include "ground/xoa/ex_4_2.ma".
include "ground/xoa/ex_4_3.ma".
include "ground/xoa/ex_5_3.ma".
-include "delayed_updating/syntax/term_constructors.ma".
+include "delayed_updating/syntax/preterm_constructors.ma".
include "delayed_updating/notation/relations/in_predicate_d_phi_1.ma".
(* BY-DEPTH DELAYED (BDD) TERM **********************************************)
-inductive bdd: predicate term ≝
+inductive bdd: predicate preterm ≝
| bdd_oref: ∀n. bdd #n
| bdd_iref: ∀t,n. bdd t → bdd 𝛗n.t
| bdd_abst: ∀t. bdd t → bdd 𝛌.t
.
interpretation
- "well-formed by-depth delayed (term)"
+ "well-formed by-depth delayed (preterm)"
'InPredicateDPhi t = (bdd t).
-(* Basic destructions *******************************************************)
+(* Basic inversions *********************************************************)
-lemma bdd_inv_in_com_gen:
+lemma bdd_inv_in_comp_gen:
∀t,p. t ϵ 𝐃𝛗 → p ϵ⬦ t →
∨∨ ∃∃n. #n = t & 𝗱❨n❩;𝐞 = p
| ∃∃u,q,n. u ϵ 𝐃𝛗 & q ϵ⬦ u & 𝛗n.u = t & 𝗱❨n❩;q = p
]
qed-.
-lemma bdd_inv_in_com_d:
+lemma bdd_inv_in_comp_d:
∀t,q,n. t ϵ 𝐃𝛗 → 𝗱❨n❩;q ϵ⬦ t →
∨∨ ∧∧ #n = t & 𝐞 = q
| ∃∃u. u ϵ 𝐃𝛗 & q ϵ⬦ u & 𝛗n.u = t
.
#t #q #n #Ht #Hq
-elim (bdd_inv_in_com_gen … Ht Hq) -Ht -Hq *
+elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
[ #n0 #H1 #H2 destruct /3 width=1 by or_introl, conj/
| #u0 #q0 #n0 #Hu0 #Hq0 #H1 #H2 destruct /3 width=4 by ex3_intro, or_intror/
| #u0 #q0 #_ #_ #_ #H0 destruct
]
qed-.
-lemma bdd_inv_in_ini_d:
+lemma bdd_inv_in_root_d:
∀t,q,n. t ϵ 𝐃𝛗 → 𝗱❨n❩;q ϵ▵ t →
∨∨ ∧∧ #n = t & 𝐞 = q
| ∃∃u. u ϵ 𝐃𝛗 & q ϵ▵ u & 𝛗n.u = t
.
#t #q #n #Ht * #r #Hq
-elim (bdd_inv_in_com_d … Ht Hq) -Ht -Hq *
+elim (bdd_inv_in_comp_d … Ht Hq) -Ht -Hq *
[ #H1 #H2
elim (eq_inv_list_empty_append … H2) -H2 #H2 #_ destruct
/3 width=1 by or_introl, conj/
-| #u #Hu #Hq #H1 destruct
+| #u #Hu #Hq #H0 destruct
/4 width=4 by ex3_intro, ex_intro, or_intror/
]
qed-.
-lemma pippo:
+lemma bdd_inv_in_comp_L:
+ ∀t,q. t ϵ 𝐃𝛗 → 𝗟;q ϵ⬦ t →
+ ∃∃u. u ϵ 𝐃𝛗 & q ϵ⬦ u & 𝛌.u = t
+.
+#t #q #Ht #Hq
+elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
+[ #n0 #_ #H0 destruct
+| #u0 #q0 #n0 #_ #_ #_ #H0 destruct
+| #u0 #q0 #Hu0 #Hq0 #H1 #H2 destruct /2 width=4 by ex3_intro/
+| #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
+| #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
+]
+qed-.
+
+lemma bdd_inv_in_root_L:
+ ∀t,q. t ϵ 𝐃𝛗 → 𝗟;q ϵ▵ t →
+ ∃∃u. u ϵ 𝐃𝛗 & q ϵ▵ u & 𝛌.u = t.
+#t #q #Ht * #r #Hq
+elim (bdd_inv_in_comp_L … Ht Hq) -Ht -Hq
+#u #Hu #Hq #H0 destruct
+/3 width=4 by ex3_intro, ex_intro/
+qed-.
+
+lemma bdd_inv_in_comp_A:
+ ∀t,q. t ϵ 𝐃𝛗 → 𝗔;q ϵ⬦ t →
+ ∃∃v,u. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ⬦ u & @v.u = t
+.
