-lemma pippo (M) (gv): is_model M → is_extensional M →
- ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ∀n. ⫯*[n] 𝐔❴1❵ = f →
- ∀lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[n←d]lv].
-#M #gv #H1M #H2M #f #T1 #T2 #H elim H -f -T1 -T2
-[ /4 width=3 by seq_trans, seq_sym, ms/
-| #f #i1 #i2 #Hi12 #n #Hn #lv #d
- @(mr … H1M) [4,5: @(seq_sym … H1M) @(ml … H1M) |1,2: skip ]
-| /4 width=3 by seq_trans, seq_sym, mg/
-| #f #p * #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #n #Hn #lv #d
- [ @(mr … H1M) [4,5: @(seq_sym … H1M) @(md … H1M) |1,2: skip ]
+(* Forward lemmas with generic relocation ***********************************)
+
+fact lifts_fwd_vpush_aux (M): is_model M → is_extensional M →
+ ∀f,T1,T2. ⇧*[f] T1 ≘ T2 → ∀m. 𝐛❨m,1❩ = f →
+ ∀gv,lv,d. ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,⫯[m←d]lv].
+#M #H1M #H2M #f #T1 #T2 #H elim H -f -T1 -T2
+[ #f #s #m #Hf #gv #lv #d
+ @(mq … H1M) [4,5: /3 width=2 by seq_sym, ms/ |1,2: skip ]
+ /2 width=1 by mr/
+| #f #i1 #i2 #Hi12 #m #Hm #gv #lv #d destruct
+ @(mq … H1M) [4,5: /3 width=2 by seq_sym, ml/ |1,2: skip ]
+ elim (lt_or_ge i1 m) #Hi1
+ [ lapply (at_basic_inv_lt … Hi12) -Hi12 // #H destruct
+ >vpush_lt /2 width=1 by mr/
+ | lapply (at_basic_inv_ge … Hi12) -Hi12 // #H destruct
+ >vpush_gt /2 width=1 by mr, le_S_S/
+ ]
+| #f #l #m #Hf #gv #lv #d
+ @(mq … H1M) [4,5: /3 width=2 by seq_sym, mg/ |1,2: skip ]
+ /2 width=1 by mr/
+| #f #p * #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #m #Hm #gv #lv #d destruct
+ [ @(mq … H1M) [4,5: /3 width=2 by seq_sym, md/ |1,2: skip ]
+ @mc [3:|*: /2 width=1 by/ ]
+ @(seq_trans … H1M)
+ [3: @ti_comp // | skip ]
+ [1,2: /2 width=2 by veq_refl/ ]
+ [2: @(vpush_comp … H1M) | skip ]
+ [1,2: /2 width=2 by/ |3,4: /2 width=2 by veq_refl/ ] -IHV