1 (* A functor for finite mappings based on Tables *) |
1 (* Title: Library/normarith.ML |
2 signature FUNC = |
2 Author: Amine Chaieb, University of Cambridge |
3 sig |
3 Description: A simple decision procedure for linear problems in euclidean space |
4 type 'a T |
4 *) |
5 type key |
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6 val apply : 'a T -> key -> 'a |
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7 val applyd :'a T -> (key -> 'a) -> key -> 'a |
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8 val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T |
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9 val defined : 'a T -> key -> bool |
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10 val dom : 'a T -> key list |
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11 val fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b |
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12 val graph : 'a T -> (key * 'a) list |
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13 val is_undefined : 'a T -> bool |
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14 val mapf : ('a -> 'b) -> 'a T -> 'b T |
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15 val tryapplyd : 'a T -> key -> 'a -> 'a |
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16 val undefine : key -> 'a T -> 'a T |
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17 val undefined : 'a T |
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18 val update : key * 'a -> 'a T -> 'a T |
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19 val updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T |
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20 val choose : 'a T -> key * 'a |
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21 val onefunc : key * 'a -> 'a T |
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22 val get_first: (key*'a -> 'a option) -> 'a T -> 'a option |
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23 val fns: |
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24 {key_ord: key*key -> order, |
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25 apply : 'a T -> key -> 'a, |
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26 applyd :'a T -> (key -> 'a) -> key -> 'a, |
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27 combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T, |
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28 defined : 'a T -> key -> bool, |
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29 dom : 'a T -> key list, |
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30 fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b, |
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31 graph : 'a T -> (key * 'a) list, |
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32 is_undefined : 'a T -> bool, |
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33 mapf : ('a -> 'b) -> 'a T -> 'b T, |
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34 tryapplyd : 'a T -> key -> 'a -> 'a, |
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35 undefine : key -> 'a T -> 'a T, |
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36 undefined : 'a T, |
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37 update : key * 'a -> 'a T -> 'a T, |
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38 updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T, |
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39 choose : 'a T -> key * 'a, |
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40 onefunc : key * 'a -> 'a T, |
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41 get_first: (key*'a -> 'a option) -> 'a T -> 'a option} |
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42 end; |
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43 |
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44 functor FuncFun(Key: KEY) : FUNC= |
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45 struct |
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46 |
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47 type key = Key.key; |
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48 structure Tab = TableFun(Key); |
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49 type 'a T = 'a Tab.table; |
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50 |
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51 val undefined = Tab.empty; |
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52 val is_undefined = Tab.is_empty; |
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53 val mapf = Tab.map; |
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54 val fold = Tab.fold; |
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55 val graph = Tab.dest; |
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56 val dom = Tab.keys; |
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57 fun applyd f d x = case Tab.lookup f x of |
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58 SOME y => y |
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59 | NONE => d x; |
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60 |
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61 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x; |
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62 fun tryapplyd f a d = applyd f (K d) a; |
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63 val defined = Tab.defined; |
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64 fun undefine x t = (Tab.delete x t handle UNDEF => t); |
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65 val update = Tab.update; |
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66 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t |
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67 fun combine f z a b = |
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68 let |
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69 fun h (k,v) t = case Tab.lookup t k of |
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70 NONE => Tab.update (k,v) t |
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71 | SOME v' => let val w = f v v' |
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72 in if z w then Tab.delete k t else Tab.update (k,w) t end; |
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73 in Tab.fold h a b end; |
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74 |
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75 fun choose f = case Tab.max_key f of |
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76 SOME k => (k,valOf (Tab.lookup f k)) |
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77 | NONE => error "FuncFun.choose : Completely undefined function" |
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78 |
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79 fun onefunc kv = update kv undefined |
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80 |
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81 local |
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82 fun find f (k,v) NONE = f (k,v) |
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83 | find f (k,v) r = r |
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84 in |
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85 fun get_first f t = fold (find f) t NONE |
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86 end |
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87 |
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88 val fns = |
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89 {key_ord = Key.ord, |
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90 apply = apply, |
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91 applyd = applyd, |
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92 combine = combine, |
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93 defined = defined, |
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94 dom = dom, |
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95 fold = fold, |
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96 graph = graph, |
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97 is_undefined = is_undefined, |
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98 mapf = mapf, |
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99 tryapplyd = tryapplyd, |
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100 undefine = undefine, |
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101 undefined = undefined, |
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102 update = update, |
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103 updatep = updatep, |
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104 choose = choose, |
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105 onefunc = onefunc, |
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106 get_first = get_first} |
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107 |
