(* *)
(**************************************************************************)
+include "ground/xoa/ex_4_5.ma".
include "basic_2/rt_transition/cpg_drops.ma".
include "basic_2/rt_transition/cpx.ma".
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS **************)
(* Advanced properties ******************************************************)
(* Basic_2A1: was: cpx_delta *)
-lemma cpx_delta_drops: ∀h,I,G,L,K,V,V2,W2,i.
- ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⬈[h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ⬈[h] W2.
-#h * #G #L #K #V #V2 #W2 #i #HLK *
+lemma cpx_delta_drops (G) (L):
+ ∀I,K,V,V2,W2,i.
+ ⇩[i] L ≘ K.ⓑ[I]V → ❨G,K❩ ⊢ V ⬈ V2 →
+ ⇧[↑i] V2 ≘ W2 → ❨G,L❩ ⊢ #i ⬈ W2.
+#G #L * #K #V #V2 #W2 #i #HLK *
/3 width=7 by cpg_ell_drops, cpg_delta_drops, ex_intro/
qed.
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: was: cpx_inv_atom1 *)
-lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[h] T2 →
- ∨∨ T2 = ⓪{I}
- | ∃∃s. T2 = ⋆(next h s) & I = Sort s
- | ∃∃J,K,V,V2,i. ⬇*[i] L ≘ K.ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 &
- ⬆*[↑i] V2 ≘ T2 & I = LRef i.
-#h #I #G #L #T2 * #c #H elim (cpg_inv_atom1_drops … H) -H *
-/4 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex2_intro, ex_intro/
+lemma cpx_inv_atom1_drops (G) (L):
+ ∀I,T2. ❨G,L❩ ⊢ ⓪[I] ⬈ T2 →
+ ∨∨ T2 = ⓪[I]
+ | ∃∃s1,s2. T2 = ⋆s2 & I = Sort s1
+ | ∃∃J,K,V,V2,i. ⇩[i] L ≘ K.ⓑ[J]V & ❨G,K❩ ⊢ V ⬈ V2 & ⇧[↑i] V2 ≘ T2 & I = LRef i.
+#G #L #I #T2 * #c #H elim (cpg_inv_atom1_drops … H) -H *
+/4 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex2_2_intro, ex_intro/
qed-.
(* Basic_2A1: was: cpx_inv_lref1 *)
-lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈[h] T2 →
- T2 = #i ∨
- ∃∃J,K,V,V2. ⬇*[i] L ≘ K. ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 &
- ⬆*[↑i] V2 ≘ T2.
-#h #G #L #T1 #i * #c #H elim (cpg_inv_lref1_drops … H) -H *
+lemma cpx_inv_lref1_drops (G) (L):
+ ∀T2,i. ❨G,L❩ ⊢ #i ⬈ T2 →
+ ∨∨ T2 = #i
+ | ∃∃J,K,V,V2. ⇩[i] L ≘ K. ⓑ[J]V & ❨G,K❩ ⊢ V ⬈ V2 & ⇧[↑i] V2 ≘ T2.
+#G #L #T1 #i * #c #H elim (cpg_inv_lref1_drops … H) -H *
/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
qed-.
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: includes: cpx_lift *)
-lemma cpx_lifts_sn: ∀h,G. d_liftable2_sn … lifts (cpx h G).
-#h #G #K #T1 #T2 * #cT #HT12 #b #f #L #HLK #U1 #HTU1
+lemma cpx_lifts_sn (G): d_liftable2_sn … lifts (cpx G).
+#G #K #T1 #T2 * #cT #HT12 #b #f #L #HLK #U1 #HTU1
elim (cpg_lifts_sn … HT12 … HLK … HTU1) -K -T1
/3 width=4 by ex2_intro, ex_intro/
qed-.
-lemma cpx_lifts_bi: ∀h,G. d_liftable2_bi … lifts (cpx h G).
-#h #G #K #T1 #T2 * /3 width=10 by cpg_lifts_bi, ex_intro/
+lemma cpx_lifts_bi (G): d_liftable2_bi … lifts (cpx G).
+#G #K #T1 #T2 * /3 width=10 by cpg_lifts_bi, ex_intro/
qed-.
(* Inversion lemmas with generic slicing for local environments *************)
(* Basic_2A1: includes: cpx_inv_lift1 *)
-lemma cpx_inv_lifts_sn: ∀h,G. d_deliftable2_sn … lifts (cpx h G).
-#h #G #L #U1 #U2 * #cU #HU12 #b #f #K #HLK #T1 #HTU1
+lemma cpx_inv_lifts_sn (G): d_deliftable2_sn … lifts (cpx G).
+#G #L #U1 #U2 * #cU #HU12 #b #f #K #HLK #T1 #HTU1
elim (cpg_inv_lifts_sn … HU12 … HLK … HTU1) -L -U1
/3 width=4 by ex2_intro, ex_intro/
qed-.
-lemma cpx_inv_lifts_bi: ∀h,G. d_deliftable2_bi …lifts (cpx h G).
-#h #G #L #U1 #U2 * /3 width=10 by cpg_inv_lifts_bi, ex_intro/
+lemma cpx_inv_lifts_bi (G): d_deliftable2_bi …lifts (cpx G).
+#G #L #U1 #U2 * /3 width=10 by cpg_inv_lifts_bi, ex_intro/
qed-.