(**************************************************************************)
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/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 **************************************************)
-theorem dfr_lift_bi (f) (p) (q) (t1) (t2): (*t1 ฯต ๐ โ *)
- t1 โก๐๐[p,q] t2 โ โ[f]t1 โก๐[โ[f]p,โ[โ[pโ๐โ๐]f]q] โ[f]t2.
-#f #p #q #t1 #t2
-* #n * #H1n #Ht1 #Ht2
-@(ex_intro โฆ ((โ[pโ๐โ๐โq]f)๏ผ โงฃโจnโฉ)) @and3_intro
-[ -Ht1 -Ht2
- >H1n >path_head_structure_depth <H1n -H1n //
-| lapply (in_comp_unwind2_path_term f โฆ 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 โฆ H1n)
- <reverse_lcons <depth_L_dx #H2n
- lapply (eq_inv_ninj_bi โฆ H2n) -H2n #H2n <H2n -H2n -H1n #Ht1 //
-| lapply (unwind2_term_eq_repl_dx f โฆ Ht2) -Ht2 #Ht2
+theorem dfr_lift_bi (f) (t1) (t2) (r):
+ t1 โก๐๐[r] t2 โ ๐ ก[f]t1 โก๐๐[๐ ก[f]r] ๐ ก[f]t2.
+#f #t1 #t2 #r
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro โฆ (๐ ก[f]p) (๐ ก[๐ ข[f](pโ๐โ๐)]q) (๐ ข[f](pโ๐โ๐โq)๏ผ ยงโจnโฉ))
+[ -Hn -Ht1 -Ht2 //
+| -Ht1 -Ht2
+ /2 width=1 by lift_path_rmap_closed_L/
+| lapply (in_comp_lift_path_term f โฆ Ht1) -Ht2 -Ht1 -Hn
+ <lift_path_d_dx #Ht1 //
+| lapply (lift_term_eq_repl_dx f โฆ Ht2) -Ht2 #Ht2 -Ht1
@(subset_eq_trans โฆ Ht2) -t2
- @(subset_eq_trans โฆ (unwind2_term_fsubst โฆ))
- [ @fsubst_eq_repl [ // | // ]
- @(subset_eq_trans โฆ (unwind2_term_iref โฆ))
- @(subset_eq_canc_sn โฆ (lift_term_eq_repl_dx โฆ))
- [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
- @(subset_eq_trans โฆ (unwind2_lift_term_after โฆ))
- @unwind2_term_eq_repl_sn
+ @(subset_eq_trans โฆ (lift_term_fsubst โฆ))
+ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans โฆ (lift_term_iref_nap โฆ))
+ @iref_eq_repl
+ @(subset_eq_canc_sn โฆ (lift_term_grafted_S โฆ))
+ @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 โฆ H1n) #H2n
- >nrplus_inj_dx in โข (???%); <H2n -H2n
- lapply (tls_unwind2_rmap_append_closed f โฆ H1n) #H2n
- <(tr_pap_eq_repl โฆ 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.