+#t #q #Ht #Hq
+elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
+[ #n0 #_ #H0 destruct
+| #u0 #q0 #n0 #_ #_ #_ #H0 destruct
+| #u0 #q0 #_ #_ #_ #H0 destruct
+| #v0 #u0 #q0 #Hv0 #Hu0 #Hq0 #H1 #H2 destruct /2 width=6 by ex4_2_intro/
+| #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
+]
+qed-.
+
+lemma bdd_inv_in_root_A:
+ ∀t,q. t ϵ 𝐃𝛗 → 𝗔;q ϵ▵ t →
+ ∃∃v,u. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ▵ u & @v.u = t
+.
+#t #q #Ht * #r #Hq
+elim (bdd_inv_in_comp_A … Ht Hq) -Ht -Hq
+#v #u #Hv #Hu #Hq #H0 destruct
+/3 width=6 by ex4_2_intro, ex_intro/
+qed-.
+
+lemma bdd_inv_in_comp_S:
+ ∀t,q. t ϵ 𝐃𝛗 → 𝗦;q ϵ⬦ t →
+ ∃∃v,u. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ⬦ v & @v.u = t
+.
+#t #q #Ht #Hq
+elim (bdd_inv_in_comp_gen … Ht Hq) -Ht -Hq *
+[ #n0 #_ #H0 destruct
+| #u0 #q0 #n0 #_ #_ #_ #H0 destruct
+| #u0 #q0 #_ #_ #_ #H0 destruct
+| #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
+| #v0 #u0 #q0 #Hv0 #Hu0 #Hq0 #H1 #H2 destruct /2 width=6 by ex4_2_intro/
+]
+qed-.
+
+lemma bdd_inv_in_root_S:
+ ∀t,q. t ϵ 𝐃𝛗 → 𝗦;q ϵ▵ t →
+ ∃∃v,u. v ϵ 𝐃𝛗 & u ϵ 𝐃𝛗 & q ϵ▵ v & @v.u = t
+.
+#t #q #Ht * #r #Hq
+elim (bdd_inv_in_comp_S … Ht Hq) -Ht -Hq
+#v #u #Hv #Hu #Hq #H0 destruct
+/3 width=6 by ex4_2_intro, ex_intro/
+qed-.
+
+(* Advanced inversions ******************************************************)
+
+lemma bbd_mono_in_root_d:
∀l,n,p,t. t ϵ 𝐃𝛗 → p,𝗱❨n❩ ϵ▵ t → p,l ϵ▵ t → 𝗱❨n❩ = l.
#l #n #p elim p -p
[ #t #Ht <list_cons_comm <list_cons_comm #Hn #Hl
- elim (bdd_inv_in_ini_d … Ht Hn) -Ht -Hn *
- [ #H1 #_ destruct
- elim (term_in_ini_inv_lcons_oref … Hl) -Hl //
- | #u #_ #_ #H1 destruct
- elim (term_in_ini_inv_lcons_iref … Hl) -Hl //
+ elim (bdd_inv_in_root_d … Ht Hn) -Ht -Hn *
+ [ #H0 #_ destruct
+ elim (preterm_in_root_inv_lcons_oref … Hl) -Hl //
+ | #u #_ #_ #H0 destruct
+ elim (preterm_in_root_inv_lcons_iref … Hl) -Hl //
]
| * [ #m ] #p #IH #t #Ht
<list_cons_shift <list_cons_shift #Hn #Hl
- [ elim (bdd_inv_in_ini_d … Ht Hn) -Ht -Hn *
- [ #_ #H
- elim (eq_inv_list_empty_rcons ??? H)
- | #u #Hu #Hp #H destruct
- elim (term_in_ini_inv_lcons_iref … Hl) -Hl #_ #Hl
- @(IH … Hu) //
+ [ elim (bdd_inv_in_root_d … Ht Hn) -Ht -Hn *
+ [ #_ #H0
+ elim (eq_inv_list_empty_rcons ??? H0)
+ | #u #Hu #Hp #H0 destruct
+ elim (preterm_in_root_inv_lcons_iref … Hl) -Hl #_ #Hl
+ /2 width=4 by/
]
- |
+ | elim (bdd_inv_in_root_L … Ht Hn) -Ht -Hn
+ #u #Hu #Hp #H0 destruct
+ elim (preterm_in_root_inv_lcons_abst … Hl) -Hl #_ #Hl
+ /2 width=4 by/
+ | elim (bdd_inv_in_root_A … Ht Hn) -Ht -Hn
+ #v #u #_ #Hu #Hp #H0 destruct
+ elim (preterm_in_root_inv_lcons_appl … Hl) -Hl * #H0 #Hl destruct
+ /2 width=4 by/
+ | elim (bdd_inv_in_root_S … Ht Hn) -Ht -Hn
+ #v #u #Hv #_ #Hp #H0 destruct
+ elim (preterm_in_root_inv_lcons_appl … Hl) -Hl * #H0 #Hl destruct
+ /2 width=4 by/
]
]
+qed-.
