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1 (* A functor for finite mappings based on Tables *) |
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2 signature FUNC = |
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3 sig |
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4 type 'a T |
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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 | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd) |
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373 fun is_ratconst t = can dest_ratconst t |
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374 |
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375 fun find_term p t = if p t then t else |
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376 case t of |
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377 a$b => (find_term p a handle TERM _ => find_term p b) |
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378 | Abs (_,_,t') => find_term p t' |
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379 | _ => raise TERM ("find_term",[t]); |
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380 |
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381 fun find_cterm p t = if p t then t else |
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382 case term_of t of |
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383 a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t)) |
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384 | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd) |
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385 | _ => raise CTERM ("find_cterm",[t]); |
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386 |
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387 |
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388 (* A general real arithmetic prover *) |
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389 |
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390 fun gen_gen_real_arith ctxt (mk_numeric, |
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391 numeric_eq_conv,numeric_ge_conv,numeric_gt_conv, |
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392 poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv, |
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393 absconv1,absconv2,prover) = |
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394 let |
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395 open Conv Thm; |
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396 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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397 val prenex_ss = HOL_basic_ss addsimps prenex_simps |
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398 val skolemize_ss = HOL_basic_ss addsimps [choice_iff] |
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399 val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss) |
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400 val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss) |
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401 val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss) |
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402 val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps |
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403 val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss) |
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404 fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI} |
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405 fun oprconv cv ct = |
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406 let val g = Thm.dest_fun2 ct |
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407 in if g aconvc @{cterm "op <= :: real => _"} |
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408 orelse g aconvc @{cterm "op < :: real => _"} |
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409 then arg_conv cv ct else arg1_conv cv ct |
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410 end |
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411 |
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412 fun real_ineq_conv th ct = |
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413 let |
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414 val th' = (instantiate (match (lhs_of th, ct)) th |
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415 handle MATCH => raise CTERM ("real_ineq_conv", [ct])) |
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416 in transitive th' (oprconv poly_conv (Thm.rhs_of th')) |
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417 end |
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418 val [real_lt_conv, real_le_conv, real_eq_conv, |
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419 real_not_lt_conv, real_not_le_conv, _] = |
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420 map real_ineq_conv pth |
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421 fun match_mp_rule ths ths' = |
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422 let |
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423 fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths) |
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424 | th::ths => (ths' MRS th handle THM _ => f ths ths') |
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425 in f ths ths' end |
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426 fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv)) |
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427 (match_mp_rule pth_mul [th, th']) |
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428 fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv)) |
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429 (match_mp_rule pth_add [th, th']) |
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430 fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) |
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431 (instantiate' [] [SOME ct] (th RS pth_emul)) |
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432 fun square_rule t = fconv_rule (arg_conv (oprconv poly_mul_conv)) |
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433 (instantiate' [] [SOME t] pth_square) |
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434 |
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435 fun hol_of_positivstellensatz(eqs,les,lts) = |
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436 let |
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437 fun translate prf = case prf of |
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438 Axiom_eq n => nth eqs n |
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439 | Axiom_le n => nth les n |
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440 | Axiom_lt n => nth lts n |
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441 | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop} |
