--- /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/rt_computation/lpxs_rdeq.ma".
+include "basic_2/rt_computation/lpxs_lpxs.ma".
+include "basic_2/rt_computation/rsx_rsx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Properties with unbound rt-computation for full local environments *******)
+
+(* Basic_2A1: uses: lsx_intro_alt *)
+lemma rsx_intro_lpxs (h) (G):
+ ∀L1,T. (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) →
+ G ⊢ ⬈*[h,T] 𝐒⦃L1⦄.
+/4 width=1 by lpx_lpxs, rsx_intro/ qed-.
+
+(* Basic_2A1: uses: lsx_lpxs_trans *)
+lemma rsx_lpxs_trans (h) (G):
+ ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄.
+#h #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2
+/2 width=3 by rsx_lpx_trans/
+qed-.
+
+(* Eliminators with unbound rt-computation for full local environments ******)
+
+lemma rsx_ind_lpxs_rdeq (h) (G) (T) (Q:predicate lenv):
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ Q L1
+ ) →
+ ∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L0. ⦃G,L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
+#h #G #T #Q #IH #L1 #H @(rsx_ind … H) -L1
+#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
+@IH -IH /3 width=3 by rsx_lpxs_trans, rsx_rdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
+lapply (rdeq_rdneq_trans … HL02 … HnLK2) -HnLK2 #H
+elim (rdeq_lpxs_trans … HLK2 … HL02) -L2 #K0 #HLK0 #HK02
+lapply (rdneq_rdeq_canc_dx … H … HK02) -H #HnLK0
+elim (rdeq_dec L1 L0 T) #H
+[ lapply (rdeq_rdneq_trans … H … HnLK0) -H -HnLK0 #Hn10
+ lapply (lpxs_trans … HL10 … HLK0) -L0 #H10
+ elim (lpxs_rdneq_inv_step_sn … H10 … Hn10) -H10 -Hn10
+ /3 width=8 by rdeq_trans/
+| elim (lpxs_rdneq_inv_step_sn … HL10 … H) -HL10 -H #L #K #HL1 #HnL1 #HLK #HKL0
+ elim (rdeq_lpxs_trans … HLK0 … HKL0) -L0
+ /3 width=8 by lpxs_trans, rdeq_trans/
+]
+qed-.
+
+(* Basic_2A1: uses: lsx_ind_alt *)
+lemma rsx_ind_lpxs (h) (G) (T) (Q:predicate lenv):
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ Q L1
+ ) →
+ ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → Q L.
+#h #G #T #Q #IH #L #HL
+@(rsx_ind_lpxs_rdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
+qed-.
+
+(* Advanced properties ******************************************************)
+
+fact rsx_bind_lpxs_aux (h) (G):
+ ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓑ{I}V → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+#h #G #p #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
+#L1 #_ #IHL1 #Y #T #H @(rsx_ind_lpxs … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct
+@rsx_intro_lpxs #L0 #HL20
+lapply (lpxs_trans … HL12 … HL20) #HL10 #H
+elim (rdneq_inv_bind … H) -H [ -IHY | -HY -IHL1 -HL12 ]
+[ #HnV elim (rdeq_dec L1 L2 V)
+ [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
+ /3 width=4 by rsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
+ | -HnV -HL10 /4 width=4 by rsx_lpxs_trans, lpxs_bind_refl_dx/
+ ]
+| /3 width=4 by lpxs_bind_refl_dx/
+]
+qed-.
+
+(* Basic_2A1: uses: lsx_bind *)
+lemma rsx_bind (h) (G):
+ ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓑ{I}V⦄ →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
+/2 width=3 by rsx_bind_lpxs_aux/ qed.
+
+(* Basic_2A1: uses: lsx_flat_lpxs *)
+lemma rsx_flat_lpxs (h) (G):
+ ∀I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀L2,T. G ⊢ ⬈*[h,T] 𝐒⦃L2⦄ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L2⦄.
+#h #G #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
+#L1 #HL1 #IHL1 #L2 #T #H @(rsx_ind_lpxs … H) -L2
+#L2 #HL2 #IHL2 #HL12 @rsx_intro_lpxs
+#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
+#HL10 #H elim (rdneq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (rdeq_dec L1 L2 V)
+ [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
+ /3 width=5 by rsx_lpxs_trans, rdeq_canc_sn/ (**) (* full auto too slow: 47s *)
+ | -HnV -HL10 /3 width=4 by rsx_lpxs_trans/
+ ]
+| /3 width=3 by/
+]
+qed-.
+
+(* Basic_2A1: uses: lsx_flat *)
+lemma rsx_flat (h) (G):
+ ∀I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄.
+/2 width=3 by rsx_flat_lpxs/ qed.
+
+fact rsx_bind_lpxs_void_aux (h) (G):
+ ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓧ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+#h #G #p #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
+#L1 #_ #IHL1 #Y #T #H @(rsx_ind_lpxs … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct
+@rsx_intro_lpxs #L0 #HL20
+lapply (lpxs_trans … HL12 … HL20) #HL10 #H
+elim (rdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ]
+[ #HnV elim (rdeq_dec L1 L2 V)
+ [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
+ /3 width=6 by rsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
+ | -HnV -HL10 /4 width=4 by rsx_lpxs_trans, lpxs_bind_refl_dx/
+ ]
+| /3 width=4 by lpxs_bind_refl_dx/
+]
+qed-.
+
+lemma rsx_bind_void (h) (G):
+ ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄ →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
+/2 width=3 by rsx_bind_lpxs_void_aux/ qed.
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