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108 end; |
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109 |
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110 structure Intfunc = FuncFun(type key = int val ord = int_ord); |
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111 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord); |
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112 structure Termfunc = FuncFun(type key = term val ord = TermOrd.fast_term_ord); |
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113 structure Ctermfunc = FuncFun(type key = cterm val ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t))); |
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114 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord); |
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115 |
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116 (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*) |
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117 structure Conv2 = |
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118 struct |
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119 open Conv |
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120 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms) |
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121 fun is_comb t = case (term_of t) of _$_ => true | _ => false; |
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122 fun is_abs t = case (term_of t) of Abs _ => true | _ => false; |
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123 |
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124 fun end_itlist f l = |
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125 case l of |
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126 [] => error "end_itlist" |
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127 | [x] => x |
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128 | (h::t) => f h (end_itlist f t); |
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129 |
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130 fun absc cv ct = case term_of ct of |
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131 Abs (v,_, _) => |
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132 let val (x,t) = Thm.dest_abs (SOME v) ct |
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133 in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t) |
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134 end |
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135 | _ => all_conv ct; |
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136 |
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137 fun cache_conv conv = |
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138 let |
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139 val tab = ref Termtab.empty |
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140 fun cconv t = |
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141 case Termtab.lookup (!tab) (term_of t) of |
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142 SOME th => th |
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143 | NONE => let val th = conv t |
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144 in ((tab := Termtab.insert Thm.eq_thm (term_of t, th) (!tab)); th) end |
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145 in cconv end; |
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146 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct')) |
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147 handle CTERM _ => false; |
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148 |
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149 local |
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150 fun thenqc conv1 conv2 tm = |
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151 case try conv1 tm of |
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152 SOME th1 => (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1) |
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153 | NONE => conv2 tm |
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154 |
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155 fun thencqc conv1 conv2 tm = |
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156 let val th1 = conv1 tm |
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157 in (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1) |
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158 end |
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159 fun comb_qconv conv tm = |
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160 let val (l,r) = Thm.dest_comb tm |
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161 in (case try conv l of |
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162 SOME th1 => (case try conv r of SOME th2 => Thm.combination th1 th2 |
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163 | NONE => Drule.fun_cong_rule th1 r) |
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164 | NONE => Drule.arg_cong_rule l (conv r)) |
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165 end |
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166 fun repeatqc conv tm = thencqc conv (repeatqc conv) tm |
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167 fun sub_qconv conv tm = if is_abs tm then absc conv tm else comb_qconv conv tm |
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168 fun once_depth_qconv conv tm = |
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169 (conv else_conv (sub_qconv (once_depth_qconv conv))) tm |
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170 fun depth_qconv conv tm = |
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171 thenqc (sub_qconv (depth_qconv conv)) |
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172 (repeatqc conv) tm |
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173 fun redepth_qconv conv tm = |
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174 thenqc (sub_qconv (redepth_qconv conv)) |
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175 (thencqc conv (redepth_qconv conv)) tm |
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176 fun top_depth_qconv conv tm = |
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177 thenqc (repeatqc conv) |
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178 (thencqc (sub_qconv (top_depth_qconv conv)) |
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179 (thencqc conv (top_depth_qconv conv))) tm |
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180 fun top_sweep_qconv conv tm = |
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181 thenqc (repeatqc conv) |
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182 (sub_qconv (top_sweep_qconv conv)) tm |
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183 in |
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184 val (once_depth_conv, depth_conv, rdepth_conv, top_depth_conv, top_sweep_conv) = |
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185 (fn c => try_conv (once_depth_qconv c), |
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186 fn c => try_conv (depth_qconv c), |
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187 fn c => try_conv (redepth_qconv c), |
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188 fn c => try_conv (top_depth_qconv c), |
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189 fn c => try_conv (top_sweep_qconv c)); |
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190 end; |
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191 end; |
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192 |
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193 |
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194 (* Some useful derived rules *) |
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195 fun deduct_antisym_rule tha thb = |
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196 equal_intr (implies_intr (cprop_of thb) tha) |
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197 (implies_intr (cprop_of tha) thb); |
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198 |
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199 fun prove_hyp tha thb = |
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200 if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb)) |
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201 then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb; |
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202 |
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203 |
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204 |
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205 signature REAL_ARITH = |
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206 sig |
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207 datatype positivstellensatz = |
