(**************************************************************************)
include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/relocation/fquq_alt.ma".
+include "basic_2/substitution/fqus_alt.ma".
include "basic_2/static/ssta.ma".
include "basic_2/reduction/cpx.ma".
/3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
[ #I #G #L #V2 #U2 #HVU2
elim (lift_total U2 0 1)
- /4 width=9 by fqu_drop, cpx_append, cpx_delta, ldrop_pair, ldrop_ldrop, ex2_intro/
+ /4 width=7 by fqu_drop, cpx_delta, ldrop_pair, ldrop_ldrop, ex2_intro/
| #G #L #K #T1 #U1 #e #HLK1 #HTU1 #T2 #HTU2
elim (lift_total T2 0 (e+1))
/3 width=11 by cpx_lift, fqu_drop, ex2_intro/
∀l. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] l+1 →
∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
/3 width=5 by fquq_cpx_trans, ssta_cpx/ qed-.
+
+lemma fqu_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
+ #U2 #HVU2 @(ex3_intro … U2)
+ [1,3: /3 width=7 by fqu_drop, cpx_delta, ldrop_pair, ldrop_ldrop/
+ | #H destruct /2 width=7 by lift_inv_lref2_be/
+ ]
+| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
+ [1,3: /2 width=4 by fqu_pair_sn, cpx_pair_sn/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
+ [1,3: /2 width=4 by fqu_bind_dx, cpx_bind/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
+ [1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #G #L #K #T1 #U1 #e #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (e+1))
+ #U2 #HTU2 @(ex3_intro … U2)
+ [1,3: /2 width=9 by cpx_lift, fqu_drop/
+ | #H0 destruct /3 width=5 by lift_inj/
+]
+qed-.
+
+lemma fquq_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
+[ #H12 elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
+
+lemma fqup_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃+ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
+ #U1 #HTU1 #H #H12 elim (fqu_cpx_trans_neq … H1 … HTU1 H) -T1
+ /3 width=8 by fqup_strap2, ex3_intro/
+]
+qed-.
+
+lemma fqus_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
+[ #H12 elim (fqup_cpx_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.