1 (* Title: HOL/ex/Term |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1992 University of Cambridge |
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5 |
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6 Terms over a given alphabet -- function applications; illustrates list functor |
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7 (essentially the same type as in Trees & Forests) |
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8 *) |
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9 |
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10 open Term; |
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11 |
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12 (*** Monotonicity and unfolding of the function ***) |
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13 |
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14 goal Term.thy "term(A) = A <*> list(term(A))"; |
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15 by (fast_tac (!claset addSIs (equalityI :: term.intrs) |
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16 addEs [term.elim]) 1); |
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17 qed "term_unfold"; |
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18 |
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19 (*This justifies using term in other recursive type definitions*) |
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20 goalw Term.thy term.defs "!!A B. A<=B ==> term(A) <= term(B)"; |
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21 by (REPEAT (ares_tac ([lfp_mono, list_mono] @ basic_monos) 1)); |
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22 qed "term_mono"; |
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23 |
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24 (** Type checking -- term creates well-founded sets **) |
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25 |
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26 goalw Term.thy term.defs "term(sexp) <= sexp"; |
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27 by (rtac lfp_lowerbound 1); |
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28 by (fast_tac (!claset addIs [sexp.SconsI, list_sexp RS subsetD]) 1); |
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29 qed "term_sexp"; |
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30 |
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31 (* A <= sexp ==> term(A) <= sexp *) |
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32 bind_thm ("term_subset_sexp", ([term_mono, term_sexp] MRS subset_trans)); |
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33 |
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34 |
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35 (** Elimination -- structural induction on the set term(A) **) |
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36 |
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37 (*Induction for the set term(A) *) |
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38 val [major,minor] = goal Term.thy |
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39 "[| M: term(A); \ |
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40 \ !!x zs. [| x: A; zs: list(term(A)); zs: list({x.R(x)}) \ |
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41 \ |] ==> R(x$zs) \ |
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42 \ |] ==> R(M)"; |
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43 by (rtac (major RS term.induct) 1); |
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44 by (REPEAT (eresolve_tac ([minor] @ |
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45 ([Int_lower1,Int_lower2] RL [list_mono RS subsetD])) 1)); |
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46 (*Proof could also use mono_Int RS subsetD RS IntE *) |
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47 qed "Term_induct"; |
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48 |
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49 (*Induction on term(A) followed by induction on list *) |
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50 val major::prems = goal Term.thy |
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51 "[| M: term(A); \ |
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52 \ !!x. [| x: A |] ==> R(x$NIL); \ |
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53 \ !!x z zs. [| x: A; z: term(A); zs: list(term(A)); R(x$zs) \ |
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54 \ |] ==> R(x $ CONS z zs) \ |
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55 \ |] ==> R(M)"; |
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56 by (rtac (major RS Term_induct) 1); |
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57 by (etac list.induct 1); |
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58 by (REPEAT (ares_tac prems 1)); |
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59 qed "Term_induct2"; |
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60 |
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61 (*** Structural Induction on the abstract type 'a term ***) |
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62 |
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63 val Rep_term_in_sexp = |
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64 Rep_term RS (range_Leaf_subset_sexp RS term_subset_sexp RS subsetD); |
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65 |
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66 (*Induction for the abstract type 'a term*) |
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67 val prems = goalw Term.thy [App_def,Rep_Tlist_def,Abs_Tlist_def] |
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68 "[| !!x ts. (ALL t: set_of_list ts. R t) ==> R(App x ts) \ |
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69 \ |] ==> R(t)"; |
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70 by (rtac (Rep_term_inverse RS subst) 1); (*types force good instantiation*) |
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71 by (res_inst_tac [("P","Rep_term(t) : sexp")] conjunct2 1); |
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72 by (rtac (Rep_term RS Term_induct) 1); |
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73 by (REPEAT (ares_tac [conjI, sexp.SconsI, term_subset_sexp RS |
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74 list_subset_sexp, range_Leaf_subset_sexp] 1 |
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75 ORELSE etac rev_subsetD 1)); |
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76 by (eres_inst_tac [("A1","term(?u)"), ("f1","Rep_term"), ("g1","Abs_term")] |
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77 (Abs_map_inverse RS subst) 1); |
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78 by (rtac (range_Leaf_subset_sexp RS term_subset_sexp) 1); |
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79 by (etac Abs_term_inverse 1); |
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80 by (etac rangeE 1); |
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81 by (hyp_subst_tac 1); |
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82 by (resolve_tac prems 1); |
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83 by (etac list.induct 1); |
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84 by (etac CollectE 2); |
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85 by (stac Abs_map_CONS 2); |
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86 by (etac conjunct1 2); |
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87 by (etac rev_subsetD 2); |
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88 by (rtac list_subset_sexp 2); |
