1 (* Title: HOL/Library/positivstellensatz.ML |
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2 Author: Amine Chaieb, University of Cambridge |
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3 |
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4 A generic arithmetic prover based on Positivstellensatz certificates |
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5 --- also implements Fourier-Motzkin elimination as a special case |
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6 Fourier-Motzkin elimination. |
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7 *) |
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8 |
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9 (* A functor for finite mappings based on Tables *) |
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10 |
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11 signature FUNC = |
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12 sig |
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13 include TABLE |
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14 val apply : 'a table -> key -> 'a |
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15 val applyd :'a table -> (key -> 'a) -> key -> 'a |
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16 val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table |
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17 val dom : 'a table -> key list |
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18 val tryapplyd : 'a table -> key -> 'a -> 'a |
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19 val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table |
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20 val choose : 'a table -> key * 'a |
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21 val onefunc : key * 'a -> 'a table |
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22 end; |
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23 |
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24 functor FuncFun(Key: KEY) : FUNC = |
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25 struct |
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26 |
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27 structure Tab = Table(Key); |
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28 |
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29 open Tab; |
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30 |
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31 fun dom a = sort Key.ord (Tab.keys a); |
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32 fun applyd f d x = case Tab.lookup f x of |
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33 SOME y => y |
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34 | NONE => d x; |
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35 |
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36 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x; |
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37 fun tryapplyd f a d = applyd f (K d) a; |
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38 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t |
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39 fun combine f z a b = |
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40 let |
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41 fun h (k,v) t = case Tab.lookup t k of |
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42 NONE => Tab.update (k,v) t |
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43 | SOME v' => let val w = f v v' |
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44 in if z w then Tab.delete k t else Tab.update (k,w) t end; |
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45 in Tab.fold h a b end; |
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46 |
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47 fun choose f = |
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48 (case Tab.min f of |
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49 SOME entry => entry |
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50 | NONE => error "FuncFun.choose : Completely empty function") |
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51 |
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52 fun onefunc kv = update kv empty |
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53 |
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54 end; |
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55 |
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56 (* Some standard functors and utility functions for them *) |
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57 |
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58 structure FuncUtil = |
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59 struct |
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60 |
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61 structure Intfunc = FuncFun(type key = int val ord = int_ord); |
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62 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord); |
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63 structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord); |
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64 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord); |
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65 structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord); |
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66 structure Ctermfunc = FuncFun(type key = cterm val ord = Thm.fast_term_ord); |
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67 |
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68 type monomial = int Ctermfunc.table; |
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69 val monomial_ord = list_ord (prod_ord Thm.fast_term_ord int_ord) o apply2 Ctermfunc.dest |
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70 structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord) |
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71 |
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72 type poly = Rat.rat Monomialfunc.table; |
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73 |
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74 (* The ordering so we can create canonical HOL polynomials. *) |
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75 |
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76 fun dest_monomial mon = sort (Thm.fast_term_ord o apply2 fst) (Ctermfunc.dest mon); |
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77 |
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78 fun monomial_order (m1,m2) = |
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79 if Ctermfunc.is_empty m2 then LESS |
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80 else if Ctermfunc.is_empty m1 then GREATER |
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81 else |
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82 let |
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83 val mon1 = dest_monomial m1 |
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84 val mon2 = dest_monomial m2 |
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85 val deg1 = fold (Integer.add o snd) mon1 0 |
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86 val deg2 = fold (Integer.add o snd) mon2 0 |
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87 in if deg1 < deg2 then GREATER |
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88 else if deg1 > deg2 then LESS |
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89 else list_ord (prod_ord Thm.fast_term_ord int_ord) (mon1,mon2) |
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90 end; |
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91 |
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92 end |
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93 |
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94 (* positivstellensatz datatype and prover generation *) |
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95 |
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96 signature REAL_ARITH = |
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97 sig |
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98 |
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99 datatype positivstellensatz = |
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100 Axiom_eq of int |
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101 | Axiom_le of int |
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102 | Axiom_lt of int |
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103 | Rational_eq of Rat.rat |
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104 | Rational_le of Rat.rat |
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105 | Rational_lt of Rat.rat |
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106 | Square of FuncUtil.poly |
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107 | Eqmul of FuncUtil.poly * positivstellensatz |
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108 | Sum of positivstellensatz * positivstellensatz |
