matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_fpcs.etc

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2/computation/fprs_fprs.ma".
  16 include "basic_2/conversion/fpc_fpc.ma".
  17 include "basic_2/equivalence/fpcs_fprs.ma".
  18
  19 (* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
  20
  21 (* Advanced inversion lemmas ************************************************)
  22
  23 lemma fpcs_inv_fprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ →
  24                      ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡* ⦃L, T⦄.
  25 #L1 #L2 #T1 #T2 #H @(fpcs_ind … H) -L2 -T2
  26 [ /3 width=4/
  27 | #L #L2 #T #T2 #_ #HT2 * #L0 #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
  28   [ elim (fprs_strip … HT2 … HT0) -L -T #L #T #HT2 #HT0
  29     lapply (fprs_strap1 … HT10 … HT0) -L0 -T0 /2 width=4/
  30   | lapply (fprs_strap2 … HT2 … HT0) -L -T /2 width=4/
  31   ]
  32 ]
  33 qed-.
  34
  35 (* Advanced properties ******************************************************)
  36
  37 lemma fpr_fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  38 #L #L1 #L2 #T #T1 #T2 #HT1 #HT2
  39 elim (fprs_strip … HT2 … HT1) /2 width=4 by fpr_fprs_div/
  40 qed-.
  41
  42 lemma fprs_fpr_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L2, T2⦄ ⬌* ⦃L1, T1⦄.
  43 #L #L1 #L2 #T #T1 #T2 #HT1 #HT2
  44 elim (fprs_strip … HT2 … HT1) /2 width=4 by fprs_fpr_div/
  45 qed-.
  46
  47 lemma fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  48 #L #L1 #L2 #T #T1 #T2 #HT1 #HT2
  49 elim (fprs_conf … HT1 … HT2) /2 width=4/
  50 qed-.
  51
  52 lemma fpcs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ →
  53                   ∀L2,T2. ⦃L0, T0⦄ ⬌* ⦃L2, T2⦄ →
  54                   ∃∃L,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄.
  55 /3 width=4/ qed.
  56
  57 (* Main properties **********************************************************)
  58
  59 theorem fpcs_trans: bi_transitive … fpcs.
  60 /2 width=4/ qed.
  61
  62 theorem fpcs_canc_sn: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ⬌* ⦃L1, T1⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  63 /3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *)
  64
  65 theorem fpcs_canc_dx: ∀L1,L2,L,T1,T2,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L2, T2⦄ ⬌* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  66 /3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *)