1 (* Title: HOL/ex/ReflectionEx.thy |
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2 Author: Amine Chaieb, TU Muenchen |
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3 *) |
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4 |
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5 header {* Examples for generic reflection and reification *} |
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6 |
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7 theory ReflectionEx |
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8 imports "~~/src/HOL/Library/Reflection" |
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9 begin |
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10 |
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11 text{* This theory presents two methods: reify and reflection *} |
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12 |
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13 text{* |
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14 Consider an HOL type 'a, the structure of which is not recongnisable on the theory level. This is the case of bool, arithmetical terms such as int, real etc \dots |
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15 In order to implement a simplification on terms of type 'a we often need its structure. |
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16 Traditionnaly such simplifications are written in ML, proofs are synthesized. |
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17 An other strategy is to declare an HOL-datatype tau and an HOL function (the interpretation) that maps elements of tau to elements of 'a. The functionality of @{text reify} is to compute a term s::tau, which is the representant of t. For this it needs equations for the interpretation. |
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18 |
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19 NB: All the interpretations supported by @{text reify} must have the type @{text "'b list \<Rightarrow> tau \<Rightarrow> 'a"}. |
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20 The method @{text reify} can also be told which subterm of the current subgoal should be reified. The general call for @{text reify} is: @{text "reify eqs (t)"}, where @{text eqs} are the defining equations of the interpretation and @{text "(t)"} is an optional parameter which specifies the subterm to which reification should be applied to. If @{text "(t)"} is abscent, @{text reify} tries to reify the whole subgoal. |
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21 |
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22 The method reflection uses @{text reify} and has a very similar signature: @{text "reflection corr_thm eqs (t)"}. Here again @{text eqs} and @{text "(t)"} are as described above and @{text corr_thm} is a thorem proving @{term "I vs (f t) = I vs t"}. We assume that @{text I} is the interpretation and @{text f} is some useful and executable simplification of type @{text "tau \<Rightarrow> tau"}. The method @{text reflection} applies reification and hence the theorem @{term "t = I xs s"} and hence using @{text corr_thm} derives @{term "t = I xs (f s)"}. It then uses normalization by evaluation to prove @{term "f s = s'"} which almost finishes the proof of @{term "t = t'"} where @{term "I xs s' = t'"}. |
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23 *} |
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24 |
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25 text{* Example 1 : Propositional formulae and NNF.*} |
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26 text{* The type @{text fm} represents simple propositional formulae: *} |
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27 |
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28 datatype form = TrueF | FalseF | Less nat nat | |
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29 And form form | Or form form | Neg form | ExQ form |
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30 |
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31 fun interp :: "form \<Rightarrow> ('a::ord) list \<Rightarrow> bool" where |
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32 "interp TrueF e = True" | |
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33 "interp FalseF e = False" | |
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34 "interp (Less i j) e = (e!i < e!j)" | |
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35 "interp (And f1 f2) e = (interp f1 e & interp f2 e)" | |
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36 "interp (Or f1 f2) e = (interp f1 e | interp f2 e)" | |
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37 "interp (Neg f) e = (~ interp f e)" | |
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38 "interp (ExQ f) e = (EX x. interp f (x#e))" |
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39 |
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40 lemmas interp_reify_eqs = interp.simps |
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41 declare interp_reify_eqs[reify] |
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42 |
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43 lemma "EX x. x < y & x < z" |
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44 apply (reify ) |
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45 oops |
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46 |
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47 datatype fm = And fm fm | Or fm fm | Imp fm fm | Iff fm fm | NOT fm | At nat |
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48 |
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49 primrec Ifm :: "fm \<Rightarrow> bool list \<Rightarrow> bool" where |
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50 "Ifm (At n) vs = vs!n" |
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51 | "Ifm (And p q) vs = (Ifm p vs \<and> Ifm q vs)" |
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52 | "Ifm (Or p q) vs = (Ifm p vs \<or> Ifm q vs)" |
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53 | "Ifm (Imp p q) vs = (Ifm p vs \<longrightarrow> Ifm q vs)" |
