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<a href="arith__proof__rules_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span> <span class="comment">/*****************************************************************************/</span><span class="comment"></span></div>
<div class="line"><a name="l00002"></a><span class="lineno"> 2</span> <span class="comment">/*!</span></div>
<div class="line"><a name="l00003"></a><span class="lineno"> 3</span> <span class="comment"> * \file arith_proof_rules.h</span></div>
<div class="line"><a name="l00004"></a><span class="lineno"> 4</span> <span class="comment"> * \brief Arithmetic proof rules</span></div>
<div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment"> * Author: Vijay Ganesh, Sergey Berezin</span></div>
<div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment"> * </span></div>
<div class="line"><a name="l00008"></a><span class="lineno"> 8</span> <span class="comment"> * Created: Dec 13 02:09:04 GMT 2002</span></div>
<div class="line"><a name="l00009"></a><span class="lineno"> 9</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00010"></a><span class="lineno"> 10</span> <span class="comment"> * <hr></span></div>
<div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="comment"> * License to use, copy, modify, sell and/or distribute this software</span></div>
<div class="line"><a name="l00013"></a><span class="lineno"> 13</span> <span class="comment"> * and its documentation for any purpose is hereby granted without</span></div>
<div class="line"><a name="l00014"></a><span class="lineno"> 14</span> <span class="comment"> * royalty, subject to the terms and conditions defined in the \ref</span></div>
<div class="line"><a name="l00015"></a><span class="lineno"> 15</span> <span class="comment"> * LICENSE file provided with this distribution.</span></div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="comment"> * </span></div>
<div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="comment"> * <hr></span></div>
<div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="comment"> * </span></div>
<div class="line"><a name="l00019"></a><span class="lineno"> 19</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00020"></a><span class="lineno"> 20</span> <span class="comment">/*****************************************************************************/</span></div>
<div class="line"><a name="l00021"></a><span class="lineno"> 21</span> </div>
<div class="line"><a name="l00022"></a><span class="lineno"> 22</span> <span class="preprocessor">#ifndef _cvc3__arith_proof_rules_h_</span></div>
<div class="line"><a name="l00023"></a><span class="lineno"> 23</span> <span class="preprocessor"></span><span class="preprocessor">#define _cvc3__arith_proof_rules_h_</span></div>
<div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="preprocessor"></span></div>
<div class="line"><a name="l00025"></a><span class="lineno"> 25</span> <span class="preprocessor">#include <vector></span></div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span> </div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span> <span class="keyword">namespace </span>CVC3 {</div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span> </div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>  <span class="keyword">class </span>Theorem;</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>  <span class="keyword">class </span>Expr;</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>  <span class="keyword">class </span>Rational;</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span> </div>
<div class="line"><a name="l00033"></a><span class="lineno"><a class="code" href="classCVC3_1_1ArithProofRules.html"> 33</a></span>  <span class="keyword">class </span><a class="code" href="classCVC3_1_1ArithProofRules.html">ArithProofRules</a> {</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>  <span class="comment">// Destructor</span></div>
<div class="line"><a name="l00036"></a><span class="lineno"><a class="code" href="classCVC3_1_1ArithProofRules.html#a410c697fc050feca9680a388397b0968"> 36</a></span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1ArithProofRules.html#a410c697fc050feca9680a388397b0968">~ArithProofRules</a>() { }</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span> <span class="comment"></span></div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span> <span class="comment"> ////////////////////////////////////////////////////////////////////</span></div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span> <span class="comment"></span> <span class="comment">// Canonization rules</span><span class="comment"></span></div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span> <span class="comment"> ////////////////////////////////////////////////////////////////////</span></div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span> <span class="comment"></span><span class="comment"></span></div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span> <span class="comment"> //! ==> e = 1 * e</span></div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a8b8db5ad0a8d4968f429e98964db93c1" title="==> e = 1 * e">varToMult</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span> <span class="comment"></span></div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span> <span class="comment"> //! ==> -(e) = (-1) * e</span></div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aa95faca93e7b4a1cf21667ff88d390e4" title="==> -(e) = (-1) * e">uMinusToMult</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span> <span class="comment"></span></div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span> <span class="comment"> //! ==> x - y = x + (-1) * y</span></div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a4319c532082c0f43794b34a2fb9df73f" title="==> x - y = x + (-1) * y">minusToPlus</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& x, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& y) = 0;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span> <span class="comment"></span></div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span> <span class="comment"> //! -(e) ==> e / (-1); takes 'e'</span></div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span> <span class="comment"></span><span class="comment"> /*! Canon Rule for unary minus: it just converts it to division by</span></div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span> <span class="comment"> * -1, the result is not yet canonical:</span></div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a0053398d4a8fc49d9a2f8ede6949fc70" title="-(e) ==> e / (-1); takes 'e'">canonUMinusToDivide</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span> <span class="comment"></span></div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span> <span class="comment"> /**</span></div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span> <span class="comment"> * Transform e = (SUM r t1 ... tn) @ 0 into (SUM t1 ... tn) @ -r. The first </span></div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span> <span class="comment"> * sum term (r) must be a rational and t1 ... tn terms must be canonised.</span></div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span> <span class="comment"> * </span></div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span> <span class="comment"> * @param e the expression to transform</span></div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span> <span class="comment"> * @return rewrite theorem representing the transformation </span></div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a18e5970bc4f670a63ca5775750f099ab">moveSumConstantRight</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span> <span class="comment"></span></div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span> <span class="comment"> //! (c) / d ==> (c/d), takes c and d</span></div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span> <span class="comment"></span><span class="comment"> /*! Canon Rules for division by constant 'd' */</span></div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a79c759f2bb7dc41ed98c5962febef804" title="(c) / d ==> (c/d), takes c and d">canonDivideConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& c, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& d) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span> <span class="comment"> //! (c * x) / d ==> (c/d) * x, takes (c*x) and d</span></div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aee487a2a1e8ce4c34075568eb70ea874" title="(c * x) / d ==> (c/d) * x, takes (c*x) and d">canonDivideMult</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& cx, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& d) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span> <span class="comment"> //! (+ c ...)