| NCicEnvironment.ObjectNotFound s as e ->
raise (Error (lazy "eq_coerc non yet defined",Some e))
| Error _ as e -> debug_print (lazy "error"); raise e
+(* FG: for now we catch TypeCheckerFailure; to be understood *)
+ | NCicTypeChecker.TypeCheckerFailure _ ->
+ debug_print (lazy "TypeCheckerFailure");
+ raise (Error (lazy "no proof found",None))
;;
let compare_statuses ~past ~present =
(**************************************************************************)
include "basic_2/notation/relations/lazybtpredstarproper_8.ma".
-include "basic_2/reduction/fpbc.ma".
+include "basic_2/reduction/fpb.ma".
+include "basic_2/computation/fpbs.ma".
(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g. tri_TC … (fpbc h g).
+ λh,g,G1,L1,T1,G2,L2,T2.
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
interpretation "'qrst' proper parallel computation (closure)"
'LazyBTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
+lemma fpb_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by ex2_3_intro/ qed.
-lemma fpbg_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_step/ qed.
-
-lemma fpbg_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_TC_strap/ qed.
-
-lemma fpbu_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h,g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/3 width=1 by fpbu_fpbc, fpbc_fpbg/ qed.
-
-(* Basic eliminators ********************************************************)
-
-lemma fpbg_ind: ∀h,g,G1,L1,T1. ∀R:relation3 ….
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
-#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
-@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
-qed-.
-
-lemma fpbg_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 ….
- (∀G1,L1,T1. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1) →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
-#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
-@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
+lemma fpbg_fpbq_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+/3 width=9 by fpbs_strap1, ex2_3_intro/
qed-.
(* *)
(**************************************************************************)
-include "basic_2/reduction/fpbc_fleq.ma".
+include "basic_2/multiple/fleq_fleq.ma".
+include "basic_2/reduction/fpbq_alt.ma".
include "basic_2/computation/fpbg.ma".
(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
lemma fpbg_fleq_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ →
∀G2,L2,T2. ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T #H @(fpbg_ind … H) -G -L -T
-[ /3 width=5 by fpbc_fpbg, fpbc_fleq_trans/
-| /4 width=9 by fpbg_strap1, fpbc_fleq_trans/
-]
-qed-.
+/3 width=5 by fpbg_fpbq_trans, fleq_fpbq/ qed-.
lemma fleq_fpbg_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
∀G1,L1,T1. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G #G2 #L #L2 #T #T2 #H @(fpbg_ind_dx … H) -G -L -T
-[ /3 width=5 by fpbc_fpbg, fleq_fpbc_trans/
-| /4 width=9 by fpbg_strap2, fleq_fpbc_trans/
+#h #g #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
+elim (fleq_fpb_trans … H1 … H0) -G -L -T
+/4 width=9 by fpbs_strap2, fleq_fpbq, ex2_3_intro/
+qed-.
+
+(* alternative definition of fpbs *******************************************)
+
+lemma fleq_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by lleq_fpbs/
+qed.
+
+lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
+ ⦃G1, L1, T1⦄ >≡[h,g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
+/3 width=5 by fpbs_strap2, fpb_fpbq/
+qed-.
+
+lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+[ /2 width=1 by or_introl/
+| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 @(fpbq_ind_alt … H2) -H2 #H2
+ [ /3 width=5 by fleq_trans, or_introl/
+ | elim (fleq_fpb_trans … H1 … H2) -G -L -T
+ /4 width=5 by ex2_3_intro, or_intror, fleq_fpbs/
+ | /3 width=5 by fpbg_fleq_trans, or_intror/
+ | /4 width=5 by fpbg_fpbq_trans, fpb_fpbq, or_intror/
+ ]
+]
+qed-.
+
+(* Advanced properties of "qrst" parallel computation on closures ***********)
+
+lemma fpbs_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, g] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
+#h #g #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_fpbg … H) -H
+[ #H12 #G2 #L2 #U2 #H2 elim (fleq_fpb_trans … H12 … H2) -F2 -K2 -T2
+ /3 width=5 by fleq_fpbs, ex2_3_intro/
+| * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
+ @(ex2_3_intro … H4) -H4 /3 width=5 by fpbs_strap1, fpb_fpbq/
]
qed-.
(* *)
(**************************************************************************)
-include "basic_2/computation/fpbs_fpbu.ma".
-include "basic_2/computation/fpbs_fpbc.ma".
-include "basic_2/computation/fpbs_fpbs.ma".
include "basic_2/computation/fpbg_fpbs.ma".
