+++ /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/notation/relations/lazysnalt_6.ma".
-include "basic_2/substitution/lleq_lleq.ma".
-include "basic_2/computation/llpxs_lleq.ma".
-include "basic_2/computation/llsx.ma".
-
-(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-
-(* alternative definition of llsx *)
-definition llsxa: ∀h. sd h → relation4 ynat term genv lenv ≝
- λh,g,d,T,G. SN … (llpxs h g G d T) (lleq d T).
-
-interpretation
- "lazy extended strong normalization (local environment) alternative"
- 'LazySNAlt h g d T G L = (llsxa h g T d G L).
-
-(* Basic eliminators ********************************************************)
-
-lemma llsxa_ind: ∀h,g,G,T,d. ∀R:predicate lenv.
- (∀L1. G ⊢ ⋕⬊⬊*[h, g, T, d] L1 →
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
- R L1
- ) →
- ∀L. G ⊢ ⋕⬊⬊*[h, g, T, d] L → R L.
-#h #g #G #T #d #R #H0 #L1 #H elim H -L1
-/5 width=1 by lleq_sym, SN_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llsxa_intro: ∀h,g,G,L1,T,d.
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
-/5 width=1 by lleq_sym, SN_intro/ qed.
-
-fact llsxa_intro_aux: ∀h,g,G,L1,T,d.
- (∀L,L2. ⦃G, L⦄ ⊢ ➡*[h, g, T, d] L2 → L1 ⋕[T, d] L → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
-/4 width=3 by llsxa_intro/ qed-.
-
-lemma llsxa_llpxs_trans: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊⬊*[h, g, T, d] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → G ⊢ ⋕⬊⬊*[h, g, T, d] L2.
-#h #g #G #L1 #T #d #H @(llsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12 @llsxa_intro
-elim (lleq_dec T L1 L2 d) /4 width=4 by lleq_llpxs_trans, lleq_canc_sn/
-qed-.
-
-lemma llsxa_intro_llpx: ∀h,g,G,L1,T,d.
- (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊⬊*[h, g, T, d] L1.
-#h #g #G #L1 #T #d #IH @llsxa_intro_aux
-#L #L2 #H @(llpxs_ind_dx … H) -L
-[ #H destruct #H elim H //
-| #L0 #L elim (lleq_dec T L1 L d)
- /4 width=3 by llsxa_llpxs_trans, lleq_llpx_trans/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem llsx_llsxa: ∀h,g,G,L,T,d. G ⊢ ⋕⬊*[h, g, T, d] L → G ⊢ ⋕⬊⬊*[h, g, T, d] L.
-#h #g #G #L #T #d #H @(llsx_ind … H) -L
-/4 width=1 by llsxa_intro_llpx/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem llsxa_inv_llsx: ∀h,g,G,L,T,d. G ⊢ ⋕⬊⬊*[h, g, T, d] L → G ⊢ ⋕⬊*[h, g, T, d] L.
-#h #g #G #L #T #d #H @(llsxa_ind … H) -L
-/4 width=1 by llsx_intro, llpx_llpxs/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma llsx_intro_alt: ∀h,g,G,L1,T,d.
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊*[h, g, T, d] L2) →
- G ⊢ ⋕⬊*[h, g, T, d] L1.
-/6 width=1 by llsxa_inv_llsx, llsx_llsxa, llsxa_intro/ qed.
-
-lemma llsx_llpxs_trans: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
-/4 width=3 by llsxa_inv_llsx, llsx_llsxa, llsxa_llpxs_trans/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma llsx_ind_alt: ∀h,g,G,T,d. ∀R:predicate lenv.
- (∀L1. G ⊢ ⋕⬊*[h, g, T, d] L1 →
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
- R L1
- ) →
- ∀L. G ⊢ ⋕⬊*[h, g, T, d] L → R L.
-#h #g #G #T #d #R #IH #L #H @(llsxa_ind h g G T d … L)
-/4 width=1 by llsxa_inv_llsx, llsx_llsxa/
-qed-.