/d ==> push division to all the coefficients.</span></div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span> <span class="comment"></span><span class="comment"> /*! The result is not canonical, but canonizing the summands will</span></div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span> <span class="comment"> * make it canonical.</span></div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span> <span class="comment"> * Takes (+ c ...) and d</span></div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a92846ee0cdab98d1759a13ff7e8d710d" title="(+ c ...)/d ==> push division to all the coefficients.">canonDividePlus</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& d) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span> <span class="comment"> //! x / d ==> (1/d) * x, takes x and d</span></div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a4ab0351b340ecb82d6ced530e4f30510" title="x / d ==> (1/d) * x, takes x and d">canonDivideVar</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& d) = 0;</div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span> </div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>  <span class="comment">// Canon Rules for multiplication</span></div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span> </div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>  <span class="comment">// TODO Deepak:</span></div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>  <span class="comment">// e == t1 * t2 where t1 and t2 are canonized expressions, i.e. it can be a</span></div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>  <span class="comment">// 1) Rational constant</span></div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>  <span class="comment">// 2) Arithmetic Leaf (var or term from another theory)</span></div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>  <span class="comment">// 3) (POW rational leaf)</span></div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  <span class="comment">// 4) (MULT rational mterm'_1 ...) where each mterm' is of type (2) or (3)</span></div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  <span class="comment">// 5) (PLUS rational sterm_1 sterm_2 ...) where each sterm is of </span></div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  <span class="comment">// type (2) or (3) or (4) </span></div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aa38d5c06ebe0b672570641fe933cfedb">canonMultMtermMterm</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span> <span class="comment"> //! Canonize (PLUS t1 ... tn)</span></div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a225fe511e6f654f313048e86c1550510" title="Canonize (PLUS t1 ... tn)">canonPlus</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a> & e) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</span> <span class="comment"> //! Canonize (MULT t1 ... tn)</span></div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aca910e6ff44389aac100b14b486c1942" title="Canonize (MULT t1 ... tn)">canonMult</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a> & e) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span> <span class="comment"> //! ==> 1/e = e' where e' is canonical; takes e.</span></div>
<div class="line"><a name="l00096"></a><span class="lineno"> 96</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a9a806d487932bbfa9a019d4d1eed1cea" title="==> 1/e = e' where e' is canonical; takes e.">canonInvert</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a> & e) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00097"></a><span class="lineno"> 97</span> <span class="comment"> //! Canonize t1/t2</span></div>
<div class="line"><a name="l00098"></a><span class="lineno"> 98</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a198b0b44145b8e33555ffd212f43b7b3" title="Canonize t1/t2.">canonDivide</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a> & e) = 0;</div>
<div class="line"><a name="l00099"></a><span class="lineno"> 99</span> <span class="comment"></span></div>
<div class="line"><a name="l00100"></a><span class="lineno"> 100</span> <span class="comment"> //! t*c ==> c*t, takes constant c and term t</span></div>
<div class="line"><a name="l00101"></a><span class="lineno"> 101</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a11870b90dbfcc61d2b9a2820df7f6057" title="t*c ==> c*t, takes constant c and term t">canonMultTermConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& c, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& t) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00102"></a><span class="lineno"> 102</span> <span class="comment"> //! t1*t2 ==> Error, takes t1 and t2 where both are non-constants</span></div>
<div class="line"><a name="l00103"></a><span class="lineno"> 103</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a8b9abf99f0b013714eaaa49e81a618b9" title="t1*t2 ==> Error, takes t1 and t2 where both are non-constants">canonMultTerm1Term2</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& t1, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& t2) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00104"></a><span class="lineno"> 104</span> <span class="comment"> //! 0*t ==> 0, takes 0*t</span></div>
<div class="line"><a name="l00105"></a><span class="lineno"> 105</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aae559b4d1558fecba5621c9ba27daa88" title="0*t ==> 0, takes 0*t">canonMultZero</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00106"></a><span class="lineno"> 106</span> <span class="comment"> //! 1*t ==> t, takes 1*t</span></div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a87f9a39ecb1bec46adff5187bc899726" title="1*t ==> t, takes 1*t">canonMultOne</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="comment"> //! c1*c2 ==> c', takes constant c1*c2 </span></div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a71be38a9fa2d89e147ac4fde35c49e21" title="c1*c2 ==> c', takes constant c1*c2">canonMultConstConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& c1, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& c2) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</span> <span class="comment"> //! c1*(c2*t) ==> c'*t, takes c1 and c2 and t</span></div>
<div class="line"><a name="l00111"></a><span class="lineno"> 111</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> </div>
<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <a class="code" href="classCVC3_1_1ArithProofRules.html#a96b1cb83272cf80c2de663de429e6a15" title="c1*(c2*t) ==> c'*t, takes c1 and c2 and t">canonMultConstTerm</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& c1, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& c2, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>&t) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span> <span class="comment"> //! c1*(+ c2 v1 ...) ==> (+ c' c1v1 ...), takes c1 and the sum</span></div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a978068749de266dbbba2000d9e0c3d39" title="c1*(+ c2 v1 ...) ==> (+ c' c1v1 ...), takes c1 and the sum">canonMultConstSum</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& c1, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& sum) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span> <span class="comment"> //! c^n = c' (compute the constant power expression)</span></div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a999f13a6fa4a6d8cbe97f02f93fc9ed3" title="c^n = c' (compute the constant power expression)">canonPowConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& <a class="code" href="namespaceCVC3.html#a28fc84bed3f9bc91ca9967de0f9cfa00" title="Raise 'base' into the power of 'pow' (pow must be an integer)">pow</a>) = 0;</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span> </div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  <span class="comment">// Rules for addition</span><span class="comment"></span></div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span> <span class="comment"> //! flattens the input. accepts a PLUS expr</span></div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#af0f4877d7326a53faec76bef54a3d400" title="flattens the input. accepts a PLUS expr">canonFlattenSum</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  <span class="comment"></span></div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span> <span class="comment"> //! combine like terms. accepts a flattened PLUS expr</span></div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a1da0c47c02ff2392ca3e0b6493ac4823" title="combine like terms. accepts a flattened PLUS expr">canonComboLikeTerms</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  </div>
