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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/topredtysnstrong_5.ma".
+include "basic_2/rt_computation/rsx.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Note: this should be an instance of a more general sex *)
+(* Basic_2A1: uses: lcosx *)
+inductive jsx (h) (G): rtmap → relation lenv ≝
+| jsx_atom: ∀f. jsx h G f (⋆) (⋆)
+| jsx_push: ∀f,I,K1,K2. jsx h G f K1 K2 →
+               jsx h G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})
+| jsx_unit: ∀f,I,K1,K2. jsx h G f K1 K2 →
+               jsx h G (↑f) (K1.ⓤ{I}) (K2.ⓧ)
+| jsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ →
+               jsx h G f K1 K2 → jsx h G (↑f) (K1.ⓑ{I}V) (K2.ⓧ)
+.
+
+interpretation
+  "strong normalization for unbound parallel rt-transition (compatibility)"
+  'ToPRedTySNStrong h f G L1 L2 = (jsx h G f L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact jsx_inv_atom_sn_aux (h) (G):
+     ∀g,L1,L2. G ⊢ L1 ⊒[h,g] L2 → L1 = ⋆ → L2 = ⋆.
+#h #G #g #L1 #L2 * -g -L1 -L2 //
+[ #f #I #K1 #K2 #_ #H destruct
+| #f #I #K1 #K2 #_ #H destruct
+| #f #I #K1 #K2 #V #_ #_ #H destruct
+]
+qed-.
+
+lemma jsx_inv_atom_sn (h) (G): ∀g,L2. G ⊢ ⋆ ⊒[h,g] L2 → L2 = ⋆.
+/2 width=7 by jsx_inv_atom_sn_aux/ qed-.
+
+fact jsx_inv_push_sn_aux (h) (G):
+     ∀g,L1,L2. G ⊢ L1 ⊒[h,g] L2 →
+     ∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} →
+     ∃∃K2. G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓘ{I}.
+#h #G #g #L1 #L2 * -g -L1 -L2
+[ #f #g #J #L1 #_ #H destruct
+| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
+  <(injective_push … H1) -g /2 width=3 by ex2_intro/
+| #f #I #K1 #K2 #_ #g #J #L1 #H
+  elim (discr_next_push … H)
+| #f #I #K1 #K2 #V #_ #_ #g #J #L1 #H
+  elim (discr_next_push … H)
+]
+qed-.
+
+lemma jsx_inv_push_sn (h) (G):
+      ∀f,I,K1,L2. G ⊢ K1.ⓘ{I} ⊒[h,⫯f] L2 →
+      ∃∃K2. G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓘ{I}.
+/2 width=5 by jsx_inv_push_sn_aux/ qed-.
+
+fact jsx_inv_unit_sn_aux (h) (G):
+     ∀g,L1,L2. G ⊢ L1 ⊒[h,g] L2 →
+     ∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} →
+     ∃∃K2. G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓧ.
+#h #G #g #L1 #L2 * -g -L1 -L2
+[ #f #g #J #L1 #_ #H destruct
+| #f #I #K1 #K2 #_ #g #J #L1 #H
+  elim (discr_push_next … H)
+| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
+  <(injective_next … H1) -g /2 width=3 by ex2_intro/
+| #f #I #K1 #K2 #V #_ #_ #g #J #L1 #_ #H destruct
+]
+qed-.
+
+lemma jsx_inv_unit_sn (h) (G):
+      ∀f,I,K1,L2. G ⊢ K1.ⓤ{I} ⊒[h,↑f] L2 →
+      ∃∃K2. G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓧ.
+/2 width=6 by jsx_inv_unit_sn_aux/ qed-.
+
+fact jsx_inv_pair_sn_aux (h) (G):
+     ∀g,L1,L2. G ⊢ L1 ⊒[h,g] L2 →
+     ∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V →
+     ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄  & G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓧ.
+#h #G #g #L1 #L2 * -g -L1 -L2
+[ #f #g #J #L1 #W #_ #H destruct
+| #f #I #K1 #K2 #_ #g #J #L1 #W #H
+  elim (discr_push_next … H)
+| #f #I #K1 #K2 #_ #g #J #L1 #W #_ #H destruct
+| #f #I #K1 #K2 #V #HV #HK12 #g #J #L1 #W #H1 #H2 destruct
+  <(injective_next … H1) -g /2 width=4 by ex3_intro/
+]
+qed-.
+
+(* Basic_2A1: uses: lcosx_inv_pair *)
+lemma jsx_inv_pair_sn (h) (G):
+      ∀f,I,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊒[h,↑f] L2 →
+      ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄  & G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓧ.
+/2 width=6 by jsx_inv_pair_sn_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma jsx_inv_pair_sn_gen (h) (G): ∀g,I,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊒[h,g] L2 →
+      ∨∨ ∃∃f,K2. G ⊢ K1 ⊒[h,f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
+       | ∃∃f,K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄  & G ⊢ K1 ⊒[h,f] K2 & g = ↑f & L2 = K2.ⓧ.
+#h #G #g #I #K1 #L2 #V #H
+elim (pn_split g) * #f #Hf destruct
+[ elim (jsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/
+| elim (jsx_inv_pair_sn … H) -H /3 width=6 by ex4_2_intro, or_intror/
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma jsx_fwd_bind_sn (h) (G):
+      ∀g,I1,K1,L2. G ⊢ K1.ⓘ{I1} ⊒[h,g] L2 →
+      ∃∃I2,K2. G ⊢ K1 ⊒[h,⫱g] K2 & L2 = K2.ⓘ{I2}.
+#h #G #g #I1 #K1 #L2
+elim (pn_split g) * #f #Hf destruct
+[ #H elim (jsx_inv_push_sn … H) -H
+| cases I1 -I1 #I1
+  [ #H elim (jsx_inv_unit_sn … H) -H
+  | #V #H elim (jsx_inv_pair_sn … H) -H
+  ]
+]
+/2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma jsx_eq_repl_back (h) (G): ∀L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊒[h,f] L2).
+#h #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
+[ #f #I #L1 #L2 #_ #IH #x #H
+  elim (eq_inv_px … H) -H /3 width=3 by jsx_push/
+| #f #I #L1 #L2 #_ #IH #x #H
+  elim (eq_inv_nx … H) -H /3 width=3 by jsx_unit/
+| #f #I #L1 #L2 #V #HV #_ #IH #x #H
+  elim (eq_inv_nx … H) -H /3 width=3 by jsx_pair/
+]
+qed-.
+
+lemma jsx_eq_repl_fwd (h) (G): ∀L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊒[h,f] L2).
+#h #G #L1 #L2 @eq_repl_sym /2 width=3 by jsx_eq_repl_back/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lcosx_O *)
+lemma jsx_refl (h) (G): ∀f. 𝐈⦃f⦄ → reflexive … (jsx h G f).
+#h #G #f #Hf #L elim L -L
+/3 width=3 by jsx_eq_repl_back, jsx_push, eq_push_inv_isid/
+qed.
+
+(* Basic_2A1: removed theorems 2:
+              lcosx_drop_trans_lt lcosx_inv_succ
+*)