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54 | "Ifm (Iff p q) vs = (Ifm p vs = Ifm q vs)" |
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55 | "Ifm (NOT p) vs = (\<not> (Ifm p vs))" |
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56 |
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57 lemma "Q \<longrightarrow> (D & F & ((~ D) & (~ F)))" |
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58 apply (reify Ifm.simps) |
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59 oops |
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60 |
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61 text{* Method @{text reify} maps a bool to an fm. For this it needs the |
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62 semantics of fm, i.e.\ the rewrite rules in @{text Ifm.simps}. *} |
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63 |
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64 |
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65 (* You can also just pick up a subterm to reify \<dots> *) |
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66 lemma "Q \<longrightarrow> (D & F & ((~ D) & (~ F)))" |
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67 apply (reify Ifm.simps ("((~ D) & (~ F))")) |
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68 oops |
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69 |
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70 text{* Let's perform NNF. This is a version that tends to generate disjunctions *} |
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71 primrec fmsize :: "fm \<Rightarrow> nat" where |
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72 "fmsize (At n) = 1" |
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73 | "fmsize (NOT p) = 1 + fmsize p" |
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74 | "fmsize (And p q) = 1 + fmsize p + fmsize q" |
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75 | "fmsize (Or p q) = 1 + fmsize p + fmsize q" |
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76 | "fmsize (Imp p q) = 2 + fmsize p + fmsize q" |
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77 | "fmsize (Iff p q) = 2 + 2* fmsize p + 2* fmsize q" |
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78 |
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79 lemma [measure_function]: "is_measure fmsize" .. |
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80 |
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81 fun nnf :: "fm \<Rightarrow> fm" |
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82 where |
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83 "nnf (At n) = At n" |
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84 | "nnf (And p q) = And (nnf p) (nnf q)" |
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85 | "nnf (Or p q) = Or (nnf p) (nnf q)" |
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86 | "nnf (Imp p q) = Or (nnf (NOT p)) (nnf q)" |
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87 | "nnf (Iff p q) = Or (And (nnf p) (nnf q)) (And (nnf (NOT p)) (nnf (NOT q)))" |
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88 | "nnf (NOT (And p q)) = Or (nnf (NOT p)) (nnf (NOT q))" |
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89 | "nnf (NOT (Or p q)) = And (nnf (NOT p)) (nnf (NOT q))" |
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90 | "nnf (NOT (Imp p q)) = And (nnf p) (nnf (NOT q))" |
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91 | "nnf (NOT (Iff p q)) = Or (And (nnf p) (nnf (NOT q))) (And (nnf (NOT p)) (nnf q))" |
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92 | "nnf (NOT (NOT p)) = nnf p" |
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93 | "nnf (NOT p) = NOT p" |
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94 |
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95 text{* The correctness theorem of nnf: it preserves the semantics of fm *} |
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96 lemma nnf[reflection]: "Ifm (nnf p) vs = Ifm p vs" |
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97 by (induct p rule: nnf.induct) auto |
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98 |
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99 text{* Now let's perform NNF using our @{term nnf} function defined above. First to the whole subgoal. *} |
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100 lemma "(\<not> (A = B)) \<and> (B \<longrightarrow> (A \<noteq> (B | C \<and> (B \<longrightarrow> A | D)))) \<longrightarrow> A \<or> B \<and> D" |
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101 apply (reflection Ifm.simps) |
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102 oops |
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103 |
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104 text{* Now we specify on which subterm it should be applied*} |
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105 lemma "(\<not> (A = B)) \<and> (B \<longrightarrow> (A \<noteq> (B | C \<and> (B \<longrightarrow> A | D)))) \<longrightarrow> A \<or> B \<and> D" |
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106 apply (reflection Ifm.simps only: "(B | C \<and> (B \<longrightarrow> A | D))") |
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107 oops |
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108 |
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109 |
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110 (* Example 2 : Simple arithmetic formulae *) |
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111 |
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112 text{* The type @{text num} reflects linear expressions over natural number *} |
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113 datatype num = C nat | Add num num | Mul nat num | Var nat | CN nat nat num |
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114 |
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115 text{* This is just technical to make recursive definitions easier. *} |
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116 primrec num_size :: "num \<Rightarrow> nat" |
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117 where |