<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  <span class="comment">// 0 = (* e1 e2 ...) <=> 0 = e1 OR 0 = e2 OR ...</span></div>
<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#af8f4f59bd12fdd8f93d1fd49ae41279e">multEqZero</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr) = 0;</div>
<div class="line"><a name="l00127"></a><span class="lineno"> 127</span> </div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>  <span class="comment">// 0 = (^ c x) <=> false if c <=0</span></div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>  <span class="comment">// <=> 0 = x if c > 0</span></div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#af17a5dbf4ebbe9d7e46d63ad1284ddaf">powEqZero</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr) = 0;</div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span> </div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  <span class="comment">// x^n = y^n <=> x = y (if n is odd)</span></div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  <span class="comment">// x^n = y^n <=> x = y OR x = -y (if n is even)</span></div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a0e6f19ba5333f667efa3a0e3d8787775">elimPower</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr) = 0;</div>
<div class="line"><a name="l00135"></a><span class="lineno"> 135</span> </div>
<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>  <span class="comment">// x^n = c <=> x = root (if n is odd and root^n = c)</span></div>
<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  <span class="comment">// x^n = c <=> x = root OR x = -root (if n is even and root^n = c)</span></div>
<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a0e240fba4eac33522b24be6581d98346">elimPowerConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Rational.html">Rational</a>& root) = 0;</div>
<div class="line"><a name="l00139"></a><span class="lineno"> 139</span> </div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  <span class="comment">// x^n = c <=> false (if n is even and c is negative)</span></div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a80f948f405f0b1a88283c97b70aab43e">evenPowerEqNegConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr) = 0;</div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</span> </div>
<div class="line"><a name="l00143"></a><span class="lineno"> 143</span>  <span class="comment">// x^n = c <=> false (if x is an integer and c is not a perfect n power)</span></div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aff57d78a7d743fffd21de7d4415df3e4">intEqIrrational</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& <a class="code" href="namespaceCVC3.html#a8b193cc1fee34cb43707171b2ae6471e">isInt</a>) = 0;</div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span> <span class="comment"></span></div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span> <span class="comment"> //! e0 \@ e1 <==> true | false, where \@ is {=,<,<=,>,>=}</span></div>
<div class="line"><a name="l00147"></a><span class="lineno"> 147</span> <span class="comment"></span><span class="comment"> /*! \param e takes e is (e0\@e1) where e0 and e1 are constants</span></div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a8dd6bb6fed9a4745469b0cad5ed4a139" title="e0 @ e1 <==> true | false, where @ is {=,<,<=,>,>=}">constPredicate</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span> <span class="comment"></span></div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span> <span class="comment"> //! e[0] @ e[1] <==> 0 @ e[1] - e[0], where @ is {=,<,<=,>,>=}</span></div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#acde77bf4aa73ff6b9a079b372ed7cc51" title="e[0] @ e[1] <==> 0 @ e[1] - e[0], where @ is {=,<,<=,>,>=}">rightMinusLeft</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00153"></a><span class="lineno"> 153</span> <span class="comment"></span></div>
<div class="line"><a name="l00154"></a><span class="lineno"> 154</span> <span class="comment"> //! e[0] @ e[1] <==> e[0] - e[1] @ 0, where @ is {=,<,<=,>,>=}</span></div>
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a0201544f06baaba23df9946f5da47978" title="e[0] @ e[1] <==> e[0] - e[1] @ 0, where @ is {=,<,<=,>,>=}">leftMinusRight</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span> <span class="comment"></span></div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span> <span class="comment"> //! x @ y <==> x + z @ y + z, where @ is {=,<,<=,>,>=} (given as 'kind')</span></div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a0fa920a2a5cfa4419dbde10b1d913687" title="x @ y <==> x + z @ y + z, where @ is {=,<,<=,>,>=} (given as 'kind')">plusPredicate</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& x, </div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& y, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& z, <span class="keywordtype">int</span> kind) = 0;</div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span> <span class="comment"></span></div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span> <span class="comment"> //! x = y <==> x * z = y * z, where z is a non-zero constant</span></div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#ad62d795eb10e67c655c608774881bde3" title="x = y <==> x * z = y * z, where z is a non-zero constant">multEqn</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& x, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& y, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& z) = 0;</div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span> </div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  <span class="comment">// x = y <==> z=0 OR x * 1/z = y * 1/z</span></div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a419dd84177b5ca1fda7a23f3052b6132">divideEqnNonConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& x, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& y, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& z) = 0;</div>
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span> <span class="comment"></span></div>
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span> <span class="comment"> //! Multiplying inequation by a non-zero constant</span></div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span> <span class="comment"></span><span class="comment"> /*!</span></div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span> <span class="comment"> * z>0 ==> e[0] @ e[1] <==> e[0]*z @ e[1]*z</span></div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span> <span class="comment"> * z<0 ==> e[0] @ e[1] <==> e[1]*z @ e[0]*z</span></div>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span> <span class="comment"> * for @ in {<,<=,>,>=}:</span></div>
<div class="line"><a name="l00174"></a><span class="lineno"> 174</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a70e83cfbc207a5d15fedd4adde7e9e76" title="Multiplying inequation by a non-zero constant.">multIneqn</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& z) = 0;</div>
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span> <span class="comment"></span></div>
<div class="line"><a name="l00177"></a><span class="lineno"> 177</span> <span class="comment"> //! x = y ==> x <= y and x >= y</span></div>
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a646e312e12d5d23dca0c4db39d2c9fb9" title="x = y ==> x <= y and x >= y">eqToIneq</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00179"></a><span class="lineno"> 179</span> <span class="comment"></span></div>
<div class="line"><a name="l00180"></a><span class="lineno"> 180</span> <span class="comment"> //! "op1 GE|GT op2" <==> op2 LE|LT op1</span></div>
