include "basic_2/rt_transition/cpx_simple.ma".
include "basic_2/rt_computation/csx_csx.ma".
-(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
+(* STRONGLY NORMALIZING TERMS FOR EXTENDED PARALLEL RT-TRANSITION ***********)
(* Properties with simple terms *********************************************)
-lemma csx_appl_simple: ∀h,G,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1.
- (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) →
- 𝐒⦃T1⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄.
-#h #G #L #V #H @(csx_ind … H) -V
+lemma csx_appl_simple (G) (L):
+ ∀V. ❨G,L❩ ⊢ ⬈*𝐒 V → ∀T1.
+ (∀T2. ❨G,L❩ ⊢ T1 ⬈ T2 → (T1 ≅ T2 → ⊥) → ❨G,L❩ ⊢ ⬈*𝐒 ⓐV.T2) →
+ 𝐒❨T1❩ → ❨G,L❩ ⊢ ⬈*𝐒 ⓐV.T1.
+#G #L #V #H @(csx_ind … H) -V
#V #_ #IHV #T1 #IHT1 #HT1
@csx_intro #X #H1 #H2
elim (cpx_inv_appl1_simple … H1) // -H1
#V0 #T0 #HLV0 #HLT10 #H destruct
-elim (tdneq_inv_pair … H2) -H2
+elim (tneqx_inv_pair … H2) -H2
[ #H elim H -H //
| #HV0 @(csx_cpx_trans … (ⓐV0.T1)) /2 width=1 by cpx_flat/ -HLT10
- @(IHV … HLV0 … HV0) -HV0 /4 width=5 by csx_cpx_trans, cpx_pair_sn/ (**) (* full auto too slow *)
+ @(IHV … HLV0 … HV0) -HV0 /4 width=5 by csx_cpx_trans, cpx_pair_sn/ (**) (* full auto too slow *)
| -IHV -HT1 /4 width=3 by csx_cpx_trans, cpx_pair_sn/
]
qed.