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109 | Product of positivstellensatz * positivstellensatz; |
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110 |
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111 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree |
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112 |
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113 datatype tree_choice = Left | Right |
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114 |
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115 type prover = tree_choice list -> |
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116 (thm list * thm list * thm list -> positivstellensatz -> thm) -> |
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117 thm list * thm list * thm list -> thm * pss_tree |
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118 type cert_conv = cterm -> thm * pss_tree |
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119 |
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120 val gen_gen_real_arith : |
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121 Proof.context -> (Rat.rat -> cterm) * conv * conv * conv * |
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122 conv * conv * conv * conv * conv * conv * prover -> cert_conv |
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123 val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) -> |
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124 thm list * thm list * thm list -> thm * pss_tree |
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125 |
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126 val gen_real_arith : Proof.context -> |
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127 (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv |
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128 |
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129 val gen_prover_real_arith : Proof.context -> prover -> cert_conv |
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130 |
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131 val is_ratconst : cterm -> bool |
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132 val dest_ratconst : cterm -> Rat.rat |
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133 val cterm_of_rat : Rat.rat -> cterm |
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134 |
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135 end |
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136 |
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137 structure RealArith : REAL_ARITH = |
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138 struct |
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139 |
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140 open Conv |
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141 (* ------------------------------------------------------------------------- *) |
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142 (* Data structure for Positivstellensatz refutations. *) |
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143 (* ------------------------------------------------------------------------- *) |
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144 |
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145 datatype positivstellensatz = |
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146 Axiom_eq of int |
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147 | Axiom_le of int |
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148 | Axiom_lt of int |
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149 | Rational_eq of Rat.rat |
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150 | Rational_le of Rat.rat |
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151 | Rational_lt of Rat.rat |
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152 | Square of FuncUtil.poly |
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153 | Eqmul of FuncUtil.poly * positivstellensatz |
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154 | Sum of positivstellensatz * positivstellensatz |
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155 | Product of positivstellensatz * positivstellensatz; |
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156 (* Theorems used in the procedure *) |
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157 |
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158 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree |
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159 datatype tree_choice = Left | Right |
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160 type prover = tree_choice list -> |
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161 (thm list * thm list * thm list -> positivstellensatz -> thm) -> |
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162 thm list * thm list * thm list -> thm * pss_tree |
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163 type cert_conv = cterm -> thm * pss_tree |
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164 |
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165 |
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166 (* Some useful derived rules *) |
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167 fun deduct_antisym_rule tha thb = |
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168 Thm.equal_intr (Thm.implies_intr (Thm.cprop_of thb) tha) |
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169 (Thm.implies_intr (Thm.cprop_of tha) thb); |
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170 |
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171 fun prove_hyp tha thb = |
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172 if exists (curry op aconv (Thm.concl_of tha)) (Thm.hyps_of thb) (* FIXME !? *) |
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173 then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb; |
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174 |
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175 val pth = @{lemma "(((x::real) < y) \<equiv> (y - x > 0))" and "((x \<le> y) \<equiv> (y - x \<ge> 0))" and |
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176 "((x = y) \<equiv> (x - y = 0))" and "((\<not>(x < y)) \<equiv> (x - y \<ge> 0))" and |
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177 "((\<not>(x \<le> y)) \<equiv> (x - y > 0))" and "((\<not>(x = y)) \<equiv> (x - y > 0 \<or> -(x - y) > 0))" |
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178 by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)}; |
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179 |
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180 val pth_final = @{lemma "(\<not>p \<Longrightarrow> False) \<Longrightarrow> p" by blast} |
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181 val pth_add = |
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182 @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x + y = 0 )" and "( x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and |
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183 "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y \<ge> 0)" and |
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184 "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and |
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185 "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y > 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y > 0)" and |
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186 "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" by simp_all}; |
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187 |
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188 val pth_mul = |
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189 @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y = 0)" and |
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190 "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y = 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and |
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191 "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y \<ge> 0)" and |
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192 "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and |
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193 "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y > 0)" |
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194 by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified] |
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195 mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])}; |
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196 |
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197 val pth_emul = @{lemma "y = (0::real) \<Longrightarrow> x * y = 0" by simp}; |
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198 val pth_square = @{lemma "x * x \<ge> (0::real)" by simp}; |
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199 |
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200 val weak_dnf_simps = |
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201 List.take (@{thms simp_thms}, 34) @ |