<div class="line"><a name="l00181"></a><span class="lineno"> 181</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aff4a01da77770b0b60089444c83c9cef" title=""op1 GE|GT op2" <==> op2 LE|LT op1">flipInequality</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span> <span class="comment"></span></div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span> <span class="comment"> //! Propagating negation over <,<=,>,>=</span></div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span> <span class="comment"></span><span class="comment"> /*! NOT (op1 < op2) <==> (op1 >= op2)</span></div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</span> <span class="comment"> * NOT (op1 <= op2) <==> (op1 > op2)</span></div>
<div class="line"><a name="l00187"></a><span class="lineno"> 187</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00188"></a><span class="lineno"> 188</span> <span class="comment"> * NOT (op1 > op2) <==> (op1 <= op2)</span></div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00190"></a><span class="lineno"> 190</span> <span class="comment"> * NOT (op1 >= op2) <==> (op1 < op2)</span></div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aee47bda46999143ea29d7d4ff0be89d4" title="Propagating negation over <,<=,>,>=.">negatedInequality</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>  <span class="comment"></span></div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span> <span class="comment"> //! Real shadow: a <(=) t, t <(=) b ==> a <(=) b</span></div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a203c7079289189cd4bf4f604c10645e5" title="Real shadow: a <(=) t, t <(=) b ==> a <(=) b.">realShadow</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& alphaLTt, </div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& tLTbeta) = 0;</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span> <span class="comment"></span></div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span> <span class="comment"> //! Projecting a tight inequality: alpha <= t <= alpha ==> t = alpha</span></div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a08dfccc28787af2a4141ed0f5a66bc9d" title="Projecting a tight inequality: alpha <= t <= alpha ==> t = alpha.">realShadowEq</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& alphaLEt, </div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& tLEalpha) = 0;</div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</span> <span class="comment"></span></div>
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span> <span class="comment"> //! Finite interval for integers: a <= t <= a + c ==> G(t, a, 0, c)</span></div>
<div class="line"><a name="l00203"></a><span class="lineno"> 203</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a5d419e48b93820624e356ba33efef27c" title="Finite interval for integers: a <= t <= a + c ==> G(t, a, 0, c)">finiteInterval</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& aLEt,</div>
<div class="line"><a name="l00204"></a><span class="lineno"> 204</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& tLEac,</div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isInta,</div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntt) = 0;</div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  <span class="comment"></span></div>
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span> <span class="comment"> //! Dark & Gray shadows when a <= b</span></div>
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span> <span class="comment"></span><span class="comment"> /*! takes two integer ineqs (i.e. all vars are ints) </span></div>
<div class="line"><a name="l00210"></a><span class="lineno"> 210</span> <span class="comment"> * "|- beta <= b.x" and "|- a.x <= alpha" and checks if "1 <= a <= b"</span></div>
<div class="line"><a name="l00211"></a><span class="lineno"> 211</span> <span class="comment"> * and returns (D or G) and (!D or !G), where</span></div>
<div class="line"><a name="l00212"></a><span class="lineno"> 212</span> <span class="comment"> * D = Dark_Shadow(ab-1, b.alpha - a.beta),</span></div>
<div class="line"><a name="l00213"></a><span class="lineno"> 213</span> <span class="comment"> * G = Gray_Shadow(a.x, alpha, -(a-1), 0).</span></div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a37835b10dd0f73b5c4cf36c1eebf0d1f" title="Dark & Gray shadows when a <= b.">darkGrayShadow2ab</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& betaLEbx, </div>
<div class="line"><a name="l00216"></a><span class="lineno"> 216</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& axLEalpha,</div>
<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntAlpha,</div>
<div class="line"><a name="l00218"></a><span class="lineno"> 218</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntBeta,</div>
<div class="line"><a name="l00219"></a><span class="lineno"> 219</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& <a class="code" href="theory__quant_8cpp.html#ac9940420b1f3427ebe821902adbb578d">isIntx</a>)=0;</div>
<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>  <span class="comment"></span></div>
<div class="line"><a name="l00221"></a><span class="lineno"> 221</span> <span class="comment"> //! Dark & Gray shadows when b <= a</span></div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span> <span class="comment"></span><span class="comment"> /*! takes two integer ineqs (i.e. all vars are ints) </span></div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span> <span class="comment"> * "|- beta <= b.x" and "|- a.x <= alpha" and checks if "1 <= b < a"</span></div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span> <span class="comment"> * and returns (D or G) and (!D or !G), where</span></div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span> <span class="comment"> * D = Dark_Shadow(ab-1, b.alpha - a.beta),</span></div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span> <span class="comment"> * G = Gray_Shadow(b.x, beta, 0, b-1).</span></div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#ab17282c2e4de9965586f4821f17bf60d" title="Dark & Gray shadows when b <= a.">darkGrayShadow2ba</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& betaLEbx, </div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& axLEalpha,</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntAlpha,</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntBeta,</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& <a class="code" href="theory__quant_8cpp.html#ac9940420b1f3427ebe821902adbb578d">isIntx</a>)=0;</div>
<div class="line"><a name="l00233"></a><span class="lineno"> 233</span>  <span class="comment"></span></div>
<div class="line"><a name="l00234"></a><span class="lineno"> 234</span> <span class="comment"> //! DARK_SHADOW(t1, t2) ==> t1 <= t2</span></div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a94ffe313b29aa63c32d5c3facb3bc917" title="DARK_SHADOW(t1, t2) ==> t1 <= t2.">expandDarkShadow</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& darkShadow)=0;</div>
<div class="line"><a name="l00236"></a><span class="lineno"> 236</span>  <span class="comment"></span></div>
<div class="line"><a name="l00237"></a><span class="lineno"> 237</span> <span class="comment"> //! GRAY_SHADOW(v, e, c, c) ==> v=e+c.</span></div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a32df170ed2ec7fb3d1c7541e6f3dda77" title="GRAY_SHADOW(v, e, c, c) ==> v=e+c.">expandGrayShadow0</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& g)=0;</div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</span> </div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span>  <span class="comment">// [used to be] GRAY_SHADOW(t1, t2, i) ==> t1 = t2+i OR</span></div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</span>  <span class="comment">// GRAY_SHADOW(t1, t2, i+/-1)</span></div>
<div class="line"><a name="l00242"></a><span class="lineno"> 242</span> <span class="comment"></span></div>
<div class="line"><a name="l00243"></a><span class="lineno"> 243</span> <span class="comment"> //! G(x, e, c1, c2) ==> (G1 or G2) and (!G1 or !G2)</span></div>
<div class="line"><a name="l00244"></a><span class="lineno"> 244</span> <span class="comment"></span><span class="comment"> /*! Here G1 = G(x,e,c1,c),</span></div>
<div class="line"><a name="l00245"></a><span class="lineno"> 245</span> <span class="comment"> * G2 = G(x,e,c+1,c2),</span></div>
<div class="line"><a name="l00246"></a><span class="lineno"> 246</span> <span class="comment"> * and c = floor((c1+c2)/2).</span></div>
