matita/matita/contribs/lambda_delta/basic_2/equivalence/lcpcs_lcpcs.ma

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  14
  15 include "basic_2/computation/lcprs_lcprs.ma".
  16 include "basic_2/conversion/lcpc_lcpc.ma".
  17 include "basic_2/equivalence/lcpcs_lcprs.ma".
  18
  19 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************)
  20
  21 (* Advanced inversion lemmas ************************************************)
  22
  23 lemma lcpcs_inv_lcprs: ∀L1,L2. L1 ⊢ ⬌* L2 →
  24                        ∃∃L. L1 ⊢ ➡* L & L2 ⊢ ➡* L.
  25 #L1 #L2 #H @(lcpcs_ind … H) -L2
  26 [ /3 width=3/
  27 | #L #L2 #_ #HL2 * #L0 #HL10 elim HL2 -HL2 #HL2 #HL0
  28   [ elim (lcprs_strip … HL0 … HL2) -L #L #HL0 #HL2
  29     lapply (lcprs_strap1 … HL10 … HL0) -L0 /2 width=3/
  30   | lapply (lcprs_strap2 … HL2 … HL0) -L /2 width=3/
  31   ]
  32 ]
  33 qed-.
  34
  35 (* Advanced properties ******************************************************)
  36
  37 lemma lcpcs_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 lcpcs_trans: ∀L1,L. L1 ⊢ ⬌* L → ∀L2. L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
  44 /2 width=3/ qed.
  45
  46 theorem lcpcs_canc_sn: ∀L,L1,L2. L ⊢ ⬌* L1 → L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
  47 /3 width=3 by lcpcs_trans, lcprs_comm/ qed.
  48
  49 theorem lcpcs_canc_dx: ∀L,L1,L2. L1 ⊢ ⬌* L → L2 ⊢ ⬌* L → L1 ⊢ ⬌* L2.
  50 /3 width=3 by lcpcs_trans, lcprs_comm/ qed.