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118 "num_size (C c) = 1" |
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119 | "num_size (Var n) = 1" |
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120 | "num_size (Add a b) = 1 + num_size a + num_size b" |
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121 | "num_size (Mul c a) = 1 + num_size a" |
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122 | "num_size (CN n c a) = 4 + num_size a " |
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123 |
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124 text{* The semantics of num *} |
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125 primrec Inum:: "num \<Rightarrow> nat list \<Rightarrow> nat" |
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126 where |
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127 Inum_C : "Inum (C i) vs = i" |
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128 | Inum_Var: "Inum (Var n) vs = vs!n" |
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129 | Inum_Add: "Inum (Add s t) vs = Inum s vs + Inum t vs " |
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130 | Inum_Mul: "Inum (Mul c t) vs = c * Inum t vs " |
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131 | Inum_CN : "Inum (CN n c t) vs = c*(vs!n) + Inum t vs " |
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132 |
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133 |
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134 text{* Let's reify some nat expressions \dots *} |
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135 lemma "4 * (2*x + (y::nat)) + f a \<noteq> 0" |
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136 apply (reify Inum.simps ("4 * (2*x + (y::nat)) + f a")) |
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137 oops |
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138 text{* We're in a bad situation!! x, y and f a have been recongnized as a constants, which is correct but does not correspond to our intuition of the constructor C. It should encapsulate constants, i.e. numbers, i.e. numerals.*} |
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139 |
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140 text{* So let's leave the @{text "Inum_C"} equation at the end and see what happens \dots*} |
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141 lemma "4 * (2*x + (y::nat)) \<noteq> 0" |
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142 apply (reify Inum_Var Inum_Add Inum_Mul Inum_CN Inum_C ("4 * (2*x + (y::nat))")) |
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143 oops |
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144 text{* Hmmm let's specialize @{text Inum_C} with numerals.*} |
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145 |
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146 lemma Inum_number: "Inum (C (numeral t)) vs = numeral t" by simp |
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147 lemmas Inum_eqs = Inum_Var Inum_Add Inum_Mul Inum_CN Inum_number |
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148 |
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149 text{* Second attempt *} |
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150 lemma "1 * (2*x + (y::nat)) \<noteq> 0" |
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151 apply (reify Inum_eqs ("1 * (2*x + (y::nat))")) |
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152 oops |
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153 text{* That was fine, so let's try another one \dots *} |
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154 |
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155 lemma "1 * (2* x + (y::nat) + 0 + 1) \<noteq> 0" |
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156 apply (reify Inum_eqs ("1 * (2*x + (y::nat) + 0 + 1)")) |
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157 oops |
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158 text{* Oh!! 0 is not a variable \dots\ Oh! 0 is not a @{text "numeral"} \dots\ thing. The same for 1. So let's add those equations too *} |
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159 |
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160 lemma Inum_01: "Inum (C 0) vs = 0" "Inum (C 1) vs = 1" "Inum (C(Suc n)) vs = Suc n" |
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161 by simp+ |
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162 |
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163 lemmas Inum_eqs'= Inum_eqs Inum_01 |
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164 |
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165 text{* Third attempt: *} |
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166 |
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167 lemma "1 * (2*x + (y::nat) + 0 + 1) \<noteq> 0" |
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168 apply (reify Inum_eqs' ("1 * (2*x + (y::nat) + 0 + 1)")) |
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169 oops |
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170 text{* Okay, let's try reflection. Some simplifications on num follow. You can skim until the main theorem @{text linum} *} |
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171 fun lin_add :: "num \<Rightarrow> num \<Rightarrow> num" |
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172 where |
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173 "lin_add (CN n1 c1 r1) (CN n2 c2 r2) = |
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174 (if n1=n2 then |
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175 (let c = c1 + c2 |
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176 in (if c=0 then lin_add r1 r2 else CN n1 c (lin_add r1 r2))) |
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177 else if n1 \<le> n2 then (CN n1 c1 (lin_add r1 (CN n2 c2 r2))) |
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178 else (CN n2 c2 (lin_add (CN n1 c1 r1) r2)))" |
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179 | "lin_add (CN n1 c1 r1) t = CN n1 c1 (lin_add r1 t)" |
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180 | "lin_add t (CN n2 c2 r2) = CN n2 c2 (lin_add t r2)" |
