' + | eq_fm (Dvd (int, num)) (Dvd (int', num')) = + ((int : IntInf.int) = int') andalso eq_num num num' + | eq_fm (NDvd (int, num)) (NDvd (int', num')) = + ((int : IntInf.int) = int') andalso eq_num num num' + | eq_fm (Nota fm) (Nota fm') = eq_fm fm fm' + | eq_fm (And (fm1, fm2)) (And (fm1', fm2')) = + eq_fm fm1 fm1' andalso eq_fm fm2 fm2' + | eq_fm (Or (fm1, fm2)) (Or (fm1', fm2')) = + eq_fm fm1 fm1' andalso eq_fm fm2 fm2' + | eq_fm (Impa (fm1, fm2)) (Impa (fm1', fm2')) = + eq_fm fm1 fm1' andalso eq_fm fm2 fm2' + | eq_fm (Iffa (fm1, fm2)) (Iffa (fm1', fm2')) = + eq_fm fm1 fm1' andalso eq_fm fm2 fm2' + | eq_fm (E fm) (E fm') = eq_fm fm fm' + | eq_fm (A fm) (A fm') = eq_fm fm fm' + | eq_fm (Closed nat) (Closed nat') = ((nat : IntInf.int) = nat') + | eq_fm (NClosed nat) (NClosed nat') = ((nat : IntInf.int) = nat') + | eq_fm T F = false + | eq_fm T (Lt a) = false + | eq_fm T (Le a) = false + | eq_fm T (Gt a) = false + | eq_fm T (Ge a) = false + | eq_fm T (Eq a) = false + | eq_fm T (NEq a) = false + | eq_fm T (Dvd (a, b)) = false + | eq_fm T (NDvd (a, b)) = false + | eq_fm T (Nota a) = false + | eq_fm T (And (a, b)) = false + | eq_fm T (Or (a, b)) = false + | eq_fm T (Impa (a, b)) = false + | eq_fm T (Iffa (a, b)) = false + | eq_fm T (E a) = false + | eq_fm T (A a) = false + | eq_fm T (Closed a) = false + | eq_fm T (NClosed a) = false + | eq_fm F (Lt a) = false + | eq_fm F (Le a) = false + | eq_fm F (Gt a) = false + | eq_fm F (Ge a) = false + | eq_fm F (Eq a) = false + | eq_fm F (NEq a) = false + | eq_fm F (Dvd (a, b)) = false + | eq_fm F (NDvd (a, b)) = false + | eq_fm F (Nota a) = false + | eq_fm F (And (a, b)) = false + | eq_fm F (Or (a, b)) = false + | eq_fm F (Impa (a, b)) = false + | eq_fm F (Iffa (a, b)) = false + | eq_fm F (E a) = false + | eq_fm F (A a) = false + | eq_fm F (Closed a) = false + | eq_fm F (NClosed a) = false + | eq_fm (Lt a) (Le b) = false + | eq_fm (Lt a) (Gt b) = false + | eq_fm (Lt a) (Ge b) = false + | eq_fm (Lt a) (Eq b) = false + | eq_fm (Lt a) (NEq b) = false + | eq_fm (Lt a) (Dvd (b, c)) = false + | eq_fm (Lt a) (NDvd (b, c)) = false + | eq_fm (Lt a) (Nota b) = false + | eq_fm (Lt a) (And (b, c)) = false + | eq_fm (Lt a) (Or (b, c)) = false + | eq_fm (Lt a) (Impa (b, c)) = false + | eq_fm (Lt a) (Iffa (b, c)) = false + | eq_fm (Lt a) (E b) = false + | eq_fm (Lt a) (A b) = false + | eq_fm (Lt a) (Closed b) = false + | eq_fm (Lt a) (NClosed b) = false + | eq_fm (Le a) (Gt b) = false + | eq_fm (Le a) (Ge b) = false + | eq_fm (Le a) (Eq b) = false + | eq_fm (Le a) (NEq b) = false + | eq_fm (Le a) (Dvd (b, c)) = false + | eq_fm (Le a) (NDvd (b, c)) = false + | eq_fm (Le a) (Nota b) = false + | eq_fm (Le a) (And (b, c)) = false + | eq_fm (Le a) (Or (b, c)) = false + | eq_fm (Le a) (Impa (b, c)) = false + | eq_fm (Le a) (Iffa (b, c)) = false + | eq_fm (Le a) (E b) = false + | eq_fm (Le a) (A b) = false + | eq_fm (Le a) (Closed b) = false + | eq_fm (Le a) (NClosed b) = false + | eq_fm (Gt a) (Ge b) = false + | eq_fm (Gt a) (Eq b) = false + | eq_fm (Gt a) (NEq b) = false + | eq_fm (Gt a) (Dvd (b, c)) = false + | eq_fm (Gt a) (NDvd (b, c)) = false + | eq_fm (Gt a) (Nota b) = false + | eq_fm (Gt a) (And (b, c)) = false + | eq_fm (Gt a) (Or (b, c)) = false + | eq_fm (Gt a) (Impa (b, c)) = false + | eq_fm (Gt a) (Iffa (b, c)) = false + | eq_fm (Gt a) (E b) = false + | eq_fm (Gt a) (A b) = false + | eq_fm (Gt a) (Closed b) = false + | eq_fm (Gt a) (NClosed b) = false + | eq_fm (Ge a) (Eq b) = false + | eq_fm (Ge a) (NEq b) = false + | eq_fm (Ge a) (Dvd (b, c)) = false + | eq_fm (Ge a) (NDvd (b, c)) = false + | eq_fm (Ge a) (Nota b) = false + | eq_fm (Ge a) (And (b, c)) = false + | eq_fm (Ge a) (Or (b, c)) = false + | eq_fm (Ge a) (Impa (b, c)) = false + | eq_fm (Ge a) (Iffa (b, c)) = false + | eq_fm (Ge a) (E b) = false + | eq_fm (Ge a) (A b) = false + | eq_fm (Ge a) (Closed b) = false + | eq_fm (Ge a) (NClosed b) = false + | eq_fm (Eq a) (NEq b) = false + | eq_fm (Eq a) (Dvd (b, c)) = false + | eq_fm (Eq a) (NDvd (b, c)) = false + | eq_fm (Eq a) (Nota b) = false + | eq_fm (Eq a) (And (b, c)) = false + | eq_fm (Eq a) (Or (b, c)) = false + | eq_fm (Eq a) (Impa (b, c)) = false + | eq_fm (Eq a) (Iffa (b, c)) = false + | eq_fm (Eq a) (E b) = false + | eq_fm (Eq a) (A b) = false + | eq_fm (Eq a) (Closed b) = false + | eq_fm (Eq a) (NClosed b) = false + | eq_fm (NEq a) (Dvd (b, c)) = false + | eq_fm (NEq a) (NDvd (b, c)) = false + | eq_fm (NEq a) (Nota b) = false + | eq_fm (NEq a) (And (b, c)) = false + | eq_fm (NEq a) (Or (b, c)) = false + | eq_fm (NEq a) (Impa (b, c)) = false + | eq_fm (NEq a) (Iffa (b, c)) = false + | eq_fm (NEq a) (E b) = false + | eq_fm (NEq a) (A b) = false + | eq_fm (NEq a) (Closed b) = false + | eq_fm (NEq a) (NClosed b) = false + | eq_fm (Dvd (a, b)) (NDvd (c, d)) = false + | eq_fm (Dvd (a, b)) (Nota c) = false + | eq_fm (Dvd (a, b)) (And (c, d)) = false + | eq_fm (Dvd (a, b)) (Or (c, d)) = false + | eq_fm (Dvd (a, b)) (Impa (c, d)) = false + | eq_fm (Dvd (a, b)) (Iffa (c, d)) = false + | eq_fm (Dvd (a, b)) (E c) = false + | eq_fm (Dvd (a, b)) (A c) = false + | eq_fm (Dvd (a, b)) (Closed c) = false + | eq_fm (Dvd (a, b)) (NClosed c) = false + | eq_fm (NDvd (a, b)) (Nota c) = false + | eq_fm (NDvd (a, b)) (And (c, d)) = false + | eq_fm (NDvd (a, b)) (Or (c, d)) = false + | eq_fm (NDvd (a, b)) (Impa (c, d)) = false + | eq_fm (NDvd (a, b)) (Iffa (c, d)) = false + | eq_fm (NDvd (a, b)) (E c) = false + | eq_fm (NDvd (a, b)) (A c) = false + | eq_fm (NDvd (a, b)) (Closed c) = false + | eq_fm (NDvd (a, b)) (NClosed c) = false + | eq_fm (Nota a) (And (b, c)) = false + | eq_fm (Nota a) (Or (b, c)) = false + | eq_fm (Nota a) (Impa (b, c)) = false + | eq_fm (Nota a) (Iffa (b, c)) = false + | eq_fm (Nota a) (E b) = false + | eq_fm (Nota a) (A b) = false + | eq_fm (Nota a) (Closed b) = false + | eq_fm (Nota a) (NClosed b) = false + | eq_fm (And (a, b)) (Or (c, d)) = false + | eq_fm (And (a, b)) (Impa (c, d)) = false + | eq_fm (And (a, b)) (Iffa (c, d)) = false + | eq_fm (And (a, b)) (E c) = false + | eq_fm (And (a, b)) (A c) = false + | eq_fm (And (a, b)) (Closed c) = false + | eq_fm (And (a, b)) (NClosed c) = false + | eq_fm (Or (a, b)) (Impa (c, d)) = false + | eq_fm (Or (a, b)) (Iffa (c, d)) = false + | eq_fm (Or (a, b)) (E c) = false + | eq_fm (Or (a, b)) (A c) = false + | eq_fm (Or (a, b)) (Closed c) = false + | eq_fm (Or (a, b)) (NClosed c) = false + | eq_fm (Impa (a, b)) (Iffa (c, d)) = false + | eq_fm (Impa (a, b)) (E c) = false + | eq_fm (Impa (a, b)) (A c) = false + | eq_fm (Impa (a, b)) (Closed c) = false + | eq_fm (Impa (a, b)) (NClosed c) = false + | eq_fm (Iffa (a, b)) (E c) = false + | eq_fm (Iffa (a, b)) (A c) = false + | eq_fm (Iffa (a, b)) (Closed c) = false + | eq_fm (Iffa (a, b)) (NClosed c) = false + | eq_fm (E a) (A b) = false + | eq_fm (E a) (Closed b) = false + | eq_fm (E a) (NClosed b) = false + | eq_fm (A a) (Closed b) = false + | eq_fm (A a) (NClosed b) = false + | eq_fm (Closed a) (NClosed b) = false + | eq_fm F T = false + | eq_fm (Lt a) T = false + | eq_fm (Le a) T = false + | eq_fm (Gt a) T = false + | eq_fm (Ge a) T = false + | eq_fm (Eq a) T = false + | eq_fm (NEq a) T = false + | eq_fm (Dvd (a, b)) T = false + | eq_fm (NDvd (a, b)) T = false + | eq_fm (Nota a) T = false + | eq_fm (And (a, b)) T = false + | eq_fm (Or (a, b)) T = false + | eq_fm (Impa (a, b)) T = false + | eq_fm (Iffa (a, b)) T = false + | eq_fm (E a) T = false + | eq_fm (A a) T = false + | eq_fm (Closed a) T = false + | eq_fm (NClosed a) T = false + | eq_fm (Lt a) F = false + | eq_fm (Le a) F = false + | eq_fm (Gt a) F = false + | eq_fm (Ge a) F = false + | eq_fm (Eq a) F = false + | eq_fm (NEq a) F = false + | eq_fm (Dvd (a, b)) F = false + | eq_fm (NDvd (a, b)) F = false + | eq_fm (Nota a) F = false + | eq_fm (And (a, b)) F = false + | eq_fm (Or (a, b)) F = false + | eq_fm (Impa (a, b)) F = false + | eq_fm (Iffa (a, b)) F = false + | eq_fm (E a) F = false + | eq_fm (A a) F = false + | eq_fm (Closed a) F = false + | eq_fm (NClosed a) F = false + | eq_fm (Le b) (Lt a) = false + | eq_fm (Gt b) (Lt a) = false + | eq_fm (Ge b) (Lt a) = false + | eq_fm (Eq b) (Lt a) = false + | eq_fm (NEq b) (Lt a) = false + | eq_fm (Dvd (b, c)) (Lt a) = false + | eq_fm (NDvd (b, c)) (Lt a) = false + | eq_fm (Nota b) (Lt a) = false + | eq_fm (And (b, c)) (Lt a) = false + | eq_fm (Or (b, c)) (Lt a) = false + | eq_fm (Impa (b, c)) (Lt a) = false + | eq_fm (Iffa (b, c)) (Lt a) = false + | eq_fm (E b) (Lt a) = false + | eq_fm (A b) (Lt a) = false + | eq_fm (Closed b) (Lt a) = false + | eq_fm (NClosed b) (Lt a) = false + | eq_fm (Gt b) (Le a) = false + | eq_fm (Ge b) (Le a) = false + | eq_fm (Eq b) (Le a) = false + | eq_fm (NEq b) (Le a) = false + | eq_fm (Dvd (b, c)) (Le a) = false + | eq_fm (NDvd (b, c)) (Le a) = false + | eq_fm (Nota b) (Le a) = false + | eq_fm (And (b, c)) (Le a) = false + | eq_fm (Or (b, c)) (Le a) = false + | eq_fm (Impa (b, c)) (Le a) = false + | eq_fm (Iffa (b, c)) (Le a) = false + | eq_fm (E b) (Le a) = false + | eq_fm (A b) (Le a) = false + | eq_fm (Closed b) (Le a) = false + | eq_fm (NClosed b) (Le a) = false + | eq_fm (Ge b) (Gt a) = false + | eq_fm (Eq b) (Gt a) = false + | eq_fm (NEq b) (Gt a) = false + | eq_fm (Dvd (b, c)) (Gt a) = false + | eq_fm (NDvd (b, c)) (Gt a) = false + | eq_fm (Nota b) (Gt a) = false + | eq_fm (And (b, c)) (Gt a) = false + | eq_fm (Or (b, c)) (Gt a) = false + | eq_fm (Impa (b, c)) (Gt a) = false + | eq_fm (Iffa (b, c)) (Gt a) = false + | eq_fm (E b) (Gt a) = false + | eq_fm (A b) (Gt a) = false + | eq_fm (Closed b) (Gt a) = false + | eq_fm (NClosed b) (Gt a) = false + | eq_fm (Eq b) (Ge a) = false + | eq_fm (NEq b) (Ge a) = false + | eq_fm (Dvd (b, c)) (Ge a) = false + | eq_fm (NDvd (b, c)) (Ge a) = false + | eq_fm (Nota b) (Ge a) = false + | eq_fm (And (b, c)) (Ge a) = false + | eq_fm (Or (b, c)) (Ge a) = false + | eq_fm (Impa (b, c)) (Ge a) = false + | eq_fm (Iffa (b, c)) (Ge a) = false + | eq_fm (E b) (Ge a) = false + | eq_fm (A b) (Ge a) = false + | eq_fm (Closed b) (Ge a) = false + | eq_fm (NClosed b) (Ge a) = false + | eq_fm (NEq b) (Eq a) = false + | eq_fm (Dvd (b, c)) (Eq a) = false + | eq_fm (NDvd (b, c)) (Eq a) = false + | eq_fm (Nota b) (Eq a) = false + | eq_fm (And (b, c)) (Eq a) = false + | eq_fm (Or (b, c)) (Eq a) = false + | eq_fm (Impa (b, c)) (Eq a) = false + | eq_fm (Iffa (b, c)) (Eq a) = false + | eq_fm (E b) (Eq a) = false + | eq_fm (A b) (Eq a) = false + | eq_fm (Closed b) (Eq a) = false + | eq_fm (NClosed b) (Eq a) = false + | eq_fm (Dvd (b, c)) (NEq a) = false + | eq_fm (NDvd (b, c)) (NEq a) = false + | eq_fm (Nota b) (NEq a) = false + | eq_fm (And (b, c)) (NEq a) = false + | eq_fm (Or (b, c)) (NEq a) = false + | eq_fm (Impa (b, c)) (NEq a) = false + | eq_fm (Iffa (b, c)) (NEq a) = false + | eq_fm (E b) (NEq a) = false + | eq_fm (A b) (NEq a) = false + | eq_fm (Closed b) (NEq a) = false + | eq_fm (NClosed b) (NEq a) = false + | eq_fm (NDvd (c, d)) (Dvd (a, b)) = false + | eq_fm (Nota c) (Dvd (a, b)) = false + | eq_fm (And (c, d)) (Dvd (a, b)) = false + | eq_fm (Or (c, d)) (Dvd (a, b)) = false + | eq_fm (Impa (c, d)) (Dvd (a, b)) = false + | eq_fm (Iffa (c, d)) (Dvd (a, b)) = false + | eq_fm (E c) (Dvd (a, b)) = false + | eq_fm (A c) (Dvd (a, b)) = false + | eq_fm (Closed c) (Dvd (a, b)) = false + | eq_fm (NClosed c) (Dvd (a, b)) = false + | eq_fm (Nota c) (NDvd (a, b)) = false + | eq_fm (And (c, d)) (NDvd (a, b)) = false + | eq_fm (Or (c, d)) (NDvd (a, b)) = false + | eq_fm (Impa (c, d)) (NDvd (a, b)) = false + | eq_fm (Iffa (c, d)) (NDvd (a, b)) = false + | eq_fm (E c) (NDvd (a, b)) = false + | eq_fm (A c) (NDvd (a, b)) = false + | eq_fm (Closed c) (NDvd (a, b)) = false + | eq_fm (NClosed c) (NDvd (a, b)) = false + | eq_fm (And (b, c)) (Nota a) = false + | eq_fm (Or (b, c)) (Nota a) = false + | eq_fm (Impa (b, c)) (Nota a) = false + | eq_fm (Iffa (b, c)) (Nota a) = false + | eq_fm (E b) (Nota a) = false + | eq_fm (A b) (Nota a) = false + | eq_fm (Closed b) (Nota a) = false + | eq_fm (NClosed b) (Nota a) = false + | eq_fm (Or (c, d)) (And (a, b)) = false + | eq_fm (Impa (c, d)) (And (a, b)) = false + | eq_fm (Iffa (c, d)) (And (a, b)) = false + | eq_fm (E c) (And (a, b)) = false + | eq_fm (A c) (And (a, b)) = false + | eq_fm (Closed c) (And (a, b)) = false + | eq_fm (NClosed c) (And (a, b)) = false + | eq_fm (Impa (c, d)) (Or (a, b)) = false + | eq_fm (Iffa (c, d)) (Or (a, b)) = false + | eq_fm (E c) (Or (a, b)) = false + | eq_fm (A c) (Or (a, b)) = false + | eq_fm (Closed c) (Or (a, b)) = false + | eq_fm (NClosed c) (Or (a, b)) = false + | eq_fm (Iffa (c, d)) (Impa (a, b)) = false + | eq_fm (E c) (Impa (a, b)) = false + | eq_fm (A c) (Impa (a, b)) = false + | eq_fm (Closed c) (Impa (a, b)) = false + | eq_fm (NClosed c) (Impa (a, b)) = false + | eq_fm (E c) (Iffa (a, b)) = false + | eq_fm (A c) (Iffa (a, b)) = false + | eq_fm (Closed c) (Iffa (a, b)) = false + | eq_fm (NClosed c) (Iffa (a, b)) = false + | eq_fm (A b) (E a) = false + | eq_fm (Closed b) (E a) = false + | eq_fm (NClosed b) (E a) = false + | eq_fm (Closed b) (A a) = false + | eq_fm (NClosed b) (A a) = false + | eq_fm (NClosed b) (Closed a) = false; + +fun djf f p q = + (if eq_fm q T then T + else (if eq_fm q F then f p + else (case f p of T => T | F => q | Lt num => Or (f p, q) + | Le num => Or (f p, q) | Gt num => Or (f p, q) + | Ge num => Or (f p, q) | Eq num => Or (f p, q) + | NEq num => Or (f p, q) | Dvd (int, num) => Or (f p, q) + | NDvd (int, num) => Or (f p, q) | Nota fm => Or (f p, q) + | And (fm1, fm2) => Or (f p, q) + | Or (fm1, fm2) => Or (f p, q) + | Impa (fm1, fm2) => Or (f p, q) + | Iffa (fm1, fm2) => Or (f p, q) | E fm => Or (f p, q) + | A fm => Or (f p, q) | Closed nat => Or (f p, q) + | NClosed nat => Or (f p, q)))); + +fun evaldjf f ps = List.foldr (djf f) ps F; + +fun dj f p = evaldjf f (disjuncts p); + +fun zsplit0 (Mul (i, a)) = + let + val (i', a') = zsplit0 a; + in + (IntInf.* (i, i'), Mul (i, a')) + end + | zsplit0 (Sub (a, b)) = + let + val (ia, a') = zsplit0 a; + val (ib, b') = zsplit0 b; + in + (IntInf.- (ia, ib), Sub (a', b')) + end + | zsplit0 (Add (a, b)) = + let + val (ia, a') = zsplit0 a; + val (ib, b') = zsplit0 b; + in + (IntInf.+ (ia, ib), Add (a', b')) + end + | zsplit0 (Neg a) = + let + val (i', a') = zsplit0 a; + in + (IntInf.~ i', Neg a') + end + | zsplit0 (Cx (i, a)) = + let + val (i', aa) = zsplit0 a; + in + (IntInf.+ (i, i'), aa) + end + | zsplit0 (Bound n) = + (if ((n : IntInf.int) = Integer.zero_nat) + then ((1 : IntInf.int), C (0 : IntInf.int)) + else ((0 : IntInf.int), Bound n)) + | zsplit0 (C c) = ((0 : IntInf.int), C c); + +fun zlfm (NClosed ar) = NClosed ar + | zlfm (Closed aq) = Closed aq + | zlfm (A ap) = A ap + | zlfm (E ao) = E ao + | zlfm (Nota (A cj)) = Nota (A cj) + | zlfm (Nota (E ci)) = Nota (E ci) + | zlfm F = F + | zlfm T = T + | zlfm (Nota (NClosed p)) = Closed p + | zlfm (Nota (Closed p)) = NClosed p + | zlfm (Nota (NDvd (i, a))) = zlfm (Dvd (i, a)) + | zlfm (Nota (Dvd (i, a))) = zlfm (NDvd (i, a)) + | zlfm (Nota (NEq a)) = zlfm (Eq a) + | zlfm (Nota (Eq a)) = zlfm (NEq a) + | zlfm (Nota (Ge a)) = zlfm (Lt a) + | zlfm (Nota (Gt a)) = zlfm (Le a) + | zlfm (Nota (Le a)) = zlfm (Gt a) + | zlfm (Nota (Lt a)) = zlfm (Ge a) + | zlfm (Nota F) = T + | zlfm (Nota T) = F + | zlfm (Nota (Nota p)) = zlfm p + | zlfm (Nota (Iffa (p, q))) = + Or (And (zlfm p, zlfm (Nota q)), And (zlfm (Nota p), zlfm q)) + | zlfm (Nota (Impa (p, q))) = And (zlfm p, zlfm (Nota q)) + | zlfm (Nota (Or (p, q))) = And (zlfm (Nota p), zlfm (Nota q)) + | zlfm (Nota (And (p, q))) = Or (zlfm (Nota p), zlfm (Nota q)) + | zlfm (NDvd (i, a)) = + (if ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (NEq a) + else let + val (c, r) = zsplit0 a; + in + (if ((c : IntInf.int) = (0 : IntInf.int)) + then NDvd (Integer.abs_int i, r) + else (if IntInf.< ((0 : IntInf.int), c) + then NDvd (Integer.abs_int i, Cx (c, r)) + else NDvd (Integer.abs_int i, Cx (IntInf.~ c, Neg r)))) + end) + | zlfm (Dvd (i, a)) = + (if ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (Eq a) + else let + val (c, r) = zsplit0 a; + in + (if ((c : IntInf.int) = (0 : IntInf.int)) + then Dvd (Integer.abs_int i, r) + else (if IntInf.< ((0 : IntInf.int), c) + then Dvd (Integer.abs_int i, Cx (c, r)) + else Dvd (Integer.abs_int i, Cx (IntInf.~ c, Neg r)))) + end) + | zlfm (NEq a) = + let + val (c, r) = zsplit0 a; + in + (if ((c : IntInf.int) = (0 : IntInf.int)) then NEq r + else (if IntInf.< ((0 : IntInf.int), c) then NEq (Cx (c, r)) + else NEq (Cx (IntInf.~ c, Neg r)))) + end + | zlfm (Eq a) = + let + val (c, r) = zsplit0 a; + in + (if ((c : IntInf.int) = (0 : IntInf.int)) then Eq r + else (if IntInf.< ((0 : IntInf.int), c) then Eq (Cx (c, r)) + else Eq (Cx (IntInf.~ c, Neg r)))) + end + | zlfm (Ge a) = + let + val (c, r) = zsplit0 a; + in + (if ((c : IntInf.int) = (0 : IntInf.int)) then Ge r + else (if IntInf.< ((0 : IntInf.int), c) then Ge (Cx (c, r)) + else Le (Cx (IntInf.~ c, Neg r)))) + end + | zlfm (Gt a) = + let + val (c, r) = zsplit0 a; + in + (if ((c : IntInf.int) = (0 : IntInf.int)) then Gt r + else (if IntInf.< ((0 : IntInf.int), c) then Gt (Cx (c, r)) + else Lt (Cx (IntInf.~ c, Neg r)))) + end + | zlfm (Le a) = + let + val (c, r) = zsplit0 a; + in + (if ((c : IntInf.int) = (0 : IntInf.int)) then Le r + else (if IntInf.< ((0 : IntInf.int), c) then Le (Cx (c, r)) + else Ge (Cx (IntInf.~ c, Neg r)))) + end + | zlfm (Lt a) = + let + val (c, r) = zsplit0 a; + in + (if ((c : IntInf.int) = (0 : IntInf.int)) then Lt r + else (if IntInf.< ((0 : IntInf.int), c) then Lt (Cx (c, r)) + else Gt (Cx (IntInf.