X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Fcpx_lift.ma;h=505eabf1aebe1a86c2437ab05e6fa2d7ec304ead;hb=b5d702735754632652b2659c425dd67d7f92f24b;hp=05ed4ce0c211655d38e88a008e924eaeccbb1687;hpb=fdb2c62b58006b82c015ba70b494d50c7860e28f;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
index 05ed4ce0c..505eabf1a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/relocation/fsupq.ma".
+include "basic_2/substitution/fqus_alt.ma".
 include "basic_2/static/ssta.ma".
 include "basic_2/reduction/cpx.ma".
 
@@ -24,16 +24,16 @@ include "basic_2/reduction/cpx.ma".
 lemma ssta_cpx: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •[h, g] T2 →
                 ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
 #h #g #G #L #T1 #T2 #l #H elim H -G -L -T1 -T2
-[ #G #L #k #H lapply (da_inv_sort … H) -H /2 width=2/
+[ /3 width=4 by cpx_sort, da_inv_sort/
 | #G #L #K #V #U #W #i #HLK #_ #HWU #IHVW #H
   elim (da_inv_lref … H) -H * #K0 #V0 [| #l0 ] #HLK0
-  lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7/
+  lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpx_delta/
 | #G #L #K #W #U #l0 #i #HLK #_ #HWU #H
   elim (da_inv_lref … H) -H * #K0 #W0 [| #l1 ] #HLK0
-  lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /2 width=7/
-| #a #I #G #L #V #T #U #_ #IHTU #H lapply (da_inv_bind … H) -H /3 width=1/
-| #G #L #V #T #U #_ #IHTU #H lapply (da_inv_flat … H) -H /3 width=1/
-| #G #L #W #T #U #_ #IHTU #H lapply (da_inv_flat … H) -H /3 width=1/
+  lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct /2 width=7 by cpx_delta/
+| /4 width=2 by cpx_bind, da_inv_bind/
+| /4 width=3 by cpx_flat, da_inv_flat/
+| /4 width=3 by cpx_tau, da_inv_flat/
 ]
 qed.
 
@@ -45,125 +45,211 @@ lemma cpx_lift: ∀h,g,G. l_liftable (cpx h g G).
   >(lift_mono … H1 … H2) -H1 -H2 //
 | #G #K #k #l #Hkl #L #d #e #_ #U1 #H1 #U2 #H2
   >(lift_inv_sort1 … H1) -U1
-  >(lift_inv_sort1 … H2) -U2 /2 width=2/
+  >(lift_inv_sort1 … H2) -U2 /2 width=2 by cpx_sort/
 | #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
   elim (lift_inv_lref1 … H) * #Hid #H destruct
   [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
-    elim (ldrop_trans_le … HLK … HKV) -K /2 width=2/ #X #HLK #H
-    elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K #Y #HKV #HVY #H destruct /3 width=9/
-  | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 // /2 width=1/ >plus_plus_comm_23 #HVU2
-    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=7/
+    elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+    elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #HKV #HVY #H destruct /3 width=9 by cpx_delta/
+  | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K /3 width=7 by cpx_delta/
   ]
 | #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
-  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5/
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5 by cpx_bind, ldrop_skip/
 | #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
-  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpx_flat/
 | #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #H #U2 #HTU2
   elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
-  elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=5/
+  elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=5 by cpx_zeta, ldrop_skip/
 | #G #K #V #T1 #T2 #_ #IHT12 #L #d #e #HLK #U1 #H #U2 #HTU2
-  elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5/
+  elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5 by cpx_tau/
 | #G #K #V1 #V2 #T #_ #IHV12 #L #d #e #HLK #U1 #H #U2 #HVU2
-  elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5/
+  elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5 by cpx_ti/
 | #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #d #e #HLK #X1 #HX1 #X2 #HX2
   elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
   elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
   elim (lift_inv_bind1 … HX2) -HX2 #X #T3 #HX #HT23 #HX2 destruct
-  elim (lift_inv_flat1 … HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=5/
+  elim (lift_inv_flat1 … HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=5 by cpx_beta, ldrop_skip/
 | #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #d #e #HLK #X1 #HX1 #X2 #HX2
   elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
   elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
   elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
   elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
-  elim (lift_trans_ge … HV2 … HV3) -V2 // /4 width=5/
+  elim (lift_trans_ge … HV2 … HV3) -V2 /4 width=5 by cpx_theta, ldrop_skip/
 ]
 qed.
 
