1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "delayed_updating/reduction/dfr.ma".
16 include "delayed_updating/reduction/ifr.ma".
18 include "delayed_updating/unwind1/unwind_fsubst.ma".
19 include "delayed_updating/unwind1/unwind_constructors.ma".
20 include "delayed_updating/unwind1/unwind_preterm_eq.ma".
21 include "delayed_updating/unwind1/unwind_structure_depth.ma".
22 include "delayed_updating/unwind1/unwind_depth.ma".
24 include "delayed_updating/substitution/fsubst_eq.ma".
25 include "delayed_updating/substitution/lift_prototerm_eq.ma".
26 include "delayed_updating/syntax/prototerm_proper_constructors.ma".
27 include "delayed_updating/syntax/path_structure_depth.ma".
28 include "ground/relocation/tr_uni_compose.ma".
29 include "ground/relocation/tr_pap_pushs.ma".
31 (* DELAYED FOCUSED REDUCTION ************************************************)
34 axiom pippo (b) (q) (n):
36 ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)@❨n+❘b❘❩.
38 lemma unwind_rmap_tls_eq_id (p) (n):
41 #p @(list_ind_rcons … p) -p
42 [ #n <depth_empty #H destruct
44 [ <depth_d_dx <unwind_rmap_pap_d_dx #H0
45 @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
46 @(stream_eq_trans … (IH …)) -IH //
48 | <depth_L_dx <unwind_rmap_L_dx
49 cases n -n [| #n ] #H0
59 (* (↑❘q❘+❘b❘=↑[b●𝗟◗q]𝐢@❨n+❘b❘❩ *)
60 (* [↑[p]𝐢@❨n❩]⫯*[❘p❘]f∘⇂*[n]↑[p]𝐢) *)
61 lemma unwind_rmap_tls_eq (f) (p) (n):
65 @(stream_eq_canc_dx … (stream_tls_eq_repl …))
66 [| @unwind_rmap_decompose | skip ]
69 @(stream_eq_canc_dx) … (unwind_rmap_decompose …))
72 lemma dfr_unwind_id_bi (p) (q) (t1) (t2): t1 ϵ 𝐓 →
73 t1 ➡𝐝𝐟[p,q] t2 → ▼[𝐢]t1 ➡𝐟[⊗p,⊗q] ▼[𝐢]t2.
75 * #b #n * #Hb #Hn #Ht1 #Ht2
76 @(ex1_2_intro … (⊗b) (↑♭⊗q)) @and4_intro
79 | lapply (in_comp_unwind_bi (𝐢) … Ht1) -Ht1 -H0t1 -Hb -Ht2
80 <unwind_path_d_empty_dx <depth_structure //
81 | lapply (unwind_term_eq_repl_dx (𝐢) … Ht2) -Ht2 #Ht2
82 @(subset_eq_trans … Ht2) -t2
83 @(subset_eq_trans … (unwind_fsubst …))
84 [ (*<unwind_rmap_append <unwind_rmap_A_sn <unwind_rmap_append <unwind_rmap_L_sn *)
85 <structure_append <structure_A_sn <structure_append <structure_L_sn
86 <depth_append <depth_L_sn <depth_structure <depth_structure
87 @fsubst_eq_repl [ // ]
88 @(subset_eq_trans … (unwind_iref …))
90 elim Hb -Hb #Hb #H0 <H0 -H0 <nrplus_zero_dx <nplus_zero_dx <nsucc_unfold
92 @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
93 [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
96 @(subset_eq_trans … (lift_term_eq_repl_dx …))
97 [ @(unwind_term_eq_repl_sn … (tls_succ_unwind q …)) | skip ]
102 [2: @unwind_term_eq_repl_dx
103 @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
105 @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
106 [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
108 @(subset_eq_trans … (unwind_term_after …))
109 @(subset_eq_canc_dx … (unwind_term_after …))
110 @unwind_term_eq_repl_sn -t1
111 @(stream_eq_trans … (tr_compose_uni_dx …))
112 lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
113 lapply (pippo … b … Hn) -Hn #Hn
115 [ <unwind_rmap_pap_le //
116 <Hn <nrplus_inj_sn //
121 | /2 width=2 by ex_intro/
127 Hn : ↑❘q❘ = ↑[p●𝗔◗b●𝗟◗q]𝐢@❨n❩
128 ---------------------------
129 ↑[𝐮❨↑❘q❘+❘b❘❩] ↑[↑[p]𝐢] t ⇔ ↑[𝐮❨↑[p●𝗔◗b●𝗟◗q]𝐢@❨n+❘b❘❩❩] t
132 (↑[𝐮❨↑❘q❘❩]▼[⇂*[↑❘q❘]▼[p●𝗟◗q]𝐢](t1⋔(p◖𝗦))⇔▼[𝐮❨↑❘q❘❩∘▼[p●𝗔◗b●𝗟◗q]𝐢](t1⋔(p◖𝗦))