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/substitution/lift.ma".
16 include "ground/relocation/tr_pap_eq.ma".
17 include "ground/relocation/tr_pn_eq.ma".
19 (* LIFT FOR PATH ***********************************************************)
21 definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
22 λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
25 "extensional equivalence (lift continuation)"
26 'RingEq A k1 k2 = (lift_exteq A k1 k2).
28 (* Constructions with lift_exteq ********************************************)
30 lemma lift_eq_repl (A) (p) (k1) (k2):
31 k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
32 #A #p elim p -p [| * [ #n ] #q #IH ]
33 #k1 #k2 #Hk #f1 #f2 #Hf
34 [ <lift_empty <lift_empty /2 width=1 by/
35 | <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
36 /3 width=1 by stream_eq_refl/
38 | /3 width=1 by tr_push_eq_repl/
44 (* Advanced constructions ***************************************************)
46 lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
47 ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
49 @lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
52 lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
53 ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
54 #A #k #f #p1 #p #l #Hk
55 @lift_eq_repl // #g1 #g2 #p2 #Hg
56 <list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
59 (* Advanced constructions with proj_path ************************************)
61 lemma proj_path_proper:
62 proj_path ≗ proj_path.
65 lemma lift_path_eq_repl (p):
66 stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
67 /2 width=1 by lift_eq_repl/ qed.
69 lemma lift_path_append_sn (p) (f) (q):
70 q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
71 #p elim p -p // * [ #n ] #p #IH #f #q
72 [ <lift_d_sn <lift_d_sn
73 | <lift_m_sn <lift_m_sn
74 | <lift_L_sn <lift_L_sn
75 | <lift_A_sn <lift_A_sn
76 | <lift_S_sn <lift_S_sn
78 >lift_lcons_alt // >lift_append_rcons_sn //
79 <IH <IH -IH <list_append_rcons_sn //
82 lemma lift_path_lcons (f) (p) (l):
83 l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
85 >lift_lcons_alt <lift_path_append_sn //
88 lemma lift_path_d_sn (f) (p) (n):
89 (𝗱(f@❨n❩)◗↑[𝐢]p) = ↑[f](𝗱n◗p).
92 lemma lift_path_m_sn (f) (p):
93 (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
96 lemma lift_path_L_sn (f) (p):
97 (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
100 lemma lift_path_A_sn (f) (p):
101 (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
104 lemma lift_path_S_sn (f) (p):
105 (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
108 lemma lift_path_id (p):
117 lemma lift_path_append (p2) (p1) (f):
118 (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
119 #p2 #p1 elim p1 -p1 //
121 [ <lift_path_d_sn <lift_path_d_sn <IH //
122 | <lift_path_m_sn <lift_path_m_sn <IH //
123 | <lift_path_L_sn <lift_path_L_sn <IH //
124 | <lift_path_A_sn <lift_path_A_sn <IH //
125 | <lift_path_S_sn <lift_path_S_sn <IH //
129 lemma lift_path_d_dx (n) (p) (f):
130 (↑[f]p)◖𝗱((↑[p]f)@❨n❩) = ↑[f](p◖𝗱n).
131 #n #p #f <lift_path_append //
134 lemma lift_path_m_dx (p) (f):
135 (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
136 #p #f <lift_path_append //
139 lemma lift_path_L_dx (p) (f):
140 (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
141 #p #f <lift_path_append //
144 lemma lift_path_A_dx (p) (f):
145 (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
146 #p #f <lift_path_append //
149 lemma lift_path_S_dx (p) (f):
150 (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
151 #p #f <lift_path_append //
156 (* Advanced constructions with proj_rmap and stream_tls *********************)
158 lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
159 ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
161 <lift_rmap_d_dx >nrplus_inj_dx
162 /2 width=1 by tr_tls_compose_uni_dx/