(* Constructions with subset_equivalence ************************************)
lemma lift_term_eq_repl_sn (f1) (f2) (t):
- f1 ≗ f2 → ↑[f1]t ⇔ ↑[f2]t.
+ f1 ≗ f2 → 🠡[f1]t ⇔ 🠡[f2]t.
/3 width=1 by subset_equivalence_ext_f1_exteq, lift_path_eq_repl/
qed.
lemma lift_term_eq_repl_dx (f) (t1) (t2):
- t1 ⇔ t2 → ↑[f]t1 ⇔ ↑[f]t2.
+ t1 ⇔ t2 → 🠡[f]t1 ⇔ 🠡[f]t2.
/2 width=1 by subset_equivalence_ext_f1_bi/
qed.
lemma lift_term_grafted_sn (f) (t) (p):
- ↑[↑[p]f](t⋔p) ⊆ (↑[f]t)⋔(↑[f]p).
+ 🠡[🠢[f]p](t⋔p) ⊆ (🠡[f]t)⋔(🠡[f]p).
#f #t #p #q * #r #Hr #H0 destruct
/2 width=3 by ex2_intro/
qed-.
lemma lift_term_grafted_dx (f) (t) (p):
- (↑[f]t)⋔(↑[f]p) ⊆ ↑[↑[p]f](t⋔p).
+ (🠡[f]t)⋔(🠡[f]p) ⊆ 🠡[🠢[f]p](t⋔p).
#f #t #p #q * #r #Hr #H0
elim (lift_path_inv_append_sn … (sym_eq … H0)) -H0
#p0 #q0 #Hp0 #Hq0 #H0 destruct
qed-.
lemma lift_term_grafted (f) (t) (p):
- ↑[↑[p]f](t⋔p) ⇔ (↑[f]t)⋔(↑[f]p).
+ 🠡[🠢[f]p](t⋔p) ⇔ (🠡[f]t)⋔(🠡[f]p).
/3 width=1 by lift_term_grafted_sn, lift_term_grafted_dx, conj/ qed.
lemma lift_term_grafted_S (f) (t) (p):
- ↑[↑[p]f](t⋔(p◖𝗦)) ⇔ (↑[f]t)⋔((↑[f]p)◖𝗦).
+ 🠡[🠢[f]p](t⋔(p◖𝗦)) ⇔ (🠡[f]t)⋔((🠡[f]p)◖𝗦).
// qed.