<div class="line"><a name="l00247"></a><span class="lineno"> 247</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00248"></a><span class="lineno"> 248</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a4b6ef5b29b05bada6ea2df72873abea6" title="G(x, e, c1, c2) ==> (G1 or G2) and (!G1 or !G2)">splitGrayShadow</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& g)=0;</div>
<div class="line"><a name="l00249"></a><span class="lineno"> 249</span>  </div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</span>  </div>
<div class="line"><a name="l00251"></a><span class="lineno"> 251</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aa33975feef2a9cdd513e5d641f14dc7a">splitGrayShadowSmall</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& g)=0;</div>
<div class="line"><a name="l00252"></a><span class="lineno"> 252</span>  <span class="comment"></span></div>
<div class="line"><a name="l00253"></a><span class="lineno"> 253</span> <span class="comment"> //! G(x, e, c1, c2) ==> e+c1 <= x AND x <= e+c2</span></div>
<div class="line"><a name="l00254"></a><span class="lineno"> 254</span> <span class="comment"></span> <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#af936b764ef75b0a6f63043d01a819044" title="G(x, e, c1, c2) ==> e+c1 <= x AND x <= e+c2.">expandGrayShadow</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& g)=0;</div>
<div class="line"><a name="l00255"></a><span class="lineno"> 255</span> <span class="comment"></span></div>
<div class="line"><a name="l00256"></a><span class="lineno"> 256</span> <span class="comment"> //! Optimized rules: GRAY_SHADOW(a*x, c, c1, c2) ==> [expansion]</span></div>
<div class="line"><a name="l00257"></a><span class="lineno"> 257</span> <span class="comment"></span><span class="comment"> /*! Implements three versions of the rule:</span></div>
<div class="line"><a name="l00258"></a><span class="lineno"> 258</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00259"></a><span class="lineno"> 259</span> <span class="comment"> * \f[\frac{\mathrm{GRAY\_SHADOW}(ax,c,c1,c2)}</span></div>
<div class="line"><a name="l00260"></a><span class="lineno"> 260</span> <span class="comment"> * {ax = c + i - \mathrm{sign}(i)\cdot j(c,i,a)</span></div>
<div class="line"><a name="l00261"></a><span class="lineno"> 261</span> <span class="comment"> * \lor GRAY\_SHADOW(ax, c, i-\mathrm{sign}(i)\cdot (a+j(c,i,a)))}\f]</span></div>
<div class="line"><a name="l00262"></a><span class="lineno"> 262</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00263"></a><span class="lineno"> 263</span> <span class="comment"> * where the conclusion may become FALSE or without the</span></div>
<div class="line"><a name="l00264"></a><span class="lineno"> 264</span> <span class="comment"> * GRAY_SHADOW part, depending on the values of a, c and i.</span></div>
<div class="line"><a name="l00265"></a><span class="lineno"> 265</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00266"></a><span class="lineno"> 266</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a3083a3c5893123f2a3ec54ce31718e09" title="Optimized rules: GRAY_SHADOW(a*x, c, c1, c2) ==> [expansion].">expandGrayShadowConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& g)=0;<span class="comment"></span></div>
<div class="line"><a name="l00267"></a><span class="lineno"> 267</span> <span class="comment"> //! |- G(ax, c, c1, c2) ==> |- G(x, 0, ceil((c1+c)/a), floor((c2+c)/a))</span></div>
<div class="line"><a name="l00268"></a><span class="lineno"> 268</span> <span class="comment"></span><span class="comment"> /*! In the case the new c1 > c2, return |- FALSE */</span></div>
<div class="line"><a name="l00269"></a><span class="lineno"> 269</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a597adbe8879c33bffefae537a5bca7c0" title="|- G(ax, c, c1, c2) ==> |- G(x, 0, ceil((c1+c)/a), floor((c2+c)/a))">grayShadowConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& g)=0;</div>
<div class="line"><a name="l00270"></a><span class="lineno"> 270</span> <span class="comment"></span></div>
<div class="line"><a name="l00271"></a><span class="lineno"> 271</span> <span class="comment"> //! a,b: INT; a < b ==> a <= b-1 [or a+1 <= b]</span></div>
<div class="line"><a name="l00272"></a><span class="lineno"> 272</span> <span class="comment"></span><span class="comment"> /*! Takes a Theorem(\\alpha < \\beta) and returns </span></div>
<div class="line"><a name="l00273"></a><span class="lineno"> 273</span> <span class="comment"> * Theorem(\\alpha < \\beta <==> \\alpha <= \\beta -1)</span></div>
<div class="line"><a name="l00274"></a><span class="lineno"> 274</span> <span class="comment"> * or Theorem(\\alpha < \\beta <==> \\alpha + 1 <= \\beta),</span></div>
<div class="line"><a name="l00275"></a><span class="lineno"> 275</span> <span class="comment"> * depending on which side must be changed (changeRight flag)</span></div>
<div class="line"><a name="l00276"></a><span class="lineno"> 276</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00277"></a><span class="lineno"> 277</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a7a0dad5cb9c2815421457dabc25bc537" title="a,b: INT; a < b ==> a <= b-1 [or a+1 <= b]">lessThanToLE</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& less, </div>
<div class="line"><a name="l00278"></a><span class="lineno"> 278</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntLHS,</div>
<div class="line"><a name="l00279"></a><span class="lineno"> 279</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntRHS,</div>
<div class="line"><a name="l00280"></a><span class="lineno"> 280</span>  <span class="keywordtype">bool</span> changeRight)=0;</div>
<div class="line"><a name="l00281"></a><span class="lineno"> 281</span> </div>
<div class="line"><a name="l00282"></a><span class="lineno"> 282</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#af5cfb693ee5ee6f3463e83f01196b31e">lessThanToLERewrite</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& ineq, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntLHS,</div>
<div class="line"><a name="l00283"></a><span class="lineno"> 283</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntRHS, <span class="keywordtype">bool</span> changeRight) = 0;</div>
<div class="line"><a name="l00284"></a><span class="lineno"> 284</span> </div>
<div class="line"><a name="l00285"></a><span class="lineno"> 285</span>  <span class="comment"></span></div>
<div class="line"><a name="l00286"></a><span class="lineno"> 286</span> <span class="comment"> /*! \param eqn is an equation 0 = a.x or 0 = c + a.x, where x is integer</span></div>
<div class="line"><a name="l00287"></a><span class="lineno"> 287</span> <span class="comment"> * \param isIntx is a proof of IS_INTEGER(x)</span></div>
<div class="line"><a name="l00288"></a><span class="lineno"> 288</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00289"></a><span class="lineno"> 289</span> <span class="comment"> * \return the theorem 0 = c + a.x <==> x=-c/a if -c/a is int else</span></div>
<div class="line"><a name="l00290"></a><span class="lineno"> 290</span> <span class="comment"> * return the theorem 0 = c + a.x <==> false.</span></div>
<div class="line"><a name="l00291"></a><span class="lineno"> 291</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00292"></a><span class="lineno"> 292</span> <span class="comment"> * It also handles the special case of 0 = a.x <==> x = 0</span></div>
<div class="line"><a name="l00293"></a><span class="lineno"> 293</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00294"></a><span class="lineno"> 294</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a62d76b2ccca8af5e1f28c5104ea2e313">intVarEqnConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& eqn,</div>
<div class="line"><a name="l00295"></a><span class="lineno"> 295</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& <a class="code" href="theory__quant_8cpp.html#ac9940420b1f3427ebe821902adbb578d">isIntx</a>) = 0;<span class="comment"></span></div>
<div class="line"><a name="l00296"></a><span class="lineno"> 296</span> <span class="comment"> /*! IS_INTEGER(x) <=> EXISTS (y : INT) y = x</span></div>
<div class="line"><a name="l00297"></a><span class="lineno"> 297</span> <span class="comment"> * where x is not already known to be an integer.</span></div>
<div class="line"><a name="l00298"></a><span class="lineno"> 298</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00299"></a><span class="lineno"> 299</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a8c3d5fdc2b0c9c11291d41c3c20067e4">IsIntegerElim</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& <a class="code" href="theory__quant_8cpp.html#ac9940420b1f3427ebe821902adbb578d">isIntx</a>) = 0;</div>
