1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basic_2/computation/lfprs_lfprs.ma".
16 include "basic_2/conversion/lfpc_lfpc.ma".
17 include "basic_2/equivalence/lfpcs_lfprs.ma".
19 (* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
21 (* Advanced inversion lemmas ************************************************)
23 lemma lfpcs_inv_lfprs: ∀L1,L2. ⦃L1⦄ ⬌* ⦃L2⦄ →
24 ∃∃L. ⦃L1⦄ ➡* ⦃L⦄ & ⦃L2⦄ ➡* ⦃L⦄.
25 #L1 #L2 #H @(lfpcs_ind … H) -L2
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/
35 (* Advanced properties ******************************************************)
37 lemma lfpcs_strip: ∀L,L1. ⦃L⦄ ⬌* ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌ ⦃L2⦄ →
38 ∃∃L0. ⦃L1⦄ ⬌ ⦃L0⦄ & ⦃L2⦄ ⬌* ⦃L0⦄.
41 (* Main properties **********************************************************)
43 theorem lfpcs_trans: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
46 theorem lfpcs_canc_sn: ∀L,L1,L2. ⦃L⦄ ⬌* ⦃L1⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
47 /3 width=3 by lfpcs_trans, lfpcs_sym/ qed.
49 theorem lfpcs_canc_dx: ∀L,L1,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L2⦄ ⬌* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
50 /3 width=3 by lfpcs_trans, lfpcs_sym/ qed.