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89 by (ALLGOALS Asm_simp_tac); |
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90 by (ALLGOALS Fast_tac); |
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91 qed "term_induct"; |
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92 |
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93 (*Induction for the abstract type 'a term*) |
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94 val prems = goal Term.thy |
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95 "[| !!x. R(App x Nil); \ |
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96 \ !!x t ts. R(App x ts) ==> R(App x (t#ts)) \ |
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97 \ |] ==> R(t)"; |
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98 by (rtac term_induct 1); (*types force good instantiation*) |
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99 by (etac rev_mp 1); |
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100 by (rtac list_induct2 1); (*types force good instantiation*) |
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101 by (ALLGOALS (asm_simp_tac (!simpset addsimps prems))); |
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102 qed "term_induct2"; |
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103 |
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104 (*Perform induction on xs. *) |
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105 fun term_ind2_tac a i = |
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106 EVERY [res_inst_tac [("t",a)] term_induct2 i, |
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107 rename_last_tac a ["1","s"] (i+1)]; |
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108 |
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109 |
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110 |
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111 (*** Term_rec -- by wf recursion on pred_sexp ***) |
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112 |
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113 goal Term.thy |
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114 "(%M. Term_rec M d) = wfrec (trancl pred_sexp) \ |
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115 \ (%g. Split(%x y. d x y (Abs_map g y)))"; |
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116 by (simp_tac (HOL_ss addsimps [Term_rec_def]) 1); |
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117 bind_thm("Term_rec_unfold", (wf_pred_sexp RS wf_trancl) RS |
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118 ((result() RS eq_reflection) RS def_wfrec)); |
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119 |
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120 (*--------------------------------------------------------------------------- |
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121 * Old: |
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122 * val Term_rec_unfold = |
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123 * wf_pred_sexp RS wf_trancl RS (Term_rec_def RS def_wfrec); |
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124 *---------------------------------------------------------------------------*) |
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125 |
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126 (** conversion rules **) |
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127 |
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128 val [prem] = goal Term.thy |
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129 "N: list(term(A)) ==> \ |
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130 \ !M. (N,M): pred_sexp^+ --> \ |
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131 \ Abs_map (cut h (pred_sexp^+) M) N = \ |
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132 \ Abs_map h N"; |
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133 by (rtac (prem RS list.induct) 1); |
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134 by (Simp_tac 1); |
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135 by (strip_tac 1); |
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136 by (etac (pred_sexp_CONS_D RS conjE) 1); |
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137 by (asm_simp_tac (!simpset addsimps [trancl_pred_sexpD1]) 1); |
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138 qed "Abs_map_lemma"; |
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139 |
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140 val [prem1,prem2,A_subset_sexp] = goal Term.thy |
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141 "[| M: sexp; N: list(term(A)); A<=sexp |] ==> \ |
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142 \ Term_rec (M$N) d = d M N (Abs_map (%Z. Term_rec Z d) N)"; |
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143 by (rtac (Term_rec_unfold RS trans) 1); |
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144 by (simp_tac (HOL_ss addsimps |
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145 [Split, |
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146 prem2 RS Abs_map_lemma RS spec RS mp, pred_sexpI2 RS r_into_trancl, |
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147 prem1, prem2 RS rev_subsetD, list_subset_sexp, |
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148 term_subset_sexp, A_subset_sexp]) 1); |
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149 qed "Term_rec"; |
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150 |
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151 (*** term_rec -- by Term_rec ***) |
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152 |
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153 local |
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154 val Rep_map_type1 = read_instantiate_sg (sign_of Term.thy) |
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155 [("f","Rep_term")] Rep_map_type; |
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156 val Rep_Tlist = Rep_term RS Rep_map_type1; |
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157 val Rep_Term_rec = range_Leaf_subset_sexp RSN (2,Rep_Tlist RSN(2,Term_rec)); |
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158 |
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159 (*Now avoids conditional rewriting with the premise N: list(term(A)), |
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160 since A will be uninstantiated and will cause rewriting to fail. *) |
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161 val term_rec_ss = HOL_ss |
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162 addsimps [Rep_Tlist RS (rangeI RS term.APP_I RS Abs_term_inverse), |
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163 Rep_term_in_sexp, Rep_Term_rec, Rep_term_inverse, inj_Leaf, |
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164 inv_f_f, Abs_Rep_map, map_ident2, sexp.LeafI] |
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165 in |
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166 |
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167 val term_rec = prove_goalw Term.thy |
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168 [term_rec_def, App_def, Rep_Tlist_def, Abs_Tlist_def] |
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169 "term_rec (App f ts) d = d f ts (map (%t. term_rec t d) ts)" |
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170 (fn _ => [simp_tac term_rec_ss 1]) |
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171 |
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172 end; |
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