<div class="line"><a name="l00300"></a><span class="lineno"> 300</span> <span class="comment"></span></div>
<div class="line"><a name="l00301"></a><span class="lineno"> 301</span> <span class="comment"> /*! @brief Equality elimination rule for integers:</span></div>
<div class="line"><a name="l00302"></a><span class="lineno"> 302</span> <span class="comment"> * \f[\frac{\mathsf{int}(a\cdot x)\quad</span></div>
<div class="line"><a name="l00303"></a><span class="lineno"> 303</span> <span class="comment"> * \mathsf{int}(C+\sum_{i=1}^{n}a_{i}\cdot x_{i})}</span></div>
<div class="line"><a name="l00304"></a><span class="lineno"> 304</span> <span class="comment"> * {a\cdot x=C+\sum_{i=1}^{n}a_{i}\cdot x_{i}</span></div>
<div class="line"><a name="l00305"></a><span class="lineno"> 305</span> <span class="comment"> * \quad\equiv\quad x=t_{2}\wedge 0=t_{3}}</span></div>
<div class="line"><a name="l00306"></a><span class="lineno"> 306</span> <span class="comment"> * \f]</span></div>
<div class="line"><a name="l00307"></a><span class="lineno"> 307</span> <span class="comment"> * See the detailed description for explanations.</span></div>
<div class="line"><a name="l00308"></a><span class="lineno"> 308</span> <span class="comment"> * </span></div>
<div class="line"><a name="l00309"></a><span class="lineno"> 309</span> <span class="comment"> * This rule implements a step in the iterative algorithm for</span></div>
<div class="line"><a name="l00310"></a><span class="lineno"> 310</span> <span class="comment"> * eliminating integer equality. The terms in the rule are</span></div>
<div class="line"><a name="l00311"></a><span class="lineno"> 311</span> <span class="comment"> * defined as follows:</span></div>
<div class="line"><a name="l00312"></a><span class="lineno"> 312</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00313"></a><span class="lineno"> 313</span> <span class="comment"> * \f[\begin{array}{rcl}</span></div>
<div class="line"><a name="l00314"></a><span class="lineno"> 314</span> <span class="comment"> * t_{2} & = & </span></div>
<div class="line"><a name="l00315"></a><span class="lineno"> 315</span> <span class="comment"> * -(C\ \mathbf{mod}\ m+\sum_{i=1}^{n}(a_{i}\ \mathbf{mod}\ m)</span></div>
<div class="line"><a name="l00316"></a><span class="lineno"> 316</span> <span class="comment"> * \cdot x_{i}-m\cdot\sigma(t))\\ & & \\</span></div>
<div class="line"><a name="l00317"></a><span class="lineno"> 317</span> <span class="comment"> * t_{3} & = & </span></div>
<div class="line"><a name="l00318"></a><span class="lineno"> 318</span> <span class="comment"> * \mathbf{f}(C,m)+\sum_{i=1}^{n}\mathbf{f}(a_{i},m)\cdot x_{i}</span></div>
<div class="line"><a name="l00319"></a><span class="lineno"> 319</span> <span class="comment"> * -a\cdot\sigma(t)\\ & & \\</span></div>
<div class="line"><a name="l00320"></a><span class="lineno"> 320</span> <span class="comment"> * t & = &</span></div>
<div class="line"><a name="l00321"></a><span class="lineno"> 321</span> <span class="comment"> * (C\ \mathbf{mod}\ m+\sum_{i=1}^{n}(a_{i}\ \mathbf{mod}\ m)</span></div>
<div class="line"><a name="l00322"></a><span class="lineno"> 322</span> <span class="comment"> * \cdot x_{i}+x)/m\\ & & \\</span></div>
<div class="line"><a name="l00323"></a><span class="lineno"> 323</span> <span class="comment"> * m & = & a+1\\ & & \\</span></div>
<div class="line"><a name="l00324"></a><span class="lineno"> 324</span> <span class="comment"> * \mathbf{f}(i,m) & = & \left\lfloor \frac{i}{m}</span></div>
<div class="line"><a name="l00325"></a><span class="lineno"> 325</span> <span class="comment"> * +\frac{1}{2}\right\rfloor +i\ \mathbf{mod}\ m\\ & & \\</span></div>
<div class="line"><a name="l00326"></a><span class="lineno"> 326</span> <span class="comment"> * i\ \mathbf{mod}\ m & = & i-m\left\lfloor\frac{i}{m}</span></div>
<div class="line"><a name="l00327"></a><span class="lineno"> 327</span> <span class="comment"> * +\frac{1}{2}\right\rfloor</span></div>
<div class="line"><a name="l00328"></a><span class="lineno"> 328</span> <span class="comment"> * \end{array}</span></div>
<div class="line"><a name="l00329"></a><span class="lineno"> 329</span> <span class="comment"> * \f]</span></div>
<div class="line"><a name="l00330"></a><span class="lineno"> 330</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00331"></a><span class="lineno"> 331</span> <span class="comment"> * All the variables and coefficients are integer, and a >= 2.</span></div>
<div class="line"><a name="l00332"></a><span class="lineno"> 332</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00333"></a><span class="lineno"> 333</span> <span class="comment"> * \param eqn is the equation</span></div>
<div class="line"><a name="l00334"></a><span class="lineno"> 334</span> <span class="comment"> * \f[a\cdot x = C + \sum_{i=1}^n a_i\cdot x_i.\f]</span></div>
<div class="line"><a name="l00335"></a><span class="lineno"> 335</span> <span class="comment"> * </span></div>
<div class="line"><a name="l00336"></a><span class="lineno"> 336</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00337"></a><span class="lineno"> 337</span> </div>
<div class="line"><a name="l00338"></a><span class="lineno"> 338</span>  <span class="comment">/*</span></div>
<div class="line"><a name="l00339"></a><span class="lineno"> 339</span> <span class="comment"> virtual Theorem eqElimIntRule(const Expr& eqn,</span></div>
<div class="line"><a name="l00340"></a><span class="lineno"> 340</span> <span class="comment"> const Theorem& isIntLHS,</span></div>
<div class="line"><a name="l00341"></a><span class="lineno"> 341</span> <span class="comment"> const Theorem& isIntRHS) = 0;</span><span class="comment"></span></div>
<div class="line"><a name="l00342"></a><span class="lineno"> 342</span> <span class="comment"> //! a <=> b ==> c AND a <=> c AND b. Takes "a <=> b" and "c".</span></div>
<div class="line"><a name="l00343"></a><span class="lineno"> 343</span> <span class="comment"></span> virtual Theorem cANDaIffcANDb(const Theorem& thm, </div>
<div class="line"><a name="l00344"></a><span class="lineno"> 344</span>  const Expr& solvedEq) = 0;</div>
<div class="line"><a name="l00345"></a><span class="lineno"> 345</span>  virtual Theorem substSolvedFormRule(const Expr& e1,</div>
<div class="line"><a name="l00346"></a><span class="lineno"> 346</span>  ExprMap<Expr>& eMap) = 0; </div>
<div class="line"><a name="l00347"></a><span class="lineno"> 347</span>  virtual Theorem aANDcIffbANDc(const Theorem& thm, const Expr& newEq) = 0;</div>
<div class="line"><a name="l00348"></a><span class="lineno"> 348</span>  */</div>
<div class="line"><a name="l00349"></a><span class="lineno"> 349</span> <span class="comment"></span></div>
<div class="line"><a name="l00350"></a><span class="lineno"> 350</span> <span class="comment"> ///////////////////////////////////////////////////////////////////////</span></div>
<div class="line"><a name="l00351"></a><span class="lineno"> 351</span> <span class="comment"></span> <span class="comment">// Alternative implementation of integer equality elimination</span><span class="comment"></span></div>
<div class="line"><a name="l00352"></a><span class="lineno"> 352</span> <span class="comment"> ///////////////////////////////////////////////////////////////////////</span></div>
<div class="line"><a name="l00353"></a><span class="lineno"> 353</span> <span class="comment"></span><span class="comment"></span></div>
<div class="line"><a name="l00354"></a><span class="lineno"> 354</span> <span class="comment"> /*! @brief</span></div>
<div class="line"><a name="l00355"></a><span class="lineno"> 355</span> <span class="comment"> * \f[\frac{\Gamma\models ax=t\quad</span></div>
<div class="line"><a name="l00356"></a><span class="lineno"> 356</span> <span class="comment"> * \Gamma'\models\mathsf{int}(x)\quad</span></div>
<div class="line"><a name="l00357"></a><span class="lineno"> 357</span> <span class="comment"> * \{\Gamma_i\models\mathsf{int}(x_i) | x_i\mbox{ is var in }t\}}</span></div>
<div class="line"><a name="l00358"></a><span class="lineno"> 358</span> <span class="comment"> * {\Gamma,\Gamma',\bigcup_i\Gamma_i\models</span></div>
<div class="line"><a name="l00359"></a><span class="lineno"> 359</span> <span class="comment"> * \exists (y:\mathrm{int}).\ x=t_2(y)\wedge 0=t_3(y)}</span></div>
<div class="line"><a name="l00360"></a><span class="lineno"> 360</span> <span class="comment"> * \f]</span></div>