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442 (capply (capply @{cterm "op =::real => _"} (mk_numeric x)) |
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443 @{cterm "0::real"}))) |
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444 | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop} |
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445 (capply (capply @{cterm "op <=::real => _"} |
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446 @{cterm "0::real"}) (mk_numeric x)))) |
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447 | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop} |
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448 (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"}) |
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449 (mk_numeric x)))) |
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450 | Square t => square_rule t |
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451 | Eqmul(t,p) => emul_rule t (translate p) |
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452 | Sum(p1,p2) => add_rule (translate p1) (translate p2) |
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453 | Product(p1,p2) => mul_rule (translate p1) (translate p2) |
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454 in fn prf => |
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455 fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) |
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456 (translate prf) |
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457 end |
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458 |
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459 val init_conv = presimp_conv then_conv |
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460 nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv |
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461 weak_dnf_conv |
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462 |
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463 val concl = dest_arg o cprop_of |
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464 fun is_binop opr ct = (dest_fun2 ct aconvc opr handle CTERM _ => false) |
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465 val is_req = is_binop @{cterm "op =:: real => _"} |
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466 val is_ge = is_binop @{cterm "op <=:: real => _"} |
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467 val is_gt = is_binop @{cterm "op <:: real => _"} |
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468 val is_conj = is_binop @{cterm "op &"} |
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469 val is_disj = is_binop @{cterm "op |"} |
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470 fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2}) |
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471 fun disj_cases th th1 th2 = |
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472 let val (p,q) = dest_binop (concl th) |
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473 val c = concl th1 |
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474 val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible" |
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475 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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476 end |
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477 fun overall dun ths = case ths of |
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478 [] => |
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479 let |
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480 val (eq,ne) = List.partition (is_req o concl) dun |
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481 val (le,nl) = List.partition (is_ge o concl) ne |
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482 val lt = filter (is_gt o concl) nl |
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483 in prover hol_of_positivstellensatz (eq,le,lt) end |
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484 | th::oths => |
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485 let |
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486 val ct = concl th |
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487 in |
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488 if is_conj ct then |
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489 let |
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490 val (th1,th2) = conj_pair th in |
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491 overall dun (th1::th2::oths) end |
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492 else if is_disj ct then |
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493 let |
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494 val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths) |
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495 val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths) |
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496 in disj_cases th th1 th2 end |
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497 else overall (th::dun) oths |
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498 end |
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499 fun dest_binary b ct = if is_binop b ct then dest_binop ct |
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500 else raise CTERM ("dest_binary",[b,ct]) |
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501 val dest_eq = dest_binary @{cterm "op = :: real => _"} |
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502 val neq_th = nth pth 5 |
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503 fun real_not_eq_conv ct = |
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504 let |
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505 val (l,r) = dest_eq (dest_arg ct) |
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506 val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th |
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507 val th_p = poly_conv(dest_arg(dest_arg1(Thm.rhs_of th))) |
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508 val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p |
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509 val th_n = fconv_rule (arg_conv poly_neg_conv) th_x |
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510 val th' = Drule.binop_cong_rule @{cterm "op |"} |
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511 (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p) |
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512 (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n) |
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513 in transitive th th' |
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514 end |
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515 fun equal_implies_1_rule PQ = |