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208 Axiom_eq of int |
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209 | Axiom_le of int |
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210 | Axiom_lt of int |
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211 | Rational_eq of Rat.rat |
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212 | Rational_le of Rat.rat |
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213 | Rational_lt of Rat.rat |
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214 | Square of cterm |
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215 | Eqmul of cterm * positivstellensatz |
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216 | Sum of positivstellensatz * positivstellensatz |
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217 | Product of positivstellensatz * positivstellensatz; |
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218 |
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219 val gen_gen_real_arith : |
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220 Proof.context -> (Rat.rat -> Thm.cterm) * conv * conv * conv * |
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221 conv * conv * conv * conv * conv * conv * |
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222 ( (thm list * thm list * thm list -> positivstellensatz -> thm) -> |
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223 thm list * thm list * thm list -> thm) -> conv |
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224 val real_linear_prover : |
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225 (thm list * thm list * thm list -> positivstellensatz -> thm) -> |
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226 thm list * thm list * thm list -> thm |
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227 |
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228 val gen_real_arith : Proof.context -> |
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229 (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * |
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230 ( (thm list * thm list * thm list -> positivstellensatz -> thm) -> |
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231 thm list * thm list * thm list -> thm) -> conv |
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232 val gen_prover_real_arith : Proof.context -> |
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233 ((thm list * thm list * thm list -> positivstellensatz -> thm) -> |
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234 thm list * thm list * thm list -> thm) -> conv |
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235 val real_arith : Proof.context -> conv |
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236 end |
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237 |
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238 structure RealArith (* : REAL_ARITH *)= |
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239 struct |
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240 |
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241 open Conv Thm Conv2;; |
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242 (* ------------------------------------------------------------------------- *) |
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243 (* Data structure for Positivstellensatz refutations. *) |
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244 (* ------------------------------------------------------------------------- *) |
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245 |
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246 datatype positivstellensatz = |
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247 Axiom_eq of int |
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248 | Axiom_le of int |
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249 | Axiom_lt of int |
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250 | Rational_eq of Rat.rat |
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251 | Rational_le of Rat.rat |
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252 | Rational_lt of Rat.rat |
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253 | Square of cterm |
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254 | Eqmul of cterm * positivstellensatz |
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255 | Sum of positivstellensatz * positivstellensatz |
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256 | Product of positivstellensatz * positivstellensatz; |
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257 (* Theorems used in the procedure *) |
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258 |
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259 fun conjunctions th = case try Conjunction.elim th of |
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260 SOME (th1,th2) => (conjunctions th1) @ conjunctions th2 |
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261 | NONE => [th]; |
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262 |
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263 val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0)) |
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264 &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0)) |
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265 &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))" |
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266 by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |> |
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267 conjunctions; |
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268 |
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269 val pth_final = @{lemma "(~p ==> False) ==> p" by blast} |
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270 val pth_add = |
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271 @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0) |
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272 &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0) |
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273 &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0) |
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274 &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0) |
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275 &&& (x > 0 ==> y > 0 ==> x + y > 0)" by simp_all} |> conjunctions ; |
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276 |
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277 val pth_mul = |
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278 @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&& |
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279 (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&& |
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280 (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&& |
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281 (x > 0 ==> y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&& |
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282 (x > 0 ==> y > 0 ==> x * y > 0)" |
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283 by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified] |
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284 mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions; |
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285 |
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286 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0" by simp}; |
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287 val pth_square = @{lemma "x * x >= (0::real)" by simp}; |
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288 |
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289 val weak_dnf_simps = List.take (simp_thms, 34) |
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290 @ conjunctions @{lemma "((P & (Q | R)) = ((P&Q) | (P&R))) &&& ((Q | R) & P) = ((Q&P) | (R&P)) &&& (P & Q) = (Q & P) &&& ((P | Q) = (Q | P))" by blast+}; |
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291 |
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292 val nnfD_simps = conjunctions @{lemma "((~(P & Q)) = (~P | ~Q)) &&& ((~(P | Q)) = (~P & ~Q) ) &&& ((P --> Q) = (~P | Q) ) &&& ((P = Q) = ((P & Q) | (~P & ~ Q))) &&& ((~(P = Q)) = ((P & ~ Q) | (~P & Q)) ) &&& ((~ ~(P)) = P)" by blast+} |
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293 |
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294 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis}; |
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295 val prenex_simps = map (fn th => th RS sym) ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @ @{thms "all_simps"(1-4)} @ @{thms "ex_simps"(1-4)}); |
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296 |
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297 val real_abs_thms1 = conjunctions @{lemma |
