(* *)
(**************************************************************************)
+include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/syntax/prototerm_equivalence.ma".
+include "delayed_updating/notation/functions/class_d_phi_0.ma".
include "ground/xoa/or_5.ma".
include "ground/xoa/ex_3_1.ma".
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/preterm_constructors.ma".
-include "delayed_updating/notation/relations/in_predicate_d_phi_1.ma".
(* BY-DEPTH DELAYED (BDD) TERM **********************************************)
-inductive bdd: predicate preterm β
-| bdd_oref: βn. bdd #n
-| bdd_iref: βt,n. bdd t β bdd πn.t
-| bdd_abst: βt. bdd t β bdd π.t
-| bdd_appl: βu,t. bdd u β bdd t β bdd @u.t
+inductive bdd: π«β¨prototermβ© β
+| bdd_oref: βn. bdd (⧣n)
+| bdd_iref: βt,n. bdd t β bdd (πn.t)
+| bdd_abst: βt. bdd t β bdd (π.t)
+| bdd_appl: βu,t. bdd u β bdd t β bdd (@u.t)
+| bdd_conv: βt1,t2. t1 β t2 β bdd t1 β bdd t2
.
interpretation
- "well-formed by-depth delayed (preterm)"
- 'InPredicateDPhi t = (bdd t).
+ "by-depth delayed (prototerm)"
+ 'ClassDPhi = (bdd).
+
+(*
(* Basic inversions *********************************************************)
lemma bdd_inv_in_comp_gen:
- βt,p. t Ο΅ ππ β p ϡ⬦ t β
- β¨β¨ ββn. #n = t & π±β¨nβ©;π = p
- | ββu,q,n. u Ο΅ ππ & q ϡ⬦ u & πn.u = t & π±β¨nβ©;q = p
- | ββu,q. u Ο΅ ππ & q ϡ⬦ u & π.u = t & π;q = p
- | ββv,u,q. v Ο΅ ππ & u Ο΅ ππ & q ϡ⬦ u & @v.u = t & π;q = p
- | ββv,u,q. v Ο΅ ππ & u Ο΅ ππ & q ϡ⬦ v & @v.u = t & π¦;q = p
+ βt,p. t Ο΅ ππ β p Ο΅ t β
+ β¨β¨ ββn. #n β t & π±nβπ = p
+ | ββu,q,n. u Ο΅ ππ & q Ο΅ u & πn.u β t & π±nβπΊβq = p
+ | ββu,q. u Ο΅ ππ & q Ο΅ u & π.u β t & πβq = p
+ | ββv,u,q. v Ο΅ ππ & u Ο΅ ππ & q Ο΅ u & @v.u β t & πβq = p
+ | ββv,u,q. v Ο΅ ππ & u Ο΅ ππ & q Ο΅ v & @v.u β t & π¦βq = p
.
-#t #p *
+#t #p #H elim H -H
[ #n * /3 width=3 by or5_intro0, ex2_intro/
-| #u #n #Hu * #q #Hq #Hp /3 width=7 by ex4_3_intro, or5_intro1/
-| #u #Hu * #q #Hq #Hp /3 width=6 by or5_intro2, ex4_2_intro/
-| #v #u #Hv #Hu * * #q #Hq #Hp /3 width=8 by ex5_3_intro, or5_intro3, or5_intro4/
+| #u #n #Hu #_ * #q #Hq #Hp /3 width=7 by ex4_3_intro, or5_intro1/
+| #u #Hu #_ * #q #Hq #Hp /3 width=6 by or5_intro2, ex4_2_intro/
+| #v #u #Hv #Hu #_ #_ * * #q #Hq #Hp /3 width=8 by ex5_3_intro, or5_intro3, or5_intro4/
+| #t1 #t2 #Ht12 #_ #IH #Ht2
+ elim IH -IH [6: /2 width=3 by subset_in_eq_repl_fwd/ ] *
+ [ /4 width=3 by subset_eq_trans, or5_intro0, ex2_intro/
+ | /4 width=7 by subset_eq_trans, ex4_3_intro, or5_intro1/
+ | /4 width=6 by subset_eq_trans, or5_intro2, ex4_2_intro/
+ | /4 width=8 by subset_eq_trans, ex5_3_intro, or5_intro3/
+ | /4 width=8 by subset_eq_trans, ex5_3_intro, or5_intro4/
+ ]
]
qed-.
lemma bdd_inv_in_comp_d:
- βt,q,n. t Ο΅ ππ β π±β¨nβ©;q ϡ⬦ t β
- β¨β¨ β§β§ #n = t & π = q
- | ββu. u Ο΅ ππ & q ϡ⬦ u & πn.u = t
+ βt,q,n. t Ο΅ ππ β π±nβq Ο΅ t β
+ β¨β¨ β§β§ #n β t & π = q
+ | ββu. u Ο΅ ππ & q Ο΅ Ι±.u & πn.u β t
.