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516 let |
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517 val P = lhs_of PQ |
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518 in implies_intr P (equal_elim PQ (assume P)) |
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519 end |
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520 (* FIXME!!! Copied from groebner.ml *) |
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521 val strip_exists = |
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522 let fun h (acc, t) = |
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523 case (term_of t) of |
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524 Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc)) |
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525 | _ => (acc,t) |
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526 in fn t => h ([],t) |
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527 end |
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528 fun name_of x = case term_of x of |
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529 Free(s,_) => s |
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530 | Var ((s,_),_) => s |
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531 | _ => "x" |
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532 |
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533 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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534 |
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535 val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec)); |
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536 |
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537 fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex} |
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538 fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t) |
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539 |
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540 fun choose v th th' = case concl_of th of |
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541 @{term Trueprop} $ (Const("Ex",_)$_) => |
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542 let |
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543 val p = (funpow 2 Thm.dest_arg o cprop_of) th |
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544 val T = (hd o Thm.dest_ctyp o ctyp_of_term) p |
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545 val th0 = fconv_rule (Thm.beta_conversion true) |
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546 (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE) |
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547 val pv = (Thm.rhs_of o Thm.beta_conversion true) |
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548 (Thm.capply @{cterm Trueprop} (Thm.capply p v)) |
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549 val th1 = forall_intr v (implies_intr pv th') |
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550 in implies_elim (implies_elim th0 th) th1 end |
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551 | _ => raise THM ("choose",0,[th, th']) |
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552 |
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553 fun simple_choose v th = |
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554 choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th |
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555 |
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556 val strip_forall = |
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557 let fun h (acc, t) = |
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558 case (term_of t) of |
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559 Const("All",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc)) |
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560 | _ => (acc,t) |
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561 in fn t => h ([],t) |
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562 end |
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563 |
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564 fun f ct = |
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565 let |
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566 val nnf_norm_conv' = |
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567 nnf_conv then_conv |
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568 literals_conv [@{term "op &"}, @{term "op |"}] [] |
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569 (cache_conv |
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570 (first_conv [real_lt_conv, real_le_conv, |
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571 real_eq_conv, real_not_lt_conv, |
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572 real_not_le_conv, real_not_eq_conv, all_conv])) |
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573 fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] [] |
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574 (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv |
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575 try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct |
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576 val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct) |
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577 val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct |
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578 val tm0 = dest_arg (Thm.rhs_of th0) |
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579 val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 else |
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580 let |
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581 val (evs,bod) = strip_exists tm0 |
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582 val (avs,ibod) = strip_forall bod |
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583 val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod)) |
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584 val th2 = overall [] [specl avs (assume (Thm.rhs_of th1))] |
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585 val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2) |
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586 in Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3) |
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587 end |
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588 in implies_elim (instantiate' [] [SOME ct] pth_final) th |
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589 end |
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590 in f |
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591 end; |
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592 |
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593 (* A linear arithmetic prover *) |
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594 local |
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595 val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero) |
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596 fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x) |
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597 val one_tm = @{cterm "1::real"} |
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598 fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse |
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599 ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(Ctermfunc.apply e one_tm))) |
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600 |
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601 fun linear_ineqs vars (les,lts) = |
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602 case find_first (contradictory (fn x => x >/ Rat.zero)) lts of |
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603 SOME r => r |
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604 | NONE => |
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605 (case find_first (contradictory (fn x => x >/ Rat.zero)) les of |
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606 SOME r => r |
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607 | NONE => |
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608 if null vars then error "linear_ineqs: no contradiction" else |
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609 let |
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610 val ineqs = les @ lts |
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611 fun blowup v = |
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612 length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) + |
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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 val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j)) |
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616 (map (fn v => (v,blowup v)) vars))) |
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617 fun addup (e1,p1) (e2,p2) acc = |
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618 let |
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619 val c1 = Ctermfunc.tryapplyd e1 v Rat.zero |
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620 val c2 = Ctermfunc.tryapplyd e2 v Rat.zero |
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621 in if c1 */ c2 >=/ Rat.zero then acc else |
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622 let |
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623 val e1' = linear_cmul (Rat.abs c2) e1 |
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624 val e2' = linear_cmul (Rat.abs c1) e2 |
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625 val p1' = Product(Rational_lt(Rat.abs c2),p1) |
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626 val p2' = Product(Rational_lt(Rat.abs c1),p2) |
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627 in (linear_add e1' e2',Sum(p1',p2'))::acc |
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628 end |
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629 end |
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630 val (les0,les1) = |
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631 List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les |
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632 val (lts0,lts1) = |
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633 List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts |
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634 val (lesp,lesn) = |
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635 List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1 |
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636 val (ltsp,ltsn) = |
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637 List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1 |
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638 val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0 |
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639 val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn |
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640 (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0) |
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641 in linear_ineqs (remove (op aconvc) v vars) (les',lts') |
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642 end) |
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643 |
|
644 fun linear_eqs(eqs,les,lts) = |
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645 case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of |
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646 SOME r => r |
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647 | NONE => (case eqs of |
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648 [] => |
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649 let val vars = remove (op aconvc) one_tm |
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650 (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) []) |
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651 in linear_ineqs vars (les,lts) end |
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652 | (e,p)::es => |
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653 if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else |
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654 let |
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655 val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e) |
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656 fun xform (inp as (t,q)) = |
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657 let val d = Ctermfunc.tryapplyd t x Rat.zero in |
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658 if d =/ Rat.zero then inp else |
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659 let |
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660 val k = (Rat.neg d) */ Rat.abs c // c |
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661 val e' = linear_cmul k e |
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662 val t' = linear_cmul (Rat.abs c) t |
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663 val p' = Eqmul(cterm_of_rat k,p) |
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664 val q' = Product(Rational_lt(Rat.abs c),q) |
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665 in (linear_add e' t',Sum(p',q')) |
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666 end |
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667 end |
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668 in linear_eqs(map xform es,map xform les,map xform lts) |
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669 end) |