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202 @{lemma "((P \<and> (Q \<or> R)) = ((P\<and>Q) \<or> (P\<and>R)))" and "((Q \<or> R) \<and> P) = ((Q\<and>P) \<or> (R\<and>P))" and |
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203 "(P \<and> Q) = (Q \<and> P)" and "((P \<or> Q) = (Q \<or> P))" by blast+}; |
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204 |
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205 (* |
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206 val nnfD_simps = |
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207 @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and |
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208 "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and |
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209 "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+}; |
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210 *) |
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211 |
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212 val choice_iff = @{lemma "(\<forall>x. \<exists>y. P x y) = (\<exists>f. \<forall>x. P x (f x))" by metis}; |
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213 val prenex_simps = |
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214 map (fn th => th RS sym) |
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215 ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @ |
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216 @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)}); |
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217 |
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218 val real_abs_thms1 = @{lemma |
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219 "((-1 * \<bar>x::real\<bar> \<ge> r) = (-1 * x \<ge> r \<and> 1 * x \<ge> r))" and |
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220 "((-1 * \<bar>x\<bar> + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and |
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221 "((a + -1 * \<bar>x\<bar> \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and |
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222 "((a + -1 * \<bar>x\<bar> + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + 1 * x + b \<ge> r))" and |
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223 "((a + b + -1 * \<bar>x\<bar> \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + 1 * x \<ge> r))" and |
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224 "((a + b + -1 * \<bar>x\<bar> + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + 1 * x + c \<ge> r))" and |
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225 "((-1 * max x y \<ge> r) = (-1 * x \<ge> r \<and> -1 * y \<ge> r))" and |
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226 "((-1 * max x y + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and |
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227 "((a + -1 * max x y \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and |
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228 "((a + -1 * max x y + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + -1 * y + b \<ge> r))" and |
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229 "((a + b + -1 * max x y \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + -1 * y \<ge> r))" and |
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230 "((a + b + -1 * max x y + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + -1 * y + c \<ge> r))" and |
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231 "((1 * min x y \<ge> r) = (1 * x \<ge> r \<and> 1 * y \<ge> r))" and |
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232 "((1 * min x y + a \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and |
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233 "((a + 1 * min x y \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and |
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234 "((a + 1 * min x y + b \<ge> r) = (a + 1 * x + b \<ge> r \<and> a + 1 * y + b \<ge> r))" and |
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235 "((a + b + 1 * min x y \<ge> r) = (a + b + 1 * x \<ge> r \<and> a + b + 1 * y \<ge> r))" and |
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236 "((a + b + 1 * min x y + c \<ge> r) = (a + b + 1 * x + c \<ge> r \<and> a + b + 1 * y + c \<ge> r))" and |
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237 "((min x y \<ge> r) = (x \<ge> r \<and> y \<ge> r))" and |
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238 "((min x y + a \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and |
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239 "((a + min x y \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and |
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240 "((a + min x y + b \<ge> r) = (a + x + b \<ge> r \<and> a + y + b \<ge> r))" and |
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241 "((a + b + min x y \<ge> r) = (a + b + x \<ge> r \<and> a + b + y \<ge> r))" and |
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242 "((a + b + min x y + c \<ge> r) = (a + b + x + c \<ge> r \<and> a + b + y + c \<ge> r))" and |
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243 "((-1 * \<bar>x\<bar> > r) = (-1 * x > r \<and> 1 * x > r))" and |
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244 "((-1 * \<bar>x\<bar> + a > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and |
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245 "((a + -1 * \<bar>x\<bar> > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and |
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246 "((a + -1 * \<bar>x\<bar> + b > r) = (a + -1 * x + b > r \<and> a + 1 * x + b > r))" and |
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247 "((a + b + -1 * \<bar>x\<bar> > r) = (a + b + -1 * x > r \<and> a + b + 1 * x > r))" and |
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248 "((a + b + -1 * \<bar>x\<bar> + c > r) = (a + b + -1 * x + c > r \<and> a + b + 1 * x + c > r))" and |
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249 "((-1 * max x y > r) = ((-1 * x > r) \<and> -1 * y > r))" and |
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250 "((-1 * max x y + a > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and |
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251 "((a + -1 * max x y > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and |
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252 "((a + -1 * max x y + b > r) = (a + -1 * x + b > r \<and> a + -1 * y + b > r))" and |
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253 "((a + b + -1 * max x y > r) = (a + b + -1 * x > r \<and> a + b + -1 * y > r))" and |
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254 "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r \<and> a + b + -1 * y + c > r))" and |
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255 "((min x y > r) = (x > r \<and> y > r))" and |
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256 "((min x y + a > r) = (a + x > r \<and> a + y > r))" and |
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257 "((a + min x y > r) = (a + x > r \<and> a + y > r))" and |
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258 "((a + min x y + b > r) = (a + x + b > r \<and> a + y + b > r))" and |
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259 "((a + b + min x y > r) = (a + b + x > r \<and> a + b + y > r))" and |
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260 "((a + b + min x y + c > r) = (a + b + x + c > r \<and> a + b + y + c > r))" |
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261 by auto}; |
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262 |
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263 val abs_split' = @{lemma "P \<bar>x::'a::linordered_idom\<bar> == (x \<ge> 0 \<and> P x \<or> x < 0 \<and> P (-x))" |
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264 by (atomize (full)) (auto split: abs_split)}; |
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265 |
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266 val max_split = @{lemma "P (max x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P y \<or> x > y \<and> P x)" |
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267 by (atomize (full)) (cases "x \<le> y", auto simp add: max_def)}; |
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268 |
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269 val min_split = @{lemma "P (min x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P x \<or> x > y \<and> P y)" |
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270 by (atomize (full)) (cases "x \<le> y", auto simp add: min_def)}; |
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271 |
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272 |
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273 (* Miscellaneous *) |
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274 fun literals_conv bops uops cv = |
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275 let |
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276 fun h t = |
