(**************************************************************************)
include "delayed_updating/reduction/dfr.ma".
-include "delayed_updating/reduction/ifr.ma".
-(*
-include "delayed_updating/unwind/unwind2_constructors.ma".
-include "delayed_updating/unwind/unwind2_preterm_fsubst.ma".
-include "delayed_updating/unwind/unwind2_preterm_eq.ma".
-include "delayed_updating/unwind/unwind2_prototerm_lift.ma".
-include "delayed_updating/unwind/unwind2_rmap_head.ma".
-*)
-include "delayed_updating/substitution/fsubst_eq.ma".
-include "delayed_updating/substitution/lift_prototerm_eq.ma".
-include "delayed_updating/substitution/lift_path_head.ma".
-
-include "delayed_updating/syntax/prototerm_proper_constructors.ma".
+include "delayed_updating/substitution/fsubst_lift.ma".
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_constructors.ma".
+include "delayed_updating/substitution/lift_path_closed.ma".
+include "delayed_updating/substitution/lift_rmap_closed.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
(* Constructions with lift **************************************************)
-(*
-lemma pippo (f) (r):
- ↑[↑[r]f](rᴿ) = (↑[f]r)ᴿ.
-#f #r @(list_ind_rcons … r) -r //
-#p * [ #n ] #IH
-[ <reverse_rcons <lift_path_d_sn <lift_rmap_d_dx
- <lift_path_d_dx <reverse_rcons
- <IH
-*)
-theorem dfr_lift_bi (f) (p) (q) (t1) (t2): (*t1 ϵ 𝐓 → *)
- t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐟[↑[f]p,↑[↑[p◖𝗔◖𝗟]f]q] ↑[f]t2.
-#f #p #q #t1 #t2
-* #n * #H1n #Ht1 #Ht2
-@(ex_intro … ((↑[p●𝗔◗𝗟◗q]f)@⧣❨n❩)) @and3_intro
-[ -Ht1 -Ht2
- <lift_rmap_L_dx >lift_path_L_sn
- >list_append_rcons_sn in H1n; <reverse_append #H1n
- <(lift_path_head … H1n) -H1n //
-(*
-| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
- <unwind2_path_d_dx <depth_structure
- >list_append_rcons_sn in H1n; <reverse_append #H1n
- lapply (unwind2_rmap_append_pap_closed f … H1n)
- <reverse_lcons <depth_L_dx #H2n
- lapply (eq_inv_ninj_bi … H2n) -H2n #H2n <H2n -H2n -H1n #Ht1 //
-| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
+theorem dfr_lift_bi (f) (t1) (t2) (r):
+ t1 ➡𝐝𝐟[r] t2 → 🠡[f]t1 ➡𝐝𝐟[🠡[f]r] 🠡[f]t2.
+#f #t1 #t2 #r
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro … (🠡[f]p) (🠡[🠢[f](p◖𝗔◖𝗟)]q) (🠢[f](p●𝗔◗𝗟◗q)@§❨n❩))
+[ -Hn -Ht1 -Ht2 //
+| -Ht1 -Ht2
+ /2 width=1 by lift_path_rmap_closed_L/
+| lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -Hn
+ <lift_path_d_dx #Ht1 //
+| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
@(subset_eq_trans … Ht2) -t2
- @(subset_eq_trans … (unwind2_term_fsubst …))
- [ @fsubst_eq_repl [ // | // ]
- @(subset_eq_trans … (unwind2_term_iref …))
- @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
- [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
- @(subset_eq_trans … (unwind2_lift_term_after …))
- @unwind2_term_eq_repl_sn
+ @(subset_eq_trans … (lift_term_fsubst …))
+ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans … (lift_term_iref_nap …))
+ @iref_eq_repl
+ @(subset_eq_canc_sn … (lift_term_grafted_S …))
+ @lift_term_eq_repl_sn
(* Note: crux of the proof begins *)
- @nstream_eq_inv_ext #m
- <tr_compose_pap <tr_compose_pap
- <tr_uni_pap <tr_uni_pap <tr_pap_plus
- >list_append_rcons_sn in H1n; <reverse_append #H1n
- lapply (unwind2_rmap_append_pap_closed f … H1n) #H2n
- >nrplus_inj_dx in ⊢ (???%); <H2n -H2n
- lapply (tls_unwind2_rmap_append_closed f … H1n) #H2n
- <(tr_pap_eq_repl … H2n) -H2n -H1n //
+ /2 width=2 by tls_succ_lift_rmap_append_closed_Lq_dx/
(* Note: crux of the proof ends *)
- | //
- | /2 width=2 by ex_intro/
- | //
- ]
]
qed.
-*)
\ No newline at end of file