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670 |
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671 fun linear_prover (eq,le,lt) = |
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672 let |
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673 val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1)) |
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674 val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1)) |
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675 val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1)) |
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676 in linear_eqs(eqs,les,lts) |
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677 end |
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678 |
|
679 fun lin_of_hol ct = |
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680 if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined |
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681 else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one) |
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682 else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct) |
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683 else |
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684 let val (lop,r) = Thm.dest_comb ct |
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685 in if not (is_comb lop) then Ctermfunc.onefunc (ct, Rat.one) |
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686 else |
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687 let val (opr,l) = Thm.dest_comb lop |
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688 in if opr aconvc @{cterm "op + :: real =>_"} |
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689 then linear_add (lin_of_hol l) (lin_of_hol r) |
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690 else if opr aconvc @{cterm "op * :: real =>_"} |
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691 andalso is_ratconst l then Ctermfunc.onefunc (r, dest_ratconst l) |
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692 else Ctermfunc.onefunc (ct, Rat.one) |
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693 end |
|
694 end |
|
695 |
|
696 fun is_alien ct = case term_of ct of |
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697 Const(@{const_name "real"}, _)$ n => |
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698 if can HOLogic.dest_number n then false else true |
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699 | _ => false |
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700 open Thm |
|
701 in |
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702 fun real_linear_prover translator (eq,le,lt) = |
|
703 let |
|
704 val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of |
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705 val rhs = lin_of_hol o dest_arg o dest_arg o cprop_of |
|
706 val eq_pols = map lhs eq |
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707 val le_pols = map rhs le |
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708 val lt_pols = map rhs lt |
|
709 val aliens = filter is_alien |
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710 (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom) |
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711 (eq_pols @ le_pols @ lt_pols) []) |
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712 val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens |
|
713 val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols) |
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714 val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens |
|
715 in (translator (eq,le',lt) proof) : thm |
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716 end |
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717 end; |
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718 |
|
719 (* A less general generic arithmetic prover dealing with abs,max and min*) |
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720 |
|
721 local |
|
722 val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1 |
|
723 fun absmaxmin_elim_conv1 ctxt = |
|
724 Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1) |
|
725 |
|
726 val absmaxmin_elim_conv2 = |
|
727 let |
|
728 val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split' |
|
729 val pth_max = instantiate' [SOME @{ctyp real}] [] max_split |
|
730 val pth_min = instantiate' [SOME @{ctyp real}] [] min_split |
|
731 val abs_tm = @{cterm "abs :: real => _"} |
|
732 val p_tm = @{cpat "?P :: real => bool"} |
|
733 val x_tm = @{cpat "?x :: real"} |
|
734 val y_tm = @{cpat "?y::real"} |
|
735 val is_max = is_binop @{cterm "max :: real => _"} |
|
736 val is_min = is_binop @{cterm "min :: real => _"} |
|
737 fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm |
|
738 fun eliminate_construct p c tm = |
|
739 let |
|
740 val t = find_cterm p tm |
|
741 val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t) |
|
742 val (p,ax) = (dest_comb o Thm.rhs_of) th0 |
|
743 in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false)))) |
|
744 (transitive th0 (c p ax)) |
|
745 end |
|
746 |
|
747 val elim_abs = eliminate_construct is_abs |
|
748 (fn p => fn ax => |
|
749 instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs) |
|
750 val elim_max = eliminate_construct is_max |
|
751 (fn p => fn ax => |
|
752 let val (ax,y) = dest_comb ax |
|
753 in instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) |
|
754 pth_max end) |
|
755 val elim_min = eliminate_construct is_min |
|
756 (fn p => fn ax => |
|
757 let val (ax,y) = dest_comb ax |
|
758 in instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) |
|
759 pth_min end) |
|
760 in first_conv [elim_abs, elim_max, elim_min, all_conv] |
|
761 end; |
|
762 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) = |
|
763 gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul, |
|
764 absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover) |
|
765 end; |
|
766 |
|
767 (* An instance for reals*) |
|
768 |
|
769 fun gen_prover_real_arith ctxt prover = |
|
770 let |
|
771 fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS |
|
772 val {add,mul,neg,pow,sub,main} = |
|
773 Normalizer.semiring_normalizers_ord_wrapper ctxt |
|
774 (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) |
|
775 simple_cterm_ord |
|
776 in gen_real_arith ctxt |
|
777 (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv, |
|
778 main,neg,add,mul, prover) |
|
779 end; |
|
780 |
|
781 fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover; |
|
782 end |
|
783 |
|
784 (* Now the norm procedure for euclidean spaces *) |
|
785 |
|
786 |
|
787 signature NORM_ARITH = |
|
788 sig |
|
789 val norm_arith : Proof.context -> conv |
|
790 val norm_arith_tac : Proof.context -> int -> tactic |
|
791 end |
|
792 |
|
793 structure NormArith : NORM_ARITH = |
|
794 struct |
|
795 |
|
796 open Conv Thm Conv2; |