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277 (case Thm.term_of t of |
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278 b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t |
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279 | u$_ => if member (op aconv) uops u then arg_conv h t else cv t |
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280 | _ => cv t) |
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281 in h end; |
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282 |
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283 fun cterm_of_rat x = |
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284 let |
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285 val (a, b) = Rat.dest x |
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286 in |
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287 if b = 1 then Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a |
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288 else Thm.apply (Thm.apply \<^cterm>\<open>(/) :: real \<Rightarrow> _\<close> |
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289 (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a)) |
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290 (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> b) |
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291 end; |
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292 |
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293 fun dest_ratconst t = |
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294 case Thm.term_of t of |
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295 Const(\<^const_name>\<open>divide\<close>, _)$a$b => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd) |
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296 | _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd) |
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297 fun is_ratconst t = can dest_ratconst t |
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298 |
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299 (* |
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300 fun find_term p t = if p t then t else |
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301 case t of |
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302 a$b => (find_term p a handle TERM _ => find_term p b) |
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303 | Abs (_,_,t') => find_term p t' |
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304 | _ => raise TERM ("find_term",[t]); |
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305 *) |
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306 |
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307 fun find_cterm p t = |
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308 if p t then t else |
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309 case Thm.term_of t of |
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310 _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t)) |
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311 | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd) |
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312 | _ => raise CTERM ("find_cterm",[t]); |
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313 |
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314 fun is_comb t = (case Thm.term_of t of _ $ _ => true | _ => false); |
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315 |
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316 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct')) |
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317 handle CTERM _ => false; |
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318 |
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319 |
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320 (* Map back polynomials to HOL. *) |
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321 |
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322 fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply \<^cterm>\<open>(^) :: real \<Rightarrow> _\<close> x) |
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323 (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> k) |
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324 |
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325 fun cterm_of_monomial m = |
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326 if FuncUtil.Ctermfunc.is_empty m then \<^cterm>\<open>1::real\<close> |
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327 else |
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328 let |
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329 val m' = FuncUtil.dest_monomial m |
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330 val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' [] |
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331 in foldr1 (fn (s, t) => Thm.apply (Thm.apply \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close> s) t) vps |
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332 end |
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333 |
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334 fun cterm_of_cmonomial (m,c) = |
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335 if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c |
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336 else if c = @1 then cterm_of_monomial m |
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337 else Thm.apply (Thm.apply \<^cterm>\<open>(*)::real \<Rightarrow> _\<close> (cterm_of_rat c)) (cterm_of_monomial m); |
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338 |
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339 fun cterm_of_poly p = |
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340 if FuncUtil.Monomialfunc.is_empty p then \<^cterm>\<open>0::real\<close> |
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341 else |
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342 let |
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343 val cms = map cterm_of_cmonomial |
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344 (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p)) |
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345 in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close> t1) t2) cms |
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346 end; |
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347 |
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348 (* A general real arithmetic prover *) |
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349 |
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350 fun gen_gen_real_arith ctxt (mk_numeric, |
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351 numeric_eq_conv,numeric_ge_conv,numeric_gt_conv, |
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352 poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv, |
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353 absconv1,absconv2,prover) = |
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354 let |
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355 val pre_ss = put_simpset HOL_basic_ss ctxt addsimps |
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356 @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib |
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357 all_conj_distrib if_bool_eq_disj} |
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358 val prenex_ss = put_simpset HOL_basic_ss ctxt addsimps prenex_simps |
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359 val skolemize_ss = put_simpset HOL_basic_ss ctxt addsimps [choice_iff] |
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360 val presimp_conv = Simplifier.rewrite pre_ss |
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361 val prenex_conv = Simplifier.rewrite prenex_ss |
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362 val skolemize_conv = Simplifier.rewrite skolemize_ss |
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363 val weak_dnf_ss = put_simpset HOL_basic_ss ctxt addsimps weak_dnf_simps |
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364 val weak_dnf_conv = Simplifier.rewrite weak_dnf_ss |
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365 fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI} |
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366 fun oprconv cv ct = |
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367 let val g = Thm.dest_fun2 ct |
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368 in if g aconvc \<^cterm>\<open>(\<le>) :: real \<Rightarrow> _\<close> |
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369 orelse g aconvc \<^cterm>\<open>(<) :: real \<Rightarrow> _\<close> |
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370 then arg_conv cv ct else arg1_conv cv ct |
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371 end |
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372 |
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373 fun real_ineq_conv th ct = |
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374 let |