(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
-(* Advanced inversion lemmas ************************************************)
-
-lemma fpbg_inv_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind_dx … H) -G1 -L1 -T1
-[ #G1 #L1 #T1 * /3 width=5 by fleq_fpbs, ex2_3_intro/
-| #G1 #G #L1 #L #T1 #T *
- #G0 #L0 #T0 #H10 #H0 #_ *
- /5 width=9 by fpbu_fpbs, fpbs_trans, fleq_fpbs, ex2_3_intro/
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind … H) -G2 -L2 -T2
-[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 @(fpbs_trans … IH1) -IH1 ] (**) (* full auto fails *)
-/2 width=1 by fpbc_fpbs/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fpbs_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, g] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
-/5 width=5 by fpbg_inv_fpbu_sn, fpbs_fpbg_trans, fpbc_fpbg, fpbu_fpbc/ qed-.
-
-(* Man properties ***********************************************************)
+(* Main properties **********************************************************)
theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
-/2 width=5 by tri_TC_transitive/ qed-.
+/3 width=5 by fpbg_fpbs_trans, fpbg_fwd_fpbs/ qed-.
(* Properties on "qrst" parallel reduction on closures **********************)
-lemma fpbg_fpb_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpb_fpbu … H2) -H2
-/3 width=5 by fpbg_fleq_trans, fpbg_strap1, fpbu_fpbc/
-qed-.
-
lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
- â¦\83G1, L1, T1â¦\84 â\89½[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ â¦\83G1, L1, T1â¦\84 â\89»[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu … H1) -H1
-/3 width=5 by fleq_fpbg_trans, fpbg_strap2, fpbu_fpbc/
+/3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
+
+lemma fpbq_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fpbq_ind_alt … H1) -H1
+/2 width=5 by fleq_fpbg_trans, fpb_fpbg_trans/
qed-.
(* Properties on "qrst" parallel compuutation on closures *******************)
lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpb_fpbg_trans/
+#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
qed-.
(* Note: this is used in the closure proof *)
lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpb_trans/
+#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
qed-.
-lemma fpbu_fpbs_fpbg: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbg_fpbs_trans, fpbu_fpbg/ qed.
-
(* Note: this is used in the closure proof *)
lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
-/3 width=5 by fqus_fpbs, fpbu_fqu, fpbu_fpbs_fpbg/
+/3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/
qed.
lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
(T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
-/4 width=5 by cpxs_fpbs, fpbu_cpx, fpbu_fpbs_fpbg/
+/4 width=5 by cpxs_fpbs, fpb_cpx, ex2_3_intro/
qed.
lemma lstas_fpbg: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, l2] T2 → (T1 = T2 → ⊥) →
lemma lpxs_fpbg: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
(L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, g] ⦃G, L2, T⦄.
#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
-/4 width=5 by fpbu_fpbs_fpbg, fpbu_lpx, lpxs_lleq_fpbs/
+/4 width=5 by fpb_lpx, lpxs_lleq_fpbs, ex2_3_intro/
qed.
-
-lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
- ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
-[ /2 width=1 by or_introl/
-| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 elim (fpb_fpbu … H2) -H2 #H2
- [ /3 width=5 by fleq_trans, or_introl/
- | /5 width=5 by fpbc_fpbg, fleq_fpbc_trans, fpbu_fpbc, or_intror/
- | /3 width=5 by fpbg_fleq_trans, or_intror/
- | /4 width=5 by fpbg_strap1, fpbu_fpbc, or_intror/
- ]
-]
-qed-.
(* *)
(**************************************************************************)
-include "basic_2/reduction/fpbu_lift.ma".
+include "basic_2/reduction/fpb_lift.ma".
include "basic_2/computation/fpbg.ma".
(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
lemma sta_fpbg: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
-/4 width=2 by fpbu_fpbg, sta_fpbu/ qed.
+/4 width=2 by fpb_fpbg, sta_fpb/ qed.
include "basic_2/notation/relations/btpredstar_8.ma".
include "basic_2/multiple/fqus.ma".
-include "basic_2/reduction/fpb.ma".
+include "basic_2/reduction/fpbq.ma".
include "basic_2/computation/cpxs.ma".
include "basic_2/computation/lpxs.ma".
(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g. tri_TC … (fpb h g).
+ λh,g. tri_TC … (fpbq h g).
interpretation "'qrst' parallel computation (closure)"
'BTPRedStar h g G1 L1 T1 G2 L2 T2 = (fpbs h g G1 L1 T1 G2 L2 T2).
lemma fpbs_refl: ∀h,g. tri_reflexive … (fpbs h g).
/2 width=1 by tri_inj/ qed.
-lemma fpb_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fpbq_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
/2 width=1 by tri_inj/ qed.
lemma fpbs_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
lemma fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/4 width=5 by fqu_fquq, fpb_fquq, tri_step/
+/4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
qed.
lemma fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
-/3 width=5 by fpb_fquq, tri_step/
+/3 width=5 by fpbq_fquq, tri_step/
qed.
lemma cpxs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpb_cpx, fpbs_strap1/
+/3 width=5 by fpbq_cpx, fpbs_strap1/
qed.
lemma lpxs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
#h #g #G #L1 #L2 #T #H @(lpxs_ind … H) -L2
-/3 width=5 by fpb_lpx, fpbs_strap1/
+/3 width=5 by fpbq_lpx, fpbs_strap1/
qed.
lemma lleq_fpbs: ∀h,g,G,L1,L2,T. L1 ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
-/3 width=1 by fpb_fpbs, fpb_lleq/ qed.
