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1 (* Title: pattern |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow and Christine Heinzelmann, TU Muenchen |
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4 Copyright 1993 TU Muenchen |
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5 |
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6 Unification of Higher-Order Patterns. |
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7 |
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8 See also: |
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9 Tobias Nipkow. Functional Unification of Higher-Order Patterns. |
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10 In Proceedings of the 8th IEEE Symposium Logic in Computer Science, 1993. |
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11 *) |
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12 |
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13 signature PATTERN = |
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14 sig |
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15 type type_sig |
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16 type sg |
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17 type env |
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18 val eta_contract: term -> term |
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19 val match: type_sig -> term * term |
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20 -> (indexname*typ)list * (indexname*term)list |
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21 val eta_matches: type_sig -> term * term -> bool |
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22 val unify: sg * env * (term * term)list -> env |
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23 exception Unif |
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24 exception MATCH |
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25 exception Pattern |
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26 end; |
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27 |
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28 functor PatternFun(structure Sign:SIGN and Envir:ENVIR): PATTERN = |
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29 struct |
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30 |
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31 structure Type = Sign.Type; |
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32 |
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33 type type_sig = Type.type_sig |
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34 type sg = Sign.sg |
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35 type env = Envir.env |
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36 |
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37 exception Unif; |
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38 exception Pattern; |
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39 |
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40 fun occurs(F,t,env) = |
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41 let fun occ(Var(G,_)) = (case Envir.lookup(env,G) of |
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42 Some(t) => occ t |
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43 | None => F=G) |
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44 | occ(t1$t2) = occ t1 orelse occ t2 |
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45 | occ(Abs(_,_,t)) = occ t |
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46 | occ _ = false |
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47 in occ t end; |
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48 |
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49 (* Something's wrong *) |
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50 fun ill_formed s = error ("Ill-formed argument in "^s); |
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51 |
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52 |
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53 fun mapbnd f = |
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54 let fun mpb d (Bound(i)) = if i < d then Bound(i) else Bound(f(i-d)+d) |
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55 | mpb d (Free(c,T)) = Free(c,T) |
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56 | mpb d (Const(c,T)) = Const(c,T) |
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57 | mpb d (Var(iname,T)) = Var(iname,T) |
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58 | mpb d (Abs(s,T,t)) = Abs(s,T,mpb(d+1) t) |
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59 | mpb d ((u1 $ u2)) = mpb d (u1)$ mpb d (u2) |
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60 in mpb 0 end; |
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61 |
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62 fun idx [] j = ~10000 |
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63 | idx(i::is) j = if i=j then length is else idx is j; |
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64 |
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65 val nth_type = snd o nth_elem; |
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66 |
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67 fun at xs i = nth_elem (i,xs); |
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68 |
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69 fun mkabs (binders,is,t) = |
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70 let fun mk(i::is) = let val (x,T) = nth_elem(i,binders) |
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71 in Abs(x,T,mk is) end |
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72 | mk [] = t |
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73 in mk is end; |
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74 |
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75 val incr = mapbnd (fn i => i+1); |
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76 |
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77 (* termlist --> intlist *) |
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78 fun ints_of [] = [] |
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79 | ints_of (Bound i ::bs) = |
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80 let val is = ints_of bs |
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81 in if i mem is then raise Pattern else i::is end |
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82 | ints_of _ = raise Pattern; |
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83 |
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84 |
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85 fun app (s,(i::is)) = app (s$Bound(i),is) |
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86 | app (s,[]) = s; |
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87 |
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88 fun red (Abs(_,_,s)) (i::is) js = red s is (i::js) |
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89 | red s is jn = app (mapbnd (at jn) s,is); |
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90 |
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91 (* split_type ([T1,....,Tn]---> T,n,[]) = ([Tn,...,T1],T) *) |
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92 fun split_type (T,0,Ts) = (Ts,T) |
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93 | split_type (Type ("fun",[T1,T2]),n,Ts) = split_type (T2,n-1,T1::Ts) |
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94 | split_type _ = ill_formed("split_type"); |