<div class="line"><a name="l00361"></a><span class="lineno"> 361</span> <span class="comment"> * See detailed docs for more information.</span></div>
<div class="line"><a name="l00362"></a><span class="lineno"> 362</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00363"></a><span class="lineno"> 363</span> <span class="comment"> * This rule implements a step in the iterative algorithm for</span></div>
<div class="line"><a name="l00364"></a><span class="lineno"> 364</span> <span class="comment"> * eliminating integer equality. The terms in the rule are</span></div>
<div class="line"><a name="l00365"></a><span class="lineno"> 365</span> <span class="comment"> * defined as follows:</span></div>
<div class="line"><a name="l00366"></a><span class="lineno"> 366</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00367"></a><span class="lineno"> 367</span> <span class="comment"> * \f[\begin{array}{rcl}</span></div>
<div class="line"><a name="l00368"></a><span class="lineno"> 368</span> <span class="comment"> * t & = & C+\sum_{i=1}^na_{i}\cdot x_{i}\\</span></div>
<div class="line"><a name="l00369"></a><span class="lineno"> 369</span> <span class="comment"> * t_{2}(y) & = & </span></div>
<div class="line"><a name="l00370"></a><span class="lineno"> 370</span> <span class="comment"> * -(C\ \mathbf{mod}\ m+\sum_{i=1}^{n}(a_{i}\ \mathbf{mod}\ m)</span></div>
<div class="line"><a name="l00371"></a><span class="lineno"> 371</span> <span class="comment"> * \cdot x_{i}-m\cdot y)\\ & & \\</span></div>
<div class="line"><a name="l00372"></a><span class="lineno"> 372</span> <span class="comment"> * t_{3}(y) & = & </span></div>
<div class="line"><a name="l00373"></a><span class="lineno"> 373</span> <span class="comment"> * \mathbf{f}(C,m)+\sum_{i=1}^{n}\mathbf{f}(a_{i},m)\cdot x_{i}</span></div>
<div class="line"><a name="l00374"></a><span class="lineno"> 374</span> <span class="comment"> * -a\cdot y\\ & & \\</span></div>
<div class="line"><a name="l00375"></a><span class="lineno"> 375</span> <span class="comment"> * m & = & a+1\\ & & \\</span></div>
<div class="line"><a name="l00376"></a><span class="lineno"> 376</span> <span class="comment"> * \mathbf{f}(i,m) & = & \left\lfloor \frac{i}{m}</span></div>
<div class="line"><a name="l00377"></a><span class="lineno"> 377</span> <span class="comment"> * +\frac{1}{2}\right\rfloor +i\ \mathbf{mod}\ m\\ & & \\</span></div>
<div class="line"><a name="l00378"></a><span class="lineno"> 378</span> <span class="comment"> * i\ \mathbf{mod}\ m & = & i-m\left\lfloor\frac{i}{m}</span></div>
<div class="line"><a name="l00379"></a><span class="lineno"> 379</span> <span class="comment"> * +\frac{1}{2}\right\rfloor</span></div>
<div class="line"><a name="l00380"></a><span class="lineno"> 380</span> <span class="comment"> * \end{array}</span></div>
<div class="line"><a name="l00381"></a><span class="lineno"> 381</span> <span class="comment"> * \f]</span></div>
<div class="line"><a name="l00382"></a><span class="lineno"> 382</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00383"></a><span class="lineno"> 383</span> <span class="comment"> * All the variables and coefficients are integer, and a >= 2.</span></div>
<div class="line"><a name="l00384"></a><span class="lineno"> 384</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00385"></a><span class="lineno"> 385</span> <span class="comment"> * \param eqn is the equation ax=t:</span></div>
<div class="line"><a name="l00386"></a><span class="lineno"> 386</span> <span class="comment"> * \f[a\cdot x = C + \sum_{i=1}^n a_i\cdot x_i.\f]</span></div>
<div class="line"><a name="l00387"></a><span class="lineno"> 387</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00388"></a><span class="lineno"> 388</span> <span class="comment"> * \param isIntx is a Theorem deriving the integrality of the</span></div>
<div class="line"><a name="l00389"></a><span class="lineno"> 389</span> <span class="comment"> * LHS variable: IS_INTEGER(x)</span></div>
<div class="line"><a name="l00390"></a><span class="lineno"> 390</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00391"></a><span class="lineno"> 391</span> <span class="comment"> * \param isIntVars is a vector of Theorems deriving the</span></div>
<div class="line"><a name="l00392"></a><span class="lineno"> 392</span> <span class="comment"> * integrality of all variables on the RHS</span></div>
<div class="line"><a name="l00393"></a><span class="lineno"> 393</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00394"></a><span class="lineno"> 394</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a08b14a730bbc6b4ca28bb808e6f6dc85" title="Equality elimination rule for integers: See the detailed description for explanations.">eqElimIntRule</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& eqn,</div>
<div class="line"><a name="l00395"></a><span class="lineno"> 395</span>  <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& <a class="code" href="theory__quant_8cpp.html#ac9940420b1f3427ebe821902adbb578d">isIntx</a>,</div>
<div class="line"><a name="l00396"></a><span class="lineno"> 396</span>  <span class="keyword">const</span> std::vector<Theorem>& isIntVars) = 0;</div>
<div class="line"><a name="l00397"></a><span class="lineno"> 397</span> <span class="comment"></span></div>
<div class="line"><a name="l00398"></a><span class="lineno"> 398</span> <span class="comment"> /*!</span></div>
<div class="line"><a name="l00399"></a><span class="lineno"> 399</span> <span class="comment"> * @brief return e <=> TRUE/FALSE for e == IS_INTEGER(c), where c</span></div>
<div class="line"><a name="l00400"></a><span class="lineno"> 400</span> <span class="comment"> * is a constant</span></div>
<div class="line"><a name="l00401"></a><span class="lineno"> 401</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00402"></a><span class="lineno"> 402</span> <span class="comment"> * \param e is a predicate IS_INTEGER(c) where c is a rational constant</span></div>
<div class="line"><a name="l00403"></a><span class="lineno"> 403</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00404"></a><span class="lineno"> 404</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a5e9bccd77000156ab34b0b42d3e3bb18" title="return e <=> TRUE/FALSE for e == IS_INTEGER(c), where c is a constant">isIntConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00405"></a><span class="lineno"> 405</span> </div>
<div class="line"><a name="l00406"></a><span class="lineno"> 406</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#ac05fd3606540a5222ebaa68f9091345b">equalLeaves1</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& thm) = 0;</div>
<div class="line"><a name="l00407"></a><span class="lineno"> 407</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a5052c783af25243987424ef66e2bf5fc">equalLeaves2</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& thm) = 0;</div>
<div class="line"><a name="l00408"></a><span class="lineno"> 408</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a893c86f0404789d229540f7a0a00df89">equalLeaves3</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& thm) = 0;</div>
<div class="line"><a name="l00409"></a><span class="lineno"> 409</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aa9018c4d16d3d0a2d8f1614f8e7cae23">equalLeaves4</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& thm) = 0;</div>
<div class="line"><a name="l00410"></a><span class="lineno"> 410</span> <span class="comment"></span></div>
<div class="line"><a name="l00411"></a><span class="lineno"> 411</span> <span class="comment"> //! x /= y ==> (x < y) OR (x > y)</span></div>
<div class="line"><a name="l00412"></a><span class="lineno"> 412</span> <span class="comment"></span><span class="comment"> /*! Used in concrete model generation */</span></div>
<div class="line"><a name="l00413"></a><span class="lineno"> 413</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#af180898d9f1d255ef3a2f528ecd8ac65" title="x /= y ==> (x < y) OR (x > y)">diseqToIneq</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& diseq) = 0;</div>
<div class="line"><a name="l00414"></a><span class="lineno"> 414</span> </div>
<div class="line"><a name="l00415"></a><span class="lineno"> 415</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a24ea340651805e648d7180debad17225">dummyTheorem</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00416"></a><span class="lineno"> 416</span>  </div>
