+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/substitution/lleq_alt.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-fact lleq_inv_S_aux: ∀L1,L2,T,d0. L1 ⋕[T, d0] L2 → ∀d. d0 = d + 1 →
- ∀K1,K2,I,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
- K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
-#L1 #L2 #T #d0 #H @(lleq_ind_alt … H) -L1 -L2 -T -d0
-/2 width=1 by lleq_gref, lleq_free, lleq_sort/
-[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V #HLK1 #HLK2 #HV destruct
- elim (yle_split_eq i d) /2 width=1 by lleq_skip, ylt_fwd_succ2/ -HL12 -Hid
- #H destruct /2 width=8 by lleq_lref/
-| #I1 #I2 #L1 #L2 #K11 #K22 #V #d0 #i #Hd0i #HLK11 #HLK22 #HV #_ #d #H #K1 #K2 #J #W #_ #_ #_ destruct
- /3 width=8 by lleq_lref, yle_pred_sn/
-| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
- /4 width=7 by lleq_bind, ldrop_drop/
-| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
- /3 width=7 by lleq_flat/
-]
-qed-.
-
-lemma lleq_inv_S: ∀T,L1,L2,d. L1 ⋕[T, d+1] L2 →
- ∀K1,K2,I,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
- K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
-/2 width=7 by lleq_inv_S_aux/ qed-.
-
-lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 →
- L1 ⋕[V, 0] L2 ∧ L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V.
-#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind … H) -H
-/3 width=7 by ldrop_pair, conj, lleq_inv_S/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lleq_fwd_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 →
- L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V.
-#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind_O … H) -H //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_ge: ∀L1,L2,T,d1. L1 ⋕[T, d1] L2 → ∀d2. d1 ≤ d2 → L1 ⋕[T, d2] L2.
-#L1 #L2 #T #d1 #H @(lleq_ind_alt … H) -L1 -L2 -T -d1
-/4 width=1 by lleq_sort, lleq_free, lleq_gref, lleq_bind, lleq_flat, yle_succ/
-[ /3 width=3 by lleq_skip, ylt_yle_trans/
-| #I1 #I2 #L1 #L2 #K1 #K2 #V #d1 #i #Hi #HLK1 #HLK2 #HV #IHV #d2 #Hd12 elim (ylt_split i d2)
- [ lapply (lleq_fwd_length … HV) #HK12 #Hid2
- lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
- normalize in ⊢ (%→%→?); -I1 -I2 -V -d1 /2 width=1 by lleq_skip/
- | /3 width=8 by lleq_lref, yle_trans/
- ]
-]
-qed-.
-
-lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[V, 0] L2 → L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
- L1 ⋕[ⓑ{a,I}V.T, 0] L2.
-/3 width=3 by lleq_ge, lleq_bind/ qed.
-
-lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ⋕[T, 0] L2.ⓑ{I2}V2 →
- ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ⋕[T, 1] L2.ⓑ{J2}W2.
-#I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (lleq_ge … HT 1 ?) // -HT
-#HT @(lleq_lsuby_repl … HT) /2 width=1 by lsuby_succ/ (**) (* full auto fails *)
-qed-.
-
-lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
- ∀J,W. L1 ⋕[W, 0] L2 → L1.ⓑ{J}W ⋕[T, 0] L2.ⓑ{J}W.
-/3 width=7 by lleq_bind_repl_SO, lleq_inv_S/ qed-.
-
-(* Inversion lemmas on negated lazy quivalence for local environments *******)
-
-lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ⋕[ⓑ{a,I}V.T, 0] L2 → ⊥) →
- (L1 ⋕[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V → ⊥).
-#a #I #L1 #L2 #V #T #H elim (lleq_dec V L1 L2 0)
-/4 width=1 by lleq_bind_O, or_intror, or_introl/
-qed-.