(* *)
(**************************************************************************)
-include "basic_2/relocation/lifts.ma".
+include "ground/relocation/rtmap_basic.ma".
+include "static_2/relocation/lifts.ma".
include "apps_2/models/veq.ma".
(* EVALUATION EQUIVALENCE **************************************************)
-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
@(seq_trans … H1M)
- [2: @(ti_comp_l … H1M) | skip ]
- [2: @(vlift_comp … lv lv) | skip ]
- [3: /2 width=1 by veq_refl/ ]
- [2: @(IHV … d) // | skip ]
- @(seq_trans … H1M) [2: @(IHT … d) // | skip ]
- /4 width=1 by seq_sym, ti_ext_l, vlift_swap/
- | @mx /2 width=1 by/ #d0 @(seq_trans … H1M)
- [3: @(seq_sym … H1M) @(ti_ext_l … H1M) | skip ]
- [2: @vlift_swap // | skip ]
+ [3: @ti_comp // | skip ]
+ [1,2: /2 width=2 by veq_refl/ ]
+ [2: @veq_sym // @vpush_swap // | skip ]
+ /2 width=1 by/
+ | @mx // [ /2 width=1 by/ ] -IHV #d0
+ @(seq_trans … H1M)
+ [3: @ti_comp // | skip ]
+ [1,2: /2 width=2 by veq_refl/ ]
+ [2: @veq_sym // @vpush_swap // | skip ]
/2 width=1 by/
]
-| #f * #V1 #v2 #T1 #T2 #_ #_ #IHV #IHT #n #Hn #lv #d
- [ /4 width=5 by seq_sym, ma, mc, mr/
- | /4 width=5 by seq_sym, me, mr/
+| #f * #V1 #v2 #T1 #T2 #_ #_ #IHV #IHT #m #Hm #gv #lv #d
+ [ /4 width=5 by seq_sym, ma, mp, mq/
+ | /4 width=5 by seq_sym, me, mq/
]
-]
+]
+qed-.
+
+lemma lifts_SO_fwd_vpush (M) (gv): is_model M → is_extensional M →
+ ∀T1,T2. ⇧[1] T1 ≘ T2 →
+ ∀lv,d. ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,⫯[0←d]lv].
+/2 width=3 by lifts_fwd_vpush_aux/ qed-.