(**************************************************************************)
include "ground/xoa/ex_4_3.ma".
-include "basic_2/notation/relations/topredtysnstrong_4.ma".
+include "basic_2/notation/relations/topredtysnstrong_3.ma".
include "basic_2/rt_computation/rsx.ma".
-(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR EXTENDED RT-TRANSITION *********)
(* Note: this should be an instance of a more general sex *)
(* Basic_2A1: uses: lcosx *)
-inductive jsx (h) (G): relation lenv ≝
-| jsx_atom: jsx h G (⋆) (⋆)
-| jsx_bind: ∀I,K1,K2. jsx h G K1 K2 →
- jsx h G (K1.ⓘ[I]) (K2.ⓘ[I])
-| jsx_pair: ∀I,K1,K2,V. jsx h G K1 K2 →
- G ⊢ ⬈*𝐒[h,V] K2 → jsx h G (K1.ⓑ[I]V) (K2.ⓧ)
+inductive jsx (G): relation lenv ≝
+| jsx_atom: jsx G (⋆) (⋆)
+| jsx_bind: ∀I,K1,K2. jsx G K1 K2 →
+ jsx G (K1.ⓘ[I]) (K2.ⓘ[I])
+| jsx_pair: ∀I,K1,K2,V. jsx G K1 K2 →
+ G ⊢ ⬈*𝐒[V] K2 → jsx G (K1.ⓑ[I]V) (K2.ⓧ)
.
interpretation
- "strong normalization for unbound parallel rt-transition (compatibility)"
- 'ToPRedTySNStrong h G L1 L2 = (jsx h G L1 L2).
+ "strong normalization for extended parallel rt-transition (compatibility)"
+ 'ToPRedTySNStrong G L1 L2 = (jsx G L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact jsx_inv_atom_sn_aux (h) (G):
- ∀L1,L2. G ⊢ L1 ⊒[h] L2 → L1 = ⋆ → L2 = ⋆.
-#h #G #L1 #L2 * -L1 -L2
+fact jsx_inv_atom_sn_aux (G):
+ ∀L1,L2. G ⊢ L1 ⊒ L2 → L1 = ⋆ → L2 = ⋆.
+#G #L1 #L2 * -L1 -L2
[ //
| #I #K1 #K2 #_ #H destruct
| #I #K1 #K2 #V #_ #_ #H destruct
]
qed-.
-lemma jsx_inv_atom_sn (h) (G):
- ∀L2. G ⊢ ⋆ ⊒[h] L2 → L2 = ⋆.
+lemma jsx_inv_atom_sn (G):
+ ∀L2. G ⊢ ⋆ ⊒ L2 → L2 = ⋆.
/2 width=5 by jsx_inv_atom_sn_aux/ qed-.
-fact jsx_inv_bind_sn_aux (h) (G):
- ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
+fact jsx_inv_bind_sn_aux (G):
+ ∀L1,L2. G ⊢ L1 ⊒ L2 →
∀I,K1. L1 = K1.ⓘ[I] →
- ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I]
- | ∃∃J,K2,V. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*𝐒[h,V] K2 & I = BPair J V & L2 = K2.ⓧ.
-#h #G #L1 #L2 * -L1 -L2
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒ K2 & L2 = K2.ⓘ[I]
+ | ∃∃J,K2,V. G ⊢ K1 ⊒ K2 & G ⊢ ⬈*𝐒[V] K2 & I = BPair J V & L2 = K2.ⓧ.
+#G #L1 #L2 * -L1 -L2
[ #J #L1 #H destruct
| #I #K1 #K2 #HK12 #J #L1 #H destruct /3 width=3 by ex2_intro, or_introl/
| #I #K1 #K2 #V #HK12 #HV #J #L1 #H destruct /3 width=7 by ex4_3_intro, or_intror/
]
qed-.
-lemma jsx_inv_bind_sn (h) (G):
- ∀I,K1,L2. G ⊢ K1.ⓘ[I] ⊒[h] L2 →
- ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I]
- | ∃∃J,K2,V. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*𝐒[h,V] K2 & I = BPair J V & L2 = K2.ⓧ.
+lemma jsx_inv_bind_sn (G):
+ ∀I,K1,L2. G ⊢ K1.ⓘ[I] ⊒ L2 →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒ K2 & L2 = K2.ⓘ[I]
+ | ∃∃J,K2,V. G ⊢ K1 ⊒ K2 & G ⊢ ⬈*𝐒[V] K2 & I = BPair J V & L2 = K2.ⓧ.
/2 width=3 by jsx_inv_bind_sn_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: uses: lcosx_inv_pair *)
-lemma jsx_inv_pair_sn (h) (G):
- ∀I,K1,L2,V. G ⊢ K1.ⓑ[I]V ⊒[h] L2 →
- ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓑ[I]V
- | ∃∃K2. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*𝐒[h,V] K2 & L2 = K2.ⓧ.
-#h #G #I #K1 #L2 #V #H elim (jsx_inv_bind_sn … H) -H *
+lemma jsx_inv_pair_sn (G):
+ ∀I,K1,L2,V. G ⊢ K1.ⓑ[I]V ⊒ L2 →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒ K2 & L2 = K2.ⓑ[I]V
+ | ∃∃K2. G ⊢ K1 ⊒ K2 & G ⊢ ⬈*𝐒[V] K2 & L2 = K2.ⓧ.
+#G #I #K1 #L2 #V #H elim (jsx_inv_bind_sn … H) -H *
[ /3 width=3 by ex2_intro, or_introl/
| #J #K2 #X #HK12 #HX #H1 #H2 destruct /3 width=4 by ex3_intro, or_intror/
]
qed-.
-lemma jsx_inv_void_sn (h) (G):
- ∀K1,L2. G ⊢ K1.ⓧ ⊒[h] L2 →
- ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓧ.
-#h #G #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
+lemma jsx_inv_void_sn (G):
+ ∀K1,L2. G ⊢ K1.ⓧ ⊒ L2 →
+ ∃∃K2. G ⊢ K1 ⊒ K2 & L2 = K2.ⓧ.
+#G #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
/2 width=3 by ex2_intro/
qed-.
(* Advanced forward lemmas **************************************************)
-lemma jsx_fwd_bind_sn (h) (G):
- ∀I1,K1,L2. G ⊢ K1.ⓘ[I1] ⊒[h] L2 →
- ∃∃I2,K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I2].
-#h #G #I1 #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
+lemma jsx_fwd_bind_sn (G):
+ ∀I1,K1,L2. G ⊢ K1.ⓘ[I1] ⊒ L2 →
+ ∃∃I2,K2. G ⊢ K1 ⊒ K2 & L2 = K2.ⓘ[I2].
+#G #I1 #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
/2 width=4 by ex2_2_intro/
qed-.
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: lcosx_O *)
-lemma jsx_refl (h) (G): reflexive … (jsx h G).
-#h #G #L elim L -L /2 width=1 by jsx_atom, jsx_bind/
+lemma jsx_refl (G):
+ reflexive … (jsx G).
+#G #L elim L -L /2 width=1 by jsx_atom, jsx_bind/
qed.
(* Basic_2A1: removed theorems 2: