helm/software/matita/contribs/CoRN-Decl/tactics/RingReflection.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 set "baseuri" "cic:/matita/CoRN-Decl/tactics/RingReflection".
  18
  19 (* $Id: RingReflection.v,v 1.4 2004/04/23 10:01:06 lcf Exp $ *)
  20
  21 (* begin hide *)
  22
  23 (* INCLUDE
  24 CRings
  25 *)
  26
  27 (* INCLUDE
  28 AlgReflection
  29 *)
  30
  31 (* UNEXPORTED
  32 Section Ring_Interpretation_Function.
  33 *)
  34
  35 inline cic:/CoRN/tactics/RingReflection/R.var.
  36
  37 inline cic:/CoRN/tactics/RingReflection/val.var.
  38
  39 inline cic:/CoRN/tactics/RingReflection/unop.var.
  40
  41 inline cic:/CoRN/tactics/RingReflection/binop.var.
  42
  43 inline cic:/CoRN/tactics/RingReflection/pfun.var.
  44
  45 inline cic:/CoRN/tactics/RingReflection/interpR.ind.
  46
  47 inline cic:/CoRN/tactics/RingReflection/wfR.con.
  48
  49 inline cic:/CoRN/tactics/RingReflection/xexprR.ind.
  50
  51 inline cic:/CoRN/tactics/RingReflection/xforgetR.con.
  52
  53 inline cic:/CoRN/tactics/RingReflection/xinterpR.con.
  54
  55 inline cic:/CoRN/tactics/RingReflection/xexprR2interpR.con.
  56
  57 inline cic:/CoRN/tactics/RingReflection/xexprR_diagram_commutes.con.
  58
  59 inline cic:/CoRN/tactics/RingReflection/xexprR2wfR.con.
  60
  61 inline cic:/CoRN/tactics/RingReflection/fexprR.ind.
  62
  63 inline cic:/CoRN/tactics/RingReflection/fexprR_var.con.
  64
  65 inline cic:/CoRN/tactics/RingReflection/fexprR_int.con.
  66
  67 inline cic:/CoRN/tactics/RingReflection/fexprR_plus.con.
  68
  69 inline cic:/CoRN/tactics/RingReflection/fexprR_mult.con.
  70
  71 inline cic:/CoRN/tactics/RingReflection/fforgetR.con.
  72
  73 inline cic:/CoRN/tactics/RingReflection/fexprR2interp.con.
  74
  75 inline cic:/CoRN/tactics/RingReflection/fexprR2wf.con.
  76
  77 (* UNEXPORTED
  78 Opaque csg_crr.
  79 *)
  80
  81 (* UNEXPORTED
  82 Opaque cm_crr.
  83 *)
  84
  85 (* UNEXPORTED
  86 Opaque cg_crr.
  87 *)
  88
  89 (* UNEXPORTED
  90 Opaque cr_crr.
  91 *)
  92
  93 (* UNEXPORTED
  94 Opaque csf_fun.
  95 *)
  96
  97 (* UNEXPORTED
  98 Opaque csbf_fun.
  99 *)
 100
 101 (* UNEXPORTED
 102 Opaque csr_rel.
 103 *)
 104
 105 (* UNEXPORTED
 106 Opaque cs_eq.
 107 *)
 108
 109 (* UNEXPORTED
 110 Opaque cs_neq.
 111 *)
 112
 113 (* UNEXPORTED
 114 Opaque cs_ap.
 115 *)
 116
 117 (* UNEXPORTED
 118 Opaque cm_unit.
 119 *)
 120
 121 (* UNEXPORTED
 122 Opaque csg_op.
 123 *)
 124
 125 (* UNEXPORTED
 126 Opaque cg_inv.
 127 *)
 128
 129 (* UNEXPORTED
 130 Opaque cg_minus.
 131 *)
 132
 133 (* UNEXPORTED
 134 Opaque cr_one.
 135 *)
 136
 137 (* UNEXPORTED
 138 Opaque cr_mult.
 139 *)
 140
 141 (* UNEXPORTED
 142 Opaque nexp_op.
 143 *)
 144
 145 inline cic:/CoRN/tactics/RingReflection/refl_interpR.con.
 146
 147 inline cic:/CoRN/tactics/RingReflection/interpR_wd.con.
 148
 149 (* UNEXPORTED
 150 End Ring_Interpretation_Function.
 151 *)
 152
 153 (* UNEXPORTED
 154 Section Ring_NormCorrect.
 155 *)
 156
 157 inline cic:/CoRN/tactics/RingReflection/R.var.
 158
 159 inline cic:/CoRN/tactics/RingReflection/val.var.
 160
 161 inline cic:/CoRN/tactics/RingReflection/unop.var.
 162
 163 inline cic:/CoRN/tactics/RingReflection/binop.var.
 164
 165 inline cic:/CoRN/tactics/RingReflection/pfun.var.
 166
 167 (*
 168 four kinds of exprs:
 169
 170   I     (expr_int _)
 171   V     (expr_var _)
 172   M     (expr_mult V M)
 173         I
 174   P     (expr_plus M P)
 175         I
 176
 177 M: sorted on V
 178 P: sorted on M, all M's not an I
 179 *)
 180
 181 (* UNEXPORTED
 182 Opaque Zmult.
 183 *)
 184
 185 inline cic:/CoRN/tactics/RingReflection/MI_mult_corr_R.con.