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181 | "lin_add (C b1) (C b2) = C (b1+b2)" |
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182 | "lin_add a b = Add a b" |
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183 lemma lin_add: "Inum (lin_add t s) bs = Inum (Add t s) bs" |
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184 apply (induct t s rule: lin_add.induct, simp_all add: Let_def) |
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185 apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all) |
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186 by (case_tac "n1 = n2", simp_all add: algebra_simps) |
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187 |
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188 fun lin_mul :: "num \<Rightarrow> nat \<Rightarrow> num" |
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189 where |
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190 "lin_mul (C j) i = C (i*j)" |
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191 | "lin_mul (CN n c a) i = (if i=0 then (C 0) else CN n (i*c) (lin_mul a i))" |
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192 | "lin_mul t i = (Mul i t)" |
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193 |
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194 lemma lin_mul: "Inum (lin_mul t i) bs = Inum (Mul i t) bs" |
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195 by (induct t i rule: lin_mul.induct, auto simp add: algebra_simps) |
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196 |
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197 lemma [measure_function]: "is_measure num_size" .. |
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198 |
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199 fun linum:: "num \<Rightarrow> num" |
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200 where |
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201 "linum (C b) = C b" |
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202 | "linum (Var n) = CN n 1 (C 0)" |
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203 | "linum (Add t s) = lin_add (linum t) (linum s)" |
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204 | "linum (Mul c t) = lin_mul (linum t) c" |
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205 | "linum (CN n c t) = lin_add (linum (Mul c (Var n))) (linum t)" |
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206 |
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207 lemma linum[reflection] : "Inum (linum t) bs = Inum t bs" |
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208 by (induct t rule: linum.induct, simp_all add: lin_mul lin_add) |
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209 |
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210 text{* Now we can use linum to simplify nat terms using reflection *} |
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211 lemma "(Suc (Suc 1)) * (x + (Suc 1)*y) = 3*x + 6*y" |
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212 apply (reflection Inum_eqs' only: "(Suc (Suc 1)) * (x + (Suc 1)*y)") |
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213 oops |
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214 |
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215 text{* Let's lift this to formulae and see what happens *} |
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216 |
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217 datatype aform = Lt num num | Eq num num | Ge num num | NEq num num | |
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218 Conj aform aform | Disj aform aform | NEG aform | T | F |
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219 |
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220 primrec linaformsize:: "aform \<Rightarrow> nat" |
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221 where |
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222 "linaformsize T = 1" |
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223 | "linaformsize F = 1" |
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224 | "linaformsize (Lt a b) = 1" |
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225 | "linaformsize (Ge a b) = 1" |
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226 | "linaformsize (Eq a b) = 1" |
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227 | "linaformsize (NEq a b) = 1" |
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228 | "linaformsize (NEG p) = 2 + linaformsize p" |
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229 | "linaformsize (Conj p q) = 1 + linaformsize p + linaformsize q" |
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230 | "linaformsize (Disj p q) = 1 + linaformsize p + linaformsize q" |
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231 |
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232 lemma [measure_function]: "is_measure linaformsize" .. |
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233 |
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234 primrec is_aform :: "aform => nat list => bool" |
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235 where |
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236 "is_aform T vs = True" |
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237 | "is_aform F vs = False" |
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238 | "is_aform (Lt a b) vs = (Inum a vs < Inum b vs)" |
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239 | "is_aform (Eq a b) vs = (Inum a vs = Inum b vs)" |
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240 | "is_aform (Ge a b) vs = (Inum a vs \<ge> Inum b vs)" |
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241 | "is_aform (NEq a b) vs = (Inum a vs \<noteq> Inum b vs)" |
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242 | "is_aform (NEG p) vs = (\<not> (is_aform p vs))" |
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243 | "is_aform (Conj p q) vs = (is_aform p vs \<and> is_aform q vs)" |
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244 | "is_aform (Disj p q) vs = (is_aform p vs \<or> is_aform q vs)" |