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298 "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&& |
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299 ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&& |
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300 ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&& |
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301 ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&& |
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302 ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&& |
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303 ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&& |
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304 ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&& |
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305 ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&& |
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306 ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&& |
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307 ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y + b >= r)) &&& |
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308 ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&& |
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309 ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y + c >= r)) &&& |
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310 ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&& |
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311 ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&& |
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312 ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&& |
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313 ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y + b >= r) )&&& |
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314 ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&& |
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315 ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y + c >= r)) &&& |
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316 ((min x y >= r) = (x >= r & y >= r)) &&& |
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317 ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&& |
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318 ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&& |
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319 ((a + min x y + b >= r) = (a + x + b >= r & a + y + b >= r)) &&& |
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320 ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&& |
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321 ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&& |
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322 ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&& |
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323 ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&& |
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324 ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&& |
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325 ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&& |
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326 ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&& |
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327 ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&& |
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328 ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&& |
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329 ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&& |
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330 ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&& |
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331 ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y + b > r)) &&& |
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332 ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&& |
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333 ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y + c > r)) &&& |
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334 ((min x y > r) = (x > r & y > r)) &&& |
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335 ((min x y + a > r) = (a + x > r & a + y > r)) &&& |
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336 ((a + min x y > r) = (a + x > r & a + y > r)) &&& |
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337 ((a + min x y + b > r) = (a + x + b > r & a + y + b > r)) &&& |
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338 ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&& |
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339 ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))" |
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340 by auto}; |
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341 |
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342 val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))" |
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343 by (atomize (full)) (auto split add: abs_split)}; |
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344 |
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345 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)" |
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346 by (atomize (full)) (cases "x <= y", auto simp add: max_def)}; |
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347 |
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348 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)" |
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349 by (atomize (full)) (cases "x <= y", auto simp add: min_def)}; |
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350 |
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351 |
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352 (* Miscalineous *) |
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353 fun literals_conv bops uops cv = |
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354 let fun h t = |
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355 case (term_of t) of |
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356 b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t |
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357 | u$_ => if member (op aconv) uops u then arg_conv h t else cv t |
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358 | _ => cv t |
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359 in h end; |
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360 |
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361 fun cterm_of_rat x = |
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362 let val (a, b) = Rat.quotient_of_rat x |
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363 in |
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364 if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a |
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365 else Thm.capply (Thm.capply @{cterm "op / :: real => _"} |
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366 (Numeral.mk_cnumber @{ctyp "real"} a)) |
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367 (Numeral.mk_cnumber @{ctyp "real"} b) |
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368 end; |
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369 |
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370 fun dest_ratconst t = case term_of t of |
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371 Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd) |
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372 | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd) |
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373 | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd) |
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374 fun is_ratconst t = can dest_ratconst t |
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375 |
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376 fun find_term p t = if p t then t else |
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377 case t of |
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378 a$b => (find_term p a handle TERM _ => find_term p b) |
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379 | Abs (_,_,t') => find_term p t' |
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380 | _ => raise TERM ("find_term",[t]); |
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381 |
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382 fun find_cterm p t = if p t then t else |
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383 case term_of t of |
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384 a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t)) |