|
797 val bool_eq = op = : bool *bool -> bool |
|
798 fun dest_ratconst t = case term_of t of |
|
799 Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd) |
|
800 | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd) |
|
801 fun is_ratconst t = can dest_ratconst t |
|
802 fun augment_norm b t acc = case term_of t of |
|
803 Const(@{const_name norm}, _) $ _ => insert (eq_pair bool_eq (op aconvc)) (b,dest_arg t) acc |
|
804 | _ => acc |
|
805 fun find_normedterms t acc = case term_of t of |
|
806 @{term "op + :: real => _"}$_$_ => |
|
807 find_normedterms (dest_arg1 t) (find_normedterms (dest_arg t) acc) |
|
808 | @{term "op * :: real => _"}$_$n => |
|
809 if not (is_ratconst (dest_arg1 t)) then acc else |
|
810 augment_norm (dest_ratconst (dest_arg1 t) >=/ Rat.zero) |
|
811 (dest_arg t) acc |
|
812 | _ => augment_norm true t acc |
|
813 |
|
814 val cterm_lincomb_neg = Ctermfunc.mapf Rat.neg |
|
815 fun cterm_lincomb_cmul c t = |
|
816 if c =/ Rat.zero then Ctermfunc.undefined else Ctermfunc.mapf (fn x => x */ c) t |
|
817 fun cterm_lincomb_add l r = Ctermfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r |
|
818 fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r) |
|
819 fun cterm_lincomb_eq l r = Ctermfunc.is_undefined (cterm_lincomb_sub l r) |
|
820 |
|
821 val int_lincomb_neg = Intfunc.mapf Rat.neg |
|
822 fun int_lincomb_cmul c t = |
|
823 if c =/ Rat.zero then Intfunc.undefined else Intfunc.mapf (fn x => x */ c) t |
|
824 fun int_lincomb_add l r = Intfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r |
|
825 fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r) |
|
826 fun int_lincomb_eq l r = Intfunc.is_undefined (int_lincomb_sub l r) |
|
827 |
|
828 fun vector_lincomb t = case term_of t of |
|
829 Const(@{const_name plus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) $ _ $ _ => |
|
830 cterm_lincomb_add (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t)) |
|
831 | Const(@{const_name minus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) $ _ $ _ => |
|
832 cterm_lincomb_sub (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t)) |
|
833 | Const(@{const_name vector_scalar_mult},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$_$_ => |
|
834 cterm_lincomb_cmul (dest_ratconst (dest_arg1 t)) (vector_lincomb (dest_arg t)) |
|
835 | Const(@{const_name uminus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$_ => |
|
836 cterm_lincomb_neg (vector_lincomb (dest_arg t)) |
|
837 | Const(@{const_name vec},_)$_ => |
|
838 let |
|
839 val b = ((snd o HOLogic.dest_number o term_of o dest_arg) t = 0 |
|
840 handle TERM _=> false) |
|
841 in if b then Ctermfunc.onefunc (t,Rat.one) |
|
842 else Ctermfunc.undefined |
|
843 end |
|
844 | _ => Ctermfunc.onefunc (t,Rat.one) |
|
845 |
|
846 fun vector_lincombs ts = |
|
847 fold_rev |
|
848 (fn t => fn fns => case AList.lookup (op aconvc) fns t of |
|
849 NONE => |
|
850 let val f = vector_lincomb t |
|
851 in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of |
|
852 SOME (_,f') => (t,f') :: fns |
|
853 | NONE => (t,f) :: fns |
|
854 end |
|
855 | SOME _ => fns) ts [] |
|
856 |
|
857 fun replacenegnorms cv t = case term_of t of |
|
858 @{term "op + :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t |
|
859 | @{term "op * :: real => _"}$_$_ => |
|
860 if dest_ratconst (dest_arg1 t) </ Rat.zero then arg_conv cv t else reflexive t |
|
861 | _ => reflexive t |
|
862 fun flip v eq = |
|
863 if Ctermfunc.defined eq v |
|
864 then Ctermfunc.update (v, Rat.neg (Ctermfunc.apply eq v)) eq else eq |
|
865 fun allsubsets s = case s of |
|
866 [] => [[]] |
|
867 |(a::t) => let val res = allsubsets t in |
|
868 map (cons a) res @ res end |
|
869 fun evaluate env lin = |
|
870 Intfunc.fold (fn (x,c) => fn s => s +/ c */ (Intfunc.apply env x)) |
|
871 lin Rat.zero |
|
872 |
|
873 fun solve (vs,eqs) = case (vs,eqs) of |
|
874 ([],[]) => SOME (Intfunc.onefunc (0,Rat.one)) |
|
875 |(_,eq::oeqs) => |
|
876 (case vs inter (Intfunc.dom eq) of |
|
877 [] => NONE |
|
878 | v::_ => |
|
879 if Intfunc.defined eq v |
|
880 then |
|
881 let |
|
882 val c = Intfunc.apply eq v |
|
883 val vdef = int_lincomb_cmul (Rat.neg (Rat.inv c)) eq |
|
884 fun eliminate eqn = if not (Intfunc.defined eqn v) then eqn |
|
885 else int_lincomb_add (int_lincomb_cmul (Intfunc.apply eqn v) vdef) eqn |
|
886 in (case solve (vs \ v,map eliminate oeqs) of |
|
887 NONE => NONE |
|
888 | SOME soln => SOME (Intfunc.update (v, evaluate soln (Intfunc.undefine v vdef)) soln)) |
|
889 end |
|
890 else NONE) |
|
891 |
|
892 fun combinations k l = if k = 0 then [[]] else |
|
893 case l of |
|
894 [] => [] |
|
895 | h::t => map (cons h) (combinations (k - 1) t) @ combinations k t |
|
896 |
|
897 |
|
898 fun forall2 p l1 l2 = case (l1,l2) of |
|
899 ([],[]) => true |
|
900 | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2 |
|
901 | _ => false; |
|
902 |
|
903 |
|
904 fun vertices vs eqs = |
|
905 let |
|
906 fun vertex cmb = case solve(vs,cmb) of |
|
907 NONE => NONE |
|
908 | SOME soln => SOME (map (fn v => Intfunc.tryapplyd soln v Rat.zero) vs) |
|
909 val rawvs = map_filter vertex (combinations (length vs) eqs) |
|
910 val unset = filter (forall (fn c => c >=/ Rat.zero)) rawvs |
|
911 in fold_rev (insert (uncurry (forall2 (curry op =/)))) unset [] |
|
912 end |
|
913 |
|
914 fun subsumes l m = forall2 (fn x => fn y => Rat.abs x <=/ Rat.abs y) l m |
|
915 |
|
916 fun subsume todo dun = case todo of |
|
917 [] => dun |
|
918 |v::ovs => |
|
919 let val dun' = if exists (fn w => subsumes w v) dun then dun |
|
920 else v::(filter (fn w => not(subsumes v w)) dun) |
|
921 in subsume ovs dun' |
|
922 end; |
|
923 |
|
924 fun match_mp PQ P = P RS PQ; |
|
925 |
|
926 fun cterm_of_rat x = |
|
927 let val (a, b) = Rat.quotient_of_rat x |
|
928 in |
|
929 if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a |
|
930 else Thm.capply (Thm.capply @{cterm "op / :: real => _"} |
|
931 (Numeral.mk_cnumber @{ctyp "real"} a)) |
|
932 (Numeral.mk_cnumber @{ctyp "real"} b) |
|
933 end; |
|
934 |
|
935 fun norm_cmul_rule c th = instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm}); |
|
936 |
|
937 fun norm_add_rule th1 th2 = [th1, th2] MRS @{thm norm_add_rule_thm}; |
|
938 |
|
939 (* I think here the static context should be sufficient!! *) |
|
940 fun inequality_canon_rule ctxt = |
|
941 let |
|
942 (* FIXME : Should be computed statically!! *) |
|
943 val real_poly_conv = |
|
944 Normalizer.semiring_normalize_wrapper ctxt |
|
945 (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) |
|
946 in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (field_comp_conv then_conv real_poly_conv))) |
|
947 end; |
|
948 |
|
949 fun absc cv ct = case term_of ct of |
|
950 Abs (v,_, _) => |
|
951 let val (x,t) = Thm.dest_abs (SOME v) ct |
|
952 in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t) |
|
953 end |
|
954 | _ => all_conv ct; |
|
955 |
|
956 fun sub_conv cv ct = (comb_conv cv else_conv absc cv) ct; |