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245 |
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246 text{* Let's reify and do reflection *} |
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247 lemma "(3::nat)*x + t < 0 \<and> (2 * x + y \<noteq> 17)" |
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248 apply (reify Inum_eqs' is_aform.simps) |
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249 oops |
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250 |
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251 text{* Note that reification handles several interpretations at the same time*} |
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252 lemma "(3::nat)*x + t < 0 & x*x + t*x + 3 + 1 = z*t*4*z | x + x + 1 < 0" |
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253 apply (reflection Inum_eqs' is_aform.simps only:"x + x + 1") |
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254 oops |
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255 |
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256 text{* For reflection we now define a simple transformation on aform: NNF + linum on atoms *} |
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257 |
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258 fun linaform:: "aform \<Rightarrow> aform" |
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259 where |
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260 "linaform (Lt s t) = Lt (linum s) (linum t)" |
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261 | "linaform (Eq s t) = Eq (linum s) (linum t)" |
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262 | "linaform (Ge s t) = Ge (linum s) (linum t)" |
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263 | "linaform (NEq s t) = NEq (linum s) (linum t)" |
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264 | "linaform (Conj p q) = Conj (linaform p) (linaform q)" |
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265 | "linaform (Disj p q) = Disj (linaform p) (linaform q)" |
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266 | "linaform (NEG T) = F" |
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267 | "linaform (NEG F) = T" |
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268 | "linaform (NEG (Lt a b)) = Ge a b" |
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269 | "linaform (NEG (Ge a b)) = Lt a b" |
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270 | "linaform (NEG (Eq a b)) = NEq a b" |
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271 | "linaform (NEG (NEq a b)) = Eq a b" |
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272 | "linaform (NEG (NEG p)) = linaform p" |
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273 | "linaform (NEG (Conj p q)) = Disj (linaform (NEG p)) (linaform (NEG q))" |
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274 | "linaform (NEG (Disj p q)) = Conj (linaform (NEG p)) (linaform (NEG q))" |
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275 | "linaform p = p" |
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276 |
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277 lemma linaform: "is_aform (linaform p) vs = is_aform p vs" |
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278 by (induct p rule: linaform.induct) (auto simp add: linum) |
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279 |
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280 lemma "(((Suc(Suc (Suc 0)))*((x::nat) + (Suc (Suc 0)))) + (Suc (Suc (Suc 0))) * ((Suc(Suc (Suc 0)))*((x::nat) + (Suc (Suc 0))))< 0) \<and> (Suc 0 + Suc 0< 0)" |
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281 apply (reflection Inum_eqs' is_aform.simps rules: linaform) |
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282 oops |
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283 |
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284 declare linaform[reflection] |
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285 lemma "(((Suc(Suc (Suc 0)))*((x::nat) + (Suc (Suc 0)))) + (Suc (Suc (Suc 0))) * ((Suc(Suc (Suc 0)))*((x::nat) + (Suc (Suc 0))))< 0) \<and> (Suc 0 + Suc 0< 0)" |
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286 apply (reflection Inum_eqs' is_aform.simps) |
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287 oops |
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288 |
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289 text{* We now give an example where Interpretaions have 0 or more than only one envornement of different types and show that automatic reification also deals with binding *} |
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290 datatype rb = BC bool| BAnd rb rb | BOr rb rb |
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291 primrec Irb :: "rb \<Rightarrow> bool" |
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292 where |
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293 "Irb (BC p) = p" |
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294 | "Irb (BAnd s t) = (Irb s \<and> Irb t)" |
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295 | "Irb (BOr s t) = (Irb s \<or> Irb t)" |
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296 |
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297 lemma "A \<and> (B \<or> D \<and> B) \<and> A \<and> (B \<or> D \<and> B) \<or> A \<and> (B \<or> D \<and> B) \<or> A \<and> (B \<or> D \<and> B)" |
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298 apply (reify Irb.simps) |
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299 oops |
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300 |
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301 |
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302 datatype rint = IC int| IVar nat | IAdd rint rint | IMult rint rint | INeg rint | ISub rint rint |
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303 primrec Irint :: "rint \<Rightarrow> int list \<Rightarrow> int" |
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304 where |
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305 Irint_Var: "Irint (IVar n) vs = vs!n" |
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306 | Irint_Neg: "Irint (INeg t) vs = - Irint t vs" |
