include "delayed_updating/reduction/dfr.ma".
include "delayed_updating/reduction/ifr.ma".
-include "delayed_updating/substitution/fsubst_lift.ma".
-include "delayed_updating/substitution/lift_structure_depth.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_rmap_head.ma".
+
+include "delayed_updating/substitution/fsubst_eq.ma".
+
include "delayed_updating/syntax/prototerm_proper_constructors.ma".
+include "delayed_updating/syntax/path_head_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
-include "ground/relocation/tr_pap_pushs.ma".
+include "delayed_updating/syntax/path_structure_reverse.ma".
+include "delayed_updating/syntax/path_depth_reverse.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
-lemma dfr_lift_bi (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
- t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐟[⊗p,⊗q] ↑[f]t2.
+(* Main destructions with ifr ***********************************************)
+
+theorem dfr_des_ifr (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
+ t1 ➡𝐝𝐟[p,q] t2 → ▼[f]t1 ➡𝐢𝐟[⊗p,⊗q] ▼[f]t2.
#f #p #q #t1 #t2 #H0t1
-* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro … (⊗b) (❘⊗q❘)) @and4_intro
-[ //
-| #g <lift_rmap_structure <depth_structure
- >tr_pushs_swap <tr_pap_pushs_le //
-| lapply (in_comp_lift_bi f … Ht1) -Ht1 -H0t1 -Hb -Ht2
- <lift_d_empty_dx //
-| lapply (eq_lift_bi f … Ht2) -Ht2 #Ht2
+* #n * #H1n #Ht1 #Ht2
+@(ex_intro … (↑♭q)) @and3_intro
+[ -H0t1 -Ht1 -Ht2
+ >structure_L_sn >structure_reverse
+ >H1n >path_head_structure_depth <H1n -H1n //
+| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
+ <unwind2_path_d_dx
+ >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
@(subset_eq_trans … Ht2) -t2
- @(subset_eq_trans … (lift_fsubst …))
- [ <structure_append <structure_A_sn <structure_append <structure_L_sn
+ @(subset_eq_trans … (unwind2_term_fsubst …))
+ [ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans … (unwind2_term_iref …))
+ @(subset_eq_canc_sn … (unwind2_term_eq_repl_dx …))
+ [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+ @(subset_eq_trans … (unwind2_term_after …))
+ @unwind2_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) -H1n #H2n
+ <(tr_pap_eq_repl … H2n) -H2n //
+(* Note: crux of the proof ends *)
| //
| /2 width=2 by ex_intro/
| //
]
]
+qed.