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375 val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th |
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376 handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct])) |
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377 in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th')) |
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378 end |
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379 val [real_lt_conv, real_le_conv, real_eq_conv, |
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380 real_not_lt_conv, real_not_le_conv, _] = |
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381 map real_ineq_conv pth |
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382 fun match_mp_rule ths ths' = |
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383 let |
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384 fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths) |
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385 | th::ths => (ths' MRS th handle THM _ => f ths ths') |
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386 in f ths ths' end |
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387 fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv)) |
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388 (match_mp_rule pth_mul [th, th']) |
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389 fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv)) |
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390 (match_mp_rule pth_add [th, th']) |
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391 fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) |
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392 (Thm.instantiate' [] [SOME ct] (th RS pth_emul)) |
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393 fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv)) |
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394 (Thm.instantiate' [] [SOME t] pth_square) |
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395 |
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396 fun hol_of_positivstellensatz(eqs,les,lts) proof = |
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397 let |
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398 fun translate prf = |
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399 case prf of |
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400 Axiom_eq n => nth eqs n |
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401 | Axiom_le n => nth les n |
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402 | Axiom_lt n => nth lts n |
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403 | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply \<^cterm>\<open>Trueprop\<close> |
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404 (Thm.apply (Thm.apply \<^cterm>\<open>(=)::real \<Rightarrow> _\<close> (mk_numeric x)) |
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405 \<^cterm>\<open>0::real\<close>))) |
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406 | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply \<^cterm>\<open>Trueprop\<close> |
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407 (Thm.apply (Thm.apply \<^cterm>\<open>(\<le>)::real \<Rightarrow> _\<close> |
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408 \<^cterm>\<open>0::real\<close>) (mk_numeric x)))) |
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409 | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply \<^cterm>\<open>Trueprop\<close> |
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410 (Thm.apply (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) |
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411 (mk_numeric x)))) |
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412 | Square pt => square_rule (cterm_of_poly pt) |
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413 | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p) |
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414 | Sum(p1,p2) => add_rule (translate p1) (translate p2) |
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415 | Product(p1,p2) => mul_rule (translate p1) (translate p2) |
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416 in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) |
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417 (translate proof) |
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418 end |
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419 |
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420 val init_conv = presimp_conv then_conv |
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421 nnf_conv ctxt then_conv skolemize_conv then_conv prenex_conv then_conv |
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422 weak_dnf_conv |
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423 |
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424 val concl = Thm.dest_arg o Thm.cprop_of |
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425 fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false) |
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426 val is_req = is_binop \<^cterm>\<open>(=):: real \<Rightarrow> _\<close> |
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427 val is_ge = is_binop \<^cterm>\<open>(\<le>):: real \<Rightarrow> _\<close> |
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428 val is_gt = is_binop \<^cterm>\<open>(<):: real \<Rightarrow> _\<close> |
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429 val is_conj = is_binop \<^cterm>\<open>HOL.conj\<close> |
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430 val is_disj = is_binop \<^cterm>\<open>HOL.disj\<close> |
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431 fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2}) |
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432 fun disj_cases th th1 th2 = |
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433 let |
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434 val (p,q) = Thm.dest_binop (concl th) |
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435 val c = concl th1 |
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436 val _ = |
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437 if c aconvc (concl th2) then () |
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438 else error "disj_cases : conclusions not alpha convertible" |
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439 in Thm.implies_elim (Thm.implies_elim |
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440 (Thm.implies_elim (Thm.instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th) |
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441 (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> p) th1)) |
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442 (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> q) th2) |
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443 end |
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444 fun overall cert_choice dun ths = |
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445 case ths of |
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446 [] => |
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447 let |
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448 val (eq,ne) = List.partition (is_req o concl) dun |
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449 val (le,nl) = List.partition (is_ge o concl) ne |
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450 val lt = filter (is_gt o concl) nl |
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451 in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end |
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452 | th::oths => |
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453 let |
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454 val ct = concl th |
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455 in |
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456 if is_conj ct then |
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457 let |
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458 val (th1,th2) = conj_pair th |
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459 in overall cert_choice dun (th1::th2::oths) end |
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460 else if is_disj ct then |
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461 let |
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462 val (th1, cert1) = |
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463 overall (Left::cert_choice) dun |
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464 (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.dest_arg1 ct))::oths) |
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465 val (th2, cert2) = |
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466 overall (Right::cert_choice) dun |
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467 (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.dest_arg ct))::oths) |
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468 in (disj_cases th th1 th2, Branch (cert1, cert2)) end |
