-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/aaa_lift.ma".
-include "basic_2/computation/lsubc.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-(* Properties concerning basic local environment slicing ********************)
-
-(* Basic_1: was: csubc_drop_conf_O *)
-(* Note: the constant 0 can not be generalized *)
-lemma lsubc_drop_O1_trans: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀K2,b,k. ⬇[b, 0, k] L2 ≡ K2 →
- ∃∃K1. ⬇[b, 0, k] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
-#RP #G #L1 #L2 #H elim H -L1 -L2
-[ #X #b #k #H elim (drop_inv_atom1 … H) -H /4 width=3 by drop_atom, ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #X #b #k #H
- elim (drop_inv_O1_pair1 … H) -H * #Hm #H destruct
- [ elim (IHL12 L2 b 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
- /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
- | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-| #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #b #k #H
- elim (drop_inv_O1_pair1 … H) -H * #Hm #H destruct
- [ elim (IHL12 L2 b 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
- /3 width=8 by lsubc_beta, drop_pair, ex2_intro/
- | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-]
-qed-.
-
-(* Basic_1: was: csubc_drop_conf_rev *)
-lemma drop_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
- ∀G,L1,K1,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
- ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇[Ⓕ, l, k] L2 ≡ K2.
-#RR #RS #RP #Hgcp #G #L1 #K1 #l #k #H elim H -L1 -K1 -l -k
-[ #l #k #Hm #X #H elim (lsubc_inv_atom1 … H) -H
- >Hm /2 width=3 by ex2_intro/
-| #L1 #I #V1 #X #H
- elim (lsubc_inv_pair1 … H) -H *
- [ #K1 #HLK1 #H destruct /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
- | #K1 #V #W1 #A #HV1 #H1W1 #H2W1 #HLK1 #H1 #H2 #H3 destruct
- /3 width=4 by lsubc_beta, drop_pair, ex2_intro/
- ]
-| #I #L1 #K1 #V1 #k #_ #IHLK1 #K2 #HK12
- elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_drop, ex2_intro/
-| #I #L1 #K1 #V1 #V2 #l #k #HLK1 #HV21 #IHLK1 #X #H
- elim (lsubc_inv_pair1 … H) -H *
- [ #K2 #HK12 #H destruct
- elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_skip, ex2_intro/
- | #K2 #V #W2 #A #HV2 #H1W2 #H2W2 #HK12 #H1 #H2 #H3 destruct
- elim (lift_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
- elim (IHLK1 … HK12) #K #HL1K #HK2
- lapply (gcr_lift … Hgcp … HV2 … HLK1 … HV3) -HV2
- lapply (gcr_lift … Hgcp … H1W2 … HLK1 … HW23) -H1W2
- /4 width=11 by lsubc_beta, aaa_lift, drop_skip, ex2_intro/
- ]
-]
-qed-.