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307 | Irint_Add: "Irint (IAdd s t) vs = Irint s vs + Irint t vs" |
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308 | Irint_Sub: "Irint (ISub s t) vs = Irint s vs - Irint t vs" |
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309 | Irint_Mult: "Irint (IMult s t) vs = Irint s vs * Irint t vs" |
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310 | Irint_C: "Irint (IC i) vs = i" |
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311 lemma Irint_C0: "Irint (IC 0) vs = 0" |
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312 by simp |
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313 lemma Irint_C1: "Irint (IC 1) vs = 1" |
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314 by simp |
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315 lemma Irint_Cnumeral: "Irint (IC (numeral x)) vs = numeral x" |
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316 by simp |
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317 lemmas Irint_simps = Irint_Var Irint_Neg Irint_Add Irint_Sub Irint_Mult Irint_C0 Irint_C1 Irint_Cnumeral |
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318 lemma "(3::int) * x + y*y - 9 + (- z) = 0" |
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319 apply (reify Irint_simps ("(3::int) * x + y*y - 9 + (- z)")) |
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320 oops |
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321 datatype rlist = LVar nat| LEmpty| LCons rint rlist | LAppend rlist rlist |
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322 primrec Irlist :: "rlist \<Rightarrow> int list \<Rightarrow> (int list) list \<Rightarrow> (int list)" |
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323 where |
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324 "Irlist (LEmpty) is vs = []" |
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325 | "Irlist (LVar n) is vs = vs!n" |
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326 | "Irlist (LCons i t) is vs = ((Irint i is)#(Irlist t is vs))" |
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327 | "Irlist (LAppend s t) is vs = (Irlist s is vs) @ (Irlist t is vs)" |
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328 lemma "[(1::int)] = []" |
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329 apply (reify Irlist.simps Irint_simps ("[1]:: int list")) |
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330 oops |
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331 |
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332 lemma "([(3::int) * x + y*y - 9 + (- z)] @ []) @ xs = [y*y - z - 9 + (3::int) * x]" |
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333 apply (reify Irlist.simps Irint_simps ("([(3::int) * x + y*y - 9 + (- z)] @ []) @ xs")) |
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334 oops |
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335 |
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336 datatype rnat = NC nat| NVar nat| NSuc rnat | NAdd rnat rnat | NMult rnat rnat | NNeg rnat | NSub rnat rnat | Nlgth rlist |
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337 primrec Irnat :: "rnat \<Rightarrow> int list \<Rightarrow> (int list) list \<Rightarrow> nat list \<Rightarrow> nat" |
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338 where |
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339 Irnat_Suc: "Irnat (NSuc t) is ls vs = Suc (Irnat t is ls vs)" |
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340 | Irnat_Var: "Irnat (NVar n) is ls vs = vs!n" |
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341 | Irnat_Neg: "Irnat (NNeg t) is ls vs = 0" |
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342 | Irnat_Add: "Irnat (NAdd s t) is ls vs = Irnat s is ls vs + Irnat t is ls vs" |
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343 | Irnat_Sub: "Irnat (NSub s t) is ls vs = Irnat s is ls vs - Irnat t is ls vs" |
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344 | Irnat_Mult: "Irnat (NMult s t) is ls vs = Irnat s is ls vs * Irnat t is ls vs" |
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345 | Irnat_lgth: "Irnat (Nlgth rxs) is ls vs = length (Irlist rxs is ls)" |
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346 | Irnat_C: "Irnat (NC i) is ls vs = i" |
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347 lemma Irnat_C0: "Irnat (NC 0) is ls vs = 0" |
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348 by simp |
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349 lemma Irnat_C1: "Irnat (NC 1) is ls vs = 1" |
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350 by simp |
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351 lemma Irnat_Cnumeral: "Irnat (NC (numeral x)) is ls vs = numeral x" |
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352 by simp |
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353 lemmas Irnat_simps = Irnat_Suc Irnat_Var Irnat_Neg Irnat_Add Irnat_Sub Irnat_Mult Irnat_lgth |
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354 Irnat_C0 Irnat_C1 Irnat_Cnumeral |
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355 lemma "(Suc n) * length (([(3::int) * x + y*y - 9 + (- z)] @ []) @ xs) = length xs" |
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356 apply (reify Irnat_simps Irlist.simps Irint_simps ("(Suc n) *length (([(3::int) * x + y*y - 9 + (- z)] @ []) @ xs)")) |
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357 oops |
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358 datatype rifm = RT | RF | RVar nat |
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359 | RNLT rnat rnat | RNILT rnat rint | RNEQ rnat rnat |
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360 |RAnd rifm rifm | ROr rifm rifm | RImp rifm rifm| RIff rifm rifm |
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361 | RNEX rifm | RIEX rifm| RLEX rifm | RNALL rifm | RIALL rifm| RLALL rifm |
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362 | RBEX rifm | RBALL rifm |
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363 |