inductive label: Type[0] ≝
| label_node_d: pnat → label
-| label_edge_l: label
-| label_edge_a: label
-| label_edge_s: label
+| label_edge_L: label
+| label_edge_A: label
+| label_edge_S: label
.
interpretation
interpretation
"name-free functional abstruction (label)"
- 'EdgeLabelL = (label_edge_l).
+ 'EdgeLabelL = (label_edge_L).
interpretation
"application (label)"
- 'EdgeLabelA = (label_edge_a).
+ 'EdgeLabelA = (label_edge_A).
interpretation
"side branch (label)"
- 'EdgeLabelS = (label_edge_s).
+ 'EdgeLabelS = (label_edge_S).
include "delayed_updating/notation/functions/semicolon_2.ma".
include "delayed_updating/notation/functions/comma_2.ma".
-
(* PATH *********************************************************************)
definition path ≝ list label.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground/relocation/tr_pap.ma".
+include "delayed_updating/syntax/path.ma".
+
+(* DEPHI FOR PATH ***********************************************************)
+
+rec definition path_dephi (f) (p) on p ≝
+match p with
+[ list_empty ⇒ 𝐞
+| list_lcons l q ⇒
+ match l with
+ [ label_node_d n ⇒ 𝗱❨f@❨n❩❩;path_dephi f q
+ | label_edge_L ⇒ 𝗟;path_dephi (𝟏⨮f) q
+ | label_edge_A ⇒ 𝗔;path_dephi f q
+ | label_edge_S ⇒ 𝗦;path_dephi f q
+ ]
+].
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/syntax/path.ma".
+include "delayed_updating/notation/relations/up_down_arrow_epsilon_2.ma".
+include "delayed_updating/notation/relations/up_arrow_epsilon_2.ma".
+
+(* PRETERM ******************************************************************)
+
+definition preterm: Type[0] ≝ predicate path.
+
+definition preterm_in_comp: relation2 path preterm ≝
+ λp,t. t p.
+
+interpretation
+ "belongs to complete (preterm)"
+ 'UpDownArrowEpsilon p t = (preterm_in_comp p t).
+
+definition preterm_in_root: relation2 path preterm ≝
+ λp,t. ∃q. p;;q ϵ⬦ t.
+
+interpretation
+ "belongs to root (preterm)"
+ 'UpArrowEpsilon p t = (preterm_in_root p t).
+
+(* Basic constructions ******************************************************)
+
+lemma preterm_in_comp_root (p) (t):
+ p ϵ⬦ t → p ϵ▵ t.
+/2 width=2 by ex_intro/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/notation/functions/hash_1.ma".
+include "delayed_updating/notation/functions/phi_2.ma".
+include "delayed_updating/notation/functions/lamda_1.ma".
+include "delayed_updating/notation/functions/at_2.ma".
+
+(* CONSTRUCTORS FOR PRETERM *************************************************)
+
+definition preterm_node_0 (l): preterm ≝
+ λp. l;𝐞 = p.
+
+definition preterm_node_1 (l): preterm → preterm ≝
+ λt,p. ∃∃q. q ϵ⬦ t & l;q = p.
+
+definition preterm_node_2 (l1) (l2): preterm → preterm → preterm ≝
+ λt1,t2,p.
+ ∨∨ ∃∃q. q ϵ⬦ t1 & l1;q = p
+ | ∃∃q. q ϵ⬦ t2 & l2;q = p.
+
+interpretation
+ "outer variable reference by depth (preterm)"
+ 'Hash n = (preterm_node_0 (label_node_d n)).
+
+interpretation
+ "inner variable reference by depth (preterm)"
+ 'Phi n t = (preterm_node_1 (label_node_d n) t).
+
+interpretation
+ "name-free functional abstraction (preterm)"
+ 'Lamda t = (preterm_node_1 label_edge_L t).
