-theorem cpys_trans_lpys: ∀G. lpx_sn_transitive (cpys G) (cpys G).
-#G0 #L0 #T0 @(fqup_wf_ind_eq … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #G1 #L1 * [|*]
-[ #I #HG #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
- elim (cpys_inv_atom1 … H1) -H1
- [ #H destruct
- elim (cpys_inv_atom1 … HT2) -HT2
- [ #H destruct //
- | * #I2 #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
- elim (lpys_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpys_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
- lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
- ]
- | * #I1 #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
- elim (lpys_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpys_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
- elim (cpys_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
- lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
- ]
-| #a #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpys_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
- elim (cpys_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- /4 width=5 by cpys_bind, lpys_pair/
-| #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpys_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
- elim (cpys_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- /3 width=5 by cpys_flat/
+(* Advanced properties ******************************************************)
+
+lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
+
+lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
+ (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
+
+lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
+
+lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
+
+lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 →
+ ∀i. l ≤ i → i ≤ l + m →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2.
+#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2
+[ /2 width=3 by ex2_intro/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+ elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02
+ elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
+ >ymax_pre_sn_comm //