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95 |
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96 fun type_of_G (T,n,is) = |
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97 let val (Ts,U) = split_type(T,n,[]) in map(at Ts)is ---> U end; |
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98 |
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99 fun mkhnf (binders,is,G,js) = mkabs (binders, is, app(G,js)); |
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100 |
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101 fun mknewhnf(env,binders,is,F as (a,_),T,js) = |
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102 let val (env',G) = Envir.genvar a (env,type_of_G(T,length is,js)) |
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103 in Envir.update((F,mkhnf(binders,is,G,js)),env') end; |
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104 |
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105 |
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106 fun devar env t = case strip_comb t of |
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107 (Var(F,_),ys) => |
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108 (case Envir.lookup(env,F) of |
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109 Some(t) => devar env (red t (ints_of ys) []) |
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110 | None => t) |
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111 | _ => t; |
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112 |
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113 |
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114 (* mk_proj_list(is) = [ |is| - k | 1 <= k <= |is| and is[k] >= 0 ] *) |
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115 fun mk_proj_list is = |
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116 let fun mk(i::is,j) = if i >= 0 then j :: mk(is,j-1) else mk(is,j-1) |
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117 | mk([],_) = [] |
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118 in mk(is,length is - 1) end; |
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119 |
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120 fun proj(s,env,binders,is) = |
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121 let fun trans d i = if i<d then i else (idx is (i-d))+d; |
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122 fun pr(s,env,d,binders) = (case devar env s of |
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123 Abs(a,T,t) => let val (t',env') = pr(t,env,d+1,((a,T)::binders)) |
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124 in (Abs(a,T,t'),env') end |
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125 | t => (case strip_comb t of |
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126 (c as Const _,ts) => |
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127 let val (ts',env') = prs(ts,env,d,binders) |
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128 in (list_comb(c,ts'),env') end |
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129 | (f as Free _,ts) => |
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130 let val (ts',env') = prs(ts,env,d,binders) |
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131 in (list_comb(f,ts'),env') end |
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132 | (Bound(i),ts) => |
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133 let val j = trans d i |
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134 in if j < 0 then raise Unif |
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135 else let val (ts',env') = prs(ts,env,d,binders) |
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136 in (list_comb(Bound j,ts'),env') end |
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137 end |
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138 | (Var(F as (a,_),Fty),ts) => |
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139 let val js = ints_of ts; |
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140 val js' = map (trans d) js; |
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141 val ks = mk_proj_list js'; |
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142 val ls = filter (fn i => i >= 0) js' |
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143 val Hty = type_of_G(Fty,length js,ks) |
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144 val (env',H) = Envir.genvar a (env,Hty) |
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145 val env'' = |
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146 Envir.update((F,mkhnf(binders,js,H,ks)),env') |
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147 in (app(H,ls),env'') end |
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148 | _ => raise Pattern)) |
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149 and prs(s::ss,env,d,binders) = |
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150 let val (s',env1) = pr(s,env,d,binders) |
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151 val (ss',env2) = prs(ss,env1,d,binders) |
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152 in (s'::ss',env2) end |
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153 | prs([],env,_,_) = ([],env) |
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154 in if downto0(is,length binders - 1) then (s,env) |
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155 else pr(s,env,0,binders) |
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156 end; |
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157 |
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158 |
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159 (* mk_ff_list(is,js) = [ length(is) - k | 1 <= k <= |is| and is[k] = js[k] ] *) |
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160 fun mk_ff_list(is,js) = |
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161 let fun mk([],[],_) = [] |
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162 | mk(i::is,j::js, k) = if i=j then k :: mk(is,js,k-1) |
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163 else mk(is,js,k-1) |
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164 | mk _ = ill_formed"mk_ff_list" |
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165 in mk(is,js,length is-1) end; |
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166 |
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167 fun flexflex1(env,binders,F,Fty,is,js) = |
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168 if is=js then env |
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169 else let val ks = mk_ff_list(is,js) |
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170 in mknewhnf(env,binders,is,F,Fty,ks) end; |
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171 |
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172 fun flexflex2(env,binders,F,Fty,is,G,Gty,js) = |
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173 let fun ff(F,Fty,is,G as (a,_),Gty,js) = |
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174 if js subset is |
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175 then let val t= mkabs(binders,is,app(Var(G,Gty),map (idx is) js)) |
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176 in Envir.update((F,t),env) end |
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177 else let val ks = is inter js |
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178 val Hty = type_of_G(Fty,length is,map (idx is) ks) |
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179 val (env',H) = Envir.genvar a (env,Hty) |
