matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs_lfpcs.etc

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  14
  15 include "basic_2/computation/lfprs_lfprs.ma".
  16 include "basic_2/conversion/lfpc_lfpc.ma".
  17 include "basic_2/equivalence/lfpcs_lfprs.ma".
  18
  19 (* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
  20
  21 (* Advanced inversion lemmas ************************************************)
  22
  23 lemma lfpcs_inv_lfprs: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
  24                        ∃∃L. ⦃L1⦄ ➡* ⦃L⦄ & ⦃L2⦄ ➡* ⦃L⦄.
  25 #L1 #L2 #H @(lfpcs_ind … H) -L2
  26 [ /3 width=3/
  27 | #L #L2 #_ #HL2 * #L0 #HL10 elim HL2 -HL2 #HL2 #HL0
  28   [ elim (lfprs_strip … HL0 … HL2) -L #L #HL0 #HL2
  29     lapply (lfprs_strap1 … HL10 … HL0) -L0 /2 width=3/
  30   | lapply (lfprs_strap2 … HL2 … HL0) -L /2 width=3/
  31   ]
  32 ]
  33 qed-.
  34
  35 (* Advanced properties ******************************************************)
  36
  37 lemma lfpcs_strip: ∀L,L1. ⦃L⦄ ⬌* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌ ⦃L2⦄ →
  38                    ∃∃L0. ⦃L1⦄ ⬌ ⦃L0⦄ & ⦃L2⦄ ⬌* ⦃L0⦄.
  39 /3 width=3/ qed.
  40
  41 (* Main properties **********************************************************)
  42
  43 theorem lfpcs_trans: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  44 /2 width=3/ qed.
  45
  46 theorem lfpcs_canc_sn: ∀L,L1,L2. ⦃L⦄ ⬌* ⦃L1⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  47 /3 width=3 by lfpcs_trans, lfpcs_sym/ qed.
  48
  49 theorem lfpcs_canc_dx: ∀L,L1,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L2⦄ ⬌* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  50 /3 width=3 by lfpcs_trans, lfpcs_sym/ qed.