(**************************************************************************)
include "delayed_updating/syntax/prototerm.ma".
+include "delayed_updating/notation/functions/m_hook_1.ma".
include "delayed_updating/notation/functions/hash_1.ma".
-include "delayed_updating/notation/functions/phi_2.ma".
+include "delayed_updating/notation/functions/tau_2.ma".
include "delayed_updating/notation/functions/lamda_1.ma".
include "delayed_updating/notation/functions/at_2.ma".
definition prototerm_node_1 (l): prototerm → prototerm ≝
λt,p. ∃∃q. q ϵ t & l◗q = p.
+definition prototerm_node_1_2 (l1) (l2): prototerm → prototerm ≝
+ λt,p. ∃∃q. q ϵ t & l1◗l2◗q = p.
+
definition prototerm_node_2 (l1) (l2): prototerm → prototerm → prototerm ≝
λt1,t2,p.
∨∨ ∃∃q. q ϵ t1 & l1◗q = p
| ∃∃q. q ϵ t2 & l2◗q = p.
+interpretation
+ "mark (prototerm)"
+ 'MHook t = (prototerm_node_1 label_m t).
+
interpretation
"outer variable reference by depth (prototerm)"
- 'Hash n = (prototerm_node_0 (label_node_d n)).
+ 'Hash k = (prototerm_node_0 (label_d k)).
interpretation
"inner variable reference by depth (prototerm)"
- 'Phi n t = (prototerm_node_1 (label_node_d n) t).
+ 'Tau k t = (prototerm_node_1_2 (label_d k) label_m t).
interpretation
"name-free functional abstraction (prototerm)"
- 'Lamda t = (prototerm_node_1 label_edge_L t).
+ 'Lamda t = (prototerm_node_1 label_L t).
interpretation
"application (prototerm)"
- 'At u t = (prototerm_node_2 label_edge_S label_edge_A u t).
+ 'At u t = (prototerm_node_2 label_S label_A u t).
-(* Basic Inversions *********************************************************)
+(* Basic constructions *******************************************************)
-lemma prototerm_in_root_inv_lcons_oref:
- ∀p,l,n. l◗p ϵ ▵#n →
- ∧∧ 𝗱n = l & 𝐞 = p.
-#p #l #n * #q
-<list_append_lcons_sn #H0 destruct -H0
-elim (eq_inv_list_empty_append … e0) -e0 #H0 #_
-/2 width=1 by conj/
-qed-.
+lemma in_comp_oref_hd (k):
+ (𝗱k◗𝐞) ϵ ⧣k.
+// qed.
+
+lemma in_comp_iref_hd (t) (q) (k):
+ q ϵ t → 𝗱k◗𝗺◗q ϵ 𝛕k.t.
+/2 width=3 by ex2_intro/ qed.
+
+lemma in_comp_abst_hd (t) (q):
+ q ϵ t → 𝗟◗q ϵ 𝛌.t.
+/2 width=3 by ex2_intro/ qed.
+
+lemma in_comp_appl_sd (u) (t) (q):
+ q ϵ u → 𝗦◗q ϵ @u.t.
+/3 width=3 by ex2_intro, or_introl/ qed.
-lemma prototerm_in_root_inv_lcons_iref:
- ∀t,p,l,n. l◗p ϵ ▵𝛗n.t →
- ∧∧ 𝗱n = l & p ϵ ▵t.
-#t #p #l #n * #q
-<list_append_lcons_sn * #r #Hr #H0 destruct
-/3 width=2 by ex_intro, conj/
+lemma in_comp_appl_hd (u) (t) (q):
+ q ϵ t → 𝗔◗q ϵ @u.t.
+/3 width=3 by ex2_intro, or_intror/ qed.
+
+(* Basic inversions *********************************************************)
+
+lemma in_comp_inv_iref (t) (p) (k):
+ p ϵ 𝛕k.t →
+ ∃∃q. 𝗱k◗𝗺◗q = p & q ϵ t.
+#t #p #k * #q #Hq #Hp
+/2 width=3 by ex2_intro/
qed-.
-lemma prototerm_in_root_inv_lcons_abst:
- ∀t,p,l. l◗p ϵ ▵𝛌.t →
- ∧∧ 𝗟 = l & p ϵ ▵t.
-#t #p #l * #q
-<list_append_lcons_sn * #r #Hr #H0 destruct
-/3 width=2 by ex_intro, conj/
+lemma in_comp_inv_abst (t) (p):
+ p ϵ 𝛌.t →
+ ∃∃q. 𝗟◗q = p & q ϵ t.
+#t #p * #q #Hq #Hp
+/2 width=3 by ex2_intro/
qed-.
-lemma prototerm_in_root_inv_lcons_appl:
- ∀u,t,p,l. l◗p ϵ ▵@u.t →
- ∨∨ ∧∧ 𝗦 = l & p ϵ ▵u
- | ∧∧ 𝗔 = l & p ϵ ▵t.
-#u #t #p #l * #q
-<list_append_lcons_sn * * #r #Hr #H0 destruct
-/4 width=2 by ex_intro, or_introl, or_intror, conj/
+lemma in_comp_inv_appl (u) (t) (p):
+ p ϵ @u.t →
+ ∨∨ ∃∃q. 𝗦◗q = p & q ϵ u
+ | ∃∃q. 𝗔◗q = p & q ϵ t.
+#u #t #p * * #q #Hq #Hp
+/3 width=3 by ex2_intro, or_introl, or_intror/
qed-.
+
+(* Advanced inversions ******************************************************)
+
+lemma in_comp_inv_abst_hd (t) (p):
+ (𝗟◗p) ϵ 𝛌.t → p ϵ t.
+#t #p #H0
+elim (in_comp_inv_abst … H0) -H0 #q #H0 #Hq
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H1 #H2 destruct //
+qed-.
+
+lemma in_comp_inv_appl_sd (u) (t) (p):
+ (𝗦◗p) ϵ @u.t → p ϵ u.
+#u #t #p #H0
+elim (in_comp_inv_appl … H0) -H0 * #q #H0 #Hq
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H1 #H2 destruct //
+qed-.
+
+lemma in_comp_inv_appl_hd (u) (t) (p):
+ (𝗔◗p) ϵ @u.t → p ϵ t.
+#u #t #p #H0
+elim (in_comp_inv_appl … H0) -H0 * #q #H0 #Hq
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H1 #H2 destruct //
+qed-.