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180 fun lam(is) = mkabs(binders,is,app(H,map (idx is) ks)); |
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181 in Envir.update((G,lam js), Envir.update((F,lam is),env')) |
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182 end; |
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183 in if xless(G,F) then ff(F,Fty,is,G,Gty,js) else ff(G,Gty,js,F,Fty,is) end |
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184 |
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185 val tsgr = ref(Type.tsig0); |
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186 |
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187 fun unify_types(T,U, env as Envir.Envir{asol,iTs,maxidx}) = |
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188 if T=U then env |
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189 else let val iTs' = Type.unify (!tsgr) ((U,T),iTs) |
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190 in Envir.Envir{asol=asol,maxidx=maxidx,iTs=iTs'} end |
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191 handle Type.TUNIFY => raise Unif; |
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192 |
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193 fun unif binders (env,(s,t)) = case (devar env s,devar env t) of |
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194 (Abs(ns,Ts,ts),Abs(nt,Tt,tt)) => |
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195 let val name = if ns = "" then nt else ns |
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196 in unif ((name,Ts)::binders) (env,(ts,tt)) end |
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197 | (Abs(ns,Ts,ts),t) => unif ((ns,Ts)::binders) (env,(ts,(incr t)$Bound(0))) |
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198 | (t,Abs(nt,Tt,tt)) => unif ((nt,Tt)::binders) (env,((incr t)$Bound(0),tt)) |
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199 | p => cases(binders,env,p) |
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200 |
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201 and cases(binders,env,(s,t)) = case (strip_comb s,strip_comb t) of |
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202 ((Var(F,Fty),ss),(Var(G,Gty),ts)) => |
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203 if F = G then flexflex1(env,binders,F,Fty,ints_of ss,ints_of ts) |
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204 else flexflex2(env,binders,F,Fty,ints_of ss,G,Gty,ints_of ts) |
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205 | ((Var(F,_),ss),_) => flexrigid(env,binders,F,ints_of ss,t) |
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206 | (_,(Var(F,_),ts)) => flexrigid(env,binders,F,ints_of ts,s) |
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207 | ((Const c,ss),(Const d,ts)) => rigidrigid(env,binders,c,d,ss,ts) |
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208 | ((Free(f),ss),(Free(g),ts)) => rigidrigid(env,binders,f,g,ss,ts) |
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209 | ((Bound(i),ss),(Bound(j),ts)) => rigidrigidB (env,binders,i,j,ss,ts) |
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210 | ((Abs(_),_),_) => raise Pattern |
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211 | (_,(Abs(_),_)) => raise Pattern |
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212 | _ => raise Unif |
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213 |
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214 and rigidrigid (env,binders,(a,Ta),(b,Tb),ss,ts) = |
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215 if a<>b then raise Unif |
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216 else foldl (unif binders) (unify_types(Ta,Tb,env), ss~~ts) |
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217 |
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218 and rigidrigidB (env,binders,i,j,ss,ts) = |
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219 if i <> j then raise Unif else foldl (unif binders) (env ,ss~~ts) |
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220 |
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221 and flexrigid (env,binders,F,is,t) = |
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222 if occurs(F,t,env) then raise Unif |
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223 else let val (u,env') = proj(t,env,binders,is) |
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224 in Envir.update((F,mkabs(binders,is,u)),env') end; |
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225 |
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226 fun unify(sg,env,tus) = (tsgr := #tsig(Sign.rep_sg sg); |
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227 foldl (unif []) (env,tus)); |
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228 |
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229 |
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230 (*Perform eta-contractions upon a term*) |
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231 fun eta_contract (Abs(a,T,body)) = |
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232 (case eta_contract body of |
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233 body' as (f $ Bound i) => |
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234 if i=0 andalso not (0 mem loose_bnos f) then incr_boundvars ~1 f |
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235 else Abs(a,T,body') |
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236 | body' => Abs(a,T,body')) |
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237 | eta_contract(f$t) = eta_contract f $ eta_contract t |
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238 | eta_contract t = t; |
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239 |
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240 |
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241 (* Pattern matching. Raises MATCH if non-pattern *) |
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242 exception MATCH; |
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243 (* something wron with types, esp in abstractions |
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244 fun typ_match args = Type.typ_match (!tsgr) args |
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245 handle Type.TYPE_MATCH => raise MATCH; |
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246 |
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247 fun match_bind(itms,binders,ixn,is,t) = |
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248 let val js = loose_bnos t |
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249 in if null is |
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250 then if null js then (ixn,t)::itms else raise MATCH |
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251 else if js subset is |
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252 then let val t' = if downto0(is,length binders - 1) then t |
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253 else mapbnd (idx is) t |
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254 in (ixn, eta_contract(mkabs(binders,is,t'))) :: itms end |
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255 else raise MATCH |
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256 end; |
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257 |
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258 fun match_rr (iTs,(a,Ta),(b,Tb)) = |
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259 if a<>b then raise MATCH else typ_match (iTs,(Ta,Tb)) |
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260 |