+/3 width=1 by fpbq_fpbs, fpbq_lleq/ qed.
lemma cprs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
/3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
lemma fpbs_fqus_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind … H) -G2 -L2 -T2
-/3 width=5 by fpbs_strap1, fpb_fquq/
+/3 width=5 by fpbs_strap1, fpbq_fquq/
qed-.
lemma fpbs_fqup_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
lemma fpbs_cpxs_trans: ∀h,g,G1,G,L1,L,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
#h #g #G1 #G #L1 #L #T1 #T #T2 #H1 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpbs_strap1, fpb_cpx/
+/3 width=5 by fpbs_strap1, fpbq_cpx/
qed-.
lemma fpbs_lpxs_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L2, T⦄.
#h #g #G1 #G #L1 #L #L2 #T1 #T #H1 #H @(lpxs_ind … H) -L2
-/3 width=5 by fpbs_strap1, fpb_lpx/
+/3 width=5 by fpbs_strap1, fpbq_lpx/
qed-.
lemma fpbs_lleq_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
L ≡[T, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L2, T⦄.
-/3 width=5 by fpbs_strap1, fpb_lleq/ qed-.
+/3 width=5 by fpbs_strap1, fpbq_lleq/ qed-.
lemma fqus_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
-/3 width=5 by fpbs_strap2, fpb_fquq/
+/3 width=5 by fpbs_strap2, fpbq_fquq/
qed-.
lemma cpxs_fpbs_trans: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T #T2 #H1 #H @(cpxs_ind_dx … H) -T1
-/3 width=5 by fpbs_strap2, fpb_cpx/
+/3 width=5 by fpbs_strap2, fpbq_cpx/
qed-.
lemma lpxs_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1⦄ ⊢ ➡*[h, g] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L #L2 #T1 #T2 #H1 #H @(lpxs_ind_dx … H) -L1
-/3 width=5 by fpbs_strap2, fpb_lpx/
+/3 width=5 by fpbs_strap2, fpbq_lpx/
qed-.
lemma lleq_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
L1 ≡[T1, 0] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_strap2, fpb_lleq/ qed-.
+/3 width=5 by fpbs_strap2, fpbq_lleq/ qed-.
lemma cpxs_fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
(* Note: this is used in the closure proof *)
lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
-/4 width=5 by fpbs_strap1, fpb_fpbs, lpr_fpb, cpr_fpb/
+/4 width=5 by fpbs_strap1, fpbq_fpbs, lpr_fpbq, cpr_fpbq/
qed.
(* *)
(**************************************************************************)
-include "basic_2/reduction/fpb_aaa.ma".
+include "basic_2/reduction/fpbq_aaa.ma".
include "basic_2/computation/fpbs.ma".
(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=2 by ex_intro/
#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A
-/2 width=8 by fpb_aaa_conf/
+/2 width=8 by fpbq_aaa_conf/
qed-.
theorem fpbsa_inv_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
-/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpb_lleq/
+/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_lleq/
qed-.
(* Advanced properties ******************************************************)
+++ /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 "basic_2/multiple/fleq.ma".
-include "basic_2/computation/fpbs.ma".
-
-(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
-
-(* Properties on lazy equivalence on closures *******************************)
-
-lemma fleq_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by lleq_fpbs/
-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 "basic_2/reduction/fpbq_alt.ma".
+include "basic_2/computation/fpbs_alt.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Properties on extended context-sensitive parallel computation for terms **)
+
+lemma fpbs_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2 elim (fpbs_inv_alt … H) -H
+#L00 #L0 #T0 #HT10 #H10 #HL00 #HL02 lapply (lleq_cpx_trans … HTU2 … HL02) -HTU2
+#HTU2 lapply (cpx_lleq_conf_sn … HTU2 … HL02) -HL02
+#HL02 lapply (lpxs_cpx_trans … HTU2 … HL00) -HTU2
+#HTU2 elim (fqus_cpxs_trans_neq … H10 … HTU2 HnTU2) -H10 -HTU2 -HnTU2
+#U0 #HTU0 #HnTU0 #HU02 elim (eq_term_dec T1 T0) #HnT10 destruct
+[ -HT10 elim (cpxs_neq_inv_step_sn … HTU0 HnTU0) -HTU0 -HnTU0
+| -HnTU0 elim (cpxs_neq_inv_step_sn … HT10 HnT10) -HT10 -HnT10
+]
+/4 width=10 by fpbs_intro_alt, cpxs_trans, ex3_intro/
+qed-.
