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_inner.ma".
-include "delayed_updating/unwind/unwind2_rmap_head.ma".
+include "delayed_updating/unwind/unwind2_prototerm_lift.ma".
+include "delayed_updating/unwind/unwind2_rmap_closed.ma".
include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
-include "delayed_updating/syntax/prototerm_proper_inner.ma".
-include "delayed_updating/syntax/path_head_structure.ma".
+include "delayed_updating/syntax/path_closed_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
-include "delayed_updating/syntax/path_structure_reverse.ma".
-include "delayed_updating/syntax/path_depth_reverse.ma".
(* IMMEDIATE FOCUSED REDUCTION **********************************************)
-(* Constructions with unwind ************************************************)
+(* Constructions with unwind2 ***********************************************)
-theorem ifr_unwind_bi (f) (p) (q) (t1) (t2):
- t1 Ο΅ π β t1β(pβπ¦) β§Έβ¬ π β
- t1 β‘π[p,q] t2 β βΌ[f]t1 β‘π[βp,βq] βΌ[f]t2.
-#f #p #q #t1 #t2 #H1t1 #H2t1
-* #n * #H1n #Ht1 #Ht2
-@(ex_intro β¦ (ββq)) @and3_intro
-[ -H1t1 -H2t1 -Ht1 -Ht2
- >structure_L_sn >structure_reverse
- >H1n >path_head_structure_depth <H1n -H1n //
-| lapply (in_comp_unwind2_path_term f β¦ Ht1) -Ht2 -Ht1 -H1t1 -H2t1
- <unwind2_path_d_dx
- >list_append_rcons_sn in H1n; <reverse_append #H1n
- lapply (unwind2_rmap_append_pap_closed f β¦ H1n)
- <reverse_lcons <depth_L_dx #H2n
- lapply (eq_inv_ninj_bi β¦ H2n) -H2n #H2n <H2n -H2n -H1n #Ht1 //
+lemma ifr_unwind_bi (f) (t1) (t2) (r):
+ t1 Ο΅ π β r Ο΅ π β
+ t1 β‘π’π[r] t2 β βΌ[f]t1 β‘π’π[βr] βΌ[f]t2.
+#f #t1 #t2 #r #H1t1 #H2r
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro β¦ (βp) (βq) (βq))
+[ -H1t1 -H2r -Hn -Ht1 -Ht2 //
+| -H1t1 -H2r -Ht1 -Ht2
+ /2 width=2 by path_closed_structure_depth/
+| lapply (in_comp_unwind2_path_term f β¦ Ht1) -Ht2 -Ht1 -H1t1 -H2r
+ <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn
+ <unwind2_rmap_append_closed_Lq_dx_nap_depth //
| lapply (unwind2_term_eq_repl_dx f β¦ Ht2) -Ht2 #Ht2
@(subset_eq_trans β¦ Ht2) -t2
- @(subset_eq_trans β¦ (unwind2_term_fsubst β¦))
+ @(subset_eq_trans β¦ (unwind2_term_fsubst_pic β¦))
[ @fsubst_eq_repl [ // | // ]
- @(subset_eq_canc_dx β¦ (unwind2_term_after β¦))
- @(subset_eq_canc_sn β¦ (unwind2_term_eq_repl_dx β¦))
+ @(subset_eq_canc_sn β¦ (lift_term_eq_repl_dx β¦))
[ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
- @(subset_eq_trans β¦ (unwind2_term_after β¦))
+ @(subset_eq_trans β¦ (lift_unwind2_term_after β¦))
+ @(subset_eq_canc_dx β¦ (unwind2_lift_term_after β¦))
@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 β¦ H1n) #H2n
- >nrplus_inj_dx in β’ (???%); <H2n -H2n
- lapply (tls_unwind2_rmap_append_closed f β¦ H1n) -H1n #H2n
- <(tr_pap_eq_repl β¦ H2n) -H2n //
+ <list_append_rcons_sn
+ @(stream_eq_trans β¦ (tr_compose_uni_dx_pap β¦)) <tr_pap_succ_nap
+ @tr_compose_eq_repl
+ [ <unwind2_rmap_append_closed_Lq_dx_nap_depth //
+ | /2 width=1 by tls_succ_unwind2_rmap_append_closed_Lq_dx/
+ ]
(* Note: crux of the proof ends *)
| //
| /2 width=2 by ex_intro/
- | @term_proper_outer #H0 (**) (* full auto does not work *)
- /3 width=2 by unwind2_term_des_inner/
+ | //
]
]
qed.