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469 else overall cert_choice (th::dun) oths |
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470 end |
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471 fun dest_binary b ct = |
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472 if is_binop b ct then Thm.dest_binop ct |
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473 else raise CTERM ("dest_binary",[b,ct]) |
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474 val dest_eq = dest_binary \<^cterm>\<open>(=) :: real \<Rightarrow> _\<close> |
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475 val neq_th = nth pth 5 |
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476 fun real_not_eq_conv ct = |
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477 let |
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478 val (l,r) = dest_eq (Thm.dest_arg ct) |
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479 val th = Thm.instantiate ([],[((("x", 0), \<^typ>\<open>real\<close>),l),((("y", 0), \<^typ>\<open>real\<close>),r)]) neq_th |
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480 val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th))) |
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481 val th_x = Drule.arg_cong_rule \<^cterm>\<open>uminus :: real \<Rightarrow> _\<close> th_p |
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482 val th_n = fconv_rule (arg_conv poly_neg_conv) th_x |
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483 val th' = Drule.binop_cong_rule \<^cterm>\<open>HOL.disj\<close> |
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484 (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_p) |
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485 (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_n) |
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486 in Thm.transitive th th' |
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487 end |
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488 fun equal_implies_1_rule PQ = |
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489 let |
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490 val P = Thm.lhs_of PQ |
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491 in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P)) |
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492 end |
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493 (*FIXME!!! Copied from groebner.ml*) |
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494 val strip_exists = |
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495 let |
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496 fun h (acc, t) = |
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497 case Thm.term_of t of |
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498 Const(\<^const_name>\<open>Ex\<close>,_)$Abs(_,_,_) => |
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499 h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc)) |
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500 | _ => (acc,t) |
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501 in fn t => h ([],t) |
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502 end |
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503 fun name_of x = |
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504 case Thm.term_of x of |
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505 Free(s,_) => s |
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506 | Var ((s,_),_) => s |
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507 | _ => "x" |
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508 |
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509 fun mk_forall x th = |
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510 let |
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511 val T = Thm.typ_of_cterm x |
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512 val all = Thm.cterm_of ctxt (Const (\<^const_name>\<open>All\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>)) |
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513 in Drule.arg_cong_rule all (Thm.abstract_rule (name_of x) x th) end |
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514 |
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515 val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec)); |
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516 |
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517 fun ext T = Thm.cterm_of ctxt (Const (\<^const_name>\<open>Ex\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>)) |
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518 fun mk_ex v t = Thm.apply (ext (Thm.typ_of_cterm v)) (Thm.lambda v t) |
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519 |
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520 fun choose v th th' = |
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521 case Thm.concl_of th of |
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522 \<^term>\<open>Trueprop\<close> $ (Const(\<^const_name>\<open>Ex\<close>,_)$_) => |
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523 let |
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524 val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th |
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525 val T = Thm.dest_ctyp0 (Thm.ctyp_of_cterm p) |
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526 val th0 = fconv_rule (Thm.beta_conversion true) |
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527 (Thm.instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o Thm.cprop_of) th'] exE) |
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528 val pv = (Thm.rhs_of o Thm.beta_conversion true) |
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529 (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply p v)) |
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530 val th1 = Thm.forall_intr v (Thm.implies_intr pv th') |
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531 in Thm.implies_elim (Thm.implies_elim th0 th) th1 end |
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532 | _ => raise THM ("choose",0,[th, th']) |
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533 |
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534 fun simple_choose v th = |
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535 choose v |
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536 (Thm.assume |
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537 ((Thm.apply \<^cterm>\<open>Trueprop\<close> o mk_ex v) (Thm.dest_arg (hd (Thm.chyps_of th))))) th |
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538 |
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539 val strip_forall = |
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540 let |
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541 fun h (acc, t) = |
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542 case Thm.term_of t of |
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543 Const(\<^const_name>\<open>All\<close>,_)$Abs(_,_,_) => |
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544 h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc)) |
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545 | _ => (acc,t) |
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546 in fn t => h ([],t) |
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547 end |
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548 |
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549 fun f ct = |
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550 let |
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551 val nnf_norm_conv' = |
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552 nnf_conv ctxt then_conv |
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553 literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] [] |
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554 (Conv.cache_conv |
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555 (first_conv [real_lt_conv, real_le_conv, |
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556 real_eq_conv, real_not_lt_conv, |
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557 real_not_le_conv, real_not_eq_conv, all_conv])) |
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558 fun absremover ct = (literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] [] |
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559 (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv |
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560 try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct |
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561 val nct = Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply \<^cterm>\<open>Not\<close> ct) |
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562 val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct |
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563 val tm0 = Thm.dest_arg (Thm.rhs_of th0) |
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564 val (th, certificates) = |