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261 (* Pre: pat and obj have same type *) |
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262 fun mtch(binders,env as (iTs,itms),pat,obj) = case pat of |
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263 Var(ixn,_) => (case assoc(itms,ixn) of |
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264 None => (iTs,match_bind(itms,binders,ixn,[],obj)) |
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265 | Some u => if obj aconv u then env else raise MATCH) |
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266 | Abs(ns,Ts,ts) => |
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267 (case obj of |
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268 Abs(nt,Tt,tt) => mtch((nt,Tt)::binders,env,ts,tt) |
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269 | _ => let val Tt = typ_subst_TVars iTs Ts |
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270 in mtch((ns,Tt)::binders,env,ts,(incr obj)$Bound(0)) end) |
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271 | _ => (case obj of |
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272 Abs(nt,Tt,tt) => |
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273 mtch((nt,Tt)::binders,env,(incr pat)$Bound(0),tt) |
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274 | _ => cases(binders,env,pat,obj)) |
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275 |
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276 and cases(binders,env as (iTs,itms),pat,obj) = |
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277 let fun structural() = case (pat,obj) of |
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278 (Const c,Const d) => (match_rr(iTs,c,d),itms) |
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279 | (Free f,Free g) => (match_rr(iTs,f,g),itms) |
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280 | (Bound i,Bound j) => if i=j then env else raise MATCH |
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281 | (f$t,g$u) => mtch(binders,mtch(binders,env,t,u),f,g) |
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282 | _ => raise MATCH |
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283 in case strip_comb pat of |
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284 (Var(ixn,_),bs) => |
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285 (let val is = ints_of bs |
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286 in case assoc(itms,ixn) of |
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287 None => (iTs,match_bind(itms,binders,ixn,is,obj)) |
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288 | Some u => if obj aconv (red u is []) then env else raise MATCH |
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289 end (* if ints_of fails: *) handle Pattern => structural()) |
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290 | _ => structural() |
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291 end; |
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292 |
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293 fun match tsg = (tsgr := tsg; |
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294 fn (pat,obj) => |
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295 let val pT = fastype_of([],pat) |
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296 and oT = fastype_of([],obj) |
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297 val iTs = typ_match([],(pT,oT)) |
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298 in mtch([], (iTs,[]), pat, eta_contract obj) |
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299 handle Pattern => raise MATCH |
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300 end) |
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301 |
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302 (*Predicate: does the pattern match the object?*) |
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303 fun matches tsig args = (match tsig args; true) |
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304 handle MATCH => false; |
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305 *) |
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306 |
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307 (*First-order matching; term_match tsig (pattern, object) |
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308 returns a (tyvar,typ)list and (var,term)list. |
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309 The pattern and object may have variables in common. |
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310 Instantiation does not affect the object, so matching ?a with ?a+1 works. |
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311 A Const does not match a Free of the same name! |
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312 Does not notice eta-equality, thus f does not match %(x)f(x) *) |
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313 fun match tsig (pat,obj) = |
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314 let fun typ_match args = (Type.typ_match tsig args) |
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315 handle Type.TYPE_MATCH => raise MATCH; |
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316 fun mtch (tyinsts,insts) = fn |
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317 (Var(ixn,T), t) => |
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318 if null (loose_bnos t) |
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319 then case assoc(insts,ixn) of |
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320 None => (typ_match (tyinsts, (T,fastype_of([],t))), |
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321 (ixn,t)::insts) |
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322 | Some u => if t aconv u then (tyinsts,insts) else raise MATCH |
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323 else raise MATCH |
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324 | (Free (a,T), Free (b,U)) => |
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325 if a=b then (typ_match (tyinsts,(T,U)), insts) else raise MATCH |
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326 | (Const (a,T), Const (b,U)) => |
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327 if a=b then (typ_match (tyinsts,(T,U)), insts) else raise MATCH |
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328 | (Bound i, Bound j) => |
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329 if i=j then (tyinsts,insts) else raise MATCH |
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330 | (Abs(_,T,t), Abs(_,U,u)) => |
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331 mtch (typ_match (tyinsts,(T,U)),insts) (t,u) |
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332 | (f$t, g$u) => mtch (mtch (tyinsts,insts) (f,g)) (t, u) |
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333 | _ => raise MATCH |
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334 in mtch([],[]) (pat,obj) end; |
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335 |
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336 (*Predicate: does the pattern match the object?*) |
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337 fun eta_matches tsig (pat,obj) = |
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338 (match tsig (eta_contract pat,eta_contract obj); true) |
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339 handle MATCH => false; |
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340 |
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341 end; |