+
+interpretation
+ "application (preterm)"
+ 'At u t = (preterm_node_2 label_edge_S label_edge_A u t).
+
+(* Basic Inversions *********************************************************)
+
+lemma preterm_in_root_inv_lcons_oref:
+ ∀p,l,n. l;p ϵ▵ #n →
+ ∧∧ 𝗱❨n❩ = 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 preterm_in_root_inv_lcons_iref:
+ ∀t,p,l,n. l;p ϵ▵ 𝛗n.t →
+ ∧∧ 𝗱❨n❩ = l & p ϵ▵ t.
+#t #p #l #n * #q
+<list_append_lcons_sn * #r #Hr #H0 destruct
+/3 width=2 by ex_intro, conj/
+qed-.
+
+lemma preterm_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/
+qed-.
+
+lemma preterm_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/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/syntax/preterm_constructors.ma".
+
+(* DEPHI FOR PRETERM ********************************************************)
+
+definition preterm_dephi (f) (t) ≝
+ λp. ∃∃q. q ϵ⬦ t & p = path_dephi f q.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "delayed_updating/syntax/path.ma".
-include "delayed_updating/notation/relations/up_down_arrow_epsilon_2.ma".
-include "delayed_updating/notation/relations/up_arrow_epsilon_2.ma".
-
-(* TERM *********************************************************************)
-
-definition term: Type[0] ≝ predicate path.
-
-definition term_in_com: relation2 path term ≝
- λp,t. t p.
-
-interpretation
- "belongs to complete (term)"
- 'UpDownArrowEpsilon p t = (term_in_com p t).
-
-definition term_in_ini: relation2 path term ≝
- λp,t. ∃q. p;;q ϵ⬦ t.
-
-interpretation
- "belongs to initial (term)"
- 'UpArrowEpsilon p t = (term_in_ini p t).
-
-(* Basic constructions ******************************************************)
-
-lemma term_in_com_ini (p) (t):
- p ϵ⬦ t → p ϵ▵ t.
-/2 width=2 by ex_intro/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "delayed_updating/syntax/term.ma".
-include "delayed_updating/notation/functions/hash_1.ma".
-include "delayed_updating/notation/functions/phi_2.ma".
-include "delayed_updating/notation/functions/lamda_1.ma".
-include "delayed_updating/notation/functions/at_2.ma".
-
-(* CONSTRUCTORS FOR TERM ****************************************************)
-
-definition term_node_0 (l): term ≝
- λp. l;𝐞 = p.
-
-definition term_node_1 (l): term → term ≝
- λt,p. ∃∃q. q ϵ⬦ t & l;q = p.
-
-definition term_node_2 (l1) (l2): term → term → term ≝
- λt1,t2,p.
- ∨∨ ∃∃q. q ϵ⬦ t1 & l1;q = p
- | ∃∃q. q ϵ⬦ t2 & l2;q = p.
-
-interpretation
- "outer variable reference by depth (term)"
- 'Hash n = (term_node_0 (label_node_d n)).
-
-interpretation
- "inner variable reference by depth (term)"
- 'Phi n t = (term_node_1 (label_node_d n) t).
-
-interpretation
- "name-free functional abstraction (term)"
- 'Lamda t = (term_node_1 label_edge_l t).
-
-interpretation
- "application (term)"
- 'At u t = (term_node_2 label_edge_s label_edge_a u t).
-
-(* Basic Inversions *********************************************************)
-
-lemma term_in_ini_inv_lcons_oref:
- ∀p,l,n. l;p ϵ▵ #n →
- ∧∧ 𝗱❨n❩ = l & 𝐞 = p.
-#p #l #n * #q
-<list_append_lcons_sn #H destruct -H
-elim (eq_inv_list_empty_append … e0) -e0 #H1 #_
-/2 width=1 by conj/
-qed-.
-
-lemma term_in_ini_inv_lcons_iref:
- ∀t,p,l,n. l;p ϵ▵ 𝛗n.t →
- ∧∧ 𝗱❨n❩ = l & p ϵ▵ t.
-#t #p #l #n * #q
-<list_append_lcons_sn * #r #Hr #H1 destruct
-/3 width=2 by ex_intro, conj/
-qed-.
(* Core notation *******************************************************)
-notation "hvbox(a break ⊆ b)" non associative with precedence 45
-for @{ 'subseteq $a $b }. (* \subseteq *)
+notation "hvbox( a ⊆ break term 46 b )"
+ non associative with precedence 45
+ for @{ 'subseteq $a $b }. (* \subseteq *)
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