<div class="line"><a name="l00417"></a><span class="lineno"> 417</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a1ebae10d65430ab1a3a14c866af93de6">oneElimination</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& x) = 0;</div>
<div class="line"><a name="l00418"></a><span class="lineno"> 418</span>  </div>
<div class="line"><a name="l00419"></a><span class="lineno"> 419</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#ad0b23fb01a7adea349c8958059b9f7fd">clashingBounds</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& lowerBound, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& upperBound) = 0;</div>
<div class="line"><a name="l00420"></a><span class="lineno"> 420</span>  </div>
<div class="line"><a name="l00421"></a><span class="lineno"> 421</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#ac06dd83948d0f3e474f073ecac592343">addInequalities</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& thm1, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& thm2) = 0;</div>
<div class="line"><a name="l00422"></a><span class="lineno"> 422</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#ac06dd83948d0f3e474f073ecac592343">addInequalities</a>(<span class="keyword">const</span> std::vector<Theorem>& thms) = 0;</div>
<div class="line"><a name="l00423"></a><span class="lineno"> 423</span>  </div>
<div class="line"><a name="l00424"></a><span class="lineno"> 424</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a9abe85f95b61d94fa308a0b15f2b2ef8">implyWeakerInequality</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr1, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr2) = 0;</div>
<div class="line"><a name="l00425"></a><span class="lineno"> 425</span>  </div>
<div class="line"><a name="l00426"></a><span class="lineno"> 426</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a5eaf4d106f7ce86227ab1d96ffc66b55">implyNegatedInequality</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr1, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr2) = 0;</div>
<div class="line"><a name="l00427"></a><span class="lineno"> 427</span> </div>
<div class="line"><a name="l00428"></a><span class="lineno"> 428</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a0090e0d4065efbf936ec92f0e99935e9">integerSplit</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& intVar, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Rational.html">Rational</a>& intPoint) = 0;</div>
<div class="line"><a name="l00429"></a><span class="lineno"> 429</span>  </div>
<div class="line"><a name="l00430"></a><span class="lineno"> 430</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a11976de555ee2e88236575893f2ed766">trustedRewrite</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr1, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& expr2) = 0;</div>
<div class="line"><a name="l00431"></a><span class="lineno"> 431</span>  </div>
<div class="line"><a name="l00432"></a><span class="lineno"> 432</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a4b61e3ea9d5ac911806de95241cf35de">rafineStrictInteger</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntConstrThm, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& constr) = 0;</div>
<div class="line"><a name="l00433"></a><span class="lineno"> 433</span> </div>
<div class="line"><a name="l00434"></a><span class="lineno"> 434</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a95bc88d9e126e6b9c938691ce92985aa">simpleIneqInt</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& ineq, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntRHS) = 0;</div>
<div class="line"><a name="l00435"></a><span class="lineno"> 435</span>  </div>
<div class="line"><a name="l00436"></a><span class="lineno"> 436</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a93489420e882a93eb308c7d99c27dc1e">intEqualityRationalConstant</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& isIntConstrThm, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& constr) = 0;</div>
<div class="line"><a name="l00437"></a><span class="lineno"> 437</span>  </div>
<div class="line"><a name="l00438"></a><span class="lineno"> 438</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a5a544980652c3f6793057b16cbc43cf9">cycleConflict</a>(<span class="keyword">const</span> std::vector<Theorem>& inequalitites) = 0;</div>
<div class="line"><a name="l00439"></a><span class="lineno"> 439</span> </div>
<div class="line"><a name="l00440"></a><span class="lineno"> 440</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a5b475214a53d70298e55f4ac5f3db71f">implyEqualities</a>(<span class="keyword">const</span> std::vector<Theorem>& inequalities) = 0;</div>
<div class="line"><a name="l00441"></a><span class="lineno"> 441</span>  </div>
<div class="line"><a name="l00442"></a><span class="lineno"> 442</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aafc12b4bdf180f3475ecf4eaa40648d9">implyWeakerInequalityDiffLogic</a>(<span class="keyword">const</span> std::vector<Theorem>& antecedentThms, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& implied) = 0;</div>
<div class="line"><a name="l00443"></a><span class="lineno"> 443</span>  </div>
<div class="line"><a name="l00444"></a><span class="lineno"> 444</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a61ecf1cfc7219c010519cd90cb31d465">implyNegatedInequalityDiffLogic</a>(<span class="keyword">const</span> std::vector<Theorem>& antecedentThms, <span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& implied) = 0;</div>
<div class="line"><a name="l00445"></a><span class="lineno"> 445</span>  </div>
<div class="line"><a name="l00446"></a><span class="lineno"> 446</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#ac48d6f8b2ad8e8f8fefac775d33825b5">expandGrayShadowRewrite</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& theShadow) = 0;</div>
<div class="line"><a name="l00447"></a><span class="lineno"> 447</span>  </div>
<div class="line"><a name="l00448"></a><span class="lineno"> 448</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a98020e790bcb60c9c2cc5a7e406f500f">compactNonLinearTerm</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& nonLinear) = 0;</div>
<div class="line"><a name="l00449"></a><span class="lineno"> 449</span>  </div>
<div class="line"><a name="l00450"></a><span class="lineno"> 450</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#aa6874fa9f5b02db0adb8149af9664b05">nonLinearIneqSignSplit</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a>& ineqThm) = 0;</div>
<div class="line"><a name="l00451"></a><span class="lineno"> 451</span>  </div>
<div class="line"><a name="l00452"></a><span class="lineno"> 452</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#af80ad28bb01d8a37e32f29c746d22dbc">implyDiffLogicBothBounds</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& x, std::vector<Theorem>& c1_le_x, <a class="code" href="classCVC3_1_1Rational.html">Rational</a> c1, </div>
<div class="line"><a name="l00453"></a><span class="lineno"> 453</span>  std::vector<Theorem>& x_le_c2, <a class="code" href="classCVC3_1_1Rational.html">Rational</a> c2) = 0;</div>
<div class="line"><a name="l00454"></a><span class="lineno"> 454</span>  </div>
<div class="line"><a name="l00455"></a><span class="lineno"> 455</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#ad3dee93f4078a4fc79ccacef33f05119">powerOfOne</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e) = 0;</div>
<div class="line"><a name="l00456"></a><span class="lineno"> 456</span>  </div>
<div class="line"><a name="l00457"></a><span class="lineno"> 457</span>  <span class="keyword">virtual</span> <a class="code" href="classCVC3_1_1Theorem.html">Theorem</a> <a class="code" href="classCVC3_1_1ArithProofRules.html#a81d9ea1e5c2d42c2b60a893c879145bd">rewriteLeavesConst</a>(<span class="keyword">const</span> <a class="code" href="classCVC3_1_1Expr.html" title="Data structure of expressions in CVC3.">Expr</a>& e);</div>
<div class="line"><a name="l00458"></a><span class="lineno"> 458</span> </div>
<div class="line"><a name="l00459"></a><span class="lineno"> 459</span>  }; <span class="comment">// end of class ArithProofRules</span></div>
<div class="line"><a name="l00460"></a><span class="lineno"> 460</span> } <span class="comment">// end of namespace CVC3</span></div>
<div class="line"><a name="l00461"></a><span class="lineno"> 461</span> </div>
<div class="line"><a name="l00462"></a><span class="lineno"> 462</span> <span class="preprocessor">#endif</span></div>
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