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385 | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd) |
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386 | _ => raise CTERM ("find_cterm",[t]); |
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387 |
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388 |
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389 (* A general real arithmetic prover *) |
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390 |
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391 fun gen_gen_real_arith ctxt (mk_numeric, |
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392 numeric_eq_conv,numeric_ge_conv,numeric_gt_conv, |
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393 poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv, |
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394 absconv1,absconv2,prover) = |
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395 let |
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396 open Conv Thm; |
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397 val pre_ss = HOL_basic_ss addsimps simp_thms@ ex_simps@ all_simps@[@{thm not_all},@{thm not_ex},ex_disj_distrib, all_conj_distrib, @{thm if_bool_eq_disj}] |
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398 val prenex_ss = HOL_basic_ss addsimps prenex_simps |
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399 val skolemize_ss = HOL_basic_ss addsimps [choice_iff] |
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400 val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss) |
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401 val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss) |
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402 val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss) |
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403 val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps |
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404 val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss) |
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405 fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI} |
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406 fun oprconv cv ct = |
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407 let val g = Thm.dest_fun2 ct |
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408 in if g aconvc @{cterm "op <= :: real => _"} |
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409 orelse g aconvc @{cterm "op < :: real => _"} |
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410 then arg_conv cv ct else arg1_conv cv ct |
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411 end |
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412 |
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413 fun real_ineq_conv th ct = |
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414 let |
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415 val th' = (instantiate (match (lhs_of th, ct)) th |
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416 handle MATCH => raise CTERM ("real_ineq_conv", [ct])) |
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417 in transitive th' (oprconv poly_conv (Thm.rhs_of th')) |
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418 end |
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419 val [real_lt_conv, real_le_conv, real_eq_conv, |
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420 real_not_lt_conv, real_not_le_conv, _] = |
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421 map real_ineq_conv pth |
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422 fun match_mp_rule ths ths' = |
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423 let |
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424 fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths) |
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425 | th::ths => (ths' MRS th handle THM _ => f ths ths') |
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426 in f ths ths' end |
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427 fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv)) |
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428 (match_mp_rule pth_mul [th, th']) |
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429 fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv)) |
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430 (match_mp_rule pth_add [th, th']) |
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431 fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) |
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432 (instantiate' [] [SOME ct] (th RS pth_emul)) |
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433 fun square_rule t = fconv_rule (arg_conv (oprconv poly_mul_conv)) |
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434 (instantiate' [] [SOME t] pth_square) |
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435 |
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436 fun hol_of_positivstellensatz(eqs,les,lts) = |
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437 let |
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438 fun translate prf = case prf of |
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439 Axiom_eq n => nth eqs n |
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440 | Axiom_le n => nth les n |
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441 | Axiom_lt n => nth lts n |
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442 | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop} |
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443 (capply (capply @{cterm "op =::real => _"} (mk_numeric x)) |
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444 @{cterm "0::real"}))) |
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445 | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop} |
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446 (capply (capply @{cterm "op <=::real => _"} |
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447 @{cterm "0::real"}) (mk_numeric x)))) |
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448 | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop} |
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449 (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"}) |
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450 (mk_numeric x)))) |
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451 | Square t => square_rule t |
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452 | Eqmul(t,p) => emul_rule t (translate p) |
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453 | Sum(p1,p2) => add_rule (translate p1) (translate p2) |
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454 | Product(p1,p2) => mul_rule (translate p1) (translate p2) |
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455 in fn prf => |
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456 fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) |
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457 (translate prf) |
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458 end |
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459 |
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460 val init_conv = presimp_conv then_conv |
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461 nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv |
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462 weak_dnf_conv |
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463 |
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464 val concl = dest_arg o cprop_of |
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465 fun is_binop opr ct = (dest_fun2 ct aconvc opr handle CTERM _ => false) |
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466 val is_req = is_binop @{cterm "op =:: real => _"} |
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467 val is_ge = is_binop @{cterm "op <=:: real => _"} |
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468 val is_gt = is_binop @{cterm "op <:: real => _"} |
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469 val is_conj = is_binop @{cterm "op &"} |
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470 val is_disj = is_binop @{cterm "op |"} |
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471 fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2}) |
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472 fun disj_cases th th1 th2 = |
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473 let val (p,q) = dest_binop (concl th) |
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474 val c = concl th1 |
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475 val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible" |