#t #q #n #Ht #Hq
elim (bdd_inv_in_comp_gen β¦ 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 #n0 #Hu0 #Hq0 #H1 #H2 destruct
+ /4 width=4 by ex3_intro, ex2_intro, or_intror/
| #u0 #q0 #_ #_ #_ #H0 destruct
| #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
| #v0 #u0 #q0 #_ #_ #_ #_ #H0 destruct
qed-.
lemma bdd_inv_in_root_d:
- βt,q,n. t Ο΅ ππ β π±β¨nβ©;q Ο΅β΅ t β
- β¨β¨ β§β§ #n = t & π = q
- | ββu. u Ο΅ ππ & q Ο΅β΅ u & πn.u = t
+ βt,q,n. t Ο΅ ππ β π±nβq Ο΅ β΅t β
+ β¨β¨ β§β§ #n β t & π = q
+ | ββu. u Ο΅ ππ & q Ο΅ β΅Ι±.u & πn.u β t
.
#t #q #n #Ht * #r #Hq
elim (bdd_inv_in_comp_d β¦ Ht Hq) -Ht -Hq *
qed-.
lemma bdd_inv_in_comp_L:
- βt,q. t Ο΅ ππ β π;q ϡ⬦ t β
- ββu. u Ο΅ ππ & q ϡ⬦ u & π.u = t
+ βt,q. t Ο΅ ππ β πβq Ο΅ t β
+ ββu. u Ο΅ ππ & q Ο΅ u & π.u β t
.
#t #q #Ht #Hq
elim (bdd_inv_in_comp_gen β¦ Ht Hq) -Ht -Hq *
qed-.
lemma bdd_inv_in_root_L:
- βt,q. t Ο΅ ππ β π;q Ο΅β΅ t β
- ββu. u Ο΅ ππ & q Ο΅β΅ u & π.u = t.
+ βt,q. t Ο΅ ππ β πβq Ο΅ β΅t β
+ ββu. u Ο΅ ππ & q Ο΅ β΅u & π.u β t.
#t #q #Ht * #r #Hq
elim (bdd_inv_in_comp_L β¦ Ht Hq) -Ht -Hq
#u #Hu #Hq #H0 destruct
qed-.
lemma bdd_inv_in_comp_A:
- βt,q. t Ο΅ ππ β π;q ϡ⬦ t β
- ββv,u. v Ο΅ ππ & u Ο΅ ππ & q ϡ⬦ u & @v.u = t
+ βt,q. t Ο΅ ππ β πβq Ο΅ t β
+ ββv,u. v Ο΅ ππ & u Ο΅ ππ & q Ο΅ u & @v.u β t
.
#t #q #Ht #Hq
elim (bdd_inv_in_comp_gen β¦ Ht Hq) -Ht -Hq *
qed-.
lemma bdd_inv_in_root_A:
- βt,q. t Ο΅ ππ β π;q Ο΅β΅ t β
- ββv,u. v Ο΅ ππ & u Ο΅ ππ & q Ο΅β΅ u & @v.u = t
+ βt,q. t Ο΅ ππ β πβq Ο΅ β΅t β
+ ββv,u. v Ο΅ ππ & u Ο΅ ππ & q Ο΅ β΅u & @v.u β t
.
#t #q #Ht * #r #Hq
elim (bdd_inv_in_comp_A β¦ Ht Hq) -Ht -Hq
qed-.
lemma bdd_inv_in_comp_S:
- βt,q. t Ο΅ ππ β π¦;q ϡ⬦ t β
- ββv,u. v Ο΅ ππ & u Ο΅ ππ & q ϡ⬦ v & @v.u = t
+ βt,q. t Ο΅ ππ β π¦βq Ο΅ t β
+ ββv,u. v Ο΅ ππ & u Ο΅ ππ & q Ο΅ v & @v.u β t
.
#t #q #Ht #Hq
elim (bdd_inv_in_comp_gen β¦ Ht Hq) -Ht -Hq *
qed-.
lemma bdd_inv_in_root_S:
- βt,q. t Ο΅ ππ β π¦;q Ο΅β΅ t β
- ββv,u. v Ο΅ ππ & u Ο΅ ππ & q Ο΅β΅ v & @v.u = t
+ βt,q. t Ο΅ ππ β π¦βq Ο΅ β΅t β
+ ββv,u. v Ο΅ ππ & u Ο΅ ππ & q Ο΅ β΅v & @v.u β t
.
