helm/software/matita/contribs/procedural/CoRN/tactics/RingReflection.mma

   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 include "CoRN.ma".
  18
  19 (* $Id: RingReflection.v,v 1.4 2004/04/23 10:01:06 lcf Exp $ *)
  20
  21 (* begin hide *)
  22
  23 include "algebra/CRings.ma".
  24
  25 include "tactics/AlgReflection.ma".
  26
  27 (* UNEXPORTED
  28 Section Ring_Interpretation_Function
  29 *)
  30
  31 (* UNEXPORTED
  32 cic:/CoRN/tactics/RingReflection/Ring_Interpretation_Function/R.var
  33 *)
  34
  35 (* UNEXPORTED
  36 cic:/CoRN/tactics/RingReflection/Ring_Interpretation_Function/val.var
  37 *)
  38
  39 (* UNEXPORTED
  40 cic:/CoRN/tactics/RingReflection/Ring_Interpretation_Function/unop.var
  41 *)
  42
  43 (* UNEXPORTED
  44 cic:/CoRN/tactics/RingReflection/Ring_Interpretation_Function/binop.var
  45 *)
  46
  47 (* UNEXPORTED
  48 cic:/CoRN/tactics/RingReflection/Ring_Interpretation_Function/pfun.var
  49 *)
  50
  51 inline procedural "cic:/CoRN/tactics/RingReflection/interpR.ind".
  52
  53 inline procedural "cic:/CoRN/tactics/RingReflection/wfR.con" as definition.
  54
  55 inline procedural "cic:/CoRN/tactics/RingReflection/xexprR.ind".
  56
  57 inline procedural "cic:/CoRN/tactics/RingReflection/xforgetR.con" as definition.
  58
  59 inline procedural "cic:/CoRN/tactics/RingReflection/xinterpR.con" as definition.
  60
  61 inline procedural "cic:/CoRN/tactics/RingReflection/xexprR2interpR.con" as lemma.
  62
  63 inline procedural "cic:/CoRN/tactics/RingReflection/xexprR_diagram_commutes.con" as definition.
  64
  65 inline procedural "cic:/CoRN/tactics/RingReflection/xexprR2wfR.con" as lemma.
  66
  67 inline procedural "cic:/CoRN/tactics/RingReflection/fexprR.ind".
  68
  69 inline procedural "cic:/CoRN/tactics/RingReflection/fexprR_var.con" as definition.
  70
  71 inline procedural "cic:/CoRN/tactics/RingReflection/fexprR_int.con" as definition.
  72
  73 inline procedural "cic:/CoRN/tactics/RingReflection/fexprR_plus.con" as definition.
  74
  75 inline procedural "cic:/CoRN/tactics/RingReflection/fexprR_mult.con" as definition.
  76
  77 inline procedural "cic:/CoRN/tactics/RingReflection/fforgetR.con" as definition.
  78
  79 inline procedural "cic:/CoRN/tactics/RingReflection/fexprR2interp.con" as lemma.
  80
  81 inline procedural "cic:/CoRN/tactics/RingReflection/fexprR2wf.con" as lemma.
  82
  83 (* UNEXPORTED
  84 Opaque csg_crr.
  85 *)
  86
  87 (* UNEXPORTED
  88 Opaque cm_crr.
  89 *)
  90
  91 (* UNEXPORTED
  92 Opaque cg_crr.
  93 *)
  94
  95 (* UNEXPORTED
  96 Opaque cr_crr.
  97 *)
  98
  99 (* UNEXPORTED
 100 Opaque csf_fun.
 101 *)
 102
 103 (* UNEXPORTED
 104 Opaque csbf_fun.
 105 *)
 106
 107 (* UNEXPORTED
 108 Opaque csr_rel.
 109 *)
 110
 111 (* UNEXPORTED
 112 Opaque cs_eq.
 113 *)
 114
 115 (* UNEXPORTED
 116 Opaque cs_neq.
 117 *)
 118
 119 (* UNEXPORTED
 120 Opaque cs_ap.
 121 *)
