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/unwind/unwind2_prototerm_eq.ma".
16 include "delayed_updating/unwind/unwind2_path_append.ma".
17 include "delayed_updating/substitution/fsubst.ma".
18 include "delayed_updating/syntax/preterm.ma".
19 include "delayed_updating/syntax/prototerm_proper.ma".
21 (* TAILED UNWIND FOR PRETERM ************************************************)
23 (* Constructions with fsubst and pic ****************************************)
25 lemma unwind2_term_fsubst_pic_sn (f) (t) (u) (p): p Ļµ š ā
26 (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u] ā ā¼[f](t[āpāu]).
27 #f #t #u #p #Hp #ql * *
28 [ #rl * #r #Hr #H1 #H2 destruct
29 >unwind2_path_append_pic_sn
30 /4 width=3 by in_comp_unwind2_path_term, or_introl, ex2_intro/
32 @(ex2_intro ā¦ H1) @or_intror @conj // *
33 [ <list_append_empty_sn #H2 destruct
34 elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
36 /3 width=2 by unwind2_path_append_pic_sn/
41 lemma unwind2_term_fsubst_pic_dx (f) (t) (u) (p): p Ļµ š ā p Ļµ āµt ā t Ļµ š ā
42 ā¼[f](t[āpāu]) ā (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u].
43 #f #t #u #p #Hp #H1p #H2p #ql * #q * *
44 [ #r #Hu #H1 #H2 destruct
45 /5 width=3 by unwind2_path_append_pic_sn, ex2_intro, or_introl/
46 | #Hq #H0 #H1 destruct
47 @or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
48 [ <list_append_empty_sn #Hr @(H0 ā¦ (š)) -H0
49 <list_append_empty_sn @H2p -H2p
50 /2 width=2 by unwind2_path_des_structure, prototerm_in_comp_root/
52 elim (unwind2_path_inv_append_ppc_dx ā¦ Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
53 lapply (H2p ā¦ Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
58 lemma unwind2_term_fsubst_pic (f) (t) (u) (p): p Ļµ š ā p Ļµ āµt ā t Ļµ š ā
59 (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u] ā ā¼[f](t[āpāu]).
60 /4 width=3 by unwind2_term_fsubst_pic_sn, conj, unwind2_term_fsubst_pic_dx/ qed.
62 (* Constructions with fsubst and ppc ****************************************)
64 lemma unwind2_term_fsubst_ppc_sn (f) (t) (u) (p): u Ļµ š ā
65 (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u] ā ā¼[f](t[āpāu]).
66 #f #t #u #p #Hu #ql * *
67 [ #rl * #r #Hr #H1 #H2 destruct
68 >unwind2_path_append_ppc_dx
69 /4 width=5 by in_comp_unwind2_path_term, in_comp_tpc_trans, or_introl, ex2_intro/
71 @(ex2_intro ā¦ H1) @or_intror @conj // *
72 [ <list_append_empty_sn #H2 destruct
73 elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
75 @H0 -H0 [| <unwind2_path_append_ppc_dx /2 width=3 by ppc_rcons/ ]
80 lemma unwind2_term_fsubst_ppc_dx (f) (t) (u) (p): u Ļµ š ā p Ļµ āµt ā t Ļµ š ā
81 ā¼[f](t[āpāu]) ā (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u].
82 #f #t #u #p #Hu #H1p #H2p #ql * #q * *
83 [ #r #Hu #H1 #H2 destruct
85 [|| <unwind2_path_append_ppc_dx /2 width=5 by in_comp_tpc_trans/ ]
86 /2 width=3 by ex2_intro/
87 | #Hq #H0 #H1 destruct
88 @or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
89 [ <list_append_empty_sn #Hr @(H0 ā¦ (š)) -H0
90 <list_append_empty_sn @H2p -H2p
91 /2 width=2 by unwind2_path_des_structure, prototerm_in_comp_root/
93 elim (unwind2_path_inv_append_ppc_dx ā¦ Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
94 lapply (H2p ā¦ Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
99 lemma unwind2_term_fsubst_ppc (f) (t) (u) (p): u Ļµ š ā p Ļµ āµt ā t Ļµ š ā
100 (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u] ā ā¼[f](t[āpāu]).
101 /4 width=3 by unwind2_term_fsubst_ppc_sn, conj, unwind2_term_fsubst_ppc_dx/ qed.