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476 in implies_elim (implies_elim (implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th) (implies_intr (capply @{cterm Trueprop} p) th1)) (implies_intr (capply @{cterm Trueprop} q) th2) |
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477 end |
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478 fun overall dun ths = case ths of |
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479 [] => |
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480 let |
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481 val (eq,ne) = List.partition (is_req o concl) dun |
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482 val (le,nl) = List.partition (is_ge o concl) ne |
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483 val lt = filter (is_gt o concl) nl |
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484 in prover hol_of_positivstellensatz (eq,le,lt) end |
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485 | th::oths => |
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486 let |
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487 val ct = concl th |
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488 in |
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489 if is_conj ct then |
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490 let |
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491 val (th1,th2) = conj_pair th in |
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492 overall dun (th1::th2::oths) end |
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493 else if is_disj ct then |
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494 let |
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495 val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths) |
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496 val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths) |
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497 in disj_cases th th1 th2 end |
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498 else overall (th::dun) oths |
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499 end |
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500 fun dest_binary b ct = if is_binop b ct then dest_binop ct |
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501 else raise CTERM ("dest_binary",[b,ct]) |
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502 val dest_eq = dest_binary @{cterm "op = :: real => _"} |
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503 val neq_th = nth pth 5 |
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504 fun real_not_eq_conv ct = |
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505 let |
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506 val (l,r) = dest_eq (dest_arg ct) |
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507 val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th |
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508 val th_p = poly_conv(dest_arg(dest_arg1(Thm.rhs_of th))) |
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509 val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p |
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510 val th_n = fconv_rule (arg_conv poly_neg_conv) th_x |
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511 val th' = Drule.binop_cong_rule @{cterm "op |"} |
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512 (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p) |
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513 (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n) |
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514 in transitive th th' |
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515 end |
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516 fun equal_implies_1_rule PQ = |
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517 let |
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518 val P = lhs_of PQ |
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519 in implies_intr P (equal_elim PQ (assume P)) |
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520 end |
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521 (* FIXME!!! Copied from groebner.ml *) |
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522 val strip_exists = |
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523 let fun h (acc, t) = |
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524 case (term_of t) of |
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525 Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc)) |
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526 | _ => (acc,t) |
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527 in fn t => h ([],t) |
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528 end |
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529 fun name_of x = case term_of x of |
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530 Free(s,_) => s |
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531 | Var ((s,_),_) => s |
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532 | _ => "x" |
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533 |
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534 fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (abstract_rule (name_of x) x th) |
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535 |
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536 val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec)); |
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537 |
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538 fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex} |
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539 fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t) |
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540 |
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541 fun choose v th th' = case concl_of th of |
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542 @{term Trueprop} $ (Const("Ex",_)$_) => |
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543 let |
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544 val p = (funpow 2 Thm.dest_arg o cprop_of) th |
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545 val T = (hd o Thm.dest_ctyp o ctyp_of_term) p |
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546 val th0 = fconv_rule (Thm.beta_conversion true) |
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547 (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE) |
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548 val pv = (Thm.rhs_of o Thm.beta_conversion true) |
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549 (Thm.capply @{cterm Trueprop} (Thm.capply p v)) |
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550 val th1 = forall_intr v (implies_intr pv th') |
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551 in implies_elim (implies_elim th0 th) th1 end |
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552 | _ => raise THM ("choose",0,[th, th']) |
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553 |
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554 fun simple_choose v th = |
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555 choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th |
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556 |
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557 val strip_forall = |
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558 let fun h (acc, t) = |
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559 case (term_of t) of |
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560 Const("All",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc)) |
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561 | _ => (acc,t) |
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562 in fn t => h ([],t) |
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563 end |
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564 |
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565 fun f ct = |
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566 let |
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567 val nnf_norm_conv' = |
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568 nnf_conv then_conv |
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569 literals_conv [@{term "op &"}, @{term "op |"}] [] |
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570 (cache_conv |
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571 (first_conv [real_lt_conv, real_le_conv, |
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572 real_eq_conv, real_not_lt_conv, |
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573 real_not_le_conv, real_not_eq_conv, all_conv])) |
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574 fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] [] |
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575 (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv |
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576 try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct |
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577 val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct) |
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578 val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct |
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579 val tm0 = dest_arg (Thm.rhs_of th0) |
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580 val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 else |
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581 let |
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582 val (evs,bod) = strip_exists tm0 |
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583 val (avs,ibod) = strip_forall bod |
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584 val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod)) |
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585 val th2 = overall [] [specl avs (assume (Thm.rhs_of th1))] |
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586 val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2) |
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587 in Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3) |
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588 end |
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589 in implies_elim (instantiate' [] [SOME ct] pth_final) th |
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590 end |
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591 in f |
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592 end; |
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593 |
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594 (* A linear arithmetic prover *) |
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595 local |
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596 val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero) |
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597 fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x) |
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598 val one_tm = @{cterm "1::real"} |
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599 fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse |
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600 ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(Ctermfunc.apply e one_tm))) |
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601 |
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602 fun linear_ineqs vars (les,lts) = |
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603 case find_first (contradictory (fn x => x >/ Rat.zero)) lts of |
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604 SOME r => r |
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605 | NONE => |
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606 (case find_first (contradictory (fn x => x >/ Rat.zero)) les of |
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607 SOME r => r |
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608 | NONE => |
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609 if null vars then error "linear_ineqs: no contradiction" else |
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610 let |
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611 val ineqs = les @ lts |
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612 fun blowup v = |
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613 length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) + |
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614 length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) * |
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615 length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs) |
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616 val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j)) |
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617 (map (fn v => (v,blowup v)) vars))) |
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618 fun addup (e1,p1) (e2,p2) acc = |
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619 let |
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620 val c1 = Ctermfunc.tryapplyd e1 v Rat.zero |
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621 val c2 = Ctermfunc.tryapplyd e2 v Rat.zero |
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622 in if c1 */ c2 >=/ Rat.zero then acc else |
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623 let |
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624 val e1' = linear_cmul (Rat.abs c2) e1 |
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625 val e2' = linear_cmul (Rat.abs c1) e2 |
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626 val p1' = Product(Rational_lt(Rat.abs c2),p1) |
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627 val p2' = Product(Rational_lt(Rat.abs c1),p2) |
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628 in (linear_add e1' e2',Sum(p1',p2'))::acc |
|
629 end |
|
630 end |
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631 val (les0,les1) = |
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632 List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les |
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633 val (lts0,lts1) = |
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634 List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts |
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635 val (lesp,lesn) = |
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636 List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1 |
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637 val (ltsp,ltsn) = |
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638 List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1 |
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639 val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0 |
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640 val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn |
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641 (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0) |
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642 in linear_ineqs (remove (op aconvc) v vars) (les',lts') |
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643 end) |
|
644 |
|
645 fun linear_eqs(eqs,les,lts) = |
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646 case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of |
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647 SOME r => r |
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648 | NONE => (case eqs of |
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649 [] => |
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650 let val vars = remove (op aconvc) one_tm |
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651 (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) []) |
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652 in linear_ineqs vars (les,lts) end |
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653 | (e,p)::es => |
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654 if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else |
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655 let |
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656 val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e) |
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657 fun xform (inp as (t,q)) = |
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658 let val d = Ctermfunc.tryapplyd t x Rat.zero in |
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659 if d =/ Rat.zero then inp else |
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660 let |
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661 val k = (Rat.neg d) */ Rat.abs c // c |
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662 val e' = linear_cmul k e |
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663 val t' = linear_cmul (Rat.abs c) t |
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664 val p' = Eqmul(cterm_of_rat k,p) |
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665 val q' = Product(Rational_lt(Rat.abs c),q) |
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666 in (linear_add e' t',Sum(p',q')) |
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667 end |
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668 end |
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669 in linear_eqs(map xform es,map xform les,map xform lts) |
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670 end) |
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671 |
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672 fun linear_prover (eq,le,lt) = |
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673 let |