+
+(* Properties on "rst" proper parallel reduction on closures ****************)
+
+lemma fpb_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=1 by fpbq_fpbs, fpb_fpbq/ 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 "basic_2/reduction/fpbc.ma".
-include "basic_2/computation/fpbs_fleq.ma".
-
-(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
-
-(* Properties on "qrst" proper parallel reduction for closures **************)
-
-lemma fpbc_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
-/3 width=5 by fpbu_fwd_fpb, fpbs_strap2, fleq_fpbs/
-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 "basic_2/multiple/fleq.ma".
-include "basic_2/reduction/fpbu.ma".
-include "basic_2/computation/fpbs_alt.ma".
-
-(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
-
-(* Properties on extended context-sensitive parallel computation for terms **)
-
-lemma fpbs_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2 elim (fpbs_inv_alt … H) -H
-#L00 #L0 #T0 #HT10 #H10 #HL00 #HL02 lapply (lleq_cpx_trans … HTU2 … HL02) -HTU2
-#HTU2 lapply (cpx_lleq_conf_sn … HTU2 … HL02) -HL02
-#HL02 lapply (lpxs_cpx_trans … HTU2 … HL00) -HTU2
-#HTU2 elim (fqus_cpxs_trans_neq … H10 … HTU2 HnTU2) -H10 -HTU2 -HnTU2
-#U0 #HTU0 #HnTU0 #HU02 elim (eq_term_dec T1 T0) #HnT10 destruct
-[ -HT10 elim (cpxs_neq_inv_step_sn … HTU0 HnTU0) -HTU0 -HnTU0
-| -HnTU0 elim (cpxs_neq_inv_step_sn … HT10 HnT10) -HT10 -HnT10
-]
-/4 width=10 by fpbs_intro_alt, cpxs_trans, ex3_intro/
-qed-.
-
-(* Properties on "rst" proper parallel reduction on closures ****************)
-
-lemma fpbu_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-/3 width=1 by fpb_fpbs, fpbu_fwd_fpb/ qed.
-
-lemma fpbs_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-(* ALTERNATIVE PROOF
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
-[ /2 width=1 by or_introl/
-| #G1 #G #L1 #L #T1 #T #H1 #_ * [ #H2 | * #G0 #L0 #T0 #H0 #H02 ]
- elim (fpb_fpbu … H1) -H1 #H1
- [ /3 width=1 by
-*)
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt … H) -H
-#L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct
-[ -HT1 elim (fqus_inv_gen … HT2) -HT2
- [ #H elim (fqup_inv_step_sn … H) -H
- /4 width=11 by fpbs_intro_alt, fpbu_fqu, ex2_3_intro, or_intror/
- | * #HG #HL #HT destruct elim (lleq_dec T2 L0 L 0) #H
- [ /4 width=3 by fleq_intro, lleq_trans, or_introl/
- | elim (lpxs_nlleq_inv_step_sn … HL0 H) -HL0 -H
- /5 width=7 by lpxs_lleq_fpbs, fpbu_lpx, lleq_trans, ex2_3_intro, or_intror/
- ]
- ]
-| elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
- /5 width=11 by fpbs_intro_alt, fpbu_cpx, ex2_3_intro, or_intror/
-]
-qed-.
(* *)
(**************************************************************************)
-include "basic_2/reduction/fpb_lift.ma".
include "basic_2/computation/cpxs_lift.ma".
include "basic_2/computation/fpbs.ma".
lemma sta_fpbs: ∀h,g,G,L,T,U,l.
⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •*[h, 1] U →
⦃G, L, T⦄ ≥[h, g] ⦃G, L, U⦄.
-/4 width=2 by fpb_fpbs, sta_fpb/ qed.
+/2 width=5 by lstas_fpbs/ qed.
(* Note: this is used in the closure proof *)
lemma cpr_lpr_sta_fpbs: ∀h,g,G,L1,L2,T1,T2,U2,l2.
⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
⦃G, L2⦄ ⊢ T2 ▪[h, g] l2+1 → ⦃G, L2⦄ ⊢ T2 •*[h, 1] U2 →
⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, U2⦄.
-/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, sta_cpx, fpb_cpx/ qed.
+/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, sta_cpx, fpbq_cpx/ qed.
(**************************************************************************)
include "basic_2/notation/relations/btsn_5.ma".
-include "basic_2/reduction/fpbu.ma".
+include "basic_2/reduction/fpb.ma".
include "basic_2/computation/csx.ma".
(* "QRST" STRONGLY NORMALIZING TERMS ****************************************)
(* Basic inversion lemmas ***************************************************)
lemma fsb_inv_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #H elim H -G -L -T /5 width=1 by csx_intro, fpbu_cpx/
+#h #g #G #L #T #H elim H -G -L -T /5 width=1 by csx_intro, fpb_cpx/
qed-.
