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4 (* ||A|| A project by Andrea Asperti *)
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15 include "delayed_updating/substitution/fsubst.ma".
16 include "delayed_updating/substitution/lift_prototerm_eq.ma".
18 (* FOCALIZED SUBSTITUTION ***************************************************)
20 (* Constructions with lift for preterm **************************************)
22 lemma lift_term_fsubst_sn (f) (t) (u) (p):
23 (↑[f]t)[⋔(↑[f]p)←↑[↑[p]f]u] ⊆ ↑[f](t[⋔p←u]).
25 [ #rl * #r #Hr #H1 #H2 destruct
27 /4 width=3 by in_comp_lift_path_term, or_introl, ex2_intro/
29 @(ex2_intro … H1) @or_intror @conj // -Hq #r #H2 destruct
34 lemma lift_term_fsubst_dx (f) (t) (u) (p):
35 ↑[f](t[⋔p←u]) ⊆ (↑[f]t)[⋔(↑[f]p)←↑[↑[p]f]u].
36 #f #t #u #p #ql * #q * *
37 [ #r #Hu #H1 #H2 destruct
39 [|| <lift_path_append // ]
40 /2 width=3 by ex2_intro/
41 | #Hq #H0 #H1 destruct
42 @or_intror @conj [ /2 width=1 by in_comp_lift_path_term/ ] -Hq #r #Hr
43 elim (lift_path_inv_append_sn … Hr) -Hr #s1 #s2 #Hs1 #_ #H1 destruct
44 lapply (lift_path_inj … Hs1) -Hs1 #H1 destruct
49 lemma lift_term_fsubst (f) (t) (u) (p):
50 (↑[f]t)[⋔(↑[f]p)←↑[↑[p]f]u] ⇔ ↑[f](t[⋔p←u]).
51 /3 width=1 by lift_term_fsubst_sn, conj, lift_term_fsubst_dx/ qed.