include "delayed_updating/reduction/ifr.ma".
include "delayed_updating/substitution/fsubst_lift.ma".
include "delayed_updating/substitution/fsubst_eq.ma".
-include "delayed_updating/substitution/lift_constructors.ma".
+include "delayed_updating/substitution/lift_constructors.ma".
+include "delayed_updating/substitution/lift_preterm_eq.ma".
include "delayed_updating/substitution/lift_structure_depth.ma".
include "delayed_updating/syntax/prototerm_proper_constructors.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
[ <structure_append <structure_A_sn <structure_append <structure_L_sn
@fsubst_eq_repl [ // ]
@(subset_eq_trans … (lift_iref …))
+ @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
+ [ @lift_grafted_S /2 width=2 by ex_intro/ | skip ]
+ @(subset_eq_trans … (lift_term_after …))
+ @(subset_eq_canc_dx … (lift_term_after …))
+ @lift_term_eq_repl_sn -t1
| //
| /2 width=2 by ex_intro/
| //
↑[↑[p]f](t⋔p) ⇔ (↑[f]t)⋔(⊗p).
/3 width=1 by lift_grafted_sn, conj, lift_grafted_dx/ qed.
-(*
--lemma lift_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ ������ →
-- (↑[f]t)⋔((⊗p)◖������) ⊆ ↑[↑[p]f](t⋔(p◖������)).
--#f #t #p #Hp #Ht #q * #r #Hr
--<list_append_rcons_sn #H0
--elim (lift_inv_append_proper_dx … (sym_eq … H0)) -H0 //
-+lemma lift_grafted_dx (f) (t) (p): p ϵ ������ → p ϵ ▵t → t ϵ ������ →
-+ (↑[f]t)⋔(⊗p) ⊆ ↑[↑[p]f](t⋔p).
-+#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
-+elim (lift_inv_append_inner_sn … (sym_eq … H0)) -H0 //
- #p0 #q0 #Hp0 #Hq0 #H0 destruct
- <(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
--elim (lift_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
--#r1 #r2 #Hr1 #Hr2 #H0 destruct
--lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
--[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
- /2 width=1 by in_comp_lift_bi/
- qed-.
-
--lemma lift_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ ������ →
-- ↑[↑[p]f](t⋔(p◖������)) ⇔ (↑[f]t)⋔((⊗p)◖������).
--/3 width=1 by lift_grafted_S_sn, conj, lift_grafted_S_dx/ qed.
-+lemma lift_grafted (f) (t) (p): p ϵ ������ → p ϵ ▵t → t ϵ ������ →
-+ ↑[↑[p]f](t⋔p) ⇔ (↑[f]t)⋔(⊗p).
-+/3 width=1 by lift_grafted_sn, conj, lift_grafted_dx/ qed.
+lemma lift_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+ (↑[f]t)⋔((⊗p)◖𝗦) ⊆ ↑[↑[p]f](t⋔(p◖𝗦)).
+#f #t #p #Hp #Ht #q * #r #Hr
+<list_append_rcons_sn #H0
+elim (lift_inv_append_proper_dx … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+elim (lift_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
+#r1 #r2 #Hr1 #Hr2 #H0 destruct
+lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
+[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
+/2 width=1 by in_comp_lift_bi/
+qed-.
-*)
+lemma lift_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+ ↑[↑[p]f](t⋔(p◖𝗦)) ⇔ (↑[f]t)⋔((⊗p)◖𝗦).
+#f #t #p #Hp #Ht
+@conj
+[ >lift_rmap_S_dx >structure_S_dx
+ @lift_grafted_sn //
+| /2 width=1 by lift_grafted_S_dx/
+]
+qed.