#h #g #R #IH #G #L #T #H @(csx_ind_fpbu … H) -G -L -T
#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h g … G2 … L2 … T2 … HTA1) -A1
-/2 width=2 by fpbu_fpbs/
+/2 width=2 by fpb_fpbs/
qed-.
lemma aaa_ind_fpbu: ∀h,g. ∀R:relation3 genv lenv term.
(**************************************************************************)
include "basic_2/notation/relations/btsnalt_5.ma".
-include "basic_2/computation/fpbg_fpbg.ma".
+include "basic_2/computation/fpbg_fpbs.ma".
include "basic_2/computation/fsb.ma".
(* "QRST" STRONGLY NORMALIZING TERMS ****************************************)
theorem fsb_fsba: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥⦥[h, g] T.
#h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
#G1 #L1 #T1 #_ #IH @fsba_intro
-#G2 #L2 #T2 #H elim (fpbg_inv_fpbu_sn … H) -H
-/3 width=5 by fsba_fpbs_trans/
+#G2 #L2 #T2 * /3 width=5 by fsba_fpbs_trans/
qed.
(* Main inversion lemmas ****************************************************)
theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
#h #g #G #L #T #H @(fsba_ind_alt … H) -G -L -T
-/5 width=1 by fsb_intro, fpbc_fpbg, fpbu_fpbc/
+/4 width=1 by fsb_intro, fpb_fpbg/
qed-.
(* Advanced properties ******************************************************)
(* *)
(**************************************************************************)
+include "basic_2/computation/fpbs_fpb.ma".
+include "basic_2/computation/fpbs_fpbs.ma".
include "basic_2/computation/csx_fpbs.ma".
include "basic_2/computation/lsx_csx.ma".
include "basic_2/computation/fsb_alt.ma".
+lemma fpbs_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+(* ALTERNATIVE PROOF
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
+[ /2 width=1 by or_introl/
+| #G1 #G #L1 #L #T1 #T #H1 #_ * [ #H2 | * #G0 #L0 #T0 #H0 #H02 ]
+ elim (fpb_fpbu … H1) -H1 #H1
+ [ /3 width=1 by
+*)
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt … H) -H
+#L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct
+[ -HT1 elim (fqus_inv_gen … HT2) -HT2
+ [ #H elim (fqup_inv_step_sn … H) -H
+ /4 width=11 by fpbs_intro_alt, fpbu_fqu, ex2_3_intro, or_intror/
+ | * #HG #HL #HT destruct elim (lleq_dec T2 L0 L 0) #H
+ [ /4 width=3 by fleq_intro, lleq_trans, or_introl/
+ | elim (lpxs_nlleq_inv_step_sn … HL0 H) -HL0 -H
+ /5 width=7 by lpxs_lleq_fpbs, fpbu_lpx, lleq_trans, ex2_3_intro, or_intror/
+ ]
+ ]
+| elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
+ /5 width=11 by fpbs_intro_alt, fpbu_cpx, ex2_3_intro, or_intror/
+]
+qed-.
+
(* alternative proof that needs decidability of bteq to go in fpbs.ma
* or lpx_fpbc_trans to go in fpbs_lift.ma (possibly)
*)
]
| #L0 #HL10 #_ * [ /3 width=3 by or_introl, lpx_bteq_trans/ ]
* #G3 #L3 #T3 #H13 #H32
+
--- /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 "basic_2/notation/relations/lazybtpredproper_8.ma".
+include "basic_2/multiple/fleq.ma".
+include "basic_2/reduction/fpbu.ma".
+
+(* "QRST" PROPER PARALLEL REDUCTION FOR CLOSURES ****************************)
+
+definition fpbc: ∀h. sd h → tri_relation genv lenv term ≝
+ λh,g,G1,L1,T1,G2,L2,T2.
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄.
+
+interpretation
+ "'qrst' proper parallel reduction (closure)"
+ 'LazyBTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbc h g G1 L1 T1 G2 L2 T2).
+
+(* Baic properties **********************************************************)
+
+lemma fpbu_fpbc: ∀h,g,G1,G2,L1,L2,T1,T2.
+ ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by ex2_3_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 "basic_2/multiple/fleq_fleq.ma".
+include "basic_2/reduction/fpbu_fleq.ma".
+include "basic_2/reduction/fpbc.ma".
+
+(* "QRST" PROPER PARALLEL REDUCTION FOR CLOSURES ****************************)
+
+(* Properties on lazy equivalence on closures *******************************)
+
+lemma fpbc_fleq_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+/3 width=9 by fleq_trans, ex2_3_intro/
+qed-.
+
+lemma fleq_fpbc_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 *
+#G0 #L0 #T0 #H0 #H02 elim (fleq_fpbu_trans … H1 … H0) -G -L -T
+/3 width=9 by fleq_trans, ex2_3_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 "basic_2/notation/relations/lazybtpredstarproper_8.ma".