|
957 fun botc1 conv ct = |
|
958 ((sub_conv (botc1 conv)) then_conv (conv else_conv all_conv)) ct; |
|
959 |
|
960 fun rewrs_conv eqs ct = first_conv (map rewr_conv eqs) ct; |
|
961 val apply_pth1 = rewr_conv @{thm pth_1}; |
|
962 val apply_pth2 = rewr_conv @{thm pth_2}; |
|
963 val apply_pth3 = rewr_conv @{thm pth_3}; |
|
964 val apply_pth4 = rewrs_conv @{thms pth_4}; |
|
965 val apply_pth5 = rewr_conv @{thm pth_5}; |
|
966 val apply_pth6 = rewr_conv @{thm pth_6}; |
|
967 val apply_pth7 = rewrs_conv @{thms pth_7}; |
|
968 val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm vector_smult_lzero}))); |
|
969 val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv field_comp_conv); |
|
970 val apply_ptha = rewr_conv @{thm pth_a}; |
|
971 val apply_pthb = rewrs_conv @{thms pth_b}; |
|
972 val apply_pthc = rewrs_conv @{thms pth_c}; |
|
973 val apply_pthd = try_conv (rewr_conv @{thm pth_d}); |
|
974 |
|
975 fun headvector t = case t of |
|
976 Const(@{const_name plus}, Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$ |
|
977 (Const(@{const_name vector_scalar_mult}, _)$l$v)$r => v |
|
978 | Const(@{const_name vector_scalar_mult}, _)$l$v => v |
|
979 | _ => error "headvector: non-canonical term" |
|
980 |
|
981 fun vector_cmul_conv ct = |
|
982 ((apply_pth5 then_conv arg1_conv field_comp_conv) else_conv |
|
983 (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct |
|
984 |
|
985 fun vector_add_conv ct = apply_pth7 ct |
|
986 handle CTERM _ => |
|
987 (apply_pth8 ct |
|
988 handle CTERM _ => |
|
989 (case term_of ct of |
|
990 Const(@{const_name plus},_)$lt$rt => |
|
991 let |
|
992 val l = headvector lt |
|
993 val r = headvector rt |
|
994 in (case TermOrd.fast_term_ord (l,r) of |
|
995 LESS => (apply_pthb then_conv arg_conv vector_add_conv |
|
996 then_conv apply_pthd) ct |
|
997 | GREATER => (apply_pthc then_conv arg_conv vector_add_conv |
|
998 then_conv apply_pthd) ct |
|
999 | EQUAL => (apply_pth9 then_conv |
|
1000 ((apply_ptha then_conv vector_add_conv) else_conv |
|
1001 arg_conv vector_add_conv then_conv apply_pthd)) ct) |
|
1002 end |
|
1003 | _ => reflexive ct)) |
|
1004 |
|
1005 fun vector_canon_conv ct = case term_of ct of |
|
1006 Const(@{const_name plus},_)$_$_ => |
|
1007 let |
|
1008 val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb |
|
1009 val lth = vector_canon_conv l |
|
1010 val rth = vector_canon_conv r |
|
1011 val th = Drule.binop_cong_rule p lth rth |
|
1012 in fconv_rule (arg_conv vector_add_conv) th end |
|
1013 |
|
1014 | Const(@{const_name vector_scalar_mult}, _)$_$_ => |
|
1015 let |
|
1016 val (p,r) = Thm.dest_comb ct |
|
1017 val rth = Drule.arg_cong_rule p (vector_canon_conv r) |
|
1018 in fconv_rule (arg_conv (apply_pth4 else_conv vector_cmul_conv)) rth |
|
1019 end |
|
1020 |
|
1021 | Const(@{const_name minus},_)$_$_ => (apply_pth2 then_conv vector_canon_conv) ct |
|
1022 |
|
1023 | Const(@{const_name uminus},_)$_ => (apply_pth3 then_conv vector_canon_conv) ct |
|
1024 |
|
1025 | Const(@{const_name vec},_)$n => |
|
1026 let val n = Thm.dest_arg ct |
|
1027 in if is_ratconst n andalso not (dest_ratconst n =/ Rat.zero) |
|
1028 then reflexive ct else apply_pth1 ct |
|
1029 end |
|
1030 |
|
1031 | _ => apply_pth1 ct |
|
1032 |
|
1033 fun norm_canon_conv ct = case term_of ct of |
|
1034 Const(@{const_name norm},_)$_ => arg_conv vector_canon_conv ct |
|
1035 | _ => raise CTERM ("norm_canon_conv", [ct]) |
|
1036 |
|
1037 fun fold_rev2 f [] [] z = z |
|
1038 | fold_rev2 f (x::xs) (y::ys) z = f x y (fold_rev2 f xs ys z) |
|
1039 | fold_rev2 f _ _ _ = raise UnequalLengths; |
|
1040 |
|
1041 fun int_flip v eq = |
|
1042 if Intfunc.defined eq v |
|
1043 then Intfunc.update (v, Rat.neg (Intfunc.apply eq v)) eq else eq; |
|
1044 |
|
1045 local |
|
1046 val pth_zero = @{thm "Vectors.norm_0"} |
|
1047 val tv_n = (hd o tl o dest_ctyp o ctyp_of_term o dest_arg o dest_arg1 o dest_arg o cprop_of) |
|
1048 pth_zero |
|
1049 val concl = dest_arg o cprop_of |
|
1050 fun real_vector_combo_prover ctxt translator (nubs,ges,gts) = |
|
1051 let |
|
1052 (* FIXME: Should be computed statically!!*) |
|
1053 val real_poly_conv = |
|
1054 Normalizer.semiring_normalize_wrapper ctxt |
|
1055 (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) |
|
1056 val sources = map (dest_arg o dest_arg1 o concl) nubs |
|
1057 val rawdests = fold_rev (find_normedterms o dest_arg o concl) (ges @ gts) [] |
|
1058 val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check" |
|
1059 else () |
|
1060 val dests = distinct (op aconvc) (map snd rawdests) |
|
1061 val srcfuns = map vector_lincomb sources |
|
1062 val destfuns = map vector_lincomb dests |
|
1063 val vvs = fold_rev (curry (gen_union op aconvc) o Ctermfunc.dom) (srcfuns @ destfuns) [] |
|
1064 val n = length srcfuns |
|
1065 val nvs = 1 upto n |
|
1066 val srccombs = srcfuns ~~ nvs |
|
1067 fun consider d = |
|
1068 let |
|
1069 fun coefficients x = |
|
1070 let |
|
1071 val inp = if Ctermfunc.defined d x then Intfunc.onefunc (0, Rat.neg(Ctermfunc.apply d x)) |
|
1072 else Intfunc.undefined |
|
1073 in fold_rev (fn (f,v) => fn g => if Ctermfunc.defined f x then Intfunc.update (v, Ctermfunc.apply f x) g else g) srccombs inp |
|
1074 end |
|
1075 val equations = map coefficients vvs |
|
1076 val inequalities = map (fn n => Intfunc.onefunc (n,Rat.one)) nvs |
|
1077 fun plausiblevertices f = |
|
1078 let |
|
1079 val flippedequations = map (fold_rev int_flip f) equations |
|
1080 val constraints = flippedequations @ inequalities |
|
1081 val rawverts = vertices nvs constraints |
|
1082 fun check_solution v = |
|
1083 let |
|
1084 val f = fold_rev2 (curry Intfunc.update) nvs v (Intfunc.onefunc (0, Rat.one)) |
|
1085 in forall (fn e => evaluate f e =/ Rat.zero) flippedequations |
|
1086 end |
|
1087 val goodverts = filter check_solution rawverts |
|
1088 val signfixups = map (fn n => if n mem_int f then ~1 else 1) nvs |
|
1089 in map (map2 (fn s => fn c => Rat.rat_of_int s */ c) signfixups) goodverts |
|
1090 end |
|
1091 val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) [] |
|
1092 in subsume allverts [] |
|
1093 end |
|
1094 fun compute_ineq v = |
|
1095 let |
|
1096 val ths = map_filter (fn (v,t) => if v =/ Rat.zero then NONE |
|
1097 else SOME(norm_cmul_rule v t)) |
|
1098 (v ~~ nubs) |
|
1099 in inequality_canon_rule ctxt (end_itlist norm_add_rule ths) |
|
1100 end |
|
1101 val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @ |
|
1102 map (inequality_canon_rule ctxt) nubs @ ges |
|
1103 val zerodests = filter |
|
1104 (fn t => null (Ctermfunc.dom (vector_lincomb t))) (map snd rawdests) |
|
1105 |
|
1106 in RealArith.real_linear_prover translator |
|
1107 (map (fn t => instantiate ([(tv_n,(hd o tl o dest_ctyp o ctyp_of_term) t)],[]) pth_zero) |
|
1108 zerodests, |
|
1109 map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv |
|
1110 arg_conv (arg_conv real_poly_conv))) ges', |
|
