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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/computation/lcprs_lcprs.ma".
16 include "basic_2/conversion/lcpc_lcpc.ma".
17 include "basic_2/equivalence/lcpcs_lcprs.ma".
19 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************)
21 (* Advanced inversion lemmas ************************************************)
23 lemma lcpcs_inv_lcprs: ∀L1,L2. L1 ⊢ ⬌* L2 →
24 ∃∃L. L1 ⊢ ➡* L & L2 ⊢ ➡* L.
25 #L1 #L2 #H @(lcpcs_ind … H) -L2
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/
35 (* Advanced properties ******************************************************)
37 lemma lcpcs_strip: ∀L,L1. L ⊢ ⬌* L1 → ∀L2. L ⊢ ⬌ L2 →
38 ∃∃L0. L1 ⊢ ⬌ L0 & L2 ⊢ ⬌* L0.
41 (* Main properties **********************************************************)
43 theorem lcpcs_trans: ∀L1,L. L1 ⊢ ⬌* L → ∀L2. L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
46 theorem lcpcs_canc_sn: ∀L,L1,L2. L ⊢ ⬌* L1 → L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
47 /3 width=3 by lcpcs_trans, lcprs_comm/ qed.
49 theorem lcpcs_canc_dx: ∀L,L1,L2. L1 ⊢ ⬌* L → L2 ⊢ ⬌* L → L1 ⊢ ⬌* L2.
50 /3 width=3 by lcpcs_trans, lcprs_comm/ qed.