 lemma cpx_inv_lift1: ∀h,g,G. l_deliftable_sn (cpx h g G).
 #h #g #G #L #U1 #U2 #H elim H -G -L -U1 -U2
 [ * #G #L #i #K #d #e #_ #T1 #H
-  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
-  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
-  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpx_atom, lift_sort, ex2_intro/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpx_atom, lift_gref, ex2_intro/
   ]
 | #G #L #k #l #Hkl #K #d #e #_ #T1 #H
-  lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3/
+  lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by cpx_sort, lift_sort, ex2_intro/
 | #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
   elim (lift_inv_lref2 … H) -H * #Hid #H destruct
   [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
     elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
-    elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m // -Hid /3 width=9/
+    elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=9 by cpx_delta, ex2_intro/
   | elim (le_inv_plus_l … Hid) #Hdie #Hei
     lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
-    elim (lift_split … HVW2 d (i - e + 1)) -HVW2 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
-    #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O /3 width=9/
+    elim (lift_split … HVW2 d (i - e + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hid -Hdie
+    #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpx_delta, ex2_intro/
   ]
 | #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5/
+  elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5 by cpx_bind, ldrop_skip, lift_bind, ex2_intro/
 | #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -V1
-  elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5/
+  elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpx_flat, lift_flat, ex2_intro/
 | #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #X #H
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHU1 (K.ⓓW1) … HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
-  elim (lift_div_le … HU2 … HTU) -U // /3 width=5/
+  elim (lift_div_le … HU2 … HTU) -U /3 width=5 by cpx_zeta, ex2_intro/
 | #G #L #V #U1 #U2 #_ #IHU12 #K #d #e #HLK #X #H
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
-  elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3/
+  elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_tau, ex2_intro/
 | #G #L #V1 #V2 #U1 #_ #IHV12 #K #d #e #HLK #X #H
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
-  elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3/
+  elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ti, ex2_intro/
 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #d #e #HLK #X #HX
   elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
   elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
   elim (IHV12 … HLK … HV01) -V1 #V3 #HV32 #HV03
-  elim (IHT12 (K.ⓛW0) … HT01) -T1 /2 width=1/ #T3 #HT32 #HT03
-  elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
-  @ex2_intro [2: /3 width=2/ | skip |3: /2 width=1/ ] (**) (* /4 width=6/ is slow *)
+  elim (IHT12 (K.ⓛW0) … HT01) -T1 /2 width=1 by ldrop_skip/ #T3 #HT32 #HT03
+  elim (IHW12 … HLK … HW01) -W1
+  /4 width=7 by cpx_beta, lift_bind, lift_flat, ex2_intro/
 | #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #d #e #HLK #X #HX
   elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
   elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
   elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
-  elim (IHT12 (K.ⓓW0) … HT01) -T1 /2 width=1/ #T3 #HT32 #HT03
+  elim (IHT12 (K.ⓓW0) … HT01) -T1 /2 width=1 by ldrop_skip/ #T3 #HT32 #HT03
   elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
-  elim (lift_trans_le … HV3 … HV2) -V // #V #HV3 #HV2
-  @ex2_intro [2: /3 width=2/ | skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
+  elim (lift_trans_le … HV3 … HV2) -V
+  /4 width=9 by cpx_theta, lift_bind, lift_flat, ex2_intro/
 ]
 qed-.
 
 (* Properties on supclosure *************************************************)
 
-lemma fsupq_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
-                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [1: /2 width=3/ |3,4,5: /3 width=3/ ]
-[ #I #G #L1 #V2 #U2 #HVU2
-  elim (lift_total U2 0 1) /4 width=9/
-| #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-  elim (IHT12 … HTU2) -IHT12 -HTU2 #T #HT1 #HT2
-  elim (lift_total T d e) #U #HTU
-  lapply (cpx_lift … HT1 … HLK1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+lemma fqu_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+                     ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                     ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 
+/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=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/
 ]
 qed-.
 
-lemma fsupq_ssta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
-                        ∀U2. ⦃G2, L2⦄ ⊢ T2 •[h, g] U2 →
-                        ∀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 fsupq_cpx_trans, ssta_cpx/ qed-.
+lemma fqu_ssta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 •[h, g] U2 →
+                      ∀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 fqu_cpx_trans, ssta_cpx/ qed-.
 
-lemma fsup_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+lemma fquq_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
-/3 width=3 by fsupq_cpx_trans, fsup_fsupq/ qed-.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
 
-lemma fsup_ssta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+lemma fquq_ssta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
                        ∀U2. ⦃G2, L2⦄ ⊢ T2 •[h, g] U2 →
                        ∀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 fsupq_ssta_trans, fsup_fsupq/ qed-.
+/3 width=5 by fquq_cpx_trans, ssta_cpx/ qed-.
+
+lemma fqup_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊃+ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
+  /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
+  elim (fqu_cpx_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
+  elim (IHT1 … HT2) -T /3 width=7 by fqup_strap1, ex2_intro/
+]
+qed-.
+
+lemma fqus_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
+[ #HT12 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+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-.