#t #q #Ht * #r #Hq
elim (bdd_inv_in_comp_S β¦ Ht Hq) -Ht -Hq
(* Advanced inversions ******************************************************)
lemma bbd_mono_in_root_d:
- βl,n,p,t. t Ο΅ ππ β p,π±β¨nβ© Ο΅β΅ t β p,l Ο΅β΅ t β π±β¨nβ© = l.
+ βl,n,p,t. t Ο΅ ππ β pβπ±n Ο΅ β΅t β pβl Ο΅ β΅t β π±n = l.
#l #n #p elim p -p
[ #t #Ht <list_cons_comm <list_cons_comm #Hn #Hl
elim (bdd_inv_in_root_d β¦ Ht Hn) -Ht -Hn *
- [ #H0 #_ destruct
- elim (preterm_in_root_inv_lcons_oref β¦ Hl) -Hl //
- | #u #_ #_ #H0 destruct
- elim (preterm_in_root_inv_lcons_iref β¦ Hl) -Hl //
+ [ #H0 #_
+ lapply (prototerm_root_eq_repl β¦ H0) -H0 #H0
+ lapply (subset_in_eq_repl_fwd ?? β¦ Hl β¦ H0) -H0 -Hl #Hl
+ elim (prototerm_in_root_inv_lcons_oref β¦ Hl) -Hl //
+ | #u #_ #_ #H0
+ lapply (prototerm_root_eq_repl β¦ H0) -H0 #H0
+ lapply (subset_in_eq_repl_fwd ?? β¦ Hl β¦ H0) -H0 -Hl #Hl
+ elim (prototerm_in_root_inv_lcons_iref β¦ Hl) -Hl //
]
| * [ #m ] #p #IH #t #Ht
<list_cons_shift <list_cons_shift #Hn #Hl
[ elim (bdd_inv_in_root_d β¦ Ht Hn) -Ht -Hn *
[ #_ #H0
elim (eq_inv_list_empty_rcons ??? H0)
- | #u #Hu #Hp #H0 destruct
- elim (preterm_in_root_inv_lcons_iref β¦ Hl) -Hl #_ #Hl
+ | #u #Hu #Hp #H0
+ lapply (prototerm_root_eq_repl β¦ H0) -H0 #H0
+ lapply (subset_in_eq_repl_fwd ?? β¦ Hl β¦ H0) -H0 -Hl #Hl
+ elim (prototerm_in_root_inv_lcons_iref β¦ Hl) -Hl #_ #Hl
/2 width=4 by/
]
| elim (bdd_inv_in_root_L β¦ Ht Hn) -Ht -Hn
- #u #Hu #Hp #H0 destruct
- elim (preterm_in_root_inv_lcons_abst β¦ Hl) -Hl #_ #Hl
+ #u #Hu #Hp #H0
+ lapply (prototerm_root_eq_repl β¦ H0) -H0 #H0
+ lapply (subset_in_eq_repl_fwd ?? β¦ Hl β¦ H0) -H0 -Hl #Hl
+ elim (prototerm_in_root_inv_lcons_abst β¦ Hl) -Hl #_ #Hl
/2 width=4 by/
| elim (bdd_inv_in_root_A β¦ Ht Hn) -Ht -Hn
- #v #u #_ #Hu #Hp #H0 destruct
- elim (preterm_in_root_inv_lcons_appl β¦ Hl) -Hl * #H0 #Hl destruct
+ #v #u #_ #Hu #Hp #H0
+ lapply (prototerm_root_eq_repl β¦ H0) -H0 #H0
+ lapply (subset_in_eq_repl_fwd ?? β¦ Hl β¦ H0) -H0 -Hl #Hl
+ elim (prototerm_in_root_inv_lcons_appl β¦ Hl) -Hl * #H0 #Hl destruct
/2 width=4 by/
| elim (bdd_inv_in_root_S β¦ Ht Hn) -Ht -Hn
- #v #u #Hv #_ #Hp #H0 destruct
- elim (preterm_in_root_inv_lcons_appl β¦ Hl) -Hl * #H0 #Hl destruct
+ #v #u #Hv #_ #Hp #H0
+ lapply (prototerm_root_eq_repl β¦ H0) -H0 #H0
+ lapply (subset_in_eq_repl_fwd ?? β¦ Hl β¦ H0) -H0 -Hl #Hl
+ elim (prototerm_in_root_inv_lcons_appl β¦ Hl) -Hl * #H0 #Hl destruct
/2 width=4 by/
]
]
qed-.
+*)
\ No newline at end of file