~ c, Neg r)))) + end + | zlfm (Iffa (p, q)) = + Or (And (zlfm p, zlfm q), And (zlfm (Nota p), zlfm (Nota q))) + | zlfm (Impa (p, q)) = Or (zlfm (Nota p), zlfm q) + | zlfm (Or (p, q)) = Or (zlfm p, zlfm q) + | zlfm (And (p, q)) = And (zlfm p, zlfm q); + +fun zeta (NClosed aq) = (1 : IntInf.int) + | zeta (Closed ap) = (1 : IntInf.int) + | zeta (A ao) = (1 : IntInf.int) + | zeta (E an) = (1 : IntInf.int) + | zeta (Iffa (al, am)) = (1 : IntInf.int) + | zeta (Impa (aj, ak)) = (1 : IntInf.int) + | zeta (Nota ae) = (1 : IntInf.int) + | zeta (NDvd (ac, Mul (hv, hw))) = (1 : IntInf.int) + | zeta (NDvd (ac, Sub (ht, hu))) = (1 : IntInf.int) + | zeta (NDvd (ac, Add (hr, hs))) = (1 : IntInf.int) + | zeta (NDvd (ac, Neg hq)) = (1 : IntInf.int) + | zeta (NDvd (ac, Bound hn)) = (1 : IntInf.int) + | zeta (NDvd (ac, C hm)) = (1 : IntInf.int) + | zeta (Dvd (aa, Mul (gz, ha))) = (1 : IntInf.int) + | zeta (Dvd (aa, Sub (gx, gy))) = (1 : IntInf.int) + | zeta (Dvd (aa, Add (gv, gw))) = (1 : IntInf.int) + | zeta (Dvd (aa, Neg gu)) = (1 : IntInf.int) + | zeta (Dvd (aa, Bound gr)) = (1 : IntInf.int) + | zeta (Dvd (aa, C gq)) = (1 : IntInf.int) + | zeta (NEq (Mul (gd, ge))) = (1 : IntInf.int) + | zeta (NEq (Sub (gb, gc))) = (1 : IntInf.int) + | zeta (NEq (Add (fz, ga))) = (1 : IntInf.int) + | zeta (NEq (Neg fy)) = (1 : IntInf.int) + | zeta (NEq (Bound fv)) = (1 : IntInf.int) + | zeta (NEq (C fu)) = (1 : IntInf.int) + | zeta (Eq (Mul (fh, fi))) = (1 : IntInf.int) + | zeta (Eq (Sub (ff, fg))) = (1 : IntInf.int) + | zeta (Eq (Add (fd, fe))) = (1 : IntInf.int) + | zeta (Eq (Neg fc)) = (1 : IntInf.int) + | zeta (Eq (Bound ez)) = (1 : IntInf.int) + | zeta (Eq (C ey)) = (1 : IntInf.int) + | zeta (Ge (Mul (el, em))) = (1 : IntInf.int) + | zeta (Ge (Sub (ej, ek))) = (1 : IntInf.int) + | zeta (Ge (Add (eh, ei))) = (1 : IntInf.int) + | zeta (Ge (Neg eg)) = (1 : IntInf.int) + | zeta (Ge (Bound ed)) = (1 : IntInf.int) + | zeta (Ge (C ec)) = (1 : IntInf.int) + | zeta (Gt (Mul (dp, dq))) = (1 : IntInf.int) + | zeta (Gt (Sub (dn, doa))) = (1 : IntInf.int) + | zeta (Gt (Add (dl, dm))) = (1 : IntInf.int) + | zeta (Gt (Neg dk)) = (1 : IntInf.int) + | zeta (Gt (Bound dh)) = (1 : IntInf.int) + | zeta (Gt (C dg)) = (1 : IntInf.int) + | zeta (Le (Mul (ct, cu))) = (1 : IntInf.int) + | zeta (Le (Sub (cr, cs))) = (1 : IntInf.int) + | zeta (Le (Add (cp, cq))) = (1 : IntInf.int) + | zeta (Le (Neg co)) = (1 : IntInf.int) + | zeta (Le (Bound cl)) = (1 : IntInf.int) + | zeta (Le (C ck)) = (1 : IntInf.int) + | zeta (Lt (Mul (bx, by))) = (1 : IntInf.int) + | zeta (Lt (Sub (bv, bw))) = (1 : IntInf.int) + | zeta (Lt (Add (bt, bu))) = (1 : IntInf.int) + | zeta (Lt (Neg bs)) = (1 : IntInf.int) + | zeta (Lt (Bound bp)) = (1 : IntInf.int) + | zeta (Lt (C bo)) = (1 : IntInf.int) + | zeta F = (1 : IntInf.int) + | zeta T = (1 : IntInf.int) + | zeta (NDvd (i, Cx (y, e))) = y + | zeta (Dvd (i, Cx (y, e))) = y + | zeta (Ge (Cx (y, e))) = y + | zeta (Gt (Cx (y, e))) = y + | zeta (Le (Cx (y, e))) = y + | zeta (Lt (Cx (y, e))) = y + | zeta (NEq (Cx (y, e))) = y + | zeta (Eq (Cx (y, e))) = y + | zeta (Or (p, q)) = GCD.ilcm (zeta p) (zeta q) + | zeta (And (p, q)) = GCD.ilcm (zeta p) (zeta q); + +fun a_beta (NClosed aq) = (fn k => NClosed aq) + | a_beta (Closed ap) = (fn k => Closed ap) + | a_beta (A ao) = (fn k => A ao) + | a_beta (E an) = (fn k => E an) + | a_beta (Iffa (al, am)) = (fn k => Iffa (al, am)) + | a_beta (Impa (aj, ak)) = (fn k => Impa (aj, ak)) + | a_beta (Nota ae) = (fn k => Nota ae) + | a_beta (NDvd (ac, Mul (hv, hw))) = (fn k => NDvd (ac, Mul (hv, hw))) + | a_beta (NDvd (ac, Sub (ht, hu))) = (fn k => NDvd (ac, Sub (ht, hu))) + | a_beta (NDvd (ac, Add (hr, hs))) = (fn k => NDvd (ac, Add (hr, hs))) + | a_beta (NDvd (ac, Neg hq)) = (fn k => NDvd (ac, Neg hq)) + | a_beta (NDvd (ac, Bound hn)) = (fn k => NDvd (ac, Bound hn)) + | a_beta (NDvd (ac, C hm)) = (fn k => NDvd (ac, C hm)) + | a_beta (Dvd (aa, Mul (gz, ha))) = (fn k => Dvd (aa, Mul (gz, ha))) + | a_beta (Dvd (aa, Sub (gx, gy))) = (fn k => Dvd (aa, Sub (gx, gy))) + | a_beta (Dvd (aa, Add (gv, gw))) = (fn k => Dvd (aa, Add (gv, gw))) + | a_beta (Dvd (aa, Neg gu)) = (fn k => Dvd (aa, Neg gu)) + | a_beta (Dvd (aa, Bound gr)) = (fn k => Dvd (aa, Bound gr)) + | a_beta (Dvd (aa, C gq)) = (fn k => Dvd (aa, C gq)) + | a_beta (NEq (Mul (gd, ge))) = (fn k => NEq (Mul (gd, ge))) + | a_beta (NEq (Sub (gb, gc))) = (fn k => NEq (Sub (gb, gc))) + | a_beta (NEq (Add (fz, ga))) = (fn k => NEq (Add (fz, ga))) + | a_beta (NEq (Neg fy)) = (fn k => NEq (Neg fy)) + | a_beta (NEq (Bound fv)) = (fn k => NEq (Bound fv)) + | a_beta (NEq (C fu)) = (fn k => NEq (C fu)) + | a_beta (Eq (Mul (fh, fi))) = (fn k => Eq (Mul (fh, fi))) + | a_beta (Eq (Sub (ff, fg))) = (fn k => Eq (Sub (ff, fg))) + | a_beta (Eq (Add (fd, fe))) = (fn k => Eq (Add (fd, fe))) + | a_beta (Eq (Neg fc)) = (fn k => Eq (Neg fc)) + | a_beta (Eq (Bound ez)) = (fn k => Eq (Bound ez)) + | a_beta (Eq (C ey)) = (fn k => Eq (C ey)) + | a_beta (Ge (Mul (el, em))) = (fn k => Ge (Mul (el, em))) + | a_beta (Ge (Sub (ej, ek))) = (fn k => Ge (Sub (ej, ek))) + | a_beta (Ge (Add (eh, ei))) = (fn k => Ge (Add (eh, ei))) + | a_beta (Ge (Neg eg)) = (fn k => Ge (Neg eg)) + | a_beta (Ge (Bound ed)) = (fn k => Ge (Bound ed)) + | a_beta (Ge (C ec)) = (fn k => Ge (C ec)) + | a_beta (Gt (Mul (dp, dq))) = (fn k => Gt (Mul (dp, dq))) + | a_beta (Gt (Sub (dn, doa))) = (fn k => Gt (Sub (dn, doa))) + | a_beta (Gt (Add (dl, dm))) = (fn k => Gt (Add (dl, dm))) + | a_beta (Gt (Neg dk)) = (fn k => Gt (Neg dk)) + | a_beta (Gt (Bound dh)) = (fn k => Gt (Bound dh)) + | a_beta (Gt (C dg)) = (fn k => Gt (C dg)) + | a_beta (Le (Mul (ct, cu))) = (fn k => Le (Mul (ct, cu))) + | a_beta (Le (Sub (cr, cs))) = (fn k => Le (Sub (cr, cs))) + | a_beta (Le (Add (cp, cq))) = (fn k => Le (Add (cp, cq))) + | a_beta (Le (Neg co)) = (fn k => Le (Neg co)) + | a_beta (Le (Bound cl)) = (fn k => Le (Bound cl)) + | a_beta (Le (C ck)) = (fn k => Le (C ck)) + | a_beta (Lt (Mul (bx, by))) = (fn k => Lt (Mul (bx, by))) + | a_beta (Lt (Sub (bv, bw))) = (fn k => Lt (Sub (bv, bw))) + | a_beta (Lt (Add (bt, bu))) = (fn k => Lt (Add (bt, bu))) + | a_beta (Lt (Neg bs)) = (fn k => Lt (Neg bs)) + | a_beta (Lt (Bound bp)) = (fn k => Lt (Bound bp)) + | a_beta (Lt (C bo)) = (fn k => Lt (C bo)) + | a_beta F = (fn k => F) + | a_beta T = (fn k => T) + | a_beta (NDvd (i, Cx (c, e))) = + (fn k => + NDvd (IntInf.* (Integer.div_int k c, i), + Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e)))) + | a_beta (Dvd (i, Cx (c, e))) = + (fn k => + Dvd (IntInf.* (Integer.div_int k c, i), + Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e)))) + | a_beta (Ge (Cx (c, e))) = + (fn k => Ge (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e)))) + | a_beta (Gt (Cx (c, e))) = + (fn k => Gt (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e)))) + | a_beta (Le (Cx (c, e))) = + (fn k => Le (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e)))) + | a_beta (Lt (Cx (c, e))) = + (fn k => Lt (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e)))) + | a_beta (NEq (Cx (c, e))) = + (fn k => NEq (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e)))) + | a_beta (Eq (Cx (c, e))) = + (fn k => Eq (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e)))) + | a_beta (Or (p, q)) = (fn k => Or (a_beta p k, a_beta q k)) + | a_beta (And (p, q)) = (fn k => And (a_beta p k, a_beta q k)); + +fun delta (NClosed aq) = (1 : IntInf.int) + | delta (Closed ap) = (1 : IntInf.int) + | delta (A ao) = (1 : IntInf.int) + | delta (E an) = (1 : IntInf.int) + | delta (Iffa (al, am)) = (1 : IntInf.int) + | delta (Impa (aj, ak)) = (1 : IntInf.int) + | delta (Nota ae) = (1 : IntInf.int) + | delta (NDvd (ac, Mul (ct, cu))) = (1 : IntInf.int) + | delta (NDvd (ac, Sub (cr, cs))) = (1 : IntInf.int) + | delta (NDvd (ac, Add (cp, cq))) = (1 : IntInf.int) + | delta (NDvd (ac, Neg co)) = (1 : IntInf.int) + | delta (NDvd (ac, Bound cl)) = (1 : IntInf.int) + | delta (NDvd (ac, C ck)) = (1 : IntInf.int) + | delta (Dvd (aa, Mul (bx, by))) = (1 : IntInf.int) + | delta (Dvd (aa, Sub (bv, bw))) = (1 : IntInf.int) + | delta (Dvd (aa, Add (bt, bu))) = (1 : IntInf.int) + | delta (Dvd (aa, Neg bs)) = (1 : IntInf.int) + | delta (Dvd (aa, Bound bp)) = (1 : IntInf.int) + | delta (Dvd (aa, C bo)) = (1 : IntInf.int) + | delta (NEq z) = (1 : IntInf.int) + | delta (Eq y) = (1 : IntInf.int) + | delta (Ge x) = (1 : IntInf.int) + | delta (Gt w) = (1 : IntInf.int) + | delta (Le v) = (1 : IntInf.int) + | delta (Lt u) = (1 : IntInf.int) + | delta F = (1 : IntInf.int) + | delta T = (1 : IntInf.int) + | delta (NDvd (y, Cx (c, e))) = y + | delta (Dvd (y, Cx (c, e))) = y + | delta (Or (p, q)) = GCD.ilcm (delta p) (delta q) + | delta (And (p, q)) = GCD.ilcm (delta p) (delta q); + +fun beta (NClosed aq) = [] + | beta (Closed ap) = [] + | beta (A ao) = [] + | beta (E an) = [] + | beta (Iffa (al, am)) = [] + | beta (Impa (aj, ak)) = [] + | beta (Nota ae) = [] + | beta (NDvd (ac, ad)) = [] + | beta (Dvd (aa, ab)) = [] + | beta (NEq (Mul (gd, ge))) = [] + | beta (NEq (Sub (gb, gc))) = [] + | beta (NEq (Add (fz, ga))) = [] + | beta (NEq (Neg fy)) = [] + | beta (NEq (Bound fv)) = [] + | beta (NEq (C fu)) = [] + | beta (Eq (Mul (fh, fi))) = [] + | beta (Eq (Sub (ff, fg))) = [] + | beta (Eq (Add (fd, fe))) = [] + | beta (Eq (Neg fc)) = [] + | beta (Eq (Bound ez)) = [] + | beta (Eq (C ey)) = [] + | beta (Ge (Mul (el, em))) = [] + | beta (Ge (Sub (ej, ek))) = [] + | beta (Ge (Add (eh, ei))) = [] + | beta (Ge (Neg eg)) = [] + | beta (Ge (Bound ed)) = [] + | beta (Ge (C ec)) = [] + | beta (Gt (Mul (dp, dq))) = [] + | beta (Gt (Sub (dn, doa))) = [] + | beta (Gt (Add (dl, dm))) = [] + | beta (Gt (Neg dk)) = [] + | beta (Gt (Bound dh)) = [] + | beta (Gt (C dg)) = [] + | beta (Le (Mul (ct, cu))) = [] + | beta (Le (Sub (cr, cs))) = [] + | beta (Le (Add (cp, cq))) = [] + | beta (Le (Neg co)) = [] + | beta (Le (Bound cl)) = [] + | beta (Le (C ck)) = [] + | beta (Lt (Mul (bx, by))) = [] + | beta (Lt (Sub (bv, bw))) = [] + | beta (Lt (Add (bt, bu))) = [] + | beta (Lt (Neg bs)) = [] + | beta (Lt (Bound bp)) = [] + | beta (Lt (C bo)) = [] + | beta F = [] + | beta T = [] + | beta (Ge (Cx (c, e))) = [Sub (C (~1 : IntInf.int), e)] + | beta (Gt (Cx (c, e))) = [Neg e] + | beta (Le (Cx (c, e))) = [] + | beta (Lt (Cx (c, e))) = [] + | beta (NEq (Cx (c, e))) = [Neg e] + | beta (Eq (Cx (c, e))) = [Sub (C (~1 : IntInf.int), e)] + | beta (Or (p, q)) = List.append (beta p) (beta q) + | beta (And (p, q)) = List.append (beta p) (beta q); + +fun alpha (NClosed aq) = [] + | alpha (Closed ap) = [] + | alpha (A ao) = [] + | alpha (E an) = [] + | alpha (Iffa (al, am)) = [] + | alpha (Impa (aj, ak)) = [] + | alpha (Nota ae) = [] + | alpha (NDvd (ac, ad)) = [] + | alpha (Dvd (aa, ab)) = [] + | alpha (NEq (Mul (gd, ge))) = [] + | alpha (NEq (Sub (gb, gc))) = [] + | alpha (NEq (Add (fz, ga))) = [] + | alpha (NEq (Neg fy)) = [] + | alpha (NEq (Bound fv)) = [] + | alpha (NEq (C fu)) = [] + | alpha (Eq (Mul (fh, fi))) = [] + | alpha (Eq (Sub (ff, fg))) = [] + | alpha (Eq (Add (fd, fe))) = [] + | alpha (Eq (Neg fc)) = [] + | alpha (Eq (Bound ez)) = [] + | alpha (Eq (C ey)) = [] + | alpha (Ge (Mul (el, em))) = [] + | alpha (Ge (Sub (ej, ek))) = [] + | alpha (Ge (Add (eh, ei))) = [] + | alpha (Ge (Neg eg)) = [] + | alpha (Ge (Bound ed)) = [] + | alpha (Ge (C ec)) = [] + | alpha (Gt (Mul (dp, dq))) = [] + | alpha (Gt (Sub (dn, doa))) = [] + | alpha (Gt (Add (dl, dm))) = [] + | alpha (Gt (Neg dk)) = [] + | alpha (Gt (Bound dh)) = [] + | alpha (Gt (C dg)) = [] + | alpha (Le (Mul (ct, cu))) = [] + | alpha (Le (Sub (cr, cs))) = [] + | alpha (Le (Add (cp, cq))) = [] + | alpha (Le (Neg co)) = [] + | alpha (Le (Bound cl)) = [] + | alpha (Le (C ck)) = [] + | alpha (Lt (Mul (bx, by))) = [] + | alpha (Lt (Sub (bv, bw))) = [] + | alpha (Lt (Add (bt, bu))) = [] + | alpha (Lt (Neg bs)) = [] + | alpha (Lt (Bound bp)) = [] + | alpha (Lt (C bo)) = [] + | alpha F = [] + | alpha T = [] + | alpha (Ge (Cx (c, e))) = [] + | alpha (Gt (Cx (c, e))) = [] + | alpha (Le (Cx (c, e))) = [Add (C (~1 : IntInf.int), e)] + | alpha (Lt (Cx (c, e))) = [e] + | alpha (NEq (Cx (c, e))) = [e] + | alpha (Eq (Cx (c, e))) = [Add (C (~1 : IntInf.int), e)] + | alpha (Or (p, q)) = List.append (alpha p) (alpha q) + | alpha (And (p, q)) = List.append (alpha p) (alpha q); + +fun numadd (Mul (ar, asa), Mul (aza, azb)) = Add (Mul (ar, asa), Mul (aza, azb)) + | numadd (Mul (ar, asa), Sub (ayy, ayz)) = Add (Mul (ar, asa), Sub (ayy, ayz)) + | numadd (Mul (ar, asa), Add (Mul (azw, Mul (bas, bat)), ayx)) = + Add (Mul (ar, asa), Add (Mul (azw, Mul (bas, bat)), ayx)) + | numadd (Mul (ar, asa), Add (Mul (azw, Sub (baq, bar)), ayx)) = + Add (Mul (ar, asa), Add (Mul (azw, Sub (baq, bar)), ayx)) + | numadd (Mul (ar, asa), Add (Mul (azw, Add (bao, bap)), ayx)) = + Add (Mul (ar, asa), Add (Mul (azw, Add (bao, bap)), ayx)) + | numadd (Mul (ar, asa), Add (Mul (azw, Neg ban), ayx)) = + Add (Mul (ar, asa), Add (Mul (azw, Neg ban), ayx)) + | numadd (Mul (ar, asa), Add (Mul (azw, Cx (bal, bam)), ayx)) = + Add (Mul (ar, asa), Add (Mul (azw, Cx (bal, bam)), ayx)) + | numadd (Mul (ar, asa), Add (Mul (azw, C baj), ayx)) = + Add (Mul (ar, asa), Add (Mul (azw, C baj), ayx)) + | numadd (Mul (ar, asa), Add (Sub (azu, azv), ayx)) = + Add (Mul (ar, asa), Add (Sub (azu, azv), ayx)) + | numadd (Mul (ar, asa), Add (Add (azs, azt), ayx)) = + Add (Mul (ar, asa), Add (Add (azs, azt), ayx)) + | numadd (Mul (ar, asa), Add (Neg azr, ayx)) = + Add (Mul (ar, asa), Add (Neg azr, ayx)) + | numadd (Mul (ar, asa), Add (Cx (azp, azq), ayx)) = + Add (Mul (ar, asa), Add (Cx (azp, azq), ayx)) + | numadd (Mul (ar, asa), Add (Bound azo, ayx)) = + Add (Mul (ar, asa), Add (Bound azo, ayx)) + | numadd (Mul (ar, asa), Add (C azn, ayx)) = + Add (Mul (ar, asa), Add (C azn, ayx)) + | numadd (Mul (ar, asa), Neg ayv) = Add (Mul (ar, asa), Neg ayv) + | numadd (Mul (ar, asa), Cx (ayt, ayu)) = Add (Mul (ar, asa), Cx (ayt, ayu)) + | numadd (Mul (ar, asa), Bound ays) = Add (Mul (ar, asa), Bound ays) + | numadd (Mul (ar, asa), C ayr) = Add (Mul (ar, asa), C ayr) + | numadd (Sub (ap, aq), Mul (awm, awn)) = Add (Sub (ap, aq), Mul (awm, awn)) + | numadd (Sub (ap, aq), Sub (awk, awl)) = Add (Sub (ap, aq), Sub (awk, awl)) + | numadd (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, Neg axz), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Neg axz), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, Cx (axx, axy)), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, Cx (axx, axy)), awj)) + | numadd (Sub (ap, aq), Add (Mul (axi, C axv), awj)) = + Add (Sub (ap, aq), Add (Mul (axi, C axv), awj)) + | numadd (Sub (ap, aq), Add (Sub (axg, axh), awj)) = + Add (Sub (ap, aq), Add (Sub (axg, axh), awj)) + | numadd (Sub (ap, aq), Add (Add (axe, axf), awj)) = + Add (Sub (ap, aq), Add (Add (axe, axf), awj)) + | numadd (Sub (ap, aq), Add (Neg axd, awj)) = + Add (Sub (ap, aq), Add (Neg axd, awj)) + | numadd (Sub (ap, aq), Add (Cx (axb, axc), awj)) = + Add (Sub (ap, aq), Add (Cx (axb, axc), awj)) + | numadd (Sub (ap, aq), Add (Bound axa, awj)) = + Add (Sub (ap, aq), Add (Bound axa, awj)) + | numadd (Sub (ap, aq), Add (C awz, awj)) = + Add (Sub (ap, aq), Add (C awz, awj)) + | numadd (Sub (ap, aq), Neg awh) = Add (Sub (ap, aq), Neg awh) + | numadd (Sub (ap, aq), Cx (awf, awg)) = Add (Sub (ap, aq), Cx (awf, awg)) + | numadd (Sub (ap, aq), Bound awe) = Add (Sub (ap, aq), Bound awe) + | numadd (Sub (ap, aq), C awd) = Add (Sub (ap, aq), C awd) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx)) | numadd - (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Mul (ain, aio)), afw)) = - Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Mul (ain, aio)), afw))) - | numadd (Add (Mul (c1, Bound n1), r1), Sub (afx, afy)) = - Add (Mul (c1, Bound n1), numadd (r1, Sub (afx, afy))) - | numadd (Add (Mul (c1, Bound n1), r1), Mul (afz, aga)) = - Add (Mul (c1, Bound n1), numadd (r1, Mul (afz, aga))) - | numadd (C w, Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (C w, r2)) - | numadd (Bound x, Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Bound x, r2)) - | numadd (CX (y, z), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (CX (y, z), r2)) - | numadd (Neg ab, Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Neg ab, r2)) - | numadd (Add (C li, ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (C li, ad), r2)) - | numadd (Add (Bound lj, ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Bound lj, ad), r2)) - | numadd (Add (CX (lk, ll), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (CX (lk, ll), ad), r2)) - | numadd (Add (Neg lm, ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Neg lm, ad), r2)) - | numadd (Add (Add (ln, lo), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Add (ln, lo), ad), r2)) - | numadd (Add (Sub (lp, lq), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Sub (lp, lq), ad), r2)) - | numadd (Add (Mul (lr, C abv), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, C abv), ad), r2)) - | numadd (Add (Mul (lr, CX (abx, aby)), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, CX (abx, aby)), ad), r2)) - | numadd (Add (Mul (lr, Neg abz), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Neg abz), ad), r2)) - | numadd (Add (Mul (lr, Add (aca, acb)), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Add (aca, acb)), ad), r2)) - | numadd (Add (Mul (lr, Sub (acc, acd)), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Sub (acc, acd)), ad), r2)) - | numadd (Add (Mul (lr, Mul (ace, acf)), ad), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Mul (ace, acf)), ad), r2)) - | numadd (Sub (ae, af), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Sub (ae, af), r2)) - | numadd (Mul (ag, ah), Add (Mul (c2, Bound n2), r2)) = - Add (Mul (c2, Bound n2), numadd (Mul (ag, ah), r2)) - | numadd (C b1, C b2) = C (b1 + b2) - | numadd (C ai, Bound bf) = Add (C ai, Bound bf) - | numadd (C ai, CX (bg, bh)) = Add (C ai, CX (bg, bh)) - | numadd (C ai, Neg bi) = Add (C ai, Neg bi) - | numadd (C ai, Add (C ca, bk)) = Add (C ai, Add (C ca, bk)) - | numadd (C ai, Add (Bound cb, bk)) = Add (C ai, Add (Bound cb, bk)) - | numadd (C ai, Add (CX (cc, cd), bk)) = Add (C ai, Add (CX (cc, cd), bk)) - | numadd (C ai, Add (Neg ce, bk)) = Add (C ai, Add (Neg ce, bk)) - | numadd (C ai, Add (Add (cf, cg), bk)) = Add (C ai, Add (Add (cf, cg), bk)) - | numadd (C ai, Add (Sub (ch, ci), bk)) = Add (C ai, Add (Sub (ch, ci), bk)) - | numadd (C ai, Add (Mul (cj, C cw), bk)) = - Add (C ai, Add (Mul (cj, C cw), bk)) - | numadd (C ai, Add (Mul (cj, CX (cy, cz)), bk)) = - Add (C ai, Add (Mul (cj, CX (cy, cz)), bk)) - | numadd (C ai, Add (Mul (cj, Neg da), bk)) = - Add (C ai, Add (Mul (cj, Neg da), bk)) - | numadd (C ai, Add (Mul (cj, Add (db, dc)), bk)) = - Add (C ai, Add (Mul (cj, Add (db, dc)), bk)) - | numadd (C ai, Add (Mul (cj, Sub (dd, de)), bk)) = - Add (C ai, Add (Mul (cj, Sub (dd, de)), bk)) - | numadd (C ai, Add (Mul (cj, Mul (df, dg)), bk)) = - Add (C ai, Add (Mul (cj, Mul (df, dg)), bk)) - | numadd (C ai, Sub (bl, bm)) = Add (C ai, Sub (bl, bm)) - | numadd (C ai, Mul (bn, bo)) = Add (C ai, Mul (bn, bo)) - | numadd (Bound aj, C ds) = Add (Bound aj, C ds) - | numadd (Bound aj, Bound dt) = Add (Bound aj, Bound dt) - | numadd (Bound aj, CX (du, dv)) = Add (Bound aj, CX (du, dv)) - | numadd (Bound aj, Neg dw) = Add (Bound aj, Neg dw) - | numadd (Bound aj, Add (C eo, dy)) = Add (Bound aj, Add (C eo, dy)) - | numadd (Bound aj, Add (Bound ep, dy)) = Add (Bound aj, Add (Bound ep, dy)) - | numadd (Bound aj, Add (CX (eq, er), dy)) = - Add (Bound aj, Add (CX (eq, er), dy)) - | numadd (Bound aj, Add (Neg es, dy)) = Add (Bound aj, Add (Neg es, dy)) - | numadd (Bound aj, Add (Add (et, eu), dy)) = - Add (Bound aj, Add (Add (et, eu), dy)) - | numadd (Bound aj, Add (Sub (ev, ew), dy)) = - Add (Bound aj, Add (Sub (ev, ew), dy)) - | numadd (Bound aj, Add (Mul (ex, C fk), dy)) = - Add (Bound aj, Add (Mul (ex, C fk), dy)) - | numadd (Bound aj, Add (Mul (ex, CX (fm, fn')), dy)) = - Add (Bound aj, Add (Mul (ex, CX (fm, fn')), dy)) - | numadd (Bound aj, Add (Mul (ex, Neg fo), dy)) = - Add (Bound aj, Add (Mul (ex, Neg fo), dy)) - | numadd (Bound aj, Add (Mul (ex, Add (fp, fq)), dy)) = - Add (Bound aj, Add (Mul (ex, Add (fp, fq)), dy)) - | numadd (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy)) = - Add (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy)) - | numadd (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) = - Add (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) - | numadd (Bound aj, Sub (dz, ea)) = Add (Bound aj, Sub (dz, ea)) - | numadd (Bound aj, Mul (eb, ec)) = Add (Bound aj, Mul (eb, ec)) - | numadd (CX (ak, al), C gg) = Add (CX (ak, al), C gg) - | numadd (CX (ak, al), Bound gh) = Add (CX (ak, al), Bound gh) - | numadd (CX (ak, al), CX (gi, gj)) = Add (CX (ak, al), CX (gi, gj)) - | numadd (CX (ak, al), Neg gk) = Add (CX (ak, al), Neg gk) - | numadd (CX (ak, al), Add (C hc, gm)) = Add (CX (ak, al), Add (C hc, gm)) - | numadd (CX (ak, al), Add (Bound hd, gm)) = - Add (CX (ak, al), Add (Bound hd, gm)) - | numadd (CX (ak, al), Add (CX (he, hf), gm)) = - Add (CX (ak, al), Add (CX (he, hf), gm)) - | numadd (CX (ak, al), Add (Neg hg, gm)) = Add (CX (ak, al), Add (Neg hg, gm)) - | numadd (CX (ak, al), Add (Add (hh, hi), gm)) = - Add (CX (ak, al), Add (Add (hh, hi), gm)) - | numadd (CX (ak, al), Add (Sub (hj, hk), gm)) = - Add (CX (ak, al), Add (Sub (hj, hk), gm)) - | numadd (CX (ak, al), Add (Mul (hl, C hy), gm)) = - Add (CX (ak, al), Add (Mul (hl, C hy), gm)) - | numadd (CX (ak, al), Add (Mul (hl, CX (ia, ib)), gm)) = - Add (CX (ak, al), Add (Mul (hl, CX (ia, ib)), gm)) - | numadd (CX (ak, al), Add (Mul (hl, Neg ic), gm)) = - Add (CX (ak, al), Add (Mul (hl, Neg ic), gm)) - | numadd (CX (ak, al), Add (Mul (hl, Add (id, ie)), gm)) = - Add (CX (ak, al), Add (Mul (hl, Add (id, ie)), gm)) - | numadd (CX (ak, al), Add (Mul (hl, Sub (if', ig)), gm)) = - Add (CX (ak, al), Add (Mul (hl, Sub (if', ig)), gm)) - | numadd (CX (ak, al), Add (Mul (hl, Mul (ih, ii)), gm)) = - Add (CX (ak, al), Add (Mul (hl, Mul (ih, ii)), gm)) - | numadd (CX (ak, al), Sub (gn, go)) = Add (CX (ak, al), Sub (gn, go)) - | numadd (CX (ak, al), Mul (gp, gq)) = Add (CX (ak, al), Mul (gp, gq)) - | numadd (Neg am, C iu) = Add (Neg am, C iu) - | numadd (Neg am, Bound iv) = Add (Neg am, Bound iv) - | numadd (Neg am, CX (iw, ix)) = Add (Neg am, CX (iw, ix)) - | numadd (Neg am, Neg iy) = Add (Neg am, Neg iy) - | numadd (Neg am, Add (C jq, ja)) = Add (Neg am, Add (C jq, ja)) - | numadd (Neg am, Add (Bound jr, ja)) = Add (Neg am, Add (Bound jr, ja)) - | numadd (Neg am, Add (CX (js, jt), ja)) = Add (Neg am, Add (CX (js, jt), ja)) - | numadd (Neg am, Add (Neg ju, ja)) = Add (Neg am, Add (Neg ju, ja)) - | numadd (Neg am, Add (Add (jv, jw), ja)) = - Add (Neg am, Add (Add (jv, jw), ja)) - | numadd (Neg am, Add (Sub (jx, jy), ja)) = - Add (Neg am, Add (Sub (jx, jy), ja)) - | numadd (Neg am, Add (Mul (jz, C km), ja)) = - Add (Neg am, Add (Mul (jz, C km), ja)) - | numadd (Neg am, Add (Mul (jz, CX (ko, kp)), ja)) = - Add (Neg am, Add (Mul (jz, CX (ko, kp)), ja)) - | numadd (Neg am, Add (Mul (jz, Neg kq), ja)) = - Add (Neg am, Add (Mul (jz, Neg kq), ja)) - | numadd (Neg am, Add (Mul (jz, Add (kr, ks)), ja)) = - Add (Neg am, Add (Mul (jz, Add (kr, ks)), ja)) - | numadd (Neg am, Add (Mul (jz, Sub (kt, ku)), ja)) = - Add (Neg am, Add (Mul (jz, Sub (kt, ku)), ja)) - | numadd (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) = - Add (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) - | numadd (Neg am, Sub (jb, jc)) = Add (Neg am, Sub (jb, jc)) - | numadd (Neg am, Mul (jd, je)) = Add (Neg am, Mul (jd, je)) - | numadd (Add (C lt, ao), C mp) = Add (Add (C lt, ao), C mp) - | numadd (Add (C lt, ao), Bound mq) = Add (Add (C lt, ao), Bound mq) - | numadd (Add (C lt, ao), CX (mr, ms)) = Add (Add (C lt, ao), CX (mr, ms)) - | numadd (Add (C lt, ao), Neg mt) = Add (Add (C lt, ao), Neg mt) - | numadd (Add (C lt, ao), Add (C nl, mv)) = - Add (Add (C lt, ao), Add (C nl, mv)) - | numadd (Add (C lt, ao), Add (Bound nm, mv)) = - Add (Add (C lt, ao), Add (Bound nm, mv)) - | numadd (Add (C lt, ao), Add (CX (nn, no), mv)) = - Add (Add (C lt, ao), Add (CX (nn, no), mv)) - | numadd (Add (C lt, ao), Add (Neg np, mv)) = - Add (Add (C lt, ao), Add (Neg np, mv)) - | numadd (Add (C lt, ao), Add (Add (nq, nr), mv)) = - Add (Add (C lt, ao), Add (Add (nq, nr), mv)) - | numadd (Add (C lt, ao), Add (Sub (ns, nt), mv)) = - Add (Add (C lt, ao), Add (Sub (ns, nt), mv)) - | numadd (Add (C lt, ao), Add (Mul (nu, C oh), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, C oh), mv)) - | numadd (Add (C lt, ao), Add (Mul (nu, CX (oj, ok)), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, CX (oj, ok)), mv)) - | numadd (Add (C lt, ao), Add (Mul (nu, Neg ol), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, Neg ol), mv)) - | numadd (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv)) - | numadd (Add (C lt, ao), Add (Mul (nu, Sub (oo, op')), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, Sub (oo, op')), mv)) - | numadd (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv)) = - Add (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv)) - | numadd (Add (C lt, ao), Sub (mw, mx)) = Add (Add (C lt, ao), Sub (mw, mx)) - | numadd (Add (C lt, ao), Mul (my, mz)) = Add (Add (C lt, ao), Mul (my, mz)) - | numadd (Add (Bound lu, ao), C pd) = Add (Add (Bound lu, ao), C pd) - | numadd (Add (Bound lu, ao), Bound pe) = Add (Add (Bound lu, ao), Bound pe) - | numadd (Add (Bound lu, ao), CX (pf, pg)) = - Add (Add (Bound lu, ao), CX (pf, pg)) - | numadd (Add (Bound lu, ao), Neg ph) = Add (Add (Bound lu, ao), Neg ph) - | numadd (Add (Bound lu, ao), Add (C pz, pj)) = - Add (Add (Bound lu, ao), Add (C pz, pj)) - | numadd (Add (Bound lu, ao), Add (Bound qa, pj)) = - Add (Add (Bound lu, ao), Add (Bound qa, pj)) - | numadd (Add (Bound lu, ao), Add (CX (qb, qc), pj)) = - Add (Add (Bound lu, ao), Add (CX (qb, qc), pj)) - | numadd (Add (Bound lu, ao), Add (Neg qd, pj)) = - Add (Add (Bound lu, ao), Add (Neg qd, pj)) - | numadd (Add (Bound lu, ao), Add (Add (qe, qf), pj)) = - Add (Add (Bound lu, ao), Add (Add (qe, qf), pj)) - | numadd (Add (Bound lu, ao), Add (Sub (qg, qh), pj)) = - Add (Add (Bound lu, ao), Add (Sub (qg, qh), pj)) - | numadd (Add (Bound lu, ao), Add (Mul (qi, C qv), pj)) = - Add (Add (Bound lu, ao), Add (Mul (qi, C qv), pj)) - | numadd (Add (Bound lu, ao), Add (Mul (qi, CX (qx, qy)), pj)) = - Add (Add (Bound lu, ao), Add (Mul (qi, CX (qx, qy)), pj)) - | numadd (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj)) = - Add (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj)) - | numadd (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj)) = - Add (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj)) - | numadd (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj)) = - Add (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj)) - | numadd (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj)) = - Add (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj)) - | numadd (Add (Bound lu, ao), Sub (pk, pl)) = - Add (Add (Bound lu, ao), Sub (pk, pl)) - | numadd (Add (Bound lu, ao), Mul (pm, pn)) = - Add (Add (Bound lu, ao), Mul (pm, pn)) - | numadd (Add (CX (lv, lw), ao), C rr) = Add (Add (CX (lv, lw), ao), C rr) - | numadd (Add (CX (lv, lw), ao), Bound rs) = - Add (Add (CX (lv, lw), ao), Bound rs) - | numadd (Add (CX (lv, lw), ao), CX (rt, ru)) = - Add (Add (CX (lv, lw), ao), CX (rt, ru)) - | numadd (Add (CX (lv, lw), ao), Neg rv) = Add (Add (CX (lv, lw), ao), Neg rv) - | numadd (Add (CX (lv, lw), ao), Add (C sn, rx)) = - Add (Add (CX (lv, lw), ao), Add (C sn, rx)) - | numadd (Add (CX (lv, lw), ao), Add (Bound so, rx)) = - Add (Add (CX (lv, lw), ao), Add (Bound so, rx)) - | numadd (Add (CX (lv, lw), ao), Add (CX (sp, sq), rx)) = - Add (Add (CX (lv, lw), ao), Add (CX (sp, sq), rx)) - | numadd (Add (CX (lv, lw), ao), Add (Neg sr, rx)) = - Add (Add (CX (lv, lw), ao), Add (Neg sr, rx)) - | numadd (Add (CX (lv, lw), ao), Add (Add (ss, st), rx)) = - Add (Add (CX (lv, lw), ao), Add (Add (ss, st), rx)) - | numadd (Add (CX (lv, lw), ao), Add (Sub (su, sv), rx)) = - Add (Add (CX (lv, lw), ao), Add (Sub (su, sv), rx)) - | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, C tj), rx)) = - Add (Add (CX (lv, lw), ao), Add (Mul (sw, C tj), rx)) - | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, CX (tl, tm)), rx)) = - Add (Add (CX (lv, lw), ao), Add (Mul (sw, CX (tl, tm)), rx)) - | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Neg tn), rx)) = - Add (Add (CX (lv, lw), ao), Add (Mul (sw, Neg tn), rx)) - | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx)) = - Add (Add (CX (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx)) - | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx)) = - Add (Add (CX (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx)) - | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx)) = - Add (Add (CX (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx)) - | numadd (Add (CX (lv, lw), ao), Sub (ry, rz)) = - Add (Add (CX (lv, lw), ao), Sub (ry, rz)) - | numadd (Add (CX (lv, lw), ao), Mul (sa, sb)) = - Add (Add (CX (lv, lw), ao), Mul (sa, sb)) - | numadd (Add (Neg lx, ao), C uf) = Add (Add (Neg lx, ao), C uf) - | numadd (Add (Neg lx, ao), Bound ug) = Add (Add (Neg lx, ao), Bound ug) - | numadd (Add (Neg lx, ao), CX (uh, ui)) = Add (Add (Neg lx, ao), CX (uh, ui)) - | numadd (Add (Neg lx, ao), Neg uj) = Add (Add (Neg lx, ao), Neg uj) - | numadd (Add (Neg lx, ao), Add (C vb, ul)) = - Add (Add (Neg lx, ao), Add (C vb, ul)) - | numadd (Add (Neg lx, ao), Add (Bound vc, ul)) = - Add (Add (Neg lx, ao), Add (Bound vc, ul)) - | numadd (Add (Neg lx, ao), Add (CX (vd, ve), ul)) = - Add (Add (Neg lx, ao), Add (CX (vd, ve), ul)) - | numadd (Add (Neg lx, ao), Add (Neg vf, ul)) = - Add (Add (Neg lx, ao), Add (Neg vf, ul)) - | numadd (Add (Neg lx, ao), Add (Add (vg, vh), ul)) = - Add (Add (Neg lx, ao), Add (Add (vg, vh), ul)) - | numadd (Add (Neg lx, ao), Add (Sub (vi, vj), ul)) = - Add (Add (Neg lx, ao), Add (Sub (vi, vj), ul)) - | numadd (Add (Neg lx, ao), Add (Mul (vk, C vx), ul)) = - Add (Add (Neg lx, ao), Add (Mul (vk, C vx), ul)) - | numadd (Add (Neg lx, ao), Add (Mul (vk, CX (vz, wa)), ul)) = - Add (Add (Neg lx, ao), Add (Mul (vk, CX (vz, wa)), ul)) - | numadd (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul)) = - Add (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul)) - | numadd (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul)) = - Add (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul)) - | numadd (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul)) = - Add (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul)) - | numadd (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul)) = - Add (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul)) - | numadd (Add (Neg lx, ao), Sub (um, un)) = - Add (Add (Neg lx, ao), Sub (um, un)) - | numadd (Add (Neg lx, ao), Mul (uo, up)) = - Add (Add (Neg lx, ao), Mul (uo, up)) - | numadd (Add (Add (ly, lz), ao), C wt) = Add (Add (Add (ly, lz), ao), C wt) - | numadd (Add (Add (ly, lz), ao), Bound wu) = - Add (Add (Add (ly, lz), ao), Bound wu) - | numadd (Add (Add (ly, lz), ao), CX (wv, ww)) = - Add (Add (Add (ly, lz), ao), CX (wv, ww)) - | numadd (Add (Add (ly, lz), ao), Neg wx) = - Add (Add (Add (ly, lz), ao), Neg wx) - | numadd (Add (Add (ly, lz), ao), Add (C xp, wz)) = - Add (Add (Add (ly, lz), ao), Add (C xp, wz)) - | numadd (Add (Add (ly, lz), ao), Add (Bound xq, wz)) = - Add (Add (Add (ly, lz), ao), Add (Bound xq, wz)) - | numadd (Add (Add (ly, lz), ao), Add (CX (xr, xs), wz)) = - Add (Add (Add (ly, lz), ao), Add (CX (xr, xs), wz)) - | numadd (Add (Add (ly, lz), ao), Add (Neg xt, wz)) = - Add (Add (Add (ly, lz), ao), Add (Neg xt, wz)) - | numadd (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz)) = - Add (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz)) - | numadd (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz)) = - Add (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz)) - | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz)) = - Add (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz)) - | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, CX (yn, yo)), wz)) = - Add (Add (Add (ly, lz), ao), Add (Mul (xy, CX (yn, yo)), wz)) - | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz)) = - Add (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz)) - | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz)) = - Add (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz)) - | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz)) = - Add (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz)) - | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz)) = - Add (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz)) - | numadd (Add (Add (ly, lz), ao), Sub (xa, xb)) = - Add (Add (Add (ly, lz), ao), Sub (xa, xb)) - | numadd (Add (Add (ly, lz), ao), Mul (xc, xd)) = - Add (Add (Add (ly, lz), ao), Mul (xc, xd)) - | numadd (Add (Sub (ma, mb), ao), C zh) = Add (Add (Sub (ma, mb), ao), C zh) - | numadd (Add (Sub (ma, mb), ao), Bound zi) = - Add (Add (Sub (ma, mb), ao), Bound zi) - | numadd (Add (Sub (ma, mb), ao), CX (zj, zk)) = - Add (Add (Sub (ma, mb), ao), CX (zj, zk)) - | numadd (Add (Sub (ma, mb), ao), Neg zl) = - Add (Add (Sub (ma, mb), ao), Neg zl) - | numadd (Add (Sub (ma, mb), ao), Add (C aad, zn)) = - Add (Add (Sub (ma, mb), ao), Add (C aad, zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Bound aae, zn)) = - Add (Add (Sub (ma, mb), ao), Add (Bound aae, zn)) - | numadd (Add (Sub (ma, mb), ao), Add (CX (aaf, aag), zn)) = - Add (Add (Sub (ma, mb), ao), Add (CX (aaf, aag), zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Neg aah, zn)) = - Add (Add (Sub (ma, mb), ao), Add (Neg aah, zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, CX (abb, abc)), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Mul (aam, CX (abb, abc)), zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn)) - | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn)) = - Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn)) - | numadd (Add (Sub (ma, mb), ao), Sub (zo, zp)) = - Add (Add (Sub (ma, mb), ao), Sub (zo, zp)) - | numadd (Add (Sub (ma, mb), ao), Mul (zq, zr)) = - Add (Add (Sub (ma, mb), ao), Mul (zq, zr)) - | numadd (Add (Mul (mc, C acg), ao), C adc) = - Add (Add (Mul (mc, C acg), ao), C adc) - | numadd (Add (Mul (mc, C acg), ao), Bound add) = - Add (Add (Mul (mc, C acg), ao), Bound add) - | numadd (Add (Mul (mc, C acg), ao), CX (ade, adf)) = - Add (Add (Mul (mc, C acg), ao), CX (ade, adf)) - | numadd (Add (Mul (mc, C acg), ao), Neg adg) = - Add (Add (Mul (mc, C acg), ao), Neg adg) - | numadd (Add (Mul (mc, C acg), ao), Add (C ady, adi)) = - Add (Add (Mul (mc, C acg), ao), Add (C ady, adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Bound adz, adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Bound adz, adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (CX (aea, aeb), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (CX (aea, aeb), adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Neg aec, adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Neg aec, adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, CX (aew, aex)), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, CX (aew, aex)), adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi)) - | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi)) = - Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi)) - | numadd (Add (Mul (mc, C acg), ao), Sub (adj, adk)) = - Add (Add (Mul (mc, C acg), ao), Sub (adj, adk)) - | numadd (Add (Mul (mc, C acg), ao), Mul (adl, adm)) = - Add (Add (Mul (mc, C acg), ao), Mul (adl, adm)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), C ajl) = - Add (Add (Mul (mc, CX (aci, acj)), ao), C ajl) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Bound ajm) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Bound ajm) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), CX (ajn, ajo)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), CX (ajn, ajo)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Neg ajp) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Neg ajp) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (C akh, ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (C akh, ajr)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Bound aki, ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Bound aki, ajr)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (CX (akj, akk), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (CX (akj, akk), ajr)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Neg akl, ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Neg akl, ajr)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Add (akm, akn), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Add (akm, akn), ajr)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Sub (ako, akp), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Sub (ako, akp), ajr)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, C ald), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, C ald), ajr)) + (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Mul (avq, avr)), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, Mul (avq, avr)), atv)) | numadd - (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, CX (alf, alg)), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, CX (alf, alg)), ajr)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr)) - | numadd - (Add (Mul (mc, CX (aci, acj)), ao), - Add (Mul (akq, Add (ali, alj)), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), - Add (Mul (akq, Add (ali, alj)), ajr)) - | numadd - (Add (Mul (mc, CX (aci, acj)), ao), - Add (Mul (akq, Sub (alk, all)), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), - Add (Mul (akq, Sub (alk, all)), ajr)) - | numadd - (Add (Mul (mc, CX (aci, acj)), ao), - Add (Mul (akq, Mul (alm, aln)), ajr)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), - Add (Mul (akq, Mul (alm, aln)), ajr)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Sub (ajs, ajt)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Sub (ajs, ajt)) - | numadd (Add (Mul (mc, CX (aci, acj)), ao), Mul (aju, ajv)) = - Add (Add (Mul (mc, CX (aci, acj)), ao), Mul (aju, ajv)) - | numadd (Add (Mul (mc, Neg ack), ao), C alz) = - Add (Add (Mul (mc, Neg ack), ao), C alz) - | numadd (Add (Mul (mc, Neg ack), ao), Bound ama) = - Add (Add (Mul (mc, Neg ack), ao), Bound ama) - | numadd (Add (Mul (mc, Neg ack), ao), CX (amb, amc)) = - Add (Add (Mul (mc, Neg ack), ao), CX (amb, amc)) - | numadd (Add (Mul (mc, Neg ack), ao), Neg amd) = - Add (Add (Mul (mc, Neg ack), ao), Neg amd) - | numadd (Add (Mul (mc, Neg ack), ao), Add (C amv, amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (C amv, amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (CX (amx, amy), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (CX (amx, amy), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, and'), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, and'), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, CX (ant, anu)), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, CX (ant, anu)), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf)) = - Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf)) - | numadd (Add (Mul (mc, Neg ack), ao), Sub (amg, amh)) = - Add (Add (Mul (mc, Neg ack), ao), Sub (amg, amh)) - | numadd (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) = - Add (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), C aon) = - Add (Add (Mul (mc, Add (acl, acm)), ao), C aon) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), CX (aop, aoq)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), CX (aop, aoq)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (CX (apl, apm), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (CX (apl, apm), aot)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot)) - | numadd - (Add (Mul (mc, Add (acl, acm)), ao), - Add (Mul (aps, CX (aqh, aqi)), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), - Add (Mul (aps, CX (aqh, aqi)), aot)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot)) - | numadd - (Add (Mul (mc, Add (acl, acm)), ao), - Add (Mul (aps, Add (aqk, aql)), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), - Add (Mul (aps, Add (aqk, aql)), aot)) - | numadd - (Add (Mul (mc, Add (acl, acm)), ao), - Add (Mul (aps, Sub (aqm, aqn)), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), - Add (Mul (aps, Sub (aqm, aqn)), aot)) - | numadd - (Add (Mul (mc, Add (acl, acm)), ao), - Add (Mul (aps, Mul (aqo, aqp)), aot)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), - Add (Mul (aps, Mul (aqo, aqp)), aot)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov)) - | numadd (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox)) = - Add (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), C arb) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), C arb) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), CX (ard, are)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), CX (ard, are)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (CX (arz, asa), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (CX (arz, asa), arh)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh)) - | numadd - (Add (Mul (mc, Sub (acn, aco)), ao), - Add (Mul (asg, CX (asv, asw)), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), - Add (Mul (asg, CX (asv, asw)), arh)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh)) - | numadd - (Add (Mul (mc, Sub (acn, aco)), ao), - Add (Mul (asg, Add (asy, asz)), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), - Add (Mul (asg, Add (asy, asz)), arh)) - | numadd - (Add (Mul (mc, Sub (acn, aco)), ao), - Add (Mul (asg, Sub (ata, atb)), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), - Add (Mul (asg, Sub (ata, atb)), arh)) - | numadd - (Add (Mul (mc, Sub (acn, aco)), ao), - Add (Mul (asg, Mul (atc, atd)), arh)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), - Add (Mul (asg, Mul (atc, atd)), arh)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj)) - | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl)) = - Add (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), C atp) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), C atp) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), CX (atr, ats)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), CX (atr, ats)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (CX (aun, auo), atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (CX (aun, auo), atv)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv)) - | numadd - (Add (Mul (mc, Mul (acp, acq)), ao), - Add (Mul (auu, CX (avj, avk)), atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), - Add (Mul (auu, CX (avj, avk)), atv)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv)) - | numadd - (Add (Mul (mc, Mul (acp, acq)), ao), - Add (Mul (auu, Add (avm, avn)), atv)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), - Add (Mul (auu, Add (avm, avn)), atv)) - | numadd - (Add (Mul (mc, Mul (acp, acq)), ao), - Add (Mul (auu, Sub (avo, avp)), atv)) = + (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Sub (avo, avp)), atv)) = Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Sub (avo, avp)), atv)) | numadd - (Add (Mul (mc, Mul (acp, acq)), ao), - Add (Mul (auu, Mul (avq, avr)), atv)) = + (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Add (avm, avn)), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), + Add (Mul (auu, Add (avm, avn)), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv)) + | numadd + (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Cx (avj, avk)), atv)) = Add (Add (Mul (mc, Mul (acp, acq)), ao), - Add (Mul (auu, Mul (avq, avr)), atv)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx)) - | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) = - Add (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) - | numadd (Sub (ap, aq), C awd) = Add (Sub (ap, aq), C awd) - | numadd (Sub (ap, aq), Bound awe) = Add (Sub (ap, aq), Bound awe) - | numadd (Sub (ap, aq), CX (awf, awg)) = Add (Sub (ap, aq), CX (awf, awg)) - | numadd (Sub (ap, aq), Neg awh) = Add (Sub (ap, aq), Neg awh) - | numadd (Sub (ap, aq), Add (C awz, awj)) = - Add (Sub (ap, aq), Add (C awz, awj)) - | numadd (Sub (ap, aq), Add (Bound axa, awj)) = - Add (Sub (ap, aq), Add (Bound axa, awj)) - | numadd (Sub (ap, aq), Add (CX (axb, axc), awj)) = - Add (Sub (ap, aq), Add (CX (axb, axc), awj)) - | numadd (Sub (ap, aq), Add (Neg axd, awj)) = - Add (Sub (ap, aq), Add (Neg axd, awj)) - | numadd (Sub (ap, aq), Add (Add (axe, axf), awj)) = - Add (Sub (ap, aq), Add (Add (axe, axf), awj)) - | numadd (Sub (ap, aq), Add (Sub (axg, axh), awj)) = - Add (Sub (ap, aq), Add (Sub (axg, axh), awj)) - | numadd (Sub (ap, aq), Add (Mul (axi, C axv), awj)) = - Add (Sub (ap, aq), Add (Mul (axi, C axv), awj)) - | numadd (Sub (ap, aq), Add (Mul (axi, CX (axx, axy)), awj)) = - Add (Sub (ap, aq), Add (Mul (axi, CX (axx, axy)), awj)) - | numadd (Sub (ap, aq), Add (Mul (axi, Neg axz), awj)) = - Add (Sub (ap, aq), Add (Mul (axi, Neg axz), awj)) - | numadd (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj)) = - Add (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj)) - | numadd (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj)) = - Add (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj)) - | numadd (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj)) = - Add (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj)) - | numadd (Sub (ap, aq), Sub (awk, awl)) = Add (Sub (ap, aq), Sub (awk, awl)) - | numadd (Sub (ap, aq), Mul (awm, awn)) = Add (Sub (ap, aq), Mul (awm, awn)) - | numadd (Mul (ar, as'), C ayr) = Add (Mul (ar, as'), C ayr) - | numadd (Mul (ar, as'), Bound ays) = Add (Mul (ar, as'), Bound ays) - | numadd (Mul (ar, as'), CX (ayt, ayu)) = Add (Mul (ar, as'), CX (ayt, ayu)) - | numadd (Mul (ar, as'), Neg ayv) = Add (Mul (ar, as'), Neg ayv) - | numadd (Mul (ar, as'), Add (C azn, ayx)) = - Add (Mul (ar, as'), Add (C azn, ayx)) - | numadd (Mul (ar, as'), Add (Bound azo, ayx)) = - Add (Mul (ar, as'), Add (Bound azo, ayx)) - | numadd (Mul (ar, as'), Add (CX (azp, azq), ayx)) = - Add (Mul (ar, as'), Add (CX (azp, azq), ayx)) - | numadd (Mul (ar, as'), Add (Neg azr, ayx)) = - Add (Mul (ar, as'), Add (Neg azr, ayx)) - | numadd (Mul (ar, as'), Add (Add (azs, azt), ayx)) = - Add (Mul (ar, as'), Add (Add (azs, azt), ayx)) - | numadd (Mul (ar, as'), Add (Sub (azu, azv), ayx)) = - Add (Mul (ar, as'), Add (Sub (azu, azv), ayx)) - | numadd (Mul (ar, as'), Add (Mul (azw, C baj), ayx)) = - Add (Mul (ar, as'), Add (Mul (azw, C baj), ayx)) - | numadd (Mul (ar, as'), Add (Mul (azw, CX (bal, bam)), ayx)) = - Add (Mul (ar, as'), Add (Mul (azw, CX (bal, bam)), ayx)) - | numadd (Mul (ar, as'), Add (Mul (azw, Neg ban), ayx)) = - Add (Mul (ar, as'), Add (Mul (azw, Neg ban), ayx)) - | numadd (Mul (ar, as'), Add (Mul (azw, Add (bao, bap)), ayx)) = - Add (Mul (ar, as'), Add (Mul (azw, Add (bao, bap)), ayx)) - | numadd (Mul (ar, as'), Add (Mul (azw, Sub (baq, bar)), ayx)) = - Add (Mul (ar, as'), Add (Mul (azw, Sub (baq, bar)), ayx)) - | numadd (Mul (ar, as'), Add (Mul (azw, Mul (bas, bat)), ayx)) = - Add (Mul (ar, as'), Add (Mul (azw, Mul (bas, bat)), ayx)) - | numadd (Mul (ar, as'), Sub (ayy, ayz)) = Add (Mul (ar, as'), Sub (ayy, ayz)) - | numadd (Mul (ar, as'), Mul (aza, azb)) = - Add (Mul (ar, as'), Mul (aza, azb)); + Add (Mul (auu, Cx (avj, avk)), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Cx (aun, auo), atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Cx (aun, auo), atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Cx (atr, ats)) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Cx (atr, ats)) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) + | numadd (Add (Mul (mc, Mul (acp, acq)), ao), C atp) = + Add (Add (Mul (mc, Mul (acp, acq)), ao), C atp) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj)) + | numadd + (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Mul (atc, atd)), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Mul (atc, atd)), arh)) + | numadd + (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Sub (ata, atb)), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Sub (ata, atb)), arh)) + | numadd + (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Add (asy, asz)), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Add (asy, asz)), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh)) + | numadd + (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Cx (asv, asw)), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), + Add (Mul (asg, Cx (asv, asw)), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Cx (arz, asa), arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Cx (arz, asa), arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Cx (ard, are)) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Cx (ard, are)) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) + | numadd (Add (Mul (mc, Sub (acn, aco)), ao), C arb) = + Add (Add (Mul (mc, Sub (acn, aco)), ao), C arb) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov)) + | numadd + (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Mul (aqo, aqp)), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Mul (aqo, aqp)), aot)) + | numadd + (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Sub (aqm, aqn)), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Sub (aqm, aqn)), aot)) + | numadd + (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Add (aqk, aql)), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Add (aqk, aql)), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot)) + | numadd + (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Cx (aqh, aqi)), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), + Add (Mul (aps, Cx (aqh, aqi)), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Cx (apl, apm), aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Cx (apl, apm), aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Cx (aop, aoq)) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Cx (aop, aoq)) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) = + Add (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) + | numadd (Add (Mul (mc, Add (acl, acm)), ao), C aon) = + Add (Add (Mul (mc, Add (acl, acm)), ao), C aon) + | numadd (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) = + Add (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) + | numadd (Add (Mul (mc, Neg ack), ao), Sub (amg, amh)) = + Add (Add (Mul (mc, Neg ack), ao), Sub (amg, amh)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Cx (ant, anu)), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Cx (ant, anu)), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, anda), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, anda), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Cx (amx, amy), amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Cx (amx, amy), amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Add (C amv, amf)) = + Add (Add (Mul (mc, Neg ack), ao), Add (C amv, amf)) + | numadd (Add (Mul (mc, Neg ack), ao), Neg amd) = + Add (Add (Mul (mc, Neg ack), ao), Neg amd) + | numadd (Add (Mul (mc, Neg ack), ao), Cx (amb, amc)) = + Add (Add (Mul (mc, Neg ack), ao), Cx (amb, amc)) + | numadd (Add (Mul (mc, Neg ack), ao), Bound ama) = + Add (Add (Mul (mc, Neg ack), ao), Bound ama) + | numadd (Add (Mul (mc, Neg ack), ao), C alz) = + Add (Add (Mul (mc, Neg ack), ao), C alz) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Mul (aju, ajv)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Mul (aju, ajv)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Sub (ajs, ajt)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Sub (ajs, ajt)) + | numadd + (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Mul (alm, aln)), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), + Add (Mul (akq, Mul (alm, aln)), ajr)) + | numadd + (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Sub (alk, all)), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), + Add (Mul (akq, Sub (alk, all)), ajr)) + | numadd + (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Add (ali, alj)), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), + Add (Mul (akq, Add (ali, alj)), ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr)) + | numadd + (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Cx (alf, alg)), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Cx (alf, alg)), ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, C ald), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, C ald), ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Sub (ako, akp), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Sub (ako, akp), ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Add (akm, akn), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Add (akm, akn), ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Neg akl, ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Neg akl, ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Cx (akj, akk), ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Cx (akj, akk), ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Bound aki, ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Bound aki, ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (C akh, ajr)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (C akh, ajr)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Neg ajp) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Neg ajp) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Cx (ajn, ajo)) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Cx (ajn, ajo)) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Bound ajm) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), Bound ajm) + | numadd (Add (Mul (mc, Cx (aci, acj)), ao), C ajl) = + Add (Add (Mul (mc, Cx (aci, acj)), ao), C ajl) + | numadd (Add (Mul (mc, C acg), ao), Mul (adl, adm)) = + Add (Add (Mul (mc, C acg), ao), Mul (adl, adm)) + | numadd (Add (Mul (mc, C acg), ao), Sub (adj, adk)) = + Add (Add (Mul (mc, C acg), ao), Sub (adj, adk)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Cx (aew, aex)), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Cx (aew, aex)), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Neg aec, adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Neg aec, adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Cx (aea, aeb), adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Cx (aea, aeb), adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (Bound adz, adi)) = + Add (Add (Mul (mc, C acg), ao), Add (Bound adz, adi)) + | numadd (Add (Mul (mc, C acg), ao), Add (C ady, adi)) = + Add (Add (Mul (mc, C acg), ao), Add (C ady, adi)) + | numadd (Add (Mul (mc, C acg), ao), Neg adg) = + Add (Add (Mul (mc, C acg), ao), Neg adg) + | numadd (Add (Mul (mc, C acg), ao), Cx (ade, adf)) = + Add (Add (Mul (mc, C acg), ao), Cx (ade, adf)) + | numadd (Add (Mul (mc, C acg), ao), Bound add) = + Add (Add (Mul (mc, C acg), ao), Bound add) + | numadd (Add (Mul (mc, C acg), ao), C adc) = + Add (Add (Mul (mc, C acg), ao), C adc) + | numadd (Add (Sub (ma, mb), ao), Mul (zq, zr)) = + Add (Add (Sub (ma, mb), ao), Mul (zq, zr)) + | numadd (Add (Sub (ma, mb), ao), Sub (zo, zp)) = + Add (Add (Sub (ma, mb), ao), Sub (zo, zp)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Cx (abb, abc)), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Cx (abb, abc)), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Neg aah, zn)) = + Add (Add (Sub (ma, mb), ao), Add (Neg aah, zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Cx (aaf, aag), zn)) = + Add (Add (Sub (ma, mb), ao), Add (Cx (aaf, aag), zn)) + | numadd (Add (Sub (ma, mb), ao), Add (Bound aae, zn)) = + Add (Add (Sub (ma, mb), ao), Add (Bound aae, zn)) + | numadd (Add (Sub (ma, mb), ao), Add (C aad, zn)) = + Add (Add (Sub (ma, mb), ao), Add (C aad, zn)) + | numadd (Add (Sub (ma, mb), ao), Neg zl) = + Add (Add (Sub (ma, mb), ao), Neg zl) + | numadd (Add (Sub (ma, mb), ao), Cx (zj, zk)) = + Add (Add (Sub (ma, mb), ao), Cx (zj, zk)) + | numadd (Add (Sub (ma, mb), ao), Bound zi) = + Add (Add (Sub (ma, mb), ao), Bound zi) + | numadd (Add (Sub (ma, mb), ao), C zh) = Add (Add (Sub (ma, mb), ao), C zh) + | numadd (Add (Add (ly, lz), ao), Mul (xc, xd)) = + Add (Add (Add (ly, lz), ao), Mul (xc, xd)) + | numadd (Add (Add (ly, lz), ao), Sub (xa, xb)) = + Add (Add (Add (ly, lz), ao), Sub (xa, xb)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Cx (yn, yo)), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, Cx (yn, yo)), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz)) = + Add (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz)) = + Add (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz)) = + Add (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Neg xt, wz)) = + Add (Add (Add (ly, lz), ao), Add (Neg xt, wz)) + | numadd (Add (Add (ly, lz), ao), Add (Cx (xr, xs), wz)) = + Add (Add (Add (ly, lz), ao), Add (Cx (xr, xs), wz)) + | numadd (Add (Add (ly, lz), ao), Add (Bound xq, wz)) = + Add (Add (Add (ly, lz), ao), Add (Bound xq, wz)) + | numadd (Add (Add (ly, lz), ao), Add (C xp, wz)) = + Add (Add (Add (ly, lz), ao), Add (C xp, wz)) + | numadd (Add (Add (ly, lz), ao), Neg wx) = + Add (Add (Add (ly, lz), ao), Neg wx) + | numadd (Add (Add (ly, lz), ao), Cx (wv, ww)) = + Add (Add (Add (ly, lz), ao), Cx (wv, ww)) + | numadd (Add (Add (ly, lz), ao), Bound wu) = + Add (Add (Add (ly, lz), ao), Bound wu) + | numadd (Add (Add (ly, lz), ao), C wt) = Add (Add (Add (ly, lz), ao), C wt) + | numadd (Add (Neg lx, ao), Mul (uo, up)) = + Add (Add (Neg lx, ao), Mul (uo, up)) + | numadd (Add (Neg lx, ao), Sub (um, un)) = + Add (Add (Neg lx, ao), Sub (um, un)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, Cx (vz, wa)), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, Cx (vz, wa)), ul)) + | numadd (Add (Neg lx, ao), Add (Mul (vk, C vx), ul)) = + Add (Add (Neg lx, ao), Add (Mul (vk, C vx), ul)) + | numadd (Add (Neg lx, ao), Add (Sub (vi, vj), ul)) = + Add (Add (Neg lx, ao), Add (Sub (vi, vj), ul)) + | numadd (Add (Neg lx, ao), Add (Add (vg, vh), ul)) = + Add (Add (Neg lx, ao), Add (Add (vg, vh), ul)) + | numadd (Add (Neg lx, ao), Add (Neg vf, ul)) = + Add (Add (Neg lx, ao), Add (Neg vf, ul)) + | numadd (Add (Neg lx, ao), Add (Cx (vd, ve), ul)) = + Add (Add (Neg lx, ao), Add (Cx (vd, ve), ul)) + | numadd (Add (Neg lx, ao), Add (Bound vc, ul)) = + Add (Add (Neg lx, ao), Add (Bound vc, ul)) + | numadd (Add (Neg lx, ao), Add (C vb, ul)) = + Add (Add (Neg lx, ao), Add (C vb, ul)) + | numadd (Add (Neg lx, ao), Neg uj) = Add (Add (Neg lx, ao), Neg uj) + | numadd (Add (Neg lx, ao), Cx (uh, ui)) = Add (Add (Neg lx, ao), Cx (uh, ui)) + | numadd (Add (Neg lx, ao), Bound ug) = Add (Add (Neg lx, ao), Bound ug) + | numadd (Add (Neg lx, ao), C uf) = Add (Add (Neg lx, ao), C uf) + | numadd (Add (Cx (lv, lw), ao), Mul (sa, sb)) = + Add (Add (Cx (lv, lw), ao), Mul (sa, sb)) + | numadd (Add (Cx (lv, lw), ao), Sub (ry, rz)) = + Add (Add (Cx (lv, lw), ao), Sub (ry, rz)) + | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Neg tn), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Neg tn), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Cx (tl, tm)), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Cx (tl, tm)), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, C tj), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Mul (sw, C tj), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Sub (su, sv), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Sub (su, sv), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Add (ss, st), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Add (ss, st), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Neg sr, rx)) = + Add (Add (Cx (lv, lw), ao), Add (Neg sr, rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Cx (sp, sq), rx)) = + Add (Add (Cx (lv, lw), ao), Add (Cx (sp, sq), rx)) + | numadd (Add (Cx (lv, lw), ao), Add (Bound so, rx)) = + Add (Add (Cx (lv, lw), ao), Add (Bound so, rx)) + | numadd (Add (Cx (lv, lw), ao), Add (C sn, rx)) = + Add (Add (Cx (lv, lw), ao), Add (C sn, rx)) + | numadd (Add (Cx (lv, lw), ao), Neg rv) = Add (Add (Cx (lv, lw), ao), Neg rv) + | numadd (Add (Cx (lv, lw), ao), Cx (rt, ru)) = + Add (Add (Cx (lv, lw), ao), Cx (rt, ru)) + | numadd (Add (Cx (lv, lw), ao), Bound rs) = + Add (Add (Cx (lv, lw), ao), Bound rs) + | numadd (Add (Cx (lv, lw), ao), C rr) = Add (Add (Cx (lv, lw), ao), C rr) + | numadd (Add (Bound lu, ao), Mul (pm, pn)) = + Add (Add (Bound lu, ao), Mul (pm, pn)) + | numadd (Add (Bound lu, ao), Sub (pk, pl)) = + Add (Add (Bound lu, ao), Sub (pk, pl)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, Cx (qx, qy)), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, Cx (qx, qy)), pj)) + | numadd (Add (Bound lu, ao), Add (Mul (qi, C qv), pj)) = + Add (Add (Bound lu, ao), Add (Mul (qi, C qv), pj)) + | numadd (Add (Bound lu, ao), Add (Sub (qg, qh), pj)) = + Add (Add (Bound lu, ao), Add (Sub (qg, qh), pj)) + | numadd (Add (Bound lu, ao), Add (Add (qe, qf), pj)) = + Add (Add (Bound lu, ao), Add (Add (qe, qf), pj)) + | numadd (Add (Bound lu, ao), Add (Neg qd, pj)) = + Add (Add (Bound lu, ao), Add (Neg qd, pj)) + | numadd (Add (Bound lu, ao), Add (Cx (qb, qc), pj)) = + Add (Add (Bound lu, ao), Add (Cx (qb, qc), pj)) + | numadd (Add (Bound lu, ao), Add (Bound qa, pj)) = + Add (Add (Bound lu, ao), Add (Bound qa, pj)) + | numadd (Add (Bound lu, ao), Add (C pz, pj)) = + Add (Add (Bound lu, ao), Add (C pz, pj)) + | numadd (Add (Bound lu, ao), Neg ph) = Add (Add (Bound lu, ao), Neg ph) + | numadd (Add (Bound lu, ao), Cx (pf, pg)) = + Add (Add (Bound lu, ao), Cx (pf, pg)) + | numadd (Add (Bound lu, ao), Bound pe) = Add (Add (Bound lu, ao), Bound pe) + | numadd (Add (Bound lu, ao), C pd) = Add (Add (Bound lu, ao), C pd) + | numadd (Add (C lt, ao), Mul (my, mz)) = Add (Add (C lt, ao), Mul (my, mz)) + | numadd (Add (C lt, ao), Sub (mw, mx)) = Add (Add (C lt, ao), Sub (mw, mx)) + | numadd (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, Sub (ooa, opa)), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Sub (ooa, opa)), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, Neg ol), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Neg ol), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, Cx (oj, ok)), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, Cx (oj, ok)), mv)) + | numadd (Add (C lt, ao), Add (Mul (nu, C oh), mv)) = + Add (Add (C lt, ao), Add (Mul (nu, C oh), mv)) + | numadd (Add (C lt, ao), Add (Sub (ns, nt), mv)) = + Add (Add (C lt, ao), Add (Sub (ns, nt), mv)) + | numadd (Add (C lt, ao), Add (Add (nq, nr), mv)) = + Add (Add (C lt, ao), Add (Add (nq, nr), mv)) + | numadd (Add (C lt, ao), Add (Neg np, mv)) = + Add (Add (C lt, ao), Add (Neg np, mv)) + | numadd (Add (C lt, ao), Add (Cx (nn, no), mv)) = + Add (Add (C lt, ao), Add (Cx (nn, no), mv)) + | numadd (Add (C lt, ao), Add (Bound nm, mv)) = + Add (Add (C lt, ao), Add (Bound nm, mv)) + | numadd (Add (C lt, ao), Add (C nl, mv)) = + Add (Add (C lt, ao), Add (C nl, mv)) + | numadd (Add (C lt, ao), Neg mt) = Add (Add (C lt, ao), Neg mt) + | numadd (Add (C lt, ao), Cx (mr, ms)) = Add (Add (C lt, ao), Cx (mr, ms)) + | numadd (Add (C lt, ao), Bound mq) = Add (Add (C lt, ao), Bound mq) + | numadd (Add (C lt, ao), C mp) = Add (Add (C lt, ao), C mp) + | numadd (Neg am, Mul (jd, je)) = Add (Neg am, Mul (jd, je)) + | numadd (Neg am, Sub (jb, jc)) = Add (Neg am, Sub (jb, jc)) + | numadd (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) = + Add (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) + | numadd (Neg am, Add (Mul (jz, Sub (kt, ku)), ja)) = + Add (Neg am, Add (Mul (jz, Sub (kt, ku)), ja)) + | numadd (Neg am, Add (Mul (jz, Add (kr, ks)), ja)) = + Add (Neg am, Add (Mul (jz, Add (kr, ks)), ja)) + | numadd (Neg am, Add (Mul (jz, Neg kq), ja)) = + Add (Neg am, Add (Mul (jz, Neg kq), ja)) + | numadd (Neg am, Add (Mul (jz, Cx (ko, kp)), ja)) = + Add (Neg am, Add (Mul (jz, Cx (ko, kp)), ja)) + | numadd (Neg am, Add (Mul (jz, C km), ja)) = + Add (Neg am, Add (Mul (jz, C km), ja)) + | numadd (Neg am, Add (Sub (jx, jy), ja)) = + Add (Neg am, Add (Sub (jx, jy), ja)) + | numadd (Neg am, Add (Add (jv, jw), ja)) = + Add (Neg am, Add (Add (jv, jw), ja)) + | numadd (Neg am, Add (Neg ju, ja)) = Add (Neg am, Add (Neg ju, ja)) + | numadd (Neg am, Add (Cx (js, jt), ja)) = Add (Neg am, Add (Cx (js, jt), ja)) + | numadd (Neg am, Add (Bound jr, ja)) = Add (Neg am, Add (Bound jr, ja)) + | numadd (Neg am, Add (C jq, ja)) = Add (Neg am, Add (C jq, ja)) + | numadd (Neg am, Neg iy) = Add (Neg am, Neg iy) + | numadd (Neg am, Cx (iw, ix)) = Add (Neg am, Cx (iw, ix)) + | numadd (Neg am, Bound iv) = Add (Neg am, Bound iv) + | numadd (Neg am, C iu) = Add (Neg am, C iu) + | numadd (Cx (ak, al), Mul (gp, gq)) = Add (Cx (ak, al), Mul (gp, gq)) + | numadd (Cx (ak, al), Sub (gn, go)) = Add (Cx (ak, al), Sub (gn, go)) + | numadd (Cx (ak, al), Add (Mul (hl, Mul (ih, ii)), gm)) = + Add (Cx (ak, al), Add (Mul (hl, Mul (ih, ii)), gm)) + | numadd (Cx (ak, al), Add (Mul (hl, Sub (ifa, ig)), gm)) = + Add (Cx (ak, al), Add (Mul (hl, Sub (ifa, ig)), gm)) + | numadd (Cx (ak, al), Add (Mul (hl, Add (id, ie)), gm)) = + Add (Cx (ak, al), Add (Mul (hl, Add (id, ie)), gm)) + | numadd (Cx (ak, al), Add (Mul (hl, Neg ic), gm)) = + Add (Cx (ak, al), Add (Mul (hl, Neg ic), gm)) + | numadd (Cx (ak, al), Add (Mul (hl, Cx (ia, ib)), gm)) = + Add (Cx (ak, al), Add (Mul (hl, Cx (ia, ib)), gm)) + | numadd (Cx (ak, al), Add (Mul (hl, C hy), gm)) = + Add (Cx (ak, al), Add (Mul (hl, C hy), gm)) + | numadd (Cx (ak, al), Add (Sub (hj, hk), gm)) = + Add (Cx (ak, al), Add (Sub (hj, hk), gm)) + | numadd (Cx (ak, al), Add (Add (hh, hi), gm)) = + Add (Cx (ak, al), Add (Add (hh, hi), gm)) + | numadd (Cx (ak, al), Add (Neg hg, gm)) = Add (Cx (ak, al), Add (Neg hg, gm)) + | numadd (Cx (ak, al), Add (Cx (he, hf), gm)) = + Add (Cx (ak, al), Add (Cx (he, hf), gm)) + | numadd (Cx (ak, al), Add (Bound hd, gm)) = + Add (Cx (ak, al), Add (Bound hd, gm)) + | numadd (Cx (ak, al), Add (C hc, gm)) = Add (Cx (ak, al), Add (C hc, gm)) + | numadd (Cx (ak, al), Neg gk) = Add (Cx (ak, al), Neg gk) + | numadd (Cx (ak, al), Cx (gi, gj)) = Add (Cx (ak, al), Cx (gi, gj)) + | numadd (Cx (ak, al), Bound gh) = Add (Cx (ak, al), Bound gh) + | numadd (Cx (ak, al), C gg) = Add (Cx (ak, al), C gg) + | numadd (Bound aj, Mul (eb, ec)) = Add (Bound aj, Mul (eb, ec)) + | numadd (Bound aj, Sub (dz, ea)) = Add (Bound aj, Sub (dz, ea)) + | numadd (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) = + Add (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) + | numadd (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy)) = + Add (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy)) + | numadd (Bound aj, Add (Mul (ex, Add (fp, fq)), dy)) = + Add (Bound aj, Add (Mul (ex, Add (fp, fq)), dy)) + | numadd (Bound aj, Add (Mul (ex, Neg fo), dy)) = + Add (Bound aj, Add (Mul (ex, Neg fo), dy)) + | numadd (Bound aj, Add (Mul (ex, Cx (fm, fna)), dy)) = + Add (Bound aj, Add (Mul (ex, Cx (fm, fna)), dy)) + | numadd (Bound aj, Add (Mul (ex, C fk), dy)) = + Add (Bound aj, Add (Mul (ex, C fk), dy)) + | numadd (Bound aj, Add (Sub (ev, ew), dy)) = + Add (Bound aj, Add (Sub (ev, ew), dy)) + | numadd (Bound aj, Add (Add (et, eu), dy)) = + Add (Bound aj, Add (Add (et, eu), dy)) + | numadd (Bound aj, Add (Neg es, dy)) = Add (Bound aj, Add (Neg es, dy)) + | numadd (Bound aj, Add (Cx (eq, er), dy)) = + Add (Bound aj, Add (Cx (eq, er), dy)) + | numadd (Bound aj, Add (Bound ep, dy)) = Add (Bound aj, Add (Bound ep, dy)) + | numadd (Bound aj, Add (C eo, dy)) = Add (Bound aj, Add (C eo, dy)) + | numadd (Bound aj, Neg dw) = Add (Bound aj, Neg dw) + | numadd (Bound aj, Cx (du, dv)) = Add (Bound aj, Cx (du, dv)) + | numadd (Bound aj, Bound dt) = Add (Bound aj, Bound dt) + | numadd (Bound aj, C ds) = Add (Bound aj, C ds) + | numadd (C ai, Mul (bn, bo)) = Add (C ai, Mul (bn, bo)) + | numadd (C ai, Sub (bl, bm)) = Add (C ai, Sub (bl, bm)) + | numadd (C ai, Add (Mul (cj, Mul (df, dg)), bk)) = + Add (C ai, Add (Mul (cj, Mul (df, dg)), bk)) + | numadd (C ai, Add (Mul (cj, Sub (dd, de)), bk)) = + Add (C ai, Add (Mul (cj, Sub (dd, de)), bk)) + | numadd (C ai, Add (Mul (cj, Add (db, dc)), bk)) = + Add (C ai, Add (Mul (cj, Add (db, dc)), bk)) + | numadd (C ai, Add (Mul (cj, Neg da), bk)) = + Add (C ai, Add (Mul (cj, Neg da), bk)) + | numadd (C ai, Add (Mul (cj, Cx (cy, cz)), bk)) = + Add (C ai, Add (Mul (cj, Cx (cy, cz)), bk)) + | numadd (C ai, Add (Mul (cj, C cw), bk)) = + Add (C ai, Add (Mul (cj, C cw), bk)) + | numadd (C ai, Add (Sub (ch, ci), bk)) = Add (C ai, Add (Sub (ch, ci), bk)) + | numadd (C ai, Add (Add (cf, cg), bk)) = Add (C ai, Add (Add (cf, cg), bk)) + | numadd (C ai, Add (Neg ce, bk)) = Add (C ai, Add (Neg ce, bk)) + | numadd (C ai, Add (Cx (cc, cd), bk)) = Add (C ai, Add (Cx (cc, cd), bk)) + | numadd (C ai, Add (Bound cb, bk)) = Add (C ai, Add (Bound cb, bk)) + | numadd (C ai, Add (C ca, bk)) = Add (C ai, Add (C ca, bk)) + | numadd (C ai, Neg bi) = Add (C ai, Neg bi) + | numadd (C ai, Cx (bg, bh)) = Add (C ai, Cx (bg, bh)) + | numadd (C ai, Bound bf) = Add (C ai, Bound bf) + | numadd (C b1, C b2) = C (IntInf.+ (b1, b2)) + | numadd (Mul (ag, ah), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Mul (ag, ah), r2)) + | numadd (Sub (ae, af), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Sub (ae, af), r2)) + | numadd (Add (Mul (lr, Mul (ace, acf)), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Mul (ace, acf)), ad), r2)) + | numadd (Add (Mul (lr, Sub (acc, acd)), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Sub (acc, acd)), ad), r2)) + | numadd (Add (Mul (lr, Add (aca, acb)), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Add (aca, acb)), ad), r2)) + | numadd (Add (Mul (lr, Neg abz), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Neg abz), ad), r2)) + | numadd (Add (Mul (lr, Cx (abx, aby)), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Cx (abx, aby)), ad), r2)) + | numadd (Add (Mul (lr, C abv), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, C abv), ad), r2)) + | numadd (Add (Sub (lp, lq), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Sub (lp, lq), ad), r2)) + | numadd (Add (Add (ln, lo), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Add (ln, lo), ad), r2)) + | numadd (Add (Neg lm, ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Neg lm, ad), r2)) + | numadd (Add (Cx (lk, ll), ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Cx (lk, ll), ad), r2)) + | numadd (Add (Bound lj, ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (Bound lj, ad), r2)) + | numadd (Add (C li, ad), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Add (C li, ad), r2)) + | numadd (Neg ab, Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Neg ab, r2)) + | numadd (Cx (y, z), Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Cx (y, z), r2)) + | numadd (Bound x, Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (Bound x, r2)) + | numadd (C w, Add (Mul (c2, Bound n2), r2)) = + Add (Mul (c2, Bound n2), numadd (C w, r2)) + | numadd (Add (Mul (c1, Bound n1), r1), Mul (afz, aga)) = + Add (Mul (c1, Bound n1), numadd (r1, Mul (afz, aga))) + | numadd (Add (Mul (c1, Bound n1), r1), Sub (afx, afy)) = + Add (Mul (c1, Bound n1), numadd (r1, Sub (afx, afy))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Mul (ain, aio)), afw)) + = Add (Mul (c1, Bound n1), + numadd (r1, Add (Mul (ahg, Mul (ain, aio)), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Sub (ail, aim)), afw)) + = Add (Mul (c1, Bound n1), + numadd (r1, Add (Mul (ahg, Sub (ail, aim)), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Add (aij, aik)), afw)) + = Add (Mul (c1, Bound n1), + numadd (r1, Add (Mul (ahg, Add (aij, aik)), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Neg aii), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Neg aii), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Cx (aig, aih)), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Cx (aig, aih)), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, C aie), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, C aie), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Sub (ahe, ahf), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Sub (ahe, ahf), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Add (ahc, ahd), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Add (ahc, ahd), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Neg ahb, afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Neg ahb, afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Cx (agz, aha), afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Cx (agz, aha), afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Bound agy, afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (Bound agy, afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Add (C agx, afw)) = + Add (Mul (c1, Bound n1), numadd (r1, Add (C agx, afw))) + | numadd (Add (Mul (c1, Bound n1), r1), Neg afu) = + Add (Mul (c1, Bound n1), numadd (r1, Neg afu)) + | numadd (Add (Mul (c1, Bound n1), r1), Cx (afs, aft)) = + Add (Mul (c1, Bound n1), numadd (r1, Cx (afs, aft))) + | numadd (Add (Mul (c1, Bound n1), r1), Bound afr) = + Add (Mul (c1, Bound n1), numadd (r1, Bound afr)) + | numadd (Add (Mul (c1, Bound n1), r1), C afq) = + Add (Mul (c1, Bound n1), numadd (r1, C afq)) + | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (c2, Bound n2), r2)) = + (if ((n1 : IntInf.int) = n2) + then let + val c = IntInf.+ (c1, c2); + in + (if ((c : IntInf.int) = (0 : IntInf.int)) then numadd (r1, r2) + else Add (Mul (c, Bound n1), numadd (r1, r2))) + end + else (if IntInf.<= (n1, n2) + then Add (Mul (c1, Bound n1), + numadd (r1, Add (Mul (c2, Bound n2), r2))) + else Add (Mul (c2, Bound n2), + numadd (Add (Mul (c1, Bound n1), r1), r2)))); -fun nummul (C j) = (fn i => C (i * j)) - | nummul (Add (a, b)) = (fn i => numadd (nummul a i, nummul b i)) - | nummul (Mul (c, t)) = (fn i => nummul t (i * c)) - | nummul (Bound v) = (fn i => Mul (i, Bound v)) - | nummul (CX (w, x)) = (fn i => Mul (i, CX (w, x))) - | nummul (Neg y) = (fn i => Mul (i, Neg y)) - | nummul (Sub (ac, ad)) = (fn i => Mul (i, Sub (ac, ad))); +fun nummul i (Sub (v, va)) = Mul (i, Sub (v, va)) + | nummul i (Neg v) = Mul (i, Neg v) + | nummul i (Cx (v, va)) = Mul (i, Cx (v, va)) + | nummul i (Bound v) = Mul (i, Bound v) + | nummul i (Mul (c, t)) = nummul (IntInf.* (i, c)) t + | nummul i (Add (a, b)) = numadd (nummul i a, nummul i b) + | nummul i (C j) = C (IntInf.* (i, j)); + +fun numneg t = nummul (IntInf.~ (1 : IntInf.int)) t; + +fun numsub s t = + (if eq_num s t then C (0 : IntInf.int) else numadd (s, numneg t)); -fun numneg t = nummul t (~ 1); +fun simpnum (Cx (v, va)) = Cx (v, va) + | simpnum (Mul (i, t)) = + (if ((i : IntInf.int) = (0 : IntInf.int)) then C (0 : IntInf.int) + else nummul i (simpnum t)) + | simpnum (Sub (t, s)) = numsub (simpnum t) (simpnum s) + | simpnum (Add (t, s)) = numadd (simpnum t, simpnum s) + | simpnum (Neg t) = numneg (simpnum t) + | simpnum (Bound n) = + Add (Mul ((1 : IntInf.int), Bound n), C (0 : IntInf.int)) + | simpnum (C j) = C j; + +val eq_numa = {eq = eq_num} : num HOL.eq; -fun numsub s t = (if (s = t) then C 0 else numadd (s, numneg t)); +fun mirror (NClosed aq) = NClosed aq + | mirror (Closed ap) = Closed ap + | mirror (A ao) = A ao + | mirror (E an) = E an + | mirror (Iffa (al, am)) = Iffa (al, am) + | mirror (Impa (aj, ak)) = Impa (aj, ak) + | mirror (Nota ae) = Nota ae + | mirror (NDvd (ac, Mul (hv, hw))) = NDvd (ac, Mul (hv, hw)) + | mirror (NDvd (ac, Sub (ht, hu))) = NDvd (ac, Sub (ht, hu)) + | mirror (NDvd (ac, Add (hr, hs))) = NDvd (ac, Add (hr, hs)) + | mirror (NDvd (ac, Neg hq)) = NDvd (ac, Neg hq) + | mirror (NDvd (ac, Bound hn)) = NDvd (ac, Bound hn) + | mirror (NDvd (ac, C hm)) = NDvd (ac, C hm) + | mirror (Dvd (aa, Mul (gz, ha))) = Dvd (aa, Mul (gz, ha)) + | mirror (Dvd (aa, Sub (gx, gy))) = Dvd (aa, Sub (gx, gy)) + | mirror (Dvd (aa, Add (gv, gw))) = Dvd (aa, Add (gv, gw)) + | mirror (Dvd (aa, Neg gu)) = Dvd (aa, Neg gu) + | mirror (Dvd (aa, Bound gr)) = Dvd (aa, Bound gr) + | mirror (Dvd (aa, C gq)) = Dvd (aa, C gq) + | mirror (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge)) + | mirror (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc)) + | mirror (NEq (Add (fz, ga))) = NEq (Add (fz, ga)) + | mirror (NEq (Neg fy)) = NEq (Neg fy) + | mirror (NEq (Bound fv)) = NEq (Bound fv) + | mirror (NEq (C fu)) = NEq (C fu) + | mirror (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi)) + | mirror (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg)) + | mirror (Eq (Add (fd, fe))) = Eq (Add (fd, fe)) + | mirror (Eq (Neg fc)) = Eq (Neg fc) + | mirror (Eq (Bound ez)) = Eq (Bound ez) + | mirror (Eq (C ey)) = Eq (C ey) + | mirror (Ge (Mul (el, em))) = Ge (Mul (el, em)) + | mirror (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek)) + | mirror (Ge (Add (eh, ei))) = Ge (Add (eh, ei)) + | mirror (Ge (Neg eg)) = Ge (Neg eg) + | mirror (Ge (Bound ed)) = Ge (Bound ed) + | mirror (Ge (C ec)) = Ge (C ec) + | mirror (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq)) + | mirror (Gt (Sub (dn, doa))) = Gt (Sub (dn, doa)) + | mirror (Gt (Add (dl, dm))) = Gt (Add (dl, dm)) + | mirror (Gt (Neg dk)) = Gt (Neg dk) + | mirror (Gt (Bound dh)) = Gt (Bound dh) + | mirror (Gt (C dg)) = Gt (C dg) + | mirror (Le (Mul (ct, cu))) = Le (Mul (ct, cu)) + | mirror (Le (Sub (cr, cs))) = Le (Sub (cr, cs)) + | mirror (Le (Add (cp, cq))) = Le (Add (cp, cq)) + | mirror (Le (Neg co)) = Le (Neg co) + | mirror (Le (Bound cl)) = Le (Bound cl) + | mirror (Le (C ck)) = Le (C ck) + | mirror (Lt (Mul (bx, by))) = Lt (Mul (bx, by)) + | mirror (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw)) + | mirror (Lt (Add (bt, bu))) = Lt (Add (bt, bu)) + | mirror (Lt (Neg bs)) = Lt (Neg bs) + | mirror (Lt (Bound bp)) = Lt (Bound bp) + | mirror (Lt (C bo)) = Lt (C bo) + | mirror F = F + | mirror T = T + | mirror (NDvd (i, Cx (c, e))) = NDvd (i, Cx (c, Neg e)) + | mirror (Dvd (i, Cx (c, e))) = Dvd (i, Cx (c, Neg e)) + | mirror (Ge (Cx (c, e))) = Le (Cx (c, Neg e)) + | mirror (Gt (Cx (c, e))) = Lt (Cx (c, Neg e)) + | mirror (Le (Cx (c, e))) = Ge (Cx (c, Neg e)) + | mirror (Lt (Cx (c, e))) = Gt (Cx (c, Neg e)) + | mirror (NEq (Cx (c, e))) = NEq (Cx (c, Neg e)) + | mirror (Eq (Cx (c, e))) = Eq (Cx (c, Neg e)) + | mirror (Or (p, q)) = Or (mirror p, mirror q) + | mirror (And (p, q)) = And (mirror p, mirror q); -fun simpnum (C j) = C j - | simpnum (Bound n) = Add (Mul (1, Bound n), C 0) - | 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) then C 0 else nummul (simpnum t) i) - | simpnum (CX (w, x)) = CX (w, x); +fun unita p = + let + val p' = zlfm p; + val l = zeta p'; + val q = + And (Dvd (l, Cx ((1 : IntInf.int), C (0 : IntInf.int))), a_beta p' l); + val d = delta q; + val b = List.remdups eq_numa (List.map simpnum (beta q)); + val a = List.remdups eq_numa (List.map simpnum (alpha q)); + in + (if IntInf.<= ((List.size_list b), (List.size_list a)) then (q, (b, d)) + else (mirror q, (a, d))) + end; + +fun iupt i j = + (if IntInf.< (j, i) then [] + else i :: iupt (IntInf.+ (i, (1 : IntInf.int))) j); -datatype fm = T | F | Lt of num | Le of num | Gt of num | Ge of num | Eq of num - | NEq of num | Dvd of 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 minusinf (NClosed aq) = NClosed aq + | minusinf (Closed ap) = Closed ap + | minusinf (A ao) = A ao + | minusinf (E an) = E an + | minusinf (Iffa (al, am)) = Iffa (al, am) + | minusinf (Impa (aj, ak)) = Impa (aj, ak) + | minusinf (Nota ae) = Nota ae + | minusinf (NDvd (ac, ad)) = NDvd (ac, ad) + | minusinf (Dvd (aa, ab)) = Dvd (aa, ab) + | minusinf (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge)) + | minusinf (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc)) + | minusinf (NEq (Add (fz, ga))) = NEq (Add (fz, ga)) + | minusinf (NEq (Neg fy)) = NEq (Neg fy) + | minusinf (NEq (Bound fv)) = NEq (Bound fv) + | minusinf (NEq (C fu)) = NEq (C fu) + | minusinf (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi)) + | minusinf (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg)) + | minusinf (Eq (Add (fd, fe))) = Eq (Add (fd, fe)) + | minusinf (Eq (Neg fc)) = Eq (Neg fc) + | minusinf (Eq (Bound ez)) = Eq (Bound ez) + | minusinf (Eq (C ey)) = Eq (C ey) + | minusinf (Ge (Mul (el, em))) = Ge (Mul (el, em)) + | minusinf (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek)) + | minusinf (Ge (Add (eh, ei))) = Ge (Add (eh, ei)) + | minusinf (Ge (Neg eg)) = Ge (Neg eg) + | minusinf (Ge (Bound ed)) = Ge (Bound ed) + | minusinf (Ge (C ec)) = Ge (C ec) + | minusinf (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq)) + | minusinf (Gt (Sub (dn, doa))) = Gt (Sub (dn, doa)) + | minusinf (Gt (Add (dl, dm))) = Gt (Add (dl, dm)) + | minusinf (Gt (Neg dk)) = Gt (Neg dk) + | minusinf (Gt (Bound dh)) = Gt (Bound dh) + | minusinf (Gt (C dg)) = Gt (C dg) + | minusinf (Le (Mul (ct, cu))) = Le (Mul (ct, cu)) + | minusinf (Le (Sub (cr, cs))) = Le (Sub (cr, cs)) + | minusinf (Le (Add (cp, cq))) = Le (Add (cp, cq)) + | minusinf (Le (Neg co)) = Le (Neg co) + | minusinf (Le (Bound cl)) = Le (Bound cl) + | minusinf (Le (C ck)) = Le (C ck) + | minusinf (Lt (Mul (bx, by))) = Lt (Mul (bx, by)) + | minusinf (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw)) + | minusinf (Lt (Add (bt, bu))) = Lt (Add (bt, bu)) + | minusinf (Lt (Neg bs)) = Lt (Neg bs) + | minusinf (Lt (Bound bp)) = Lt (Bound bp) + | minusinf (Lt (C bo)) = Lt (C bo) + | minusinf F = F + | minusinf T = T + | minusinf (Ge (Cx (c, e))) = F + | minusinf (Gt (Cx (c, e))) = F + | minusinf (Le (Cx (c, e))) = T + | minusinf (Lt (Cx (c, e))) = T + | minusinf (NEq (Cx (c, e))) = T + | minusinf (Eq (Cx (c, e))) = F + | minusinf (Or (p, q)) = Or (minusinf p, minusinf q) + | minusinf (And (p, q)) = And (minusinf p, minusinf q); -fun not (NOT p) = p - | not T = F - | not F = T - | not (Lt u) = NOT (Lt u) - | not (Le v) = NOT (Le v) - | not (Gt w) = NOT (Gt w) - | not (Ge x) = NOT (Ge x) - | not (Eq y) = NOT (Eq y) - | not (NEq z) = NOT (NEq z) - | not (Dvd (aa, ab)) = NOT (Dvd (aa, ab)) - | not (NDvd (ac, ad)) = NOT (NDvd (ac, ad)) - | not (And (af, ag)) = NOT (And (af, ag)) - | not (Or (ah, ai)) = NOT (Or (ah, ai)) - | not (Imp (aj, ak)) = NOT (Imp (aj, ak)) - | not (Iff (al, am)) = NOT (Iff (al, am)) - | not (E an) = NOT (E an) - | not (A ao) = NOT (A ao) - | not (Closed ap) = NOT (Closed ap) - | not (NClosed aq) = NOT (NClosed aq); +fun numsubst0 t (Mul (i, a)) = Mul (i, numsubst0 t a) + | numsubst0 t (Sub (a, b)) = Sub (numsubst0 t a, numsubst0 t b) + | numsubst0 t (Add (a, b)) = Add (numsubst0 t a, numsubst0 t b) + | numsubst0 t (Neg a) = Neg (numsubst0 t a) + | numsubst0 t (Cx (i, a)) = Add (Mul (i, t), numsubst0 t a) + | numsubst0 t (Bound n) = + (if ((n : IntInf.int) = Integer.zero_nat) then t else Bound n) + | numsubst0 t (C c) = C c; + +fun subst0 t (NClosed p) = NClosed p + | subst0 t (Closed p) = Closed p + | subst0 t (Iffa (p, q)) = Iffa (subst0 t p, subst0 t q) + | subst0 t (Impa (p, q)) = Impa (subst0 t p, subst0 t q) + | subst0 t (Or (p, q)) = Or (subst0 t p, subst0 t q) + | subst0 t (And (p, q)) = And (subst0 t p, subst0 t q) + | subst0 t (Nota p) = Nota (subst0 t p) + | subst0 t (NDvd (i, a)) = NDvd (i, numsubst0 t a) + | subst0 t (Dvd (i, a)) = Dvd (i, numsubst0 