 122
 123 (* UNEXPORTED
 124 Opaque cm_unit.
 125 *)
 126
 127 (* UNEXPORTED
 128 Opaque csg_op.
 129 *)
 130
 131 (* UNEXPORTED
 132 Opaque cg_inv.
 133 *)
 134
 135 (* UNEXPORTED
 136 Opaque cg_minus.
 137 *)
 138
 139 (* UNEXPORTED
 140 Opaque cr_one.
 141 *)
 142
 143 (* UNEXPORTED
 144 Opaque cr_mult.
 145 *)
 146
 147 (* UNEXPORTED
 148 Opaque nexp_op.
 149 *)
 150
 151 inline procedural "cic:/CoRN/tactics/RingReflection/refl_interpR.con" as lemma.
 152
 153 inline procedural "cic:/CoRN/tactics/RingReflection/interpR_wd.con" as lemma.
 154
 155 (* UNEXPORTED
 156 End Ring_Interpretation_Function
 157 *)
 158
 159 (* UNEXPORTED
 160 Section Ring_NormCorrect
 161 *)
 162
 163 (* UNEXPORTED
 164 cic:/CoRN/tactics/RingReflection/Ring_NormCorrect/R.var
 165 *)
 166
 167 (* UNEXPORTED
 168 cic:/CoRN/tactics/RingReflection/Ring_NormCorrect/val.var
 169 *)
 170
 171 (* UNEXPORTED
 172 cic:/CoRN/tactics/RingReflection/Ring_NormCorrect/unop.var
 173 *)
 174
 175 (* UNEXPORTED
 176 cic:/CoRN/tactics/RingReflection/Ring_NormCorrect/binop.var
 177 *)
 178
 179 (* UNEXPORTED
 180 cic:/CoRN/tactics/RingReflection/Ring_NormCorrect/pfun.var
 181 *)
 182
 183 (* NOTATION
 184 Notation II := (interpR R val unop binop pfun).
 185 *)
 186
 187 (*
 188 four kinds of exprs:
 189
 190   I     (expr_int _)
 191   V     (expr_var _)
 192   M     (expr_mult V M)
 193         I
 194   P     (expr_plus M P)
 195         I
 196
 197 M: sorted on V
 198 P: sorted on M, all M's not an I
 199 *)
 200
 201 (* UNEXPORTED
 202 Opaque Zmult.
 203 *)
 204
 205 inline procedural "cic:/CoRN/tactics/RingReflection/MI_mult_corr_R.con" as lemma.
 206
 207 (* UNEXPORTED
 208 Transparent Zmult.
 209 *)
 210
 211 (* UNEXPORTED
 212 Opaque MI_mult.
 213 *)
 214
 215 inline procedural "cic:/CoRN/tactics/RingReflection/MV_mult_corr_R.con" as lemma.
 216
 217 (* UNEXPORTED
 218 Transparent MI_mult.
 219 *)
 220
 221 (* UNEXPORTED
 222 Opaque MV_mult MI_mult.
 223 *)
 224
 225 inline procedural "cic:/CoRN/tactics/RingReflection/MM_mult_corr_R.con" as lemma.
 226
 227 (* UNEXPORTED
 228 Transparent MV_mult MI_mult.
 229 *)
 230
 231 (* UNEXPORTED
 232 Opaque MV_mult.
 233 *)
 234
 235 inline procedural "cic:/CoRN/tactics/RingReflection/MM_plus_corr_R.con" as lemma.
 236
 237 (* UNEXPORTED
 238 Transparent MV_mult.
 239 *)
 240
 241 (* UNEXPORTED
 242 Opaque MM_plus.
 243 *)
 244
 245 inline procedural "cic:/CoRN/tactics/RingReflection/PM_plus_corr_R.con" as lemma.
 246
 247 (* UNEXPORTED
 248 Transparent MM_plus.
 249 *)
 250
 251 (* UNEXPORTED
 252 Opaque PM_plus.
 253 *)
 254
 255 inline procedural "cic:/CoRN/tactics/RingReflection/PP_plus_corr_R.con" as lemma.