1111 map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv |
|
1112 arg_conv (arg_conv real_poly_conv))) gts) |
|
1113 end |
|
1114 in val real_vector_combo_prover = real_vector_combo_prover |
|
1115 end; |
|
1116 |
|
1117 local |
|
1118 val pth = @{thm norm_imp_pos_and_ge} |
|
1119 val norm_mp = match_mp pth |
|
1120 val concl = dest_arg o cprop_of |
|
1121 fun conjunct1 th = th RS @{thm conjunct1} |
|
1122 fun conjunct2 th = th RS @{thm conjunct2} |
|
1123 fun C f x y = f y x |
|
1124 fun real_vector_ineq_prover ctxt translator (ges,gts) = |
|
1125 let |
|
1126 (* val _ = error "real_vector_ineq_prover: pause" *) |
|
1127 val ntms = fold_rev find_normedterms (map (dest_arg o concl) (ges @ gts)) [] |
|
1128 val lctab = vector_lincombs (map snd (filter (not o fst) ntms)) |
|
1129 val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt |
|
1130 fun mk_norm t = capply (instantiate_cterm' [SOME (ctyp_of_term t)] [] @{cpat "norm :: (?'a :: norm) => real"}) t |
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1131 fun mk_equals l r = capply (capply (instantiate_cterm' [SOME (ctyp_of_term l)] [] @{cpat "op == :: ?'a =>_"}) l) r |
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1132 val asl = map2 (fn (t,_) => fn n => assume (mk_equals (mk_norm t) (cterm_of (ProofContext.theory_of ctxt') (Free(n,@{typ real}))))) lctab fxns |
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1133 val replace_conv = try_conv (rewrs_conv asl) |
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1134 val replace_rule = fconv_rule (funpow 2 arg_conv (replacenegnorms replace_conv)) |
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1135 val ges' = |
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1136 fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths) |
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1137 asl (map replace_rule ges) |
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1138 val gts' = map replace_rule gts |
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1139 val nubs = map (conjunct2 o norm_mp) asl |
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1140 val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts') |
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1141 val shs = filter (member (fn (t,th) => t aconvc cprop_of th) asl) (#hyps (crep_thm th1)) |
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1142 val th11 = hd (Variable.export ctxt' ctxt [fold implies_intr shs th1]) |
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1143 val cps = map (swap o dest_equals) (cprems_of th11) |
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1144 val th12 = instantiate ([], cps) th11 |
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1145 val th13 = fold (C implies_elim) (map (reflexive o snd) cps) th12; |
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1146 in hd (Variable.export ctxt' ctxt [th13]) |
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1147 end |
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1148 in val real_vector_ineq_prover = real_vector_ineq_prover |
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1149 end; |
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1150 |
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1151 local |
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1152 val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0})) |
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1153 fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2}) |
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1154 fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS; |
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1155 (* FIXME: Lookup in the context every time!!! Fix this !!!*) |
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1156 fun splitequation ctxt th acc = |
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1157 let |
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1158 val real_poly_neg_conv = #neg |
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1159 (Normalizer.semiring_normalizers_ord_wrapper ctxt |
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1160 (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord) |
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1161 val (th1,th2) = conj_pair(rawrule th) |
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1162 in th1::fconv_rule (arg_conv (arg_conv real_poly_neg_conv)) th2::acc |
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1163 end |
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1164 in fun real_vector_prover ctxt translator (eqs,ges,gts) = |
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1165 real_vector_ineq_prover ctxt translator |
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1166 (fold_rev (splitequation ctxt) eqs ges,gts) |
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1167 end; |
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1168 |
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1169 fun init_conv ctxt = |
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1170 Simplifier.rewrite (Simplifier.context ctxt |
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1171 (HOL_basic_ss addsimps ([@{thm vec_0}, @{thm vec_1}, @{thm dist_def}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_0}] @ @{thms arithmetic_simps} @ @{thms norm_pths}))) |
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1172 then_conv field_comp_conv |
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1173 then_conv nnf_conv |
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1174 |
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1175 fun pure ctxt = RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt); |
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1176 fun norm_arith ctxt ct = |
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1177 let |
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1178 val ctxt' = Variable.declare_term (term_of ct) ctxt |
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1179 val th = init_conv ctxt' ct |
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1180 in equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (symmetric th)) |
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1181 (pure ctxt' (rhs_of th)) |
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1182 end |
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1183 |
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1184 fun norm_arith_tac ctxt = |
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1185 clarify_tac HOL_cs THEN' |
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1186 ObjectLogic.full_atomize_tac THEN' |
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1187 CSUBGOAL ( fn (p,i) => rtac (norm_arith ctxt (Thm.dest_arg p )) i); |
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1188 |
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1189 end; |