t a) + | subst0 t (NEq a) = NEq (numsubst0 t a) + | subst0 t (Eq a) = Eq (numsubst0 t a) + | subst0 t (Ge a) = Ge (numsubst0 t a) + | subst0 t (Gt a) = Gt (numsubst0 t a) + | subst0 t (Le a) = Le (numsubst0 t a) + | subst0 t (Lt a) = Lt (numsubst0 t a) + | subst0 t F = F + | subst0 t T = T; + +fun conj p q = + (if eq_fm p F orelse eq_fm q F then F + else (if eq_fm p T then q else (if eq_fm q T then p else And (p, q)))); + +fun disj p q = + (if eq_fm p T orelse eq_fm q T then T + else (if eq_fm p F then q else (if eq_fm q F then p else Or (p, q)))); + +fun nota (NClosed v) = Nota (NClosed v) + | nota (Closed v) = Nota (Closed v) + | nota (A v) = Nota (A v) + | nota (E v) = Nota (E v) + | nota (Iffa (v, va)) = Nota (Iffa (v, va)) + | nota (Impa (v, va)) = Nota (Impa (v, va)) + | nota (Or (v, va)) = Nota (Or (v, va)) + | nota (And (v, va)) = Nota (And (v, va)) + | nota (NDvd (v, va)) = Nota (NDvd (v, va)) + | nota (Dvd (v, va)) = Nota (Dvd (v, va)) + | nota (NEq v) = Nota (NEq v) + | nota (Eq v) = Nota (Eq v) + | nota (Ge v) = Nota (Ge v) + | nota (Gt v) = Nota (Gt v) + | nota (Le v) = Nota (Le v) + | nota (Lt v) = Nota (Lt v) + | nota F = T + | nota T = F + | nota (Nota y) = y; + +fun imp p q = + (if eq_fm p F orelse eq_fm q T then T + else (if eq_fm p T then q + else (if eq_fm q F then nota p else Impa (p, q)))); fun iff p q = - (if (p = q) then T - else (if ((p = not q) orelse (not p = q)) then F - else (if (p = F) then not q - else (if (q = F) then not p - else (if (p = T) then q - else (if (q = T) then p else Iff (p, q))))))); - -fun imp p q = - (if ((p = F) orelse (q = T)) then T - else (if (p = T) then q else (if (q = F) then not p else Imp (p, q)))); - -fun disj p q = - (if ((p = T) orelse (q = T)) then T - else (if (p = F) then q else (if (q = F) then p else Or (p, q)))); - -fun conj p q = - (if ((p = F) orelse (q = F)) then F - else (if (p = T) then q else (if (q = T) then p else And (p, q)))); + (if eq_fm p q then T + else (if eq_fm p (nota q) orelse eq_fm (nota p) q then F + else (if eq_fm p F then nota q + else (if eq_fm q F then nota p + else (if eq_fm p T then q + else (if eq_fm q T then p + else Iffa (p, q))))))); -fun simpfm (And (p, q)) = conj (simpfm p) (simpfm q) - | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q) - | simpfm (Imp (p, q)) = imp (simpfm p) (simpfm q) - | simpfm (Iff (p, q)) = iff (simpfm p) (simpfm q) - | simpfm (NOT p) = not (simpfm p) - | simpfm (Lt a) = - let val a' = simpnum a - in (case a' of C x => (if (x < 0) then T else F) | Bound x => Lt a' - | CX (x, xa) => Lt a' | Neg x => Lt a' | Add (x, xa) => Lt a' - | Sub (x, xa) => Lt a' | Mul (x, xa) => Lt a') - end - | simpfm (Le a) = - let val a' = simpnum a - in (case a' of C x => (if (x <= 0) then T else F) | Bound x => Le a' - | CX (x, xa) => Le a' | Neg x => Le a' | Add (x, xa) => Le a' - | Sub (x, xa) => Le a' | Mul (x, xa) => Le a') - end - | simpfm (Gt a) = - let val a' = simpnum a - in (case a' of C x => (if (0 < x) then T else F) | Bound x => Gt a' - | CX (x, xa) => Gt a' | Neg x => Gt a' | Add (x, xa) => Gt a' - | Sub (x, xa) => Gt a' | Mul (x, xa) => Gt a') - end - | simpfm (Ge a) = - let val a' = simpnum a - in (case a' of C x => (if (0 <= x) then T else F) | Bound x => Ge a' - | CX (x, xa) => Ge a' | Neg x => Ge a' | Add (x, xa) => Ge a' - | Sub (x, xa) => Ge a' | Mul (x, xa) => Ge a') +fun simpfm (NClosed v) = NClosed v + | simpfm (Closed v) = Closed v + | simpfm (A v) = A v + | simpfm (E v) = E v + | simpfm F = F + | simpfm T = T + | simpfm (NDvd (i, a)) = + (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (NEq a) + else (if (((Integer.abs_int i) : IntInf.int) = (1 : IntInf.int)) then F + else let + val a' = simpnum a; + in + (case a' + of C v => (if not (Integer.dvd_int i v) then T else F) + | Bound nat => NDvd (i, a') + | Cx (int, num) => NDvd (i, a') | Neg num => NDvd (i, a') + | Add (num1, num2) => NDvd (i, a') + | Sub (num1, num2) => NDvd (i, a') + | Mul (int, num) => NDvd (i, a')) + end)) + | simpfm (Dvd (i, a)) = + (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (Eq a) + else (if (((Integer.abs_int i) : IntInf.int) = (1 : IntInf.int)) then T + else let + val a' = simpnum a; + in + (case a' of C v => (if Integer.dvd_int i v then T else F) + | Bound nat => Dvd (i, a') | Cx (int, num) => Dvd (i, a') + | Neg num => Dvd (i, a') + | Add (num1, num2) => Dvd (i, a') + | Sub (num1, num2) => Dvd (i, a') + | Mul (int, num) => Dvd (i, a')) + end)) + | simpfm (NEq a) = + let + val a' = simpnum a; + in + (case a' + of C v => (if not ((v : IntInf.int) = (0 : IntInf.int)) then T else F) + | Bound nat => NEq a' | Cx (int, num) => NEq a' | Neg num => NEq a' + | Add (num1, num2) => NEq a' | Sub (num1, num2) => NEq a' + | Mul (int, num) => NEq a') end | simpfm (Eq a) = - let val a' = simpnum a - in (case a' of C x => (if (x = 0) then T else F) | Bound x => Eq a' - | CX (x, xa) => Eq a' | Neg x => Eq a' | Add (x, xa) => Eq a' - | Sub (x, xa) => Eq a' | Mul (x, xa) => Eq a') + let + val a' = simpnum a; + in + (case a' + of C v => (if ((v : IntInf.int) = (0 : IntInf.int)) then T else F) + | Bound nat => Eq a' | Cx (int, num) => Eq a' | Neg num => Eq a' + | Add (num1, num2) => Eq a' | Sub (num1, num2) => Eq a' + | Mul (int, num) => Eq a') end - | simpfm (NEq a) = - let val a' = simpnum a - in (case a' of C x => (if Bool.not (x = 0) then T else F) - | Bound x => NEq a' | CX (x, xa) => NEq a' | Neg x => NEq a' - | Add (x, xa) => NEq a' | Sub (x, xa) => NEq a' - | Mul (x, xa) => NEq a') + | simpfm (Ge a) = + let + val a' = simpnum a; + in + (case a' of C v => (if IntInf.<= ((0 : IntInf.int), v) then T else F) + | Bound nat => Ge a' | Cx (int, num) => Ge a' | Neg num => Ge a' + | Add (num1, num2) => Ge a' | Sub (num1, num2) => Ge a' + | Mul (int, num) => Ge a') end - | simpfm (Dvd (i, a)) = - (if (i = 0) then simpfm (Eq a) - else (if (abs i = 1) then T - else let val a' = simpnum a - in (case a' of C x => (if dvd i x then T else F) - | Bound x => Dvd (i, a') | CX (x, xa) => Dvd (i, a') - | Neg x => Dvd (i, a') | Add (x, xa) => Dvd (i, a') - | Sub (x, xa) => Dvd (i, a') - | Mul (x, xa) => Dvd (i, a')) - end)) - | simpfm (NDvd (i, a)) = - (if (i = 0) then simpfm (NEq a) - else (if (abs i = 1) then F - else let val a' = simpnum a - in (case a' of C x => (if Bool.not (dvd i x) then T else F) - | Bound x => NDvd (i, a') | CX (x, xa) => NDvd (i, a') - | Neg x => NDvd (i, a') | Add (x, xa) => NDvd (i, a') - | Sub (x, xa) => NDvd (i, a') - | Mul (x, xa) => NDvd (i, a')) - end)) - | simpfm T = T - | simpfm F = F - | simpfm (E ao) = E ao - | simpfm (A ap) = A ap - | simpfm (Closed aq) = Closed aq - | simpfm (NClosed ar) = NClosed ar; - -fun foldr f [] a = a - | foldr f (x :: xs) a = f x (foldr f xs a); - -fun djf f p q = - (if (q = T) then T - else (if (q = F) then f p - else let val fp = f p - in (case fp of T => T | F => q | Lt x => Or (f p, q) - | Le x => Or (f p, q) | Gt x => Or (f p, q) - | Ge x => Or (f p, q) | Eq x => Or (f p, q) - | NEq x => Or (f p, q) | Dvd (x, xa) => Or (f p, q) - | NDvd (x, xa) => Or (f p, q) | NOT x => Or (f p, q) - | And (x, xa) => Or (f p, q) | Or (x, xa) => Or (f p, q) - | Imp (x, xa) => Or (f p, q) | Iff (x, xa) => Or (f p, q) - | E x => Or (f p, q) | A x => Or (f p, q) - | Closed x => Or (f p, q) | NClosed x => Or (f p, q)) - end)); - -fun evaldjf f ps = foldr (djf f) ps F; - -fun append [] ys = ys - | append (x :: xs) ys = (x :: append xs ys); - -fun disjuncts (Or (p, q)) = append (disjuncts p) (disjuncts q) - | disjuncts F = [] - | disjuncts T = [T] - | disjuncts (Lt u) = [Lt u] - | disjuncts (Le v) = [Le v] - | disjuncts (Gt w) = [Gt w] - | disjuncts (Ge x) = [Ge x] - | disjuncts (Eq y) = [Eq y] - | disjuncts (NEq z) = [NEq z] - | disjuncts (Dvd (aa, ab)) = [Dvd (aa, ab)] - | disjuncts (NDvd (ac, ad)) = [NDvd (ac, ad)] - | disjuncts (NOT ae) = [NOT ae] - | disjuncts (And (af, ag)) = [And (af, ag)] - | disjuncts (Imp (aj, ak)) = [Imp (aj, ak)] - | disjuncts (Iff (al, am)) = [Iff (al, am)] - | disjuncts (E an) = [E an] - | disjuncts (A ao) = [A ao] - | disjuncts (Closed ap) = [Closed ap] - | disjuncts (NClosed aq) = [NClosed aq]; - -fun DJ f p = evaldjf f (disjuncts p); - -fun qelim (E p) = (fn qe => DJ qe (qelim p qe)) - | qelim (A p) = (fn qe => not (qe (qelim (NOT p) qe))) - | qelim (NOT p) = (fn qe => not (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 => imp (qelim p qe) (qelim q qe)) - | qelim (Iff (p, q)) = (fn qe => iff (qelim p qe) (qelim q qe)) - | qelim T = (fn y => simpfm T) - | qelim F = (fn y => simpfm F) - | qelim (Lt u) = (fn y => simpfm (Lt u)) - | qelim (Le v) = (fn y => simpfm (Le v)) - | qelim (Gt w) = (fn y => simpfm (Gt w)) - | qelim (Ge x) = (fn y => simpfm (Ge x)) - | qelim (Eq y) = (fn ya => simpfm (Eq y)) - | qelim (NEq z) = (fn y => simpfm (NEq z)) - | qelim (Dvd (aa, ab)) = (fn y => simpfm (Dvd (aa, ab))) - | qelim (NDvd (ac, ad)) = (fn y => simpfm (NDvd (ac, ad))) - | qelim (Closed ap) = (fn y => simpfm (Closed ap)) - | qelim (NClosed aq) = (fn y => simpfm (NClosed aq)); - -fun minus_def1 m n = nat (minus_def2 (m) (n)); - -fun decrnum (Bound n) = Bound (minus_def1 n one_def0) - | 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 (C u) = C u - | decrnum (CX (w, x)) = CX (w, x); - -fun decr (Lt a) = Lt (decrnum a) - | decr (Le a) = Le (decrnum a) - | decr (Gt a) = Gt (decrnum a) - | decr (Ge a) = Ge (decrnum a) - | decr (Eq a) = Eq (decrnum a) - | decr (NEq a) = NEq (decrnum a) - | decr (Dvd (i, a)) = Dvd (i, decrnum a) - | decr (NDvd (i, a)) = NDvd (i, decrnum a) - | decr (NOT p) = NOT (decr p) - | decr (And (p, q)) = And (decr p, decr q) - | decr (Or (p, q)) = Or (decr p, decr q) - | decr (Imp (p, q)) = Imp (decr p, decr q) - | decr (Iff (p, q)) = Iff (decr p, decr q) - | decr T = T - | decr F = F - | decr (E ao) = E ao - | decr (A ap) = A ap - | decr (Closed aq) = Closed aq - | decr (NClosed ar) = NClosed ar; - -fun map f [] = [] - | map f (x :: xs) = (f x :: map f xs); - -fun allpairs f [] ys = [] - | allpairs f (x :: xs) ys = append (map (f x) ys) (allpairs f xs ys); - -fun numsubst0 t (C c) = C c - | numsubst0 t (Bound n) = (if (n = 0) then t else Bound n) - | numsubst0 t (CX (i, a)) = Add (Mul (i, t), numsubst0 t a) - | 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); - -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 (Eq (CX (c, e))) = F - | minusinf (NEq (CX (c, e))) = T - | minusinf (Lt (CX (c, e))) = T - | minusinf (Le (CX (c, e))) = T - | minusinf (Gt (CX (c, e))) = F - | minusinf (Ge (CX (c, e))) = F - | minusinf T = T - | minusinf F = F - | minusinf (Lt (C bo)) = Lt (C bo) - | minusinf (Lt (Bound bp)) = Lt (Bound bp) - | minusinf (Lt (Neg bs)) = Lt (Neg bs) - | minusinf (Lt (Add (bt, bu))) = Lt (Add (bt, bu)) - | minusinf (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw)) - | minusinf (Lt (Mul (bx, by))) = Lt (Mul (bx, by)) - | minusinf (Le (C ck)) = Le (C ck) - | minusinf (Le (Bound cl)) = Le (Bound cl) - | minusinf (Le (Neg co)) = Le (Neg co) - | minusinf (Le (Add (cp, cq))) = Le (Add (cp, cq)) - | minusinf (Le (Sub (cr, cs))) = Le (Sub (cr, cs)) - | minusinf (Le (Mul (ct, cu))) = Le (Mul (ct, cu)) - | minusinf (Gt (C dg)) = Gt (C dg) - | minusinf (Gt (Bound dh)) = Gt (Bound dh) - | minusinf (Gt (Neg dk)) = Gt (Neg dk) - | minusinf (Gt (Add (dl, dm))) = Gt (Add (dl, dm)) - | minusinf (Gt (Sub (dn, do'))) = Gt (Sub (dn, do')) - | minusinf (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq)) - | minusinf (Ge (C ec)) = Ge (C ec) - | minusinf (Ge (Bound ed)) = Ge (Bound ed) - | minusinf (Ge (Neg eg)) = Ge (Neg eg) - | minusinf (Ge (Add (eh, ei))) = Ge (Add (eh, ei)) - | minusinf (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek)) - | minusinf (Ge (Mul (el, em))) = Ge (Mul (el, em)) - | minusinf (Eq (C ey)) = Eq (C ey) - | minusinf (Eq (Bound ez)) = Eq (Bound ez) - | minusinf (Eq (Neg fc)) = Eq (Neg fc) - | minusinf (Eq (Add (fd, fe))) = Eq (Add (fd, fe)) - | minusinf (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg)) - | minusinf (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi)) - | minusinf (NEq (C fu)) = NEq (C fu) - | minusinf (NEq (Bound fv)) = NEq (Bound fv) - | minusinf (NEq (Neg fy)) = NEq (Neg fy) - | minusinf (NEq (Add (fz, ga))) = NEq (Add (fz, ga)) - | minusinf (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc)) - | minusinf (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge)) - | 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; - -fun iupt (i, j) = (if (j < i) then [] else (i :: iupt ((i + 1), j))); - -fun mirror (And (p, q)) = And (mirror p, mirror q) - | mirror (Or (p, q)) = Or (mirror p, mirror q) - | mirror (Eq (CX (c, e))) = Eq (CX (c, Neg e)) - | mirror (NEq (CX (c, e))) = NEq (CX (c, Neg e)) - | mirror (Lt (CX (c, e))) = Gt (CX (c, Neg e)) - | mirror (Le (CX (c, e))) = Ge (CX (c, Neg e)) - | mirror (Gt (CX (c, e))) = Lt (CX (c, Neg e)) - | mirror (Ge (CX (c, e))) = Le (CX (c, Neg e)) - | mirror (Dvd (i, CX (c, e))) = Dvd (i, CX (c, Neg e)) - | mirror (NDvd (i, CX (c, e))) = NDvd (i, CX (c, Neg e)) - | mirror T = T - | mirror F = F - | mirror (Lt (C bo)) = Lt (C bo) - | mirror (Lt (Bound bp)) = Lt (Bound bp) - | mirror (Lt (Neg bs)) = Lt (Neg bs) - | mirror (Lt (Add (bt, bu))) = Lt (Add (bt, bu)) - | mirror (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw)) - | mirror (Lt (Mul (bx, by))) = Lt (Mul (bx, by)) - | mirror (Le (C ck)) = Le (C ck) - | mirror (Le (Bound cl)) = Le (Bound cl) - | mirror (Le (Neg co)) = Le (Neg co) - | mirror (Le (Add (cp, cq))) = Le (Add (cp, cq)) - | mirror (Le (Sub (cr, cs))) = Le (Sub (cr, cs)) - | mirror (Le (Mul (ct, cu))) = Le (Mul (ct, cu)) - | mirror (Gt (C dg)) = Gt (C dg) - | mirror (Gt (Bound dh)) = Gt (Bound dh) - | mirror (Gt (Neg dk)) = Gt (Neg dk) - | mirror (Gt (Add (dl, dm))) = Gt (Add (dl, dm)) - | mirror (Gt (Sub (dn, do'))) = Gt (Sub (dn, do')) - | mirror (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq)) - | mirror (Ge (C ec)) = Ge (C ec) - | mirror (Ge (Bound ed)) = Ge (Bound ed) - | mirror (Ge (Neg eg)) = Ge (Neg eg) - | mirror (Ge (Add (eh, ei))) = Ge (Add (eh, ei)) - | mirror (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek)) - | mirror (Ge (Mul (el, em))) = Ge (Mul (el, em)) - | mirror (Eq (C ey)) = Eq (C ey) - | mirror (Eq (Bound ez)) = Eq (Bound ez) - | mirror (Eq (Neg fc)) = Eq (Neg fc) - | mirror (Eq (Add (fd, fe))) = Eq (Add (fd, fe)) - | mirror (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg)) - | mirror (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi)) - | mirror (NEq (C fu)) = NEq (C fu) - | mirror (NEq (Bound fv)) = NEq (Bound fv) - | mirror (NEq (Neg fy)) = NEq (Neg fy) - | mirror (NEq (Add (fz, ga))) = NEq (Add (fz, ga)) - | mirror (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc)) - | mirror (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge)) - | mirror (Dvd (aa, C gq)) = Dvd (aa, C gq) - | mirror (Dvd (aa, Bound gr)) = Dvd (aa, Bound gr) - | mirror (Dvd (aa, Neg gu)) = Dvd (aa, Neg gu) - | mirror (Dvd (aa, Add (gv, gw))) = Dvd (aa, Add (gv, gw)) - | mirror (Dvd (aa, Sub (gx, gy))) = Dvd (aa, Sub (gx, gy)) - | mirror (Dvd (aa, Mul (gz, ha))) = Dvd (aa, Mul (gz, ha)) - | mirror (NDvd (ac, C hm)) = NDvd (ac, C hm) - | mirror (NDvd (ac, Bound hn)) = NDvd (ac, Bound hn) - | mirror (NDvd (ac, Neg hq)) = NDvd (ac, Neg hq) - | mirror (NDvd (ac, Add (hr, hs))) = NDvd (ac, Add (hr, hs)) - | mirror (NDvd (ac, Sub (ht, hu))) = NDvd (ac, Sub (ht, hu)) - | mirror (NDvd (ac, Mul (hv, hw))) = NDvd (ac, Mul (hv, hw)) - | 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; - -fun plus_def0 m n = nat ((m) + (n)); - -fun size_def9 [] = 0 - | size_def9 (a :: list) = plus_def0 (size_def9 list) (0 + 1); - -fun alpha (And (p, q)) = append (alpha p) (alpha q) - | alpha (Or (p, q)) = append (alpha p) (alpha q) - | alpha (Eq (CX (c, e))) = [Add (C ~1, e)] - | alpha (NEq (CX (c, e))) = [e] - | alpha (Lt (CX (c, e))) = [e] - | alpha (Le (CX (c, e))) = [Add (C ~1, e)] - | alpha (Gt (CX (c, e))) = [] - | alpha (Ge (CX (c, e))) = [] - | alpha T = [] - | alpha F = [] - | alpha (Lt (C bo)) = [] - | alpha (Lt (Bound bp)) = [] - | alpha (Lt (Neg bs)) = [] - | alpha (Lt (Add (bt, bu))) = [] - | alpha (Lt (Sub (bv, bw))) = [] - | alpha (Lt (Mul (bx, by))) = [] - | alpha (Le (C ck)) = [] - | alpha (Le (Bound cl)) = [] - | alpha (Le (Neg co)) = [] - | alpha (Le (Add (cp, cq))) = [] - | alpha (Le (Sub (cr, cs))) = [] - | alpha (Le (Mul (ct, cu))) = [] - | alpha (Gt (C dg)) = [] - | alpha (Gt (Bound dh)) = [] - | alpha (Gt (Neg dk)) = [] - | alpha (Gt (Add (dl, dm))) = [] - | alpha (Gt (Sub (dn, do'))) = [] - | alpha (Gt (Mul (dp, dq))) = [] - | alpha (Ge (C ec)) = [] - | alpha (Ge (Bound ed)) = [] - | alpha (Ge (Neg eg)) = [] - | alpha (Ge (Add (eh, ei))) = [] - | alpha (Ge (Sub (ej, ek))) = [] - | alpha (Ge (Mul (el, em))) = [] - | alpha (Eq (C ey)) = [] - | alpha (Eq (Bound ez)) = [] - | alpha (Eq (Neg fc)) = [] - | alpha (Eq (Add (fd, fe))) = [] - | alpha (Eq (Sub (ff, fg))) = [] - | alpha (Eq (Mul (fh, fi))) = [] - | alpha (NEq (C fu)) = [] - | alpha (NEq (Bound fv)) = [] - | alpha (NEq (Neg fy)) = [] - | alpha (NEq (Add (fz, ga))) = [] - | alpha (NEq (Sub (gb, gc))) = [] - | alpha (NEq (Mul (gd, ge))) = [] - | 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) = []; - -fun memberl x [] = false - | memberl x (y :: ys) = ((x = y) orelse memberl x ys); + | simpfm (Gt a) = + let + val a' = simpnum a; + in + (case a' of C v => (if IntInf.< ((0 : IntInf.int), v) then T else F) + | Bound nat => Gt a' | Cx (int, num) => Gt a' | Neg num => Gt a' + | Add (num1, num2) => Gt a' | Sub (num1, num2) => Gt a' + | Mul (int, num) => Gt a') + end + | simpfm (Le a) = + let + val a' = simpnum a; + in + (case a' of C v => (if IntInf.<= (v, (0 : IntInf.int)) then T else F) + | Bound nat => Le a' | Cx (int, num) => Le a' | Neg num => Le a' + | Add (num1, num2) => Le a' | Sub (num1, num2) => Le a' + | Mul (int, num) => Le a') + end + | simpfm (Lt a) = + let + val a' = simpnum a; + in + (case a' of C v => (if IntInf.