 256
 257 (* UNEXPORTED
 258 Transparent PM_plus.
 259 *)
 260
 261 (* UNEXPORTED
 262 Opaque PM_plus MM_mult MI_mult.
 263 *)
 264
 265 inline procedural "cic:/CoRN/tactics/RingReflection/PM_mult_corr_R.con" as lemma.
 266
 267 (* UNEXPORTED
 268 Opaque PM_mult.
 269 *)
 270
 271 inline procedural "cic:/CoRN/tactics/RingReflection/PP_mult_corr_R.con" as lemma.
 272
 273 (*
 274 Transparent PP_plus PM_mult PP_mult PM_plus MI_mult.
 275 Lemma FF_plus_corr_R : (e,f:expr; x,y:R)
 276   (II e x)->(II f y)->(II (FF_plus e f) x[+]y).
 277 Cut (e1,e2,f1,f2:expr; x,y:R)
 278      (II (expr_div e1 e2) x)
 279      ->(II (expr_div f1 f2) y)
 280      ->(II
 281          (expr_div (PP_plus (PP_mult e1 f2) (PP_mult e2 f1))
 282            (PP_mult e2 f2)) x[+]y).
 283 Cut (e,f:expr; x,y:R)(II e x)->(II f y)->(II (expr_plus e f) x[+]y).
 284 Intros H H0 e f.
 285 Elim e; Elim f; Intros; Simpl; Auto.
 286 Intros. Apply interpR_plus with x y; Algebra.
 287 Intros. Inversion H. Inversion H0.
 288 Apply interpR_div_one with x[+]y.
 289 Algebra.
 290 Apply interpR_wd with x0[*]One[+]One[*]x1.
 291 Apply PP_plus_corr_R; Apply PP_mult_corr_R; Auto;
 292   Apply interpR_int with k:=`1`; Algebra.
 293 Step_final x0[+]x1.
 294 Apply interpR_wd with (One::R)[*]One; Algebra.
 295 Apply PP_mult_corr_R; Auto.
 296 Qed.
 297
 298 Lemma FF_mult_corr_R : (e,f:expr; x,y:R)
 299   (II e x)->(II f y)->(II (FF_mult e f) x[*]y).
 300 Cut (e1,e2,f1,f2:expr; x,y:R)
 301      (II (expr_div e1 e2) x)
 302      ->(II (expr_div f1 f2) y)
 303      ->(II (expr_div (PP_mult e1 f1) (PP_mult e2 f2)) x[*]y).
 304 Cut (e,f:expr; x,y:R)(II e x)->(II f y)->(II (expr_mult e f) x[*]y).
 305 Intros H H0 e f.
 306 Elim e; Elim f; Intros; Simpl; Auto.
 307 Intros. Apply interpR_mult with x y; Algebra.
 308 Intros. Inversion H. Inversion H0.
 309 Apply interpR_div_one with x0[*]x1.
 310 Algebra.
 311 Apply PP_mult_corr_R; Auto.
 312 Apply interpR_wd with (One::R)[*]One; Algebra.
 313 Apply PP_mult_corr_R; Auto.
 314 Qed.
 315
 316 Transparent FF_div.
 317 Lemma FF_div_corr_R : (e,f:expr; x:R)
 318   (II (expr_div e f) x)->(II (FF_div e f) x).
 319 Intro e; Case e; Simpl; Auto.
 320 Intros e0 e1 f; Case f; Simpl; Auto.
 321 Intros.
 322 Inversion H; Simpl.
 323 Inversion H3; Inversion H5.
 324 Apply interpR_div_one with x1[*]One.
 325 astepl x1. Step_final x0.
 326 Apply PP_mult_corr_R; Auto.
 327 Apply interpR_wd with One[*]x2.
 328 Apply PP_mult_corr_R; Auto.
 329 Step_final x2.
 330 Qed.
 331 *)
 332
 333 inline procedural "cic:/CoRN/tactics/RingReflection/NormR_corr.con" as lemma.
 334
 335 inline procedural "cic:/CoRN/tactics/RingReflection/Tactic_lemmaR.con" as lemma.
 336
 337 (* UNEXPORTED
 338 End Ring_NormCorrect
 339 *)
 340
 341 (* end hide *)
 342