+include "basic_2/reduction/fpbc.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
+ λh,g. tri_TC … (fpbc h g).
+
+interpretation "'qrst' proper parallel computation (closure)"
+ 'LazyBTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fpbg_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed.
+
+lemma fpbg_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed.
+
+lemma fpbu_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h,g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/3 width=1 by fpbu_fpbc, fpbc_fpbg/ qed.
+
+(* Basic eliminators ********************************************************)
+
+lemma fpbg_ind: ∀h,g,G1,L1,T1. ∀R:relation3 ….
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
+qed-.
+
+lemma fpbg_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 ….
+ (∀G1,L1,T1. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1) →
+ (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#h #g #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
+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 "basic_2/reduction/fpbc_fleq.ma".
+include "basic_2/computation/fpbg.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Properties on lazy equivalence for closures ******************************)
+
+lemma fpbg_fleq_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #H @(fpbg_ind … H) -G -L -T
+[ /3 width=5 by fpbc_fpbg, fpbc_fleq_trans/
+| /4 width=9 by fpbg_strap1, fpbc_fleq_trans/
+]
+qed-.
+
+lemma fleq_fpbg_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G #G2 #L #L2 #T #T2 #H @(fpbg_ind_dx … H) -G -L -T
+[ /3 width=5 by fpbc_fpbg, fleq_fpbc_trans/
+| /4 width=9 by fpbg_strap2, fleq_fpbc_trans/
+]
+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 "basic_2/computation/fpbs_fpbu.ma".
+include "basic_2/computation/fpbs_fpbc.ma".
+include "basic_2/computation/fpbs_fpbs.ma".
+include "basic_2/computation/fpbg_fpbs.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpbg_inv_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 * /3 width=5 by fleq_fpbs, ex2_3_intro/
+| #G1 #G #L1 #L #T1 #T *
+ #G0 #L0 #T0 #H10 #H0 #_ *
+ /5 width=9 by fpbu_fpbs, fpbs_trans, fleq_fpbs, ex2_3_intro/
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind … H) -G2 -L2 -T2
+[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 @(fpbs_trans … IH1) -IH1 ] (**) (* full auto fails *)
+/2 width=1 by fpbc_fpbs/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpbs_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, g] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
+/5 width=5 by fpbg_inv_fpbu_sn, fpbs_fpbg_trans, fpbc_fpbg, fpbu_fpbc/ qed-.
+
+(* Man properties ***********************************************************)
+
+theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
+/2 width=5 by tri_TC_transitive/ 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 "basic_2/computation/lpxs_lleq.ma".
+include "basic_2/computation/fpbs_lift.ma".
+include "basic_2/computation/fpbg_fleq.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Properties on "qrst" parallel reduction on closures **********************)
+
+lemma fpbg_fpb_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpb_fpbu … H2) -H2
+/3 width=5 by fpbg_fleq_trans, fpbg_strap1, fpbu_fpbc/
+qed-.
+
+lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu … H1) -H1
+/3 width=5 by fleq_fpbg_trans, fpbg_strap2, fpbu_fpbc/
+qed-.
+
+(* Properties on "qrst" parallel compuutation on closures *******************)
+
+lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpb_fpbg_trans/
+qed-.
+
+(* Note: this is used in the closure proof *)
+lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpb_trans/
+qed-.
+
+lemma fpbu_fpbs_fpbg: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbg_fpbs_trans, fpbu_fpbg/ qed.
+
+(* Note: this is used in the closure proof *)
+lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
+/3 width=5 by fqus_fpbs, fpbu_fqu, fpbu_fpbs_fpbg/
+qed.
+
+lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+ (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
+/4 width=5 by cpxs_fpbs, fpbu_cpx, fpbu_fpbs_fpbg/
+qed.
+
+lemma lstas_fpbg: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, l2] T2 → (T1 = T2 → ⊥) →
+ ∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
+
+lemma lpxs_fpbg: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+ (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
+/4 width=5 by fpbu_fpbs_fpbg, fpbu_lpx, lpxs_lleq_fpbs/
+qed.
+
+lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
+ ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+[ /2 width=1 by or_introl/
+| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 elim (fpb_fpbu … H2) -H2 #H2
+ [ /3 width=5 by fleq_trans, or_introl/
+ | /5 width=5 by fpbc_fpbg, fleq_fpbc_trans, fpbu_fpbc, or_intror/
+ | /3 width=5 by fpbg_fleq_trans, or_intror/
+ | /4 width=5 by fpbg_strap1, fpbu_fpbc, or_intror/
+ ]
+]
+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 "basic_2/reduction/fpbu_lift.ma".