< (v, (0 : IntInf.int)) then T else F) + | Bound nat => Lt a' | Cx (int, num) => Lt a' | Neg num => Lt a' + | Add (num1, num2) => Lt a' | Sub (num1, num2) => Lt a' + | Mul (int, num) => Lt a') + end + | simpfm (Nota p) = nota (simpfm p) + | simpfm (Iffa (p, q)) = iff (simpfm p) (simpfm q) + | simpfm (Impa (p, q)) = imp (simpfm p) (simpfm q) + | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q) + | simpfm (And (p, q)) = conj (simpfm p) (simpfm q); -fun remdups [] = [] - | remdups (x :: xs) = - (if memberl x xs then remdups xs else (x :: remdups xs)); - -fun beta (And (p, q)) = append (beta p) (beta q) - | beta (Or (p, q)) = append (beta p) (beta q) - | beta (Eq (CX (c, e))) = [Sub (C ~1, e)] - | beta (NEq (CX (c, e))) = [Neg e] - | beta (Lt (CX (c, e))) = [] - | beta (Le (CX (c, e))) = [] - | beta (Gt (CX (c, e))) = [Neg e] - | beta (Ge (CX (c, e))) = [Sub (C ~1, e)] - | beta T = [] - | beta F = [] - | beta (Lt (C bo)) = [] - | beta (Lt (Bound bp)) = [] - | beta (Lt (Neg bs)) = [] - | beta (Lt (Add (bt, bu))) = [] - | beta (Lt (Sub (bv, bw))) = [] - | beta (Lt (Mul (bx, by))) = [] - | beta (Le (C ck)) = [] - | beta (Le (Bound cl)) = [] - | beta (Le (Neg co)) = [] - | beta (Le (Add (cp, cq))) = [] - | beta (Le (Sub (cr, cs))) = [] - | beta (Le (Mul (ct, cu))) = [] - | beta (Gt (C dg)) = [] - | beta (Gt (Bound dh)) = [] - | beta (Gt (Neg dk)) = [] - | beta (Gt (Add (dl, dm))) = [] - | beta (Gt (Sub (dn, do'))) = [] - | beta (Gt (Mul (dp, dq))) = [] - | beta (Ge (C ec)) = [] - | beta (Ge (Bound ed)) = [] - | beta (Ge (Neg eg)) = [] - | beta (Ge (Add (eh, ei))) = [] - | beta (Ge (Sub (ej, ek))) = [] - | beta (Ge (Mul (el, em))) = [] - | beta (Eq (C ey)) = [] - | beta (Eq (Bound ez)) = [] - | beta (Eq (Neg fc)) = [] - | beta (Eq (Add (fd, fe))) = [] - | beta (Eq (Sub (ff, fg))) = [] - | beta (Eq (Mul (fh, fi))) = [] - | beta (NEq (C fu)) = [] - | beta (NEq (Bound fv)) = [] - | beta (NEq (Neg fy)) = [] - | beta (NEq (Add (fz, ga))) = [] - | beta (NEq (Sub (gb, gc))) = [] - | beta (NEq (Mul (gd, ge))) = [] - | 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) = []; - -fun fst (a, b) = a; - -fun div_def1 a b = fst (divAlg (a, b)); - -fun div_def0 m n = nat (div_def1 (m) (n)); - -fun mod_def0 m n = nat (mod_def1 (m) (n)); - -fun gcd (m, n) = (if (n = 0) then m else gcd (n, mod_def0 m n)); - -fun times_def0 m n = nat ((m) * (n)); - -fun lcm x = (fn (m, n) => div_def0 (times_def0 m n) (gcd (m, n))) x; - -fun ilcm x = (fn j => (lcm (nat (abs x), nat (abs j)))); - -fun delta (And (p, q)) = ilcm (delta p) (delta q) - | delta (Or (p, q)) = ilcm (delta p) (delta q) - | delta (Dvd (i, CX (c, e))) = i - | delta (NDvd (i, CX (c, e))) = i - | delta T = 1 - | delta F = 1 - | delta (Lt u) = 1 - | delta (Le v) = 1 - | delta (Gt w) = 1 - | delta (Ge x) = 1 - | delta (Eq y) = 1 - | delta (NEq z) = 1 - | delta (Dvd (aa, C bo)) = 1 - | delta (Dvd (aa, Bound bp)) = 1 - | delta (Dvd (aa, Neg bs)) = 1 - | delta (Dvd (aa, Add (bt, bu))) = 1 - | delta (Dvd (aa, Sub (bv, bw))) = 1 - | delta (Dvd (aa, Mul (bx, by))) = 1 - | delta (NDvd (ac, C ck)) = 1 - | delta (NDvd (ac, Bound cl)) = 1 - | delta (NDvd (ac, Neg co)) = 1 - | delta (NDvd (ac, Add (cp, cq))) = 1 - | delta (NDvd (ac, Sub (cr, cs))) = 1 - | delta (NDvd (ac, Mul (ct, cu))) = 1 - | delta (NOT ae) = 1 - | delta (Imp (aj, ak)) = 1 - | delta (Iff (al, am)) = 1 - | delta (E an) = 1 - | delta (A ao) = 1 - | delta (Closed ap) = 1 - | delta (NClosed aq) = 1; - -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 (Eq (CX (c, e))) = (fn k => Eq (CX (1, Mul (div_def1 k c, e)))) - | a_beta (NEq (CX (c, e))) = (fn k => NEq (CX (1, Mul (div_def1 k c, e)))) - | a_beta (Lt (CX (c, e))) = (fn k => Lt (CX (1, Mul (div_def1 k c, e)))) - | a_beta (Le (CX (c, e))) = (fn k => Le (CX (1, Mul (div_def1 k c, e)))) - | a_beta (Gt (CX (c, e))) = (fn k => Gt (CX (1, Mul (div_def1 k c, e)))) - | a_beta (Ge (CX (c, e))) = (fn k => Ge (CX (1, Mul (div_def1 k c, e)))) - | a_beta (Dvd (i, CX (c, e))) = - (fn k => Dvd ((div_def1 k c * i), CX (1, Mul (div_def1 k c, e)))) - | a_beta (NDvd (i, CX (c, e))) = - (fn k => NDvd ((div_def1 k c * i), CX (1, Mul (div_def1 k c, e)))) - | a_beta T = (fn k => T) - | a_beta F = (fn k => F) - | a_beta (Lt (C bo)) = (fn k => Lt (C bo)) - | a_beta (Lt (Bound bp)) = (fn k => Lt (Bound bp)) - | a_beta (Lt (Neg bs)) = (fn k => Lt (Neg bs)) - | a_beta (Lt (Add (bt, bu))) = (fn k => Lt (Add (bt, bu))) - | a_beta (Lt (Sub (bv, bw))) = (fn k => Lt (Sub (bv, bw))) - | a_beta (Lt (Mul (bx, by))) = (fn k => Lt (Mul (bx, by))) - | a_beta (Le (C ck)) = (fn k => Le (C ck)) - | a_beta (Le (Bound cl)) = (fn k => Le (Bound cl)) - | a_beta (Le (Neg co)) = (fn k => Le (Neg co)) - | a_beta (Le (Add (cp, cq))) = (fn k => Le (Add (cp, cq))) - | a_beta (Le (Sub (cr, cs))) = (fn k => Le (Sub (cr, cs))) - | a_beta (Le (Mul (ct, cu))) = (fn k => Le (Mul (ct, cu))) - | a_beta (Gt (C dg)) = (fn k => Gt (C dg)) - | a_beta (Gt (Bound dh)) = (fn k => Gt (Bound dh)) - | a_beta (Gt (Neg dk)) = (fn k => Gt (Neg dk)) - | a_beta (Gt (Add (dl, dm))) = (fn k => Gt (Add (dl, dm))) - | a_beta (Gt (Sub (dn, do'))) = (fn k => Gt (Sub (dn, do'))) - | a_beta (Gt (Mul (dp, dq))) = (fn k => Gt (Mul (dp, dq))) - | a_beta (Ge (C ec)) = (fn k => Ge (C ec)) - | a_beta (Ge (Bound ed)) = (fn k => Ge (Bound ed)) - | a_beta (Ge (Neg eg)) = (fn k => Ge (Neg eg)) - | a_beta (Ge (Add (eh, ei))) = (fn k => Ge (Add (eh, ei))) - | a_beta (Ge (Sub (ej, ek))) = (fn k => Ge (Sub (ej, ek))) - | a_beta (Ge (Mul (el, em))) = (fn k => Ge (Mul (el, em))) - | a_beta (Eq (C ey)) = (fn k => Eq (C ey)) - | a_beta (Eq (Bound ez)) = (fn k => Eq (Bound ez)) - | a_beta (Eq (Neg fc)) = (fn k => Eq (Neg fc)) - | a_beta (Eq (Add (fd, fe))) = (fn k => Eq (Add (fd, fe))) - | a_beta (Eq (Sub (ff, fg))) = (fn k => Eq (Sub (ff, fg))) - | a_beta (Eq (Mul (fh, fi))) = (fn k => Eq (Mul (fh, fi))) - | a_beta (NEq (C fu)) = (fn k => NEq (C fu)) - | a_beta (NEq (Bound fv)) = (fn k => NEq (Bound fv)) - | a_beta (NEq (Neg fy)) = (fn k => NEq (Neg fy)) - | a_beta (NEq (Add (fz, ga))) = (fn k => NEq (Add (fz, ga))) - | a_beta (NEq (Sub (gb, gc))) = (fn k => NEq (Sub (gb, gc))) - | a_beta (NEq (Mul (gd, ge))) = (fn k => NEq (Mul (gd, ge))) - | a_beta (Dvd (aa, C gq)) = (fn k => Dvd (aa, C gq)) - | a_beta (Dvd (aa, Bound gr)) = (fn k => Dvd (aa, Bound gr)) - | a_beta (Dvd (aa, Neg gu)) = (fn k => Dvd (aa, Neg gu)) - | a_beta (Dvd (aa, Add (gv, gw))) = (fn k => Dvd (aa, Add (gv, gw))) - | a_beta (Dvd (aa, Sub (gx, gy))) = (fn k => Dvd (aa, Sub (gx, gy))) - | a_beta (Dvd (aa, Mul (gz, ha))) = (fn k => Dvd (aa, Mul (gz, ha))) - | a_beta (NDvd (ac, C hm)) = (fn k => NDvd (ac, C hm)) - | a_beta (NDvd (ac, Bound hn)) = (fn k => NDvd (ac, Bound hn)) - | a_beta (NDvd (ac, Neg hq)) = (fn k => NDvd (ac, Neg hq)) - | a_beta (NDvd (ac, Add (hr, hs))) = (fn k => NDvd (ac, Add (hr, hs))) - | a_beta (NDvd (ac, Sub (ht, hu))) = (fn k => NDvd (ac, Sub (ht, hu))) - | a_beta (NDvd (ac, Mul (hv, hw))) = (fn k => NDvd (ac, Mul (hv, hw))) - | a_beta (NOT ae) = (fn k => NOT ae) - | a_beta (Imp (aj, ak)) = (fn k => Imp (aj, ak)) - | a_beta (Iff (al, am)) = (fn k => Iff (al, am)) - | a_beta (E an) = (fn k => E an) - | a_beta (A ao) = (fn k => A ao) - | a_beta (Closed ap) = (fn k => Closed ap) - | a_beta (NClosed aq) = (fn k => NClosed aq); +fun decrnum (Cx (w, x)) = Cx (w, x) + | decrnum (C u) = C u + | decrnum (Mul (c, a)) = Mul (c, decrnum a) + | decrnum (Sub (a, b)) = Sub (decrnum a, decrnum b) + | decrnum (Add (a, b)) = Add (decrnum a, decrnum b) + | decrnum (Neg a) = Neg (decrnum a) + | decrnum (Bound n) = Bound (Integer.nat (IntInf.- (n, (1 : IntInf.int)))); -fun zeta (And (p, q)) = ilcm (zeta p) (zeta q) - | zeta (Or (p, q)) = ilcm (zeta p) (zeta q) - | zeta (Eq (CX (c, e))) = c - | zeta (NEq (CX (c, e))) = c - | zeta (Lt (CX (c, e))) = c - | zeta (Le (CX (c, e))) = c - | zeta (Gt (CX (c, e))) = c - | zeta (Ge (CX (c, e))) = c - | zeta (Dvd (i, CX (c, e))) = c - | zeta (NDvd (i, CX (c, e))) = c - | zeta T = 1 - | zeta F = 1 - | zeta (Lt (C bo)) = 1 - | zeta (Lt (Bound bp)) = 1 - | zeta (Lt (Neg bs)) = 1 - | zeta (Lt (Add (bt, bu))) = 1 - | zeta (Lt (Sub (bv, bw))) = 1 - | zeta (Lt (Mul (bx, by))) = 1 - | zeta (Le (C ck)) = 1 - | zeta (Le (Bound cl)) = 1 - | zeta (Le (Neg co)) = 1 - | zeta (Le (Add (cp, cq))) = 1 - | zeta (Le (Sub (cr, cs))) = 1 - | zeta (Le (Mul (ct, cu))) = 1 - | zeta (Gt (C dg)) = 1 - | zeta (Gt (Bound dh)) = 1 - | zeta (Gt (Neg dk)) = 1 - | zeta (Gt (Add (dl, dm))) = 1 - | zeta (Gt (Sub (dn, do'))) = 1 - | zeta (Gt (Mul (dp, dq))) = 1 - | zeta (Ge (C ec)) = 1 - | zeta (Ge (Bound ed)) = 1 - | zeta (Ge (Neg eg)) = 1 - | zeta (Ge (Add (eh, ei))) = 1 - | zeta (Ge (Sub (ej, ek))) = 1 - | zeta (Ge (Mul (el, em))) = 1 - | zeta (Eq (C ey)) = 1 - | zeta (Eq (Bound ez)) = 1 - | zeta (Eq (Neg fc)) = 1 - | zeta (Eq (Add (fd, fe))) = 1 - | zeta (Eq (Sub (ff, fg))) = 1 - | zeta (Eq (Mul (fh, fi))) = 1 - | zeta (NEq (C fu)) = 1 - | zeta (NEq (Bound fv)) = 1 - | zeta (NEq (Neg fy)) = 1 - | zeta (NEq (Add (fz, ga))) = 1 - | zeta (NEq (Sub (gb, gc))) = 1 - | zeta (NEq (Mul (gd, ge))) = 1 - | zeta (Dvd (aa, C gq)) = 1 - | zeta (Dvd (aa, Bound gr)) = 1 - | zeta (Dvd (aa, Neg gu)) = 1 - | zeta (Dvd (aa, Add (gv, gw))) = 1 - | zeta (Dvd (aa, Sub (gx, gy))) = 1 - | zeta (Dvd (aa, Mul (gz, ha))) = 1 - | zeta (NDvd (ac, C hm)) = 1 - | zeta (NDvd (ac, Bound hn)) = 1 - | zeta (NDvd (ac, Neg hq)) = 1 - | zeta (NDvd (ac, Add (hr, hs))) = 1 - | zeta (NDvd (ac, Sub (ht, hu))) = 1 - | zeta (NDvd (ac, Mul (hv, hw))) = 1 - | zeta (NOT ae) = 1 - | zeta (Imp (aj, ak)) = 1 - | zeta (Iff (al, am)) = 1 - | zeta (E an) = 1 - | zeta (A ao) = 1 - | zeta (Closed ap) = 1 - | zeta (NClosed aq) = 1; - -fun split x = (fn p => x (fst p) (snd p)); - -fun zsplit0 (C c) = (0, C c) - | zsplit0 (Bound n) = (if (n = 0) then (1, C 0) else (0, Bound n)) - | zsplit0 (CX (i, a)) = split (fn i' => (fn x => ((i + i'), x))) (zsplit0 a) - | zsplit0 (Neg a) = (fn (i', a') => (~ i', Neg a')) (zsplit0 a) - | zsplit0 (Add (a, b)) = - (fn (ia, a') => (fn (ib, b') => ((ia + ib), Add (a', b'))) (zsplit0 b)) - (zsplit0 a) - | zsplit0 (Sub (a, b)) = - (fn (ia, a') => - (fn (ib, b') => (minus_def2 ia ib, Sub (a', b'))) (zsplit0 b)) - (zsplit0 a) - | zsplit0 (Mul (i, a)) = (fn (i', a') => ((i * i'), Mul (i, a'))) (zsplit0 a); +fun decr (NClosed ar) = NClosed ar + | decr (Closed aq) = Closed aq + | decr (A ap) = A ap + | decr (E ao) = E ao + | decr F = F + | decr T = T + | decr (Iffa (p, q)) = Iffa (decr p, decr q) + | decr (Impa (p, q)) = Impa (decr p, decr q) + | decr (Or (p, q)) = Or (decr p, decr q) + | decr (And (p, q)) = And (decr p, decr q) + | decr (Nota p) = Nota (decr p) + | decr (NDvd (i, a)) = NDvd (i, decrnum a) + | decr (Dvd (i, a)) = Dvd (i, decrnum a) + | decr (NEq a) = NEq (decrnum a) + | decr (Eq a) = Eq (decrnum a) + | decr (Ge a) = Ge (decrnum a) + | decr (Gt a) = Gt (decrnum a) + | decr (Le a) = Le (decrnum a) + | decr (Lt a) = Lt (decrnum a); -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 x = zsplit0 a - in (fn (c, r) => - (if (c = 0) then Lt r - else (if (0 < c) then Lt (CX (c, r)) else Gt (CX (~ c, Neg r))))) - x - end - | zlfm (Le a) = - let val x = zsplit0 a - in (fn (c, r) => - (if (c = 0) then Le r - else (if (0 < c) then Le (CX (c, r)) else Ge (CX (~ c, Neg r))))) - x - end - | zlfm (Gt a) = - let val x = zsplit0 a - in (fn (c, r) => - (if (c = 0) then Gt r - else (if (0 < c) then Gt (CX (c, r)) else Lt (CX (~ c, Neg r))))) - x - end - | zlfm (Ge a) = - let val x = zsplit0 a - in (fn (c, r) => - (if (c = 0) then Ge r - else (if (0 < c) then Ge (CX (c, r)) else Le (CX (~ c, Neg r))))) - x - end - | zlfm (Eq a) = - let val x = zsplit0 a - in (fn (c, r) => - (if (c = 0) then Eq r - else (if (0 < c) then Eq (CX (c, r)) else Eq (CX (~ c, Neg r))))) - x - end - | zlfm (NEq a) = - let val x = zsplit0 a - in (fn (c, r) => - (if (c = 0) then NEq r - else (if (0 < c) then NEq (CX (c, r)) else NEq (CX (~ c, Neg r))))) - x - end - | zlfm (Dvd (i, a)) = - (if (i = 0) then zlfm (Eq a) - else let val x = zsplit0 a - in (fn (c, r) => - (if (c = 0) then Dvd (abs i, r) - else (if (0 < c) then Dvd (abs i, CX (c, r)) - else Dvd (abs i, CX (~ c, Neg r))))) - x +fun cooper p = + let + val (q, a) = unita p; + val (b, d) = a; + val js = iupt (1 : IntInf.int) d; + val mq = simpfm (minusinf q); + val md = evaldjf (fn j => simpfm (subst0 (C j) mq)) js; + in + (if eq_fm md T then T + else let + val qd = + evaldjf (fn aa as (ba, j) => simpfm (subst0 (Add (ba, C j)) q)) + (List.allpairs (fn aa => fn ba => (aa, ba)) b js); + in + decr (disj md qd) end) - | zlfm (NDvd (i, a)) = - (if (i = 0) then zlfm (NEq a) - else let val x = zsplit0 a - in (fn (c, r) => - (if (c = 0) then NDvd (abs i, r) - else (if (0 < c) then NDvd (abs i, CX (c, r)) - else NDvd (abs i, CX (~ c, Neg r))))) - x - 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 unit p = - let val p' = zlfm p; val l = zeta p'; - val q = And (Dvd (l, CX (1, C 0)), a_beta p' l); val d = delta q; - val B = remdups (map simpnum (beta q)); - val a = remdups (map simpnum (alpha q)) - in (if less_eq_def3 (size_def9 B) (size_def9 a) then (q, (B, d)) - else (mirror q, (a, d))) end; -fun cooper p = - let val (q, (B, d)) = unit p; val js = iupt (1, d); - val mq = simpfm (minusinf q); - val md = evaldjf (fn j => simpfm (subst0 (C j) mq)) js - in (if (md = T) then T - else let val qd = - evaldjf (fn (b, j) => simpfm (subst0 (Add (b, C j)) q)) - (allpairs (fn x => fn xa => (x, xa)) B js) - in decr (disj md qd) end) - end; - -fun prep (E T) = T - | prep (E F) = F - | prep (E (Or (p, q))) = Or (prep (E p), prep (E q)) - | prep (E (Imp (p, q))) = Or (prep (E (NOT p)), prep (E q)) - | prep (E (Iff (p, q))) = - Or (prep (E (And (p, q))), prep (E (And (NOT p, NOT q)))) - | prep (E (NOT (And (p, q)))) = Or (prep (E (NOT p)), prep (E (NOT q))) - | prep (E (NOT (Imp (p, q)))) = prep (E (And (p, NOT q))) - | prep (E (NOT (Iff (p, q)))) = - Or (prep (E (And (p, NOT q))), prep (E (And (NOT p, q)))) - | prep (E (Lt ef)) = E (prep (Lt ef)) - | prep (E (Le eg)) = E (prep (Le eg)) - | prep (E (Gt eh)) = E (prep (Gt eh)) - | prep (E (Ge ei)) = E (prep (Ge ei)) - | prep (E (Eq ej)) = E (prep (Eq ej)) - | prep (E (NEq ek)) = E (prep (NEq ek)) - | prep (E (Dvd (el, em))) = E (prep (Dvd (el, em))) - | prep (E (NDvd (en, eo))) = E (prep (NDvd (en, eo))) - | prep (E (NOT T)) = E (prep (NOT T)) - | prep (E (NOT F)) = E (prep (NOT F)) - | prep (E (NOT (Lt gw))) = E (prep (NOT (Lt gw))) - | prep (E (NOT (Le gx))) = E (prep (NOT (Le gx))) - | prep (E (NOT (Gt gy))) = E (prep (NOT (Gt gy))) - | prep (E (NOT (Ge gz))) = E (prep (NOT (Ge gz))) - | prep (E (NOT (Eq ha))) = E (prep (NOT (Eq ha))) - | prep (E (NOT (NEq hb))) = E (prep (NOT (NEq hb))) - | prep (E (NOT (Dvd (hc, hd)))) = E (prep (NOT (Dvd (hc, hd)))) - | prep (E (NOT (NDvd (he, hf)))) = E (prep (NOT (NDvd (he, hf)))) - | prep (E (NOT (NOT hg))) = E (prep (NOT (NOT hg))) - | prep (E (NOT (Or (hj, hk)))) = E (prep (NOT (Or (hj, hk)))) - | prep (E (NOT (E hp))) = E (prep (NOT (E hp))) - | prep (E (NOT (A hq))) = E (prep (NOT (A hq))) - | prep (E (NOT (Closed hr))) = E (prep (NOT (Closed hr))) - | prep (E (NOT (NClosed hs))) = E (prep (NOT (NClosed hs))) - | prep (E (And (eq, er))) = E (prep (And (eq, er))) - | prep (E (E ey)) = E (prep (E ey)) - | prep (E (A ez)) = E (prep (A ez)) - | prep (E (Closed fa)) = E (prep (Closed fa)) - | prep (E (NClosed fb)) = E (prep (NClosed fb)) +fun prep (NClosed aq) = NClosed aq + | prep (Closed ap) = Closed ap + | prep (NDvd (ac, ad)) = NDvd (ac, ad) + | prep (Dvd (aa, ab)) = Dvd (aa, ab) + | prep (NEq z) = NEq z + | prep (Eq y) = Eq y + | prep (Ge x) = Ge x + | prep (Gt w) = Gt w + | prep (Le v) = Le v + | prep (Lt u) = Lt u + | prep F = F + | prep T = T + | prep (Iffa (p, q)) = Or (prep (And (p, q)), prep (And (Nota p, Nota q))) + | prep (Impa (p, q)) = prep (Or (Nota p, q)) + | prep (And (p, q)) = And (prep p, prep q) + | prep (Or (p, q)) = Or (prep p, prep q) + | prep (Nota (NClosed ck)) = Nota (prep (NClosed ck)) + | prep (Nota (Closed cj)) = Nota (prep (Closed cj)) + | prep (Nota (E ch)) = Nota (prep (E ch)) + | prep (Nota (NDvd (bw, bx))) = Nota (prep (NDvd (bw, bx))) + | prep (Nota (Dvd (bu, bv))) = Nota (prep (Dvd (bu, bv))) + | prep (Nota (NEq bt)) = Nota (prep (NEq bt)) + | prep (Nota (Eq bs)) = Nota (prep (Eq bs)) + | prep (Nota (Ge br)) = Nota (prep (Ge br)) + | prep (Nota (Gt bq)) = Nota (prep (Gt bq)) + | prep (Nota (Le bp)) = Nota (prep (Le bp)) + | prep (Nota (Lt bo)) = Nota (prep (Lt bo)) + | prep (Nota F) = Nota (prep F) + | prep (Nota T) = Nota (prep T) + | prep (Nota (Iffa (p, q))) = + Or (prep (And (p, Nota q)), prep (And (Nota p, q))) + | prep (Nota (Impa (p, q))) = And (prep p, prep (Nota q)) + | prep (Nota (Or (p, q))) = And (prep (Nota p), prep (Nota q)) + | prep (Nota (A p)) = prep (E (Nota p)) + | prep (Nota (And (p, q))) = Or (prep (Nota p), prep (Nota q)) + | prep (Nota (Nota p)) = prep p + | prep (A (NClosed kj)) = prep (Nota (E (Nota (NClosed kj)))) + | prep (A (Closed ki)) = prep (Nota (E (Nota (Closed ki)))) + | prep (A (A kh)) = prep (Nota (E (Nota (A kh)))) + | prep (A (E kg)) = prep (Nota (E (Nota (E kg)))) + | prep (A (Iffa (ke, kf))) = prep (Nota (E (Nota (Iffa (ke, kf))))) + | prep (A (Impa (kc, kd))) = prep (Nota (E (Nota (Impa (kc, kd))))) + | prep (A (Or (ka, kb))) = prep (Nota (E (Nota (Or (ka, kb))))) + | prep (A (Nota jx)) = prep (Nota (E (Nota (Nota jx)))) + | prep (A (NDvd (jv, jw))) = prep (Nota (E (Nota (NDvd (jv, jw))))) + | prep (A (Dvd (jt, ju))) = prep (Nota (E (Nota (Dvd (jt, ju))))) + | prep (A (NEq js)) = prep (Nota (E (Nota (NEq js)))) + | prep (A (Eq jr)) = prep (Nota (E (Nota (Eq jr)))) + | prep (A (Ge jq)) = prep (Nota (E (Nota (Ge jq)))) + | prep (A (Gt jp)) = prep (Nota (E (Nota (Gt jp)))) + | prep (A (Le jo)) = prep (Nota (E (Nota (Le jo)))) + | prep (A (Lt jn)) = prep (Nota (E (Nota (Lt jn)))) + | prep (A F) = prep (Nota (E (Nota F))) + | prep (A T) = prep (Nota (E (Nota T))) | prep (A (And (p, q))) = And (prep (A p), prep (A q)) - | prep (A T) = prep (NOT (E (NOT T))) - | prep (A F) = prep (NOT (E (NOT F))) - | prep (A (Lt jn)) = prep (NOT (E (NOT (Lt jn)))) - | prep (A (Le jo)) = prep (NOT (E (NOT (Le jo)))) - | prep (A (Gt jp)) = prep (NOT (E (NOT (Gt jp)))) - | prep (A (Ge jq)) = prep (NOT (E (NOT (Ge jq)))) - | prep (A (Eq jr)) = prep (NOT (E (NOT (Eq jr)))) - | prep (A (NEq js)) = prep (NOT (E (NOT (NEq js)))) - | prep (A (Dvd (jt, ju))) = prep (NOT (E (NOT (Dvd (jt, ju))))) - | prep (A (NDvd (jv, jw))) = prep (NOT (E (NOT (NDvd (jv, jw))))) - | prep (A (NOT jx)) = prep (NOT (E (NOT (NOT jx)))) - | prep (A (Or (ka, kb))) = prep (NOT (E (NOT (Or (ka, kb))))) - | prep (A (Imp (kc, kd))) = prep (NOT (E (NOT (Imp (kc, kd))))) - | prep (A (Iff (ke, kf))) = prep (NOT (E (NOT (Iff (ke, kf))))) - | prep (A (E kg)) = prep (NOT (E (NOT (E kg)))) - | prep (A (A kh)) = prep (NOT (E (NOT (A kh)))) - | prep (A (Closed ki)) = prep (NOT (E (NOT (Closed ki)))) - | prep (A (NClosed kj)) = prep (NOT (E (NOT (NClosed kj)))) - | prep (NOT (NOT p)) = prep p - | prep (NOT (And (p, q))) = Or (prep (NOT p), prep (NOT q)) - | prep (NOT (A p)) = prep (E (NOT p)) - | prep (NOT (Or (p, q))) = And (prep (NOT p), prep (NOT q)) - | prep (NOT (Imp (p, q))) = And (prep p, prep (NOT q)) - | prep (NOT (Iff (p, q))) = Or (prep (And (p, NOT q)), prep (And (NOT p, q))) - | prep (NOT T) = NOT (prep T) - | prep (NOT F) = NOT (prep F) - | prep (NOT (Lt bo)) = NOT (prep (Lt bo)) - | prep (NOT (Le bp)) = NOT (prep (Le bp)) - | prep (NOT (Gt bq)) = NOT (prep (Gt bq)) - | prep (NOT (Ge br)) = NOT (prep (Ge br)) - | prep (NOT (Eq bs)) = NOT (prep (Eq bs)) - | prep (NOT (NEq bt)) = NOT (prep (NEq bt)) - | prep (NOT (Dvd (bu, bv))) = NOT (prep (Dvd (bu, bv))) - | prep (NOT (NDvd (bw, bx))) = NOT (prep (NDvd (bw, bx))) - | prep (NOT (E ch)) = NOT (prep (E ch)) - | prep (NOT (Closed cj)) = NOT (prep (Closed cj)) - | prep (NOT (NClosed ck)) = NOT (prep (NClosed ck)) - | prep (Or (p, q)) = Or (prep p, prep q) - | prep (And (p, q)) = And (prep p, prep q) - | prep (Imp (p, q)) = prep (Or (NOT p, q)) - | prep (Iff (p, q)) = Or (prep (And (p, q)), prep (And (NOT p, NOT q))) - | prep T = T - | prep F = F - | prep (Lt u) = Lt u - | prep (Le v) = Le v - | prep (Gt w) = Gt w - | prep (Ge x) = Ge x - | prep (Eq y) = Eq y - | prep (NEq z) = NEq z - | prep (Dvd (aa, ab)) = Dvd (aa, ab) - | prep (NDvd (ac, ad)) = NDvd (ac, ad) - | prep (Closed ap) = Closed ap - | prep (NClosed aq) = NClosed aq; + | prep (E (NClosed fb)) = E (prep (NClosed fb)) + | prep (E (Closed fa)) = E (prep (Closed fa)) + | prep (E (A ez)) = E (prep (A ez)) + | prep (E (E ey)) = E (prep (E ey)) + | prep (E (And (eq, er))) = E (prep (And (eq, er))) + | prep (E (Nota (NClosed hs))) = E (prep (Nota (NClosed hs))) + | prep (E (Nota (Closed hr))) = E (prep (Nota (Closed hr))) + | prep (E (Nota (A hq))) = E (prep (Nota (A hq))) + | prep (E (Nota (E hp))) = E (prep (Nota (E hp))) + | prep (E (Nota (Or (hj, hk)))) = E (prep (Nota (Or (hj, hk)))) + | prep (E (Nota (Nota hg))) = E (prep (Nota (Nota hg))) + | prep (E (Nota (NDvd (he, hf)))) = E (prep (Nota (NDvd (he, hf)))) + | prep (E (Nota (Dvd (hc, hd)))) = E (prep (Nota (Dvd (hc, hd)))) + | prep (E (Nota (NEq hb))) = E (prep (Nota (NEq hb))) + | prep (E (Nota (Eq ha))) = E (prep (Nota (Eq ha))) + | prep (E (Nota (Ge gz))) = E (prep (Nota (Ge gz))) + | prep (E (Nota (Gt gy))) = E (prep (Nota (Gt gy))) + | prep (E (Nota (Le gx))) = E (prep (Nota (Le gx))) + | prep (E (Nota (Lt gw))) = E (prep (Nota (Lt gw))) + | prep (E (Nota F)) = E (prep (Nota F)) + | prep (E (Nota T)) = E (prep (Nota T)) + | prep (E (NDvd (en, eo))) = E (prep (NDvd (en, eo))) + | prep (E (Dvd (el, em))) = E (prep (Dvd (el, em))) + | prep (E (NEq ek)) = E (prep (NEq ek)) + | prep (E (Eq ej)) = E (prep (Eq ej)) + | prep (E (Ge ei)) = E (prep (Ge ei)) + | prep (E (Gt eh)) = E (prep (Gt eh)) + | prep (E (Le eg)) = E (prep (Le eg)) + | prep (E (Lt ef)) = E (prep (Lt ef)) + | prep (E (Nota (Iffa (p, q)))) = + Or (prep (E (And (p, Nota q))), prep (E (And (Nota p, q)))) + | prep (E (Nota (Impa (p, q)))) = prep (E (And (p, Nota q))) + | prep (E (Nota (And (p, q)))) = Or (prep (E (Nota p)), prep (E (Nota q))) + | prep (E (Iffa (p, q))) = + Or (prep (E (And (p, q))), prep (E (And (Nota p, Nota q)))) + | prep (E (Impa (p, q))) = Or (prep (E (Nota p)), prep (E q)) + | prep (E (Or (p, q))) = Or (prep (E p), prep (E q)) + | prep (E F) = F + | prep (E T) = T; -fun pa x = qelim (prep x) cooper; - -val pa = (fn x => pa