include "delayed_updating/reduction/dfr.ma".
include "delayed_updating/reduction/ifr.ma".
-include "delayed_updating/unwind1/unwind_fsubst.ma".
-include "delayed_updating/unwind1/unwind_constructors.ma".
-include "delayed_updating/unwind1/unwind_preterm_eq.ma".
-include "delayed_updating/unwind1/unwind_structure_depth.ma".
-include "delayed_updating/unwind1/unwind_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_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_constructors.ma".
+include "delayed_updating/syntax/path_closed_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
-include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_pap_pushs.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
-(* COMMENT
-axiom pippo (b) (q) (n):
- ↑❘q❘ = (↑[q]𝐢)＠⧣❨n❩ →
- ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)＠⧣❨n+❘b❘❩.
-
-lemma unwind_rmap_tls_eq_id (p) (n):
- ❘p❘ = ↑[p]𝐢＠⧣❨n❩ →
- (𝐢) ≗ ⇂*[n]↑[p]𝐢.
-#p @(list_ind_rcons … p) -p
-[ #n <depth_empty #H destruct
-| #p * [ #m ] #IH #n
- [ <depth_d_dx <unwind_rmap_pap_d_dx #H0
- @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
- @(stream_eq_trans … (IH …)) -IH //
- | /2 width=1 by/
- | <depth_L_dx <unwind_rmap_L_dx
- cases n -n [| #n ] #H0
- [
- |
- ]
- | /2 width=1 by/
- | /2 width=1 by/
- ]
-]
-
+(* Main destructions with ifr ***********************************************)
-(* (↑❘q❘+❘b❘=↑[b●𝗟◗q]𝐢＠⧣❨n+❘b❘❩ *)
-(* [↑[p]𝐢＠⧣❨n❩]⫯*[❘p❘]f∘⇂*[n]↑[p]𝐢) *)
-lemma unwind_rmap_tls_eq (f) (p) (n):
- ❘p❘ = ↑[p]𝐢＠⧣❨n❩ →
- f ≗ ⇂*[n]↑[p]f.
-#f #p #n #Hp
-@(stream_eq_canc_dx … (stream_tls_eq_repl …))
-[| @unwind_rmap_decompose | skip ]
-<tr_compose_tls <Hp
-
-@(stream_eq_canc_dx) … (unwind_rmap_decompose …))
-
-*)
-lemma dfr_unwind_id_bi (p) (q) (t1) (t2): t1 ϵ 𝐓 →
- t1 ➡𝐝𝐟[p,q] t2 → ▼[𝐢]t1 ➡𝐟[⊗p,⊗q] ▼[𝐢]t2.
-#p #q #t1 #t2 #H0t1
-* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro … (⊗b) (↑♭⊗q)) @and4_intro
-[ //
-| (*//*)
-| lapply (in_comp_unwind_bi (𝐢) … Ht1) -Ht1 -H0t1 -Hb -Ht2
- <unwind_path_d_empty_dx <depth_structure //
-| lapply (unwind_term_eq_repl_dx (𝐢) … Ht2) -Ht2 #Ht2
+theorem dfr_des_ifr (f) (t1) (t2) (r): t1 ϵ 𝐓 →
+ t1 ➡𝐝𝐟[r] t2 → ▼[f]t1 ➡𝐢𝐟[⊗r] ▼[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro … (⊗p) (⊗q) (♭q))
+[ -H0t1 -Hn -Ht1 -Ht2 //
+| -H0t1 -Ht1 -Ht2
+ /2 width=2 by path_closed_structure_depth/
+| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
+ <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn
+ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
+| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
@(subset_eq_trans … Ht2) -t2
- @(subset_eq_trans … (unwind_fsubst …))
- [ (*<unwind_rmap_append <unwind_rmap_A_sn <unwind_rmap_append <unwind_rmap_L_sn *)
- <structure_append <structure_A_sn <structure_append <structure_L_sn
- <depth_append <depth_L_sn <depth_structure <depth_structure
- @fsubst_eq_repl [ // ]
- @(subset_eq_trans … (unwind_iref …))
-
- elim Hb -Hb #Hb #H0 <H0 -H0 <nrplus_zero_dx <nplus_zero_dx <nsucc_unfold
- >Hn
+ @(subset_eq_trans … (unwind2_term_fsubst_ppc …))
+ [ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans … (unwind2_term_iref…))
@(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
- [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
- <Hn <Hn
-(*
- @(subset_eq_trans … (lift_term_eq_repl_dx …))
- [ @(unwind_term_eq_repl_sn … (tls_succ_unwind q …)) | skip ]
-*)
-(*
-
- @subset_eq_trans
- [2: @unwind_term_eq_repl_dx
- @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
-
- @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
- [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
-
- @(subset_eq_trans … (unwind_term_after …))
- @(subset_eq_canc_dx … (unwind_term_after …))
- @unwind_term_eq_repl_sn -t1
- @(stream_eq_trans … (tr_compose_uni_dx …))
- lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
- lapply (pippo … b … Hn) -Hn #Hn
+ [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+ @(subset_eq_trans … (lift_unwind2_term_after …))
+ @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+ <list_append_rcons_sn
+ @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
@tr_compose_eq_repl
- [ <unwind_rmap_pap_le //
- <Hn <nrplus_inj_sn //
- |
+ [ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
+ | /2 width=2 by tls_succ_unwind2_rmap_append_closed_Lq_dx/
]
-*)
+(* Note: crux of the proof ends *)
| //
| /2 width=2 by ex_intro/
| //
]
]
-
-(*
-Hn : ↑❘q❘ = ↑[p●𝗔◗b●𝗟◗q]𝐢＠⧣❨n❩
----------------------------
-↑[𝐮❨↑❘q❘+❘b❘❩] ↑[↑[p]𝐢] t ⇔ ↑[𝐮❨↑[p●𝗔◗b●𝗟◗q]𝐢＠⧣❨n+❘b❘❩❩] t
-*)
-(*
-(↑[𝐮❨↑❘q❘❩]▼[⇂*[↑❘q❘]▼[p●𝗟◗q]𝐢](t1⋔(p◖𝗦))⇔▼[𝐮❨↑❘q❘❩∘▼[p●𝗔◗b●𝗟◗q]𝐢](t1⋔(p◖𝗦))
-*)
+qed.
projects / helm.git / blobdiff