 186
 187 (* UNEXPORTED
 188 Transparent Zmult.
 189 *)
 190
 191 (* UNEXPORTED
 192 Opaque MI_mult.
 193 *)
 194
 195 inline cic:/CoRN/tactics/RingReflection/MV_mult_corr_R.con.
 196
 197 (* UNEXPORTED
 198 Transparent MI_mult.
 199 *)
 200
 201 (* UNEXPORTED
 202 Opaque MV_mult MI_mult.
 203 *)
 204
 205 inline cic:/CoRN/tactics/RingReflection/MM_mult_corr_R.con.
 206
 207 (* UNEXPORTED
 208 Transparent MV_mult MI_mult.
 209 *)
 210
 211 (* UNEXPORTED
 212 Opaque MV_mult.
 213 *)
 214
 215 inline cic:/CoRN/tactics/RingReflection/MM_plus_corr_R.con.
 216
 217 (* UNEXPORTED
 218 Transparent MV_mult.
 219 *)
 220
 221 (* UNEXPORTED
 222 Opaque MM_plus.
 223 *)
 224
 225 inline cic:/CoRN/tactics/RingReflection/PM_plus_corr_R.con.
 226
 227 (* UNEXPORTED
 228 Transparent MM_plus.
 229 *)
 230
 231 (* UNEXPORTED
 232 Opaque PM_plus.
 233 *)
 234
 235 inline cic:/CoRN/tactics/RingReflection/PP_plus_corr_R.con.
 236
 237 (* UNEXPORTED
 238 Transparent PM_plus.
 239 *)
 240
 241 (* UNEXPORTED
 242 Opaque PM_plus MM_mult MI_mult.
 243 *)
 244
 245 inline cic:/CoRN/tactics/RingReflection/PM_mult_corr_R.con.
 246
 247 (* UNEXPORTED
 248 Opaque PM_mult.
 249 *)
 250
 251 inline cic:/CoRN/tactics/RingReflection/PP_mult_corr_R.con.
 252
 253 (*
 254 Transparent PP_plus PM_mult PP_mult PM_plus MI_mult.
 255 Lemma FF_plus_corr_R : (e,f:expr; x,y:R)
 256   (II e x)->(II f y)->(II (FF_plus e f) x[+]y).
 257 Cut (e1,e2,f1,f2:expr; x,y:R)
 258      (II (expr_div e1 e2) x)
 259      ->(II (expr_div f1 f2) y)
 260      ->(II
 261          (expr_div (PP_plus (PP_mult e1 f2) (PP_mult e2 f1))
 262            (PP_mult e2 f2)) x[+]y).
 263 Cut (e,f:expr; x,y:R)(II e x)->(II f y)->(II (expr_plus e f) x[+]y).
 264 Intros H H0 e f.
 265 Elim e; Elim f; Intros; Simpl; Auto.
 266 Intros. Apply interpR_plus with x y; Algebra.
 267 Intros. Inversion H. Inversion H0.
 268 Apply interpR_div_one with x[+]y.
 269 Algebra.
 270 Apply interpR_wd with x0[*]One[+]One[*]x1.
 271 Apply PP_plus_corr_R; Apply PP_mult_corr_R; Auto;
 272   Apply interpR_int with k:=`1`; Algebra.
 273 Step_final x0[+]x1.
 274 Apply interpR_wd with (One::R)[*]One; Algebra.
 275 Apply PP_mult_corr_R; Auto.
 276 Qed.
 277
 278 Lemma FF_mult_corr_R : (e,f:expr; x,y:R)
 279   (II e x)->(II f y)->(II (FF_mult e f) x[*]y).
 280 Cut (e1,e2,f1,f2:expr; x,y:R)
 281      (II (expr_div e1 e2) x)
 282      ->(II (expr_div f1 f2) y)
 283      ->(II (expr_div (PP_mult e1 f1) (PP_mult e2 f2)) x[*]y).
 284 Cut (e,f:expr; x,y:R)(II e x)->(II f y)->(II (expr_mult e f) x[*]y).
 285 Intros H H0 e f.
 286 Elim e; Elim f; Intros; Simpl; Auto.
 287 Intros. Apply interpR_mult with x y; Algebra.
 288 Intros. Inversion H. Inversion H0.
 289 Apply interpR_div_one with x0[*]x1.
 290 Algebra.
 291 Apply PP_mult_corr_R; Auto.
 292 Apply interpR_wd with (One::R)[*]One; Algebra.
 293 Apply PP_mult_corr_R; Auto.
 294 Qed.
 295
 296 Transparent FF_div.
 297 Lemma FF_div_corr_R : (e,f:expr; x:R)
 298   (II (expr_div e f) x)->(II (FF_div e f) x).
 299 Intro e; Case e; Simpl; Auto.
 300 Intros e0 e1 f; Case f; Simpl; Auto.
 301 Intros.
 302 Inversion H; Simpl.
 303 Inversion H3; Inversion H5.
 304 Apply interpR_div_one with x1[*]One.
 305 astepl x1. Step_final x0.
 306 Apply PP_mult_corr_R; Auto.
 307 Apply interpR_wd with One[*]x2.
 308 Apply PP_mult_corr_R; Auto.
 309 Step_final x2.
 310 Qed.
 311 *)
 312
 313 inline cic:/CoRN/tactics/RingReflection/NormR_corr.con.
 314
 315 inline cic:/CoRN/tactics/RingReflection/Tactic_lemmaR.con.
 316
 317 (* UNEXPORTED
 318 End Ring_NormCorrect.
 319 *)
 320
 321 (* end hide *)
 322