x); +fun qelim (NClosed aq) = (fn y => simpfm (NClosed aq)) + | qelim (Closed ap) = (fn y => simpfm (Closed ap)) + | qelim (NDvd (ac, ad)) = (fn y => simpfm (NDvd (ac, ad))) + | qelim (Dvd (aa, ab)) = (fn y => simpfm (Dvd (aa, ab))) + | qelim (NEq z) = (fn y => simpfm (NEq z)) + | qelim (Eq y) = (fn ya => simpfm (Eq y)) + | qelim (Ge x) = (fn y => simpfm (Ge x)) + | qelim (Gt w) = (fn y => simpfm (Gt w)) + | qelim (Le v) = (fn y => simpfm (Le v)) + | qelim (Lt u) = (fn y => simpfm (Lt u)) + | qelim F = (fn y => simpfm F) + | qelim T = (fn y => simpfm T) + | qelim (Iffa (p, q)) = (fn qe => iff (qelim p qe) (qelim q qe)) + | qelim (Impa (p, q)) = (fn qe => imp (qelim p qe) (qelim q qe)) + | qelim (Or (p, q)) = (fn qe => disj (qelim p qe) (qelim q qe)) + | qelim (And (p, q)) = (fn qe => conj (qelim p qe) (qelim q qe)) + | qelim (Nota p) = (fn qe => nota (qelim p qe)) + | qelim (A p) = (fn qe => nota (qe (qelim (Nota p) qe))) + | qelim (E p) = (fn qe => dj qe (qelim p qe)); -val test = - (fn x => - pa (E (A (Imp (Ge (Sub (Bound 0, Bound one_def0)), - E (E (Eq (Sub (Add (Mul (3, Bound one_def0), - Mul (5, Bound 0)), - Bound (nat 2)))))))))); +val pa : fm -> fm = (fn p => qelim (prep p) cooper); -end; +end; (*struct Reflected_Presburger*) + +end; (*struct ROOT*) diff -r 7cd68def72b2 -r 0410269099dc src/HOL/ex/Reflected_Presburger.thy --- a/src/HOL/ex/Reflected_Presburger.thy Tue Jul 10 09:23:14 2007 +0200 +++ b/src/HOL/ex/Reflected_Presburger.thy Tue Jul 10 09:23:15 2007 +0200 @@ -3,62 +3,18 @@ uses ("coopereif.ML") ("coopertac.ML") begin -lemma allpairs_set: "set (allpairs Pair xs ys) = {(x,y). x\ set xs \ y \ set ys}" -by (induct xs) auto - - - (* generate a list from i to j*) -consts iupt :: "int \ int \ int list" -recdef iupt "measure (\ (i,j). nat (j-i +1))" - "iupt (i,j) = (if j (x#xs) ! n = xs ! (n - 1)" -using Nat.gr0_conv_Suc -by clarsimp - +function + iupt :: "int \ int \ int list" +where + "iupt i j = (if j < i then [] else i # iupt (i+1) j)" +by pat_completeness auto +termination by (relation "measure (\ (i, j). nat (j-i+1))") auto -lemma myl: "\ (a::'a::{pordered_ab_group_add}) (b::'a). (a \ b) = (0 \ b - a)" -proof(clarify) - fix x y ::"'a" - have "(x \ y) = (x - y \ 0)" by (simp only: le_iff_diff_le_0[where a="x" and b="y"]) - also have "\ = (- (y - x) \ 0)" by simp - also have "\ = (0 \ y - x)" by (simp only: neg_le_0_iff_le[where a="y-x"]) - finally show "(x \ y) = (0 \ y - x)" . -qed - -lemma myless: "\ (a::'a::{pordered_ab_group_add}) (b::'a). (a < b) = (0 < b - a)" -proof(clarify) - fix x y ::"'a" - have "(x < y) = (x - y < 0)" by (simp only: less_iff_diff_less_0[where a="x" and b="y"]) - also have "\ = (- (y - x) < 0)" by simp - also have "\ = (0 < y - x)" by (simp only: neg_less_0_iff_less[where a="y-x"]) - finally show "(x < y) = (0 < y - x)" . -qed - -lemma myeq: "\ (a::'a::{pordered_ab_group_add}) (b::'a). (a = b) = (0 = b - a)" - by auto +lemma iupt_set: "set (iupt i j) = {i..j}" + by (induct rule: iupt.induct) (simp add: simp_from_to) (* Periodicity of dvd *) -lemma dvd_period: - assumes advdd: "(a::int) dvd d" - shows "(a dvd (x + t)) = (a dvd ((x+ c*d) + t))" - using advdd -proof- - {fix x k - from inf_period(3)[OF advdd, rule_format, where x=x and k="-k"] - have " ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp} - hence "\x.\k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp - then show ?thesis by simp -qed - (*********************************************************************************) (**** SHADOW SYNTAX AND SEMANTICS ****) (*********************************************************************************) @@ -198,7 +154,7 @@ assumes nb: "numbound0 a" shows "Inum (b#bs) a = Inum (b'#bs) a" using nb -by (induct a rule: numbound0.induct) (auto simp add: nth_pos2) +by (induct a rule: numbound0.induct) (auto simp add: gr0_conv_Suc) primrec "bound0 T = True" @@ -224,7 +180,7 @@ assumes bp: "bound0 p" shows "Ifm bbs (b#bs) p = Ifm bbs (b'#bs) p" using bp numbound0_I[where b="b" and bs="bs" and b'="b'"] -by (induct p rule: bound0.induct) (auto simp add: nth_pos2) +by (induct p rule: bound0.induct) (auto simp add: gr0_conv_Suc) primrec "numsubst0 t (C c) = (C c)" @@ -237,12 +193,12 @@ lemma numsubst0_I: shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t" - by (induct t) (simp_all add: nth_pos2) + by (induct t) (auto simp add: gr0_conv_Suc) lemma numsubst0_I': assumes nb: "numbound0 a" shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t" - by (induct t) (simp_all add: nth_pos2 numbound0_I[OF nb, where b="b" and b'="b'"]) + by (induct t) (auto simp add: gr0_conv_Suc numbound0_I[OF nb, where b="b" and b'="b'"]) primrec @@ -267,7 +223,7 @@ lemma subst0_I: assumes qfp: "qfree p" shows "Ifm bbs (b#bs) (subst0 a p) = Ifm bbs ((Inum (b#bs) a)#bs) p" using qfp numsubst0_I[where b="b" and bs="bs" and a="a"] - by (induct p) (simp_all add: nth_pos2 ) + by (induct p) (simp_all add: gr0_conv_Suc) consts @@ -300,12 +256,12 @@ lemma decrnum: assumes nb: "numbound0 t" shows "Inum (x#bs) t = Inum bs (decrnum t)" - using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2) + using nb by (induct t rule: decrnum.induct, auto simp add: gr0_conv_Suc) lemma decr: assumes nb: "bound0 p" shows "Ifm bbs (x#bs) p = Ifm bbs bs (decr p)" using nb - by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum) + by (induct p rule: decr.induct, simp_all add: gr0_conv_Suc decrnum) lemma decr_qf: "bound0 p \ qfree (decr p)" by (induct p, simp_all) @@ -444,10 +400,8 @@ constdefs lex_bnd :: "num \ num \ bool" "lex_bnd t s \ lex_ns (bnds t, bnds s)" -consts simpnum:: "num \ num" +consts numadd:: "num \ num \ num" - nummul:: "num \ int \ num" - numfloor:: "num \ num" recdef numadd "measure (\ (t,s). size t + size s)" "numadd (Add (Mul c1 (Bound n1)) r1,Add (Mul c2 (Bound n2)) r2) = (if n1=n2 then @@ -460,6 +414,25 @@ "numadd (C b1, C b2) = C (b1+b2)" "numadd (a,b) = Add a b" +(*function (sequential) + numadd :: "num \ num \ num" +where + "numadd (Add (Mul c1 (Bound n1)) r1) (Add (Mul c2 (Bound n2)) r2) = + (if n1 = n2 then (let c = c1 + c2 + in (if c = 0 then numadd r1 r2 else + Add (Mul c (Bound n1)) (numadd r1 r2))) + else if n1 \ n2 then + Add (Mul c1 (Bound n1)) (numadd r1 (Add (Mul c2 (Bound n2)) r2)) + else + Add (Mul c2 (Bound n2)) (numadd (Add (Mul c1 (Bound n1)) r1) r2))" + | "numadd (Add (Mul c1 (Bound n1)) r1) t = + Add (Mul c1 (Bound n1)) (numadd r1 t)" + | "numadd t (Add (Mul c2 (Bound n2)) r2) = + Add (Mul c2 (Bound n2)) (numadd t r2)" + | "numadd (C b1) (C b2) = C (b1 + b2)" + | "numadd a b = Add a b" +apply pat_completeness apply auto*) + lemma numadd: "Inum bs (numadd (t,s)) = Inum bs (Add t s)" apply (induct t s rule: numadd.induct, simp_all add: Let_def) apply (case_tac "c1+c2 = 0",case_tac "n1 \ n2", simp_all) @@ -471,23 +444,25 @@ lemma numadd_nb: "\ numbound0 t ; numbound0 s\ \ numbound0 (numadd (t,s))" by (induct t s rule: numadd.induct, auto simp add: Let_def) -recdef nummul "measure size" - "nummul (C j) = (\ i. C (i*j))" - "nummul (Add a b) = (\ i. numadd (nummul a i, nummul b i))" - "nummul (Mul c t) = (\ i. nummul t (i*c))" - "nummul t = (\ i. Mul i t)" +fun + nummul :: "int \ num \ num" +where + "nummul i (C j) = C (i * j)" + | "nummul i (Add a b) = numadd (nummul i a, nummul i b)" + | "nummul i (Mul c t) = nummul (i * c) t" + | "nummul i t = Mul i t" -lemma nummul: "\ i. Inum bs (nummul t i) = Inum bs (Mul i t)" +lemma nummul: "\ i. Inum bs (nummul i t) = Inum bs (Mul i t)" by (induct t rule: nummul.induct, auto simp add: ring_simps numadd) -lemma nummul_nb: "\ i. numbound0 t \ numbound0 (nummul t i)" +lemma nummul_nb: "\ i. numbound0 t \ numbound0 (nummul i t)" by (induct t rule: nummul.induct, auto simp add: numadd_nb) constdefs numneg :: "num \ num" - "numneg t \ nummul t (- 1)" + "numneg t \ nummul (- 1) t" constdefs numsub :: "num \ num \ num" - "numsub s t \ (if s = t then C 0 else numadd (s,numneg t))" + "numsub s t \ (if s = t then C 0 else numadd (s, numneg t))" lemma numneg: "Inum bs (numneg t) = Inum bs (Neg t)" using numneg_def nummul by simp @@ -501,14 +476,16 @@ lemma numsub_nb: "\ numbound0 t ; numbound0 s\ \ numbound0 (numsub t s)" using numsub_def numadd_nb numneg_nb by simp -recdef simpnum "measure size" +fun + simpnum :: "num \ num" +where "simpnum (C j) = C j" - "simpnum (Bound n) = Add (Mul 1 (Bound n)) (C 0)" - "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 then (C 0) else nummul (simpnum t) i)" - "simpnum t = t" + | "simpnum (Bound n) = Add (Mul 1 (Bound n)) (C 0)" + | "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 then C 0 else nummul i (simpnum t))" + | "simpnum t = t" lemma simpnum_ci: "Inum bs (simpnum t) = Inum bs t" by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul) @@ -517,12 +494,13 @@ "numbound0 t \ numbound0 (simpnum t)" by (induct t rule: simpnum.induct, auto simp add: numadd_nb numsub_nb nummul_nb numneg_nb) -consts not:: "fm \ fm" -recdef not "measure size" +fun + not :: "fm \ fm" +where "not (NOT p) = p" - "not T = F" - "not F = T" - "not p = NOT p" + | "not T = F" + | "not F = T" + | "not p = NOT p" lemma not: "Ifm bbs bs (not p) = Ifm bbs bs (NOT p)" by (cases p) auto lemma not_qf: "qfree p \ qfree (not p)" @@ -571,27 +549,31 @@ lemma iff_nb: "\bound0 p ; bound0 q\ \ bound0 (iff p q)" using iff_def by (unfold iff_def,cases "p=q", auto simp add: not_bn) -consts simpfm :: "fm \ fm" -recdef simpfm "measure fmsize" +function (sequential) + simpfm :: "fm \ fm" +where "simpfm (And p q) = conj (simpfm p) (simpfm q)" - "simpfm (Or p q) = disj (simpfm p) (simpfm q)" - "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" - "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" - "simpfm (NOT p) = not (simpfm p)" - "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \ if (v < 0) then T else F - | _ \ Lt a')" - "simpfm (Le a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Le a')" - "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \ if (v > 0) then T else F | _ \ Gt a')" - "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Ge a')" - "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \ if (v = 0) then T else F | _ \ Eq a')" - "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ NEq a')" - "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a) + | "simpfm (Or p q) = disj (simpfm p) (simpfm q)" + | "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" + | "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" + | "simpfm (NOT p) = not (simpfm p)" + | "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \ if (v < 0) then T else F + | _ \ Lt a')" + | "simpfm (Le a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Le a')" + | "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \ if (v > 0) then T else F | _ \ Gt a')" + | "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Ge a')" + | "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \ if (v = 0) then T else F | _ \ Eq a')" + | "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ NEq a')" + | "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a) else if (abs i = 1) then T else let a' = simpnum a in case a' of C v \ if (i dvd v) then T else F | _ \ Dvd i a')" - "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) + | "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) else if (abs i = 1) then F else let a' = simpnum a in case a' of C v \ if (\(i dvd v)) then T else F | _ \ NDvd i a')" - "simpfm p = p" + | "simpfm p = p" +by pat_completeness auto +termination by (relation "measure fmsize") auto + lemma simpfm: "Ifm bbs bs (simpfm p) = Ifm bbs bs p" proof(induct p rule: simpfm.induct) case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp @@ -727,6 +709,20 @@ "qelim (Iff p q) = (\ qe. iff (qelim p qe) (qelim q qe))" "qelim p = (\ y. simpfm p)" +(*function (sequential) + qelim :: "(fm \ fm) \ fm \ fm" +where + "qelim qe (E p) = DJ qe (qelim qe p)" + | "qelim qe (A p) = not (qe ((qelim qe (NOT p))))" + | "qelim qe (NOT p) = not (qelim qe p)" + | "qelim qe (And p q) = conj (qelim qe p) (qelim qe q)" + | "qelim qe (Or p q) = disj (qelim qe p) (qelim qe q)" + | "qelim qe (Imp p q) = imp (qelim qe p) (qelim qe q)" + | "qelim qe (Iff p q) = iff (qelim qe p) (qelim qe q)" + | "qelim qe p = simpfm p" +by pat_completeness auto +termination by (relation "measure (fmsize o snd)") auto*) + lemma qelim_ci: assumes qe_inv: "\ bs p. qfree p \ qfree (qe p) \ (Ifm bbs bs (qe p) = Ifm bbs bs (E p))" shows "\ bs. qfree (qelim p qe) \ (Ifm bbs bs (qelim p qe) = Ifm bbs bs p)" @@ -735,21 +731,21 @@ (auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf simpfm simpfm_qf simp del: simpfm.simps) (* Linearity for fm where Bound 0 ranges over \ *) -consts + +fun zsplit0 :: "num \ int \ num" (* splits the bounded from the unbounded part*) -recdef zsplit0 "measure size" +where "zsplit0 (C c) = (0,C c)" - "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))" - "zsplit0 (CX i a) = (let (i',a') = zsplit0 a in (i+i', a'))" - "zsplit0 (Neg a) = (let (i',a') = zsplit0 a in (-i', Neg a'))" - "zsplit0 (Add a b) = (let (ia,a') = zsplit0 a ; + | "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))" + | "zsplit0 (CX i a) = (let (i',a') = zsplit0 a in (i+i', a'))" + | "zsplit0 (Neg a) = (let (i',a') = zsplit0 a in (-i', Neg a'))" + | "zsplit0 (Add a b) = (let (ia,a') = zsplit0 a ; (ib,b') = zsplit0 b in (ia+ib, Add a' b'))" - "zsplit0 (Sub a b) = (let (ia,a') = zsplit0 a ; + | "zsplit0 (Sub a b) = (let (ia,a') = zsplit0 a ; (ib,b') = zsplit0 b in (ia-ib, Sub a' b'))" - "zsplit0 (Mul i a) = (let (i',a') = zsplit0 a in (i*i', Mul i a'))" -(hints simp add: Let_def) + | "zsplit0 (Mul i a) = (let (i',a') = zsplit0 a in (i*i', Mul i a'))" lemma zsplit0_I: shows "\ n a. zsplit0 t = (n,a) \ (Inum ((x::int) #bs) (CX n a) = Inum (x #bs) t) \ numbound0 a" @@ -788,7 +784,7 @@ ultimately have th: "a=Add ?as ?at \ n=?ns + ?nt" using prems by (simp add: Let_def split_def) from abjs[symmetric] have bluddy: "\ x y. (x,y) = zsplit0 s" by blast - from prems have "(\ x y. (x,y) = zsplit0 s) \ (\xa xb. zsplit0 t = (xa, xb) \ Inum (x # bs) (CX xa xb) = Inum (x # bs) t \ numbound0 xb)" by simp + from prems have "(\ x y. (x,y) = zsplit0 s) \ (\xa xb. zsplit0 t = (xa, xb) \ Inum (x # bs) (CX xa xb) = Inum (x # bs) t \ numbound0 xb)" by auto with bluddy abjt have th3: "(?I x (CX ?nt ?at) = ?I x t) \ ?N ?at" by blast from abjs prems have th2: "(?I x (CX ?ns ?as) = ?I x s) \ ?N ?as" by blast from th3[simplified] th2[simplified] th[simplified] show ?case @@ -804,7 +800,7 @@ ultimately have th: "a=Sub ?as ?at \ n=?ns - ?nt" using prems by (simp add: Let_def split_def) from abjs[symmetric] have bluddy: "\ x y. (x,y) = zsplit0 s" by blast - from prems have "(\ x y. (x,y) = zsplit0 s) \ (\xa xb. zsplit0 t = (xa, xb) \ Inum (x # bs) (CX xa xb) = Inum (x # bs) t \ numbound0 xb)" by simp + from prems have "(\ x y. (x,y) = zsplit0 s) \ (\xa xb. zsplit0 t = (xa, xb) \ Inum (x # bs) (CX xa xb) = Inum (x # bs) t \ numbound0 xb)" by auto with bluddy abjt have th3: "(?I x (CX ?nt ?at) = ?I x t) \ ?N ?at" by blast from abjs prems have th2: "(?I x (CX ?ns ?as) = ?I x s) \ ?N ?as" by blast from th3[simplified] th2[simplified] th[simplified] show ?case @@ -823,7 +819,6 @@ consts iszlfm :: "fm \ bool" (* Linearity test for fm *) - zlfm :: "fm \ fm" (* Linearity transformation for fm *) recdef iszlfm "measure size" "iszlfm (And p q) = (iszlfm p \ iszlfm q)" "iszlfm (Or p q) = (iszlfm p \ iszlfm q)" @@ -842,7 +837,8 @@ lemma zlin_qfree: "iszlfm p \ qfree p" by (induct p rule: iszlfm.induct) auto - +consts + zlfm :: "fm \ fm" (* Linearity transformation for fm *) recdef zlfm "measure fmsize" "zlfm (And p q) = And (zlfm p) (zlfm q)" "zlfm (Or p q) = Or (zlfm p) (zlfm q)" @@ -1309,7 +1305,7 @@ by blast thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def) qed -qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="x - k*d" and b'="x"]) +qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="x - k*d" and b'="x"]) (* Is'nt this beautiful?*) lemma minusinf_ex: @@ -1374,7 +1370,7 @@ using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"] by simp finally show ?case by simp -qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] nth_pos2) +qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] gr0_conv_Suc) lemma mirror_l: "iszlfm p \ d\ p 1 \ iszlfm (mirror p) \ d\ (mirror p) 1" @@ -1571,7 +1567,7 @@ using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp also have "\ = (\ (k::int). c * x + Inum (x # bs) e = j * k)" by simp finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def) -qed (simp_all add: nth_pos2 numbound0_I[where bs="bs" and b="(l * x)" and b'="x"]) +qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="(l * x)" and b'="x"]) lemma a\_ex: assumes linp: "iszlfm p" and d: "d\ p l" and lp: "l>0" shows "(\ x. l dvd x \ Ifm bbs (x #bs) (a\ p l)) = (\ (x::int). Ifm bbs (x#bs) p)" @@ -1665,7 +1661,7 @@ from prems have id: "j dvd d" by simp from c1 have "?p x = (j dvd (x+ ?e))" by simp also have "\ = (j dvd x - d + ?e)" - using dvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp + using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp finally show ?case using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp next @@ -1674,9 +1670,9 @@ from prems have id: "j dvd d" by simp from c1 have "?p x = (\ j dvd (x+ ?e))" by simp also have "\ = (\ j dvd x - d + ?e)" - using dvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp + using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp finally show ?case using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp -qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] nth_pos2) +qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] gr0_conv_Suc) lemma \': assumes lp: "iszlfm p" @@ -1820,7 +1816,7 @@ constdefs cooper :: "fm \ fm" "cooper p \ - (let (q,B,d) = unit p; js = iupt (1,d); + (let (q,B,d) = unit p; js = iupt 1 d; mq = simpfm (minusinf q); md = evaldjf (\ j. simpfm (subst0 (C j) mq)) js in if md = T then T else @@ -1836,7 +1832,7 @@ let ?q = "fst (unit p)" let ?B = "fst (snd(unit p))" let ?d = "snd (snd (unit p))" - let ?js = "iupt (1,?d)" + let ?js = "iupt 1 ?d" let ?mq = "minusinf ?q" let ?smq = "simpfm ?mq" let ?md = "evaldjf (\ j. simpfm (subst0 (C j) ?smq)) ?js" @@ -1857,7 +1853,7 @@ by (auto simp add: simpfm_bound0) from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp from Bn jsnb have "\ (b,j) \ set (allpairs Pair ?B ?js). numbound0 (Add b (C j))" - by (simp add: allpairs_set) + by simp hence "\ (b,j) \ set (allpairs Pair ?B ?js). bound0 (subst0 (Add b (C j)) ?q)" using subst0_bound0[OF qfq] by blast hence "\ (b,j) \ set (allpairs Pair ?B ?js). bound0 (simpfm (subst0 (Add b (C j)) ?q))" @@ -1877,9 +1873,9 @@ also have "\ = (?I i (evaldjf (\ j. simpfm (subst0 (C j) ?smq)) ?js) \ (\ j\ set ?js. \ b\ set ?B. ?I i (subst0 (Add b (C j)) ?q)))" by (simp only: evaldjf_ex subst0_I[OF qfq]) also have "\= (?I i ?md \ (\ (b,j) \ set (allpairs Pair ?B ?js). (\ (b,j). ?I i (simpfm (subst0 (Add b (C j)) ?q))) (b,j)))" - by (simp only: simpfm allpairs_set) blast + by (simp only: simpfm set_allpairs) blast also have "\ = (?I i ?md \ (?I i (evaldjf (\ (b,j). simpfm (subst0 (Add b (C j)) ?q)) (allpairs Pair ?B ?js))))" - by (simp only: evaldjf_ex[where bs="i#bs" and f="\ (b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="allpairs Pair ?B ?js"]) (auto simp add: split_def) + by (simp only: evaldjf_ex[where bs="i#bs" and f="\ (b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="allpairs Pair ?B ?js"]) (auto simp add: split_def) finally have mdqd: "?lhs = (?I i ?md \ ?I i ?qd)" by simp also have "\ = (?I i (disj ?md ?qd))" by (simp add: disj) also have "\ = (Ifm bbs bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) @@ -1910,10 +1906,10 @@ (E (E (Eq (Sub (Add (Mul 3 (Bound 1)) (Mul 5 (Bound 0))) (Bound 2))))))))" -code_gen pa cooper_test in SML +code_gen pa cooper_test in SML to GeneratedCooper +(*code_reserved SML oo code_gen pa in SML to GeneratedCooper file "~~/src/HOL/Tools/Qelim/raw_generated_cooper.ML"*) -ML {* structure GeneratedCooper = struct open ROOT end *} -ML {* GeneratedCooper.Reflected_Presburger.cooper_test () *} +ML {* GeneratedCooper.cooper_test () *} use "coopereif.ML" oracle linzqe_oracle ("term") = Coopereif.cooper_oracle use "coopertac.ML"