+include "basic_2/computation/fpbg.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma sta_fpbg: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
+ ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+/4 width=2 by fpbu_fpbg, sta_fpbu/ 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 "basic_2/reduction/fpbc.ma".
+include "basic_2/computation/fpbs_fleq.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Properties on "qrst" proper parallel reduction for closures **************)
+
+lemma fpbc_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
+/3 width=5 by fpbu_fwd_fpb, fpbs_strap2, fleq_fpbs/
+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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻≡ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'LazyBTPRedProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ > break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'BTPRedStarProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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 "basic_2/notation/relations/btpredstarproper_8.ma".
-include "basic_2/computation/fpbc.ma".
-
-(* GENEARAL "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES *************)
-
-(* Note: this is not transitive *)
-inductive fpbg (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpbg_cpxs: ∀L2,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G1, L1⦄ ⊢ ➡*[h, g] L2 →
- fpbg h g G1 L1 T1 G1 L2 T2
-| fpbg_fqup: ∀G2,L,L2,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊃+ ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 →
- fpbg h g G1 L1 T1 G2 L2 T2
-| fpbg_lpxs: ∀G2,L,L0,L2,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊃* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L0 →
- (L ⋕[0, T2] L0 → ⊥) → ⦃G2, L0⦄ ⊢ ➡*[h, g] L2 → L0 ⋕[0, T2] L2 →
- fpbg h g G1 L1 T1 G2 L2 T2
-.
-
-interpretation "'big tree' proper parallel computation (closure)"
- 'BTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=9 by fpbg_fqup, fpbg_cpxs, fpbg_lpxs/
-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 "basic_2/computation/lpxs_lpxs.ma".
-include "basic_2/computation/fpbs_alt.ma".
-include "basic_2/computation/fpbg.ma".
-
-(* GENERAL "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **************)
-
-(* Advanced forward lemmas **************************************************)
-
-lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
-/3 width=5 by cpxs_fqus_lpxs_fpbs, cpxs_fqup_fpbs, fpbs_trans, lpxs_fpbs, cpxs_fpbs/
-qed-.
-
-lemma fpbg_fwd_fpbc_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
-[ /4 width=5 by fpbc_cpxs, lpxs_fpbs, ex2_3_intro/
-| #G2 #L #L2 #T #T2 #HT1 #HT2 #HL2 elim (eq_term_dec T1 T) #H destruct
- [ -HT1 /3 width=5 by fpbc_fqup, lpxs_fpbs, ex2_3_intro/
- | /5 width=9 by fpbc_cpxs, fpbsa_inv_fpbs, fqup_fqus, ex3_2_intro, ex2_3_intro/
- ]
-| #G2 #L #L0 #L2 #T #T2 #HT1 #HT2 #HL0 #H0 #HL02 #H02
- lapply (lpxs_trans … HL0 … HL02) #HL2
- elim (eq_term_dec T1 T) #H destruct
- [ -HT1 elim (fqus_inv_gen … HT2) -HT2
- [ /3 width=5 by fpbc_fqup, lpxs_fpbs, ex2_3_intro/
- | * #H1 #H2 #H3 destruct
- /4 width=5 by fpbc_lpxs, lpxs_fpbs, ex2_3_intro/
- ]
- | /4 width=9 by fpbc_cpxs, fpbsa_inv_fpbs, ex3_2_intro, ex2_3_intro/
- ]
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fqu_fpbs_fpbg: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H elim(fpbs_fpbsa … H) -H
-#L0 #T0 #HT0 #HT02 #HL02 elim (fqu_cpxs_trans … HT0 … H1) -T
-/3 width=7 by fpbg_fqup, fqus_strap2_fqu/
-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 "basic_2/computation/fpns.ma".
-include "basic_2/computation/fpbg.ma".
-
-(* GENEARAL "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES *************)
-
-(* Properties on parallel computation for "big tree" normal forms ***********)
-
-axiom fpns_fpbg_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ⊢ ⋕➡*[h, g] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ >[h, g] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ >[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, U2⦄.
-(*
-#h #g #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
-#K2 #HK12 #HT1 #G2 #L2 #U2 * -G2 -L2 -U2
-[ /4 width=9 by fpbc_cpxs, fpns_intro, lpxs_cpxs_trans, lleq_cpxs_conf_dx, ex2_3_intro/
-| #G2 #L2 #U2 #H12 elim (lpxs_lleq_fqup_trans … H12 … HK12 HT1) -K2
- /3 width=5 by fpbc_fqup, fpns_intro, ex2_3_intro/
-| /5 width=5 by fpbc_lpxs, lpxs_trans, lleq_canc_sn, ex2_3_intro/
-]
-qed-.
-*)
\ No newline at end of file
+++ /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 "basic_2/computation/fpbc_lift.ma".
-include "basic_2/computation/fpbg.ma".