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364 primrec Irifm :: "rifm \<Rightarrow> bool list \<Rightarrow> int list \<Rightarrow> (int list) list \<Rightarrow> nat list \<Rightarrow> bool" |
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365 where |
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366 "Irifm RT ps is ls ns = True" |
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367 | "Irifm RF ps is ls ns = False" |
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368 | "Irifm (RVar n) ps is ls ns = ps!n" |
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369 | "Irifm (RNLT s t) ps is ls ns = (Irnat s is ls ns < Irnat t is ls ns)" |
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370 | "Irifm (RNILT s t) ps is ls ns = (int (Irnat s is ls ns) < Irint t is)" |
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371 | "Irifm (RNEQ s t) ps is ls ns = (Irnat s is ls ns = Irnat t is ls ns)" |
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372 | "Irifm (RAnd p q) ps is ls ns = (Irifm p ps is ls ns \<and> Irifm q ps is ls ns)" |
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373 | "Irifm (ROr p q) ps is ls ns = (Irifm p ps is ls ns \<or> Irifm q ps is ls ns)" |
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374 | "Irifm (RImp p q) ps is ls ns = (Irifm p ps is ls ns \<longrightarrow> Irifm q ps is ls ns)" |
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375 | "Irifm (RIff p q) ps is ls ns = (Irifm p ps is ls ns = Irifm q ps is ls ns)" |
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376 | "Irifm (RNEX p) ps is ls ns = (\<exists>x. Irifm p ps is ls (x#ns))" |
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377 | "Irifm (RIEX p) ps is ls ns = (\<exists>x. Irifm p ps (x#is) ls ns)" |
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378 | "Irifm (RLEX p) ps is ls ns = (\<exists>x. Irifm p ps is (x#ls) ns)" |
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379 | "Irifm (RBEX p) ps is ls ns = (\<exists>x. Irifm p (x#ps) is ls ns)" |
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380 | "Irifm (RNALL p) ps is ls ns = (\<forall>x. Irifm p ps is ls (x#ns))" |
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381 | "Irifm (RIALL p) ps is ls ns = (\<forall>x. Irifm p ps (x#is) ls ns)" |
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382 | "Irifm (RLALL p) ps is ls ns = (\<forall>x. Irifm p ps is (x#ls) ns)" |
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383 | "Irifm (RBALL p) ps is ls ns = (\<forall>x. Irifm p (x#ps) is ls ns)" |
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384 |
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385 lemma " \<forall>x. \<exists>n. ((Suc n) * length (([(3::int) * x + (f t)*y - 9 + (- z)] @ []) @ xs) = length xs) \<and> m < 5*n - length (xs @ [2,3,4,x*z + 8 - y]) \<longrightarrow> (\<exists>p. \<forall>q. p \<and> q \<longrightarrow> r)" |
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386 apply (reify Irifm.simps Irnat_simps Irlist.simps Irint_simps) |
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387 oops |
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388 |
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389 |
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390 (* An example for equations containing type variables *) |
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391 datatype prod = Zero | One | Var nat | Mul prod prod |
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392 | Pw prod nat | PNM nat nat prod |
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393 primrec Iprod :: " prod \<Rightarrow> ('a::{linordered_idom}) list \<Rightarrow>'a" |
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394 where |
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395 "Iprod Zero vs = 0" |
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396 | "Iprod One vs = 1" |
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397 | "Iprod (Var n) vs = vs!n" |
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398 | "Iprod (Mul a b) vs = (Iprod a vs * Iprod b vs)" |
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399 | "Iprod (Pw a n) vs = ((Iprod a vs) ^ n)" |
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400 | "Iprod (PNM n k t) vs = (vs ! n)^k * Iprod t vs" |
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401 |
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402 datatype sgn = Pos prod | Neg prod | ZeroEq prod | NZeroEq prod | Tr | F |
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403 | Or sgn sgn | And sgn sgn |
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404 |
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405 primrec Isgn :: " sgn \<Rightarrow> ('a::{linordered_idom}) list \<Rightarrow>bool" |
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406 where |
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407 "Isgn Tr vs = True" |
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408 | "Isgn F vs = False" |
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409 | "Isgn (ZeroEq t) vs = (Iprod t vs = 0)" |
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410 | "Isgn (NZeroEq t) vs = (Iprod t vs \<noteq> 0)" |
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411 | "Isgn (Pos t) vs = (Iprod t vs > 0)" |
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412 | "Isgn (Neg t) vs = (Iprod t vs < 0)" |
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413 | "Isgn (And p q) vs = (Isgn p vs \<and> Isgn q vs)" |
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414 | "Isgn (Or p q) vs = (Isgn p vs \<or> Isgn q vs)" |
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415 |
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416 lemmas eqs = Isgn.simps Iprod.simps |
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417 |
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418 lemma "(x::'a::{linordered_idom})^4 * y * z * y^2 * z^23 > 0" |
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419 apply (reify eqs) |
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420 oops |
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421 |
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422 end |
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