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674 val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1)) |
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675 val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1)) |
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676 val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1)) |
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677 in linear_eqs(eqs,les,lts) |
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678 end |
|
679 |
|
680 fun lin_of_hol ct = |
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681 if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined |
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682 else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one) |
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683 else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct) |
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684 else |
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685 let val (lop,r) = Thm.dest_comb ct |
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686 in if not (is_comb lop) then Ctermfunc.onefunc (ct, Rat.one) |
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687 else |
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688 let val (opr,l) = Thm.dest_comb lop |
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689 in if opr aconvc @{cterm "op + :: real =>_"} |
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690 then linear_add (lin_of_hol l) (lin_of_hol r) |
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691 else if opr aconvc @{cterm "op * :: real =>_"} |
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692 andalso is_ratconst l then Ctermfunc.onefunc (r, dest_ratconst l) |
|
693 else Ctermfunc.onefunc (ct, Rat.one) |
|
694 end |
|
695 end |
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696 |
|
697 fun is_alien ct = case term_of ct of |
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698 Const(@{const_name "real"}, _)$ n => |
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699 if can HOLogic.dest_number n then false else true |
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700 | _ => false |
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701 open Thm |
|
702 in |
|
703 fun real_linear_prover translator (eq,le,lt) = |
|
704 let |
|
705 val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of |
|
706 val rhs = lin_of_hol o dest_arg o dest_arg o cprop_of |
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707 val eq_pols = map lhs eq |
|
708 val le_pols = map rhs le |
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709 val lt_pols = map rhs lt |
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710 val aliens = filter is_alien |
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711 (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom) |
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712 (eq_pols @ le_pols @ lt_pols) []) |
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713 val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens |
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714 val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols) |
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715 val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens |
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716 in (translator (eq,le',lt) proof) : thm |
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717 end |
|
718 end; |
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719 |
|
720 (* A less general generic arithmetic prover dealing with abs,max and min*) |
|
721 |
|
722 local |
|
723 val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1 |
|
724 fun absmaxmin_elim_conv1 ctxt = |
|
725 Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1) |
|
726 |
|
727 val absmaxmin_elim_conv2 = |
|
728 let |
|
729 val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split' |
|
730 val pth_max = instantiate' [SOME @{ctyp real}] [] max_split |
|
731 val pth_min = instantiate' [SOME @{ctyp real}] [] min_split |
|
732 val abs_tm = @{cterm "abs :: real => _"} |
|
733 val p_tm = @{cpat "?P :: real => bool"} |
|
734 val x_tm = @{cpat "?x :: real"} |
|
735 val y_tm = @{cpat "?y::real"} |
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736 val is_max = is_binop @{cterm "max :: real => _"} |
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737 val is_min = is_binop @{cterm "min :: real => _"} |
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738 fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm |
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739 fun eliminate_construct p c tm = |
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740 let |
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741 val t = find_cterm p tm |
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742 val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t) |
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743 val (p,ax) = (dest_comb o Thm.rhs_of) th0 |
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744 in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false)))) |
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745 (transitive th0 (c p ax)) |
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746 end |
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747 |
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748 val elim_abs = eliminate_construct is_abs |
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749 (fn p => fn ax => |
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750 instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs) |
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751 val elim_max = eliminate_construct is_max |
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752 (fn p => fn ax => |
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753 let val (ax,y) = dest_comb ax |
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754 in instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) |
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755 pth_max end) |
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756 val elim_min = eliminate_construct is_min |
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757 (fn p => fn ax => |
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758 let val (ax,y) = dest_comb ax |
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759 in instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) |
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760 pth_min end) |
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761 in first_conv [elim_abs, elim_max, elim_min, all_conv] |
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762 end; |
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763 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) = |
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764 gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul, |
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765 absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover) |
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766 end; |
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767 |
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768 (* An instance for reals*) |
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769 |
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770 fun gen_prover_real_arith ctxt prover = |
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771 let |
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772 fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS |
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773 val {add,mul,neg,pow,sub,main} = |
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774 Normalizer.semiring_normalizers_ord_wrapper ctxt |
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775 (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) |
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776 simple_cterm_ord |
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777 in gen_real_arith ctxt |
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778 (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv, |
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779 main,neg,add,mul, prover) |
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780 end; |
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781 |
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782 fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover; |
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783 end |
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784 |
5 |
785 (* Now the norm procedure for euclidean spaces *) |
6 (* Now the norm procedure for euclidean spaces *) |
786 |
7 |
787 |
8 |
788 signature NORM_ARITH = |
9 signature NORM_ARITH = |