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565 if tm0 aconvc \<^cterm>\<open>False\<close> then (equal_implies_1_rule th0, Trivial) else |
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566 let |
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567 val (evs,bod) = strip_exists tm0 |
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568 val (avs,ibod) = strip_forall bod |
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569 val th1 = Drule.arg_cong_rule \<^cterm>\<open>Trueprop\<close> (fold mk_forall avs (absremover ibod)) |
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570 val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))] |
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571 val th3 = |
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572 fold simple_choose evs |
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573 (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> bod))) th2) |
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574 in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs) |
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575 end |
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576 in (Thm.implies_elim (Thm.instantiate' [] [SOME ct] pth_final) th, certificates) |
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577 end |
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578 in f |
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579 end; |
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580 |
|
581 (* A linear arithmetic prover *) |
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582 local |
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583 val linear_add = FuncUtil.Ctermfunc.combine (curry op +) (fn z => z = @0) |
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584 fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c * x) |
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585 val one_tm = \<^cterm>\<open>1::real\<close> |
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586 fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p @0)) orelse |
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587 ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso |
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588 not(p(FuncUtil.Ctermfunc.apply e one_tm))) |
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589 |
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590 fun linear_ineqs vars (les,lts) = |
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591 case find_first (contradictory (fn x => x > @0)) lts of |
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592 SOME r => r |
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593 | NONE => |
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594 (case find_first (contradictory (fn x => x > @0)) les of |
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595 SOME r => r |
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596 | NONE => |
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597 if null vars then error "linear_ineqs: no contradiction" else |
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598 let |
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599 val ineqs = les @ lts |
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600 fun blowup v = |
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601 length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) ineqs) + |
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602 length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) ineqs) * |
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603 length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 < @0) ineqs) |
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604 val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j)) |
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605 (map (fn v => (v,blowup v)) vars))) |
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606 fun addup (e1,p1) (e2,p2) acc = |
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607 let |
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608 val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v @0 |
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609 val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v @0 |
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610 in |
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611 if c1 * c2 >= @0 then acc else |
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612 let |
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613 val e1' = linear_cmul (abs c2) e1 |
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614 val e2' = linear_cmul (abs c1) e2 |
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615 val p1' = Product(Rational_lt (abs c2), p1) |
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616 val p2' = Product(Rational_lt (abs c1), p2) |
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617 in (linear_add e1' e2',Sum(p1',p2'))::acc |
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618 end |
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619 end |
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620 val (les0,les1) = |
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621 List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) les |
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622 val (lts0,lts1) = |
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623 List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) lts |
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624 val (lesp,lesn) = |
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625 List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) les1 |
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626 val (ltsp,ltsn) = |
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627 List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) lts1 |
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628 val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0 |
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629 val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn |
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630 (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0) |
|
631 in linear_ineqs (remove (op aconvc) v vars) (les',lts') |
|
632 end) |
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633 |
|
634 fun linear_eqs(eqs,les,lts) = |
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635 case find_first (contradictory (fn x => x = @0)) eqs of |
|
636 SOME r => r |
|
637 | NONE => |
|
638 (case eqs of |
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639 [] => |
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640 let val vars = remove (op aconvc) one_tm |
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641 (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) []) |
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642 in linear_ineqs vars (les,lts) end |
|
643 | (e,p)::es => |
|
644 if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else |
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645 let |
|
646 val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e) |
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647 fun xform (inp as (t,q)) = |
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648 let val d = FuncUtil.Ctermfunc.tryapplyd t x @0 in |
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649 if d = @0 then inp else |
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650 let |
|
651 val k = ~ d * abs c / c |
|
652 val e' = linear_cmul k e |
|
653 val t' = linear_cmul (abs c) t |
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654 val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p) |
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655 val q' = Product(Rational_lt (abs c), q) |
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656 in (linear_add e' t',Sum(p',q')) |
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657 end |
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658 end |
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659 in linear_eqs(map xform es,map xform les,map xform lts) |
|
660 end) |
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661 |
|
662 fun linear_prover (eq,le,lt) = |
|
663 let |
|
664 val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq |
|
665 val les = map_index (fn (n, p) => (p,Axiom_le n)) le |
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666 val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt |
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667 in linear_eqs(eqs,les,lts) |
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668 end |
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669 |
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670 fun lin_of_hol ct = |
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671 if ct aconvc \<^cterm>\<open>0::real\<close> then FuncUtil.Ctermfunc.empty |
|
672 else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, @1) |
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673 else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct) |
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674 else |
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675 let val (lop,r) = Thm.dest_comb ct |
|
676 in |
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677 if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, @1) |
|
678 else |
|
679 let val (opr,l) = Thm.dest_comb lop |
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680 in |
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681 if opr aconvc \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close> |
|
682 then linear_add (lin_of_hol l) (lin_of_hol r) |
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683 else if opr aconvc \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close> |
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684 andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l) |
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685 else FuncUtil.Ctermfunc.onefunc (ct, @1) |
|
686 end |
|
687 end |
|
688 |
|
689 fun is_alien ct = |
|
690 case Thm.term_of ct of |
|
691 Const(\<^const_name>\<open>of_nat\<close>, _)$ n => not (can HOLogic.dest_number n) |
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692 | Const(\<^const_name>\<open>of_int\<close>, _)$ n => not (can HOLogic.dest_number n) |
|
693 | _ => false |
|
694 in |
|
695 fun real_linear_prover translator (eq,le,lt) = |
|
696 let |
|
697 val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o Thm.cprop_of |
|
698 val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o Thm.cprop_of |
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699 val eq_pols = map lhs eq |
|
700 val le_pols = map rhs le |
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701 val lt_pols = map rhs lt |
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702 val aliens = filter is_alien |
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703 (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) |
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704 (eq_pols @ le_pols @ lt_pols) []) |
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705 val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,@1)) aliens |
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706 val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols) |
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707 val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) aliens |
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708 in ((translator (eq,le',lt) proof), Trivial) |
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709 end |
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710 end; |
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711 |
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712 (* A less general generic arithmetic prover dealing with abs,max and min*) |
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713 |
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714 local |
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715 val absmaxmin_elim_ss1 = |
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716 simpset_of (put_simpset HOL_basic_ss \<^context> addsimps real_abs_thms1) |
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717 fun absmaxmin_elim_conv1 ctxt = |
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718 Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt) |
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719 |
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720 val absmaxmin_elim_conv2 = |
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721 let |
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722 val pth_abs = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] abs_split' |
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723 val pth_max = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] max_split |
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724 val pth_min = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] min_split |
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725 val abs_tm = \<^cterm>\<open>abs :: real \<Rightarrow> _\<close> |
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726 val p_v = (("P", 0), \<^typ>\<open>real \<Rightarrow> bool\<close>) |
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727 val x_v = (("x", 0), \<^typ>\<open>real\<close>) |
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728 val y_v = (("y", 0), \<^typ>\<open>real\<close>) |
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729 val is_max = is_binop \<^cterm>\<open>max :: real \<Rightarrow> _\<close> |
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730 val is_min = is_binop \<^cterm>\<open>min :: real \<Rightarrow> _\<close> |
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731 fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm |
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732 fun eliminate_construct p c tm = |
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733 let |
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734 val t = find_cterm p tm |
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735 val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t) |
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736 val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0 |
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737 in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false)))) |
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738 (Thm.transitive th0 (c p ax)) |
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739 end |
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740 |
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741 val elim_abs = eliminate_construct is_abs |
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742 (fn p => fn ax => |
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743 Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax)]) pth_abs) |
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744 val elim_max = eliminate_construct is_max |
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745 (fn p => fn ax => |
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746 let val (ax,y) = Thm.dest_comb ax |
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747 in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)]) |
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748 pth_max end) |
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749 val elim_min = eliminate_construct is_min |
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750 (fn p => fn ax => |
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751 let val (ax,y) = Thm.dest_comb ax |
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752 in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)]) |
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753 pth_min end) |
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754 in first_conv [elim_abs, elim_max, elim_min, all_conv] |
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755 end; |
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756 in |
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757 fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) = |
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758 gen_gen_real_arith ctxt |
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759 (mkconst,eq,ge,gt,norm,neg,add,mul, |
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760 absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover) |
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761 end; |
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762 |
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763 (* An instance for reals*) |
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764 |
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765 fun gen_prover_real_arith ctxt prover = |
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766 let |
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767 val {add, mul, neg, pow = _, sub = _, main} = |
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768 Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt |
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769 (the (Semiring_Normalizer.match ctxt \<^cterm>\<open>(0::real) + 1\<close>)) |
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770 Thm.term_ord |
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771 in gen_real_arith ctxt |
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772 (cterm_of_rat, |
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773 Numeral_Simprocs.field_comp_conv ctxt, |
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774 Numeral_Simprocs.field_comp_conv ctxt, |
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775 Numeral_Simprocs.field_comp_conv ctxt, |
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776 main ctxt, neg ctxt, add ctxt, mul ctxt, prover) |
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777 end; |
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778 |
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779 end |
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