(* *)
(**************************************************************************)
-include "basic_2/substitution/fsups_fsups.ma".
+include "basic_2/multiple/fqus_fqus.ma".
+include "basic_2/unfold/lstas_da.ma".
include "basic_2/reduction/cpx_lift.ma".
include "basic_2/computation/cpxs.ma".
(* Advanced properties ******************************************************)
-lemma cpxs_delta: ∀h,g,I,L,K,V,V2,i.
- ⇩[0, i] L ≡ K. ⓑ{I}V → ⦃h, K⦄ ⊢ V ➡*[g] V2 →
- ∀W2. ⇧[0, i + 1] V2 ≡ W2 → ⦃h, L⦄ ⊢ #i ➡*[g] W2.
-#h #g #I #L #K #V #V2 #i #HLK #H elim H -V2 [ /3 width=9/ ]
-#V1 #V2 #_ #HV12 #IHV1 #W2 #HVW2
-lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
-elim (lift_total V1 0 (i+1)) /4 width=11 by cpx_lift, cpxs_strap1/
+lemma lstas_cpxs: ∀h,g,G,L,T1,T2,l1. ⦃G, L⦄ ⊢ T1 •* [h, l1] T2 →
+ ∀l2. ⦃G, L⦄ ⊢ T1 ▪ [h, g] l2 → l1 ≤ l2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #l1 #H @(lstas_ind_dx … H) -T2 -l1 //
+#l1 #T #T2 #HT1 #HT2 #IHT1 #l2 #Hl2 #Hl12
+lapply (lstas_da_conf … HT1 … Hl2) -HT1
+>(plus_minus_m_m (l2-l1) 1 ?)
+[ /4 width=5 by cpxs_strap1, sta_cpx, lt_to_le/
+| /2 width=1 by monotonic_le_minus_r/
+]
+qed.
+
+lemma cpxs_delta: ∀h,g,I,G,L,K,V,V2,i.
+ ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ➡*[h, g] V2 →
+ ∀W2. ⇧[0, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡*[h, g] W2.
+#h #g #I #G #L #K #V #V2 #i #HLK #H elim H -V2
+[ /3 width=9 by cpx_cpxs, cpx_delta/
+| #V1 lapply (ldrop_fwd_drop2 … HLK) -HLK
+ elim (lift_total V1 0 (i+1)) /4 width=12 by cpx_lift, cpxs_strap1/
+]
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpxs_inv_lref1: ∀h,g,L,T2,i. ⦃h, L⦄ ⊢ #i ➡*[g] T2 →
+lemma cpxs_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h, g] T2 →
T2 = #i ∨
- ∃∃I,K,V1,T1. ⇩[0, i] L ≡ K.ⓑ{I}V1 & ⦃h, K⦄ ⊢ V1 ➡*[g] T1 &
- ⇧[0, i + 1] T1 ≡ T2.
-#h #g #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1/
+ ∃∃I,K,V1,T1. ⇩[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ➡*[h, g] T1 &
+ ⇧[0, i+1] T1 ≡ T2.
+#h #g #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
[ #H destruct
- elim (cpx_inv_lref1 … HT2) -HT2 /2 width=1/
- * /4 width=7/
+ elim (cpx_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
+ * /4 width=7 by cpx_cpxs, ex3_4_intro, or_intror/
| * #I #K #V1 #T1 #HLK #HVT1 #HT1
- lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
- elim (cpx_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T /4 width=7/
+ lapply (ldrop_fwd_drop2 … HLK) #H0LK
+ elim (cpx_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
+ /4 width=7 by cpxs_strap1, ex3_4_intro, or_intror/
]
qed-.
(* Relocation properties ****************************************************)
-lemma cpxs_lift: ∀h,g. l_liftable (cpxs h g).
-/3 width=9/ qed.
+lemma cpxs_lift: ∀h,g,G. l_liftable (cpxs h g G).
+/3 width=10 by cpx_lift, cpxs_strap1, l_liftable_LTC/ qed.
-lemma cpxs_inv_lift1: ∀h,g. l_deliftable_sn (cpxs h g).
-/3 width=5 by l_deliftable_sn_LTC, cpx_inv_lift1/
+lemma cpxs_inv_lift1: ∀h,g,G. l_deliftable_sn (cpxs h g G).
+/3 width=6 by l_deliftable_sn_LTC, cpx_inv_lift1/
qed-.
(* Properties on supclosure *************************************************)
-lemma fsupq_cpxs_trans: ∀h,g,L1,L2,T2,U2. ⦃h, L2⦄ ⊢ T2 ➡*[g] U2 →
- ∀T1. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ →
- ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡*[g] U1 & ⦃L1, U1⦄ ⊃* ⦃L2, U2⦄.
-#h #g #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 [ /3 width=3/ ]
-#T #T2 #HT2 #_ #IHTU2 #T1 #HT1
-elim (fsupq_cpx_trans … HT1 … HT2) -T #T #HT1 #HT2
-elim (IHTU2 … HT2) -T2 /3 width=3/
+lemma fqu_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
+#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
-lemma fsups_cpxs_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ →
- ∀U2. ⦃h, L2⦄ ⊢ T2 ➡*[g] U2 →
- ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡*[g] U1 & ⦃L1, U1⦄ ⊃* ⦃L2, U2⦄.
-#h #g #L1 #L2 #T1 #T2 #H @(fsups_ind … H) -L2 -T2 [ /2 width=3/ ]
-#L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
-elim (fsupq_cpxs_trans … HTU2 … HT2) -T2 #T2 #HT2 #HTU2
-elim (IHT1 … HT2) -T #T #HT1 #HT2
-lapply (fsups_trans … HT2 … HTU2) -L -T2 /2 width=3/
+lemma fquq_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpxs_trans … HTU2 … HT12) /3 width=3 by fqu_fquq, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fquq_lstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀U2,l1. ⦃G2, L2⦄ ⊢ T2 •*[h, l1] U2 →
+ ∀l2. ⦃G2, L2⦄ ⊢ T2 ▪ [h, g] l2 → l1 ≤ l2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+/3 width=5 by fquq_cpxs_trans, lstas_cpxs/ qed-.
+
+lemma fqup_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
+#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
+qed-.
+
+lemma fqus_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
+[ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
qed-.
-lemma fsup_ssta_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
- ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
- ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
-/3 width=4 by fsup_cpx_trans, ssta_cpx/ qed-.
+lemma fqus_lstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀U2,l1. ⦃G2, L2⦄ ⊢ T2 •*[h, l1] U2 →
+ ∀l2. ⦃G2, L2⦄ ⊢ T2 ▪ [h, g] l2 → l1 ≤ l2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+/3 width=6 by fqus_cpxs_trans, lstas_cpxs/ qed-.
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