-
-(* GENERAL "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *********************)
-
-(* Advanced properties ******************************************************)
-
-lemma lsstas_fpbg: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, g, l2] T2 → (T1 = T2 → ⊥) →
- ∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
-/4 width=5 by fpbc_fpbg, lsstas_fpbc/ qed.
-
-lemma ssta_fpbg: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
- ⦃G, L⦄ ⊢ T1 •[h, g] T2 → ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
-/3 width=2 by fpbc_fpbg, ssta_fpbc/ 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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≽ ≽ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTPRedAlt $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻≡ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'LazyBTPRedProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/btpred_8.ma".
-include "basic_2/substitution/fquq.ma".
+include "basic_2/notation/relations/btpredproper_8.ma".
+include "basic_2/substitution/fqu.ma".
include "basic_2/multiple/lleq.ma".
include "basic_2/reduction/lpx.ma".
-(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
inductive fpb (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpb_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpb h g G1 L1 T1 G2 L2 T2
-| fpb_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → fpb h g G1 L1 T1 G1 L1 T2
-| fpb_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → fpb h g G1 L1 T1 G1 L2 T1
-| fpb_lleq: ∀L2. L1 ≡[T1, 0] L2 → fpb h g G1 L1 T1 G1 L2 T1
+| fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h g G1 L1 T1 G2 L2 T2
+| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → fpb h g G1 L1 T1 G1 L1 T2
+| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T1, 0] L2 → ⊥) → fpb h g G1 L1 T1 G1 L2 T1
.
interpretation
- "'qrst' parallel reduction (closure)"
- 'BTPRed h g G1 L1 T1 G2 L2 T2 = (fpb h g G1 L1 T1 G2 L2 T2).
+ "'rst' proper parallel reduction (closure)"
+ 'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpb h g G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fpb_refl: ∀h,g. tri_reflexive … (fpb h g).
-/2 width=1 by fpb_cpx/ qed.
-
-lemma cpr_fpb: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
+lemma cpr_fpb: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) →
+ ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
/3 width=1 by fpb_cpx, cpr_cpx/ qed.
-lemma lpr_fpb: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1, T⦄ ≽[h, g] ⦃G, L2, T⦄.
+lemma lpr_fpb: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → (L1 ≡[T, 0] L2 → ⊥) →
+ ⦃G, L1, T⦄ ≻[h, g] ⦃G, L2, T⦄.
/3 width=1 by fpb_lpx, lpr_lpx/ 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 "basic_2/static/aaa_fqus.ma".
-include "basic_2/static/aaa_lleq.ma".
-include "basic_2/reduction/lpx_aaa.ma".
-include "basic_2/reduction/fpb.ma".
-
-(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-lemma fpb_aaa_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=6 by aaa_lleq_conf, lpx_aaa_conf, cpx_aaa_conf, aaa_fquq_conf, 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 "basic_2/multiple/fleq.ma".
+include "basic_2/reduction/fpb_lleq.ma".
+
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
+
+(* Properties on lazy equivalence for closures ******************************)
+
+lemma fleq_fpb_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≡[0] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≡[0] ⦃G2, L2, U2⦄.
+#h #g #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
+#K2 #HK12 #G2 #L2 #U2 #H12 elim (lleq_fpb_trans … HK12 … H12) -K2
+/3 width=5 by fleq_intro, ex2_3_intro/
+qed-.
+
+(* Inversion lemmas on lazy equivalence for closures ************************)
+
+lemma fpb_inv_fleq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #H elim (fleq_inv_gen … H) -H
+ /3 width=8 by lleq_fwd_length, fqu_inv_eq/
+| #T2 #_ #HnT #H elim (fleq_inv_gen … H) -H /2 width=1 by/
+| #L2 #_ #HnL #H elim (fleq_inv_gen … H) -H /2 width=1 by/
+]
+qed-.
+
(* *)
(**************************************************************************)
+include "basic_2/unfold/lstas_da.ma".
include "basic_2/reduction/cpx_lift.ma".
include "basic_2/reduction/fpb.ma".
-(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
(* Advanced properties ******************************************************)
-lemma sta_fpb: ∀h,g,G,L,T1,T2,l.
- ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 →
- ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
-/3 width=4 by fpb_cpx, sta_cpx/ qed.
+lemma sta_fpb: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
+ ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #l #HT1 #HT12 elim (eq_term_dec T1 T2)
+/3 width=2 by fpb_cpx, sta_cpx/ #H destruct
+elim (lstas_inv_refl_pos h G L T2 0) //
+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 "basic_2/multiple/lleq_fqus.ma".
+include "basic_2/multiple/lleq_lleq.ma".
+include "basic_2/reduction/lpx_lleq.ma".
+include "basic_2/reduction/fpb.ma".
+
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_fpb_trans: ∀h,g,F,K1,K2,T. K1 ≡[T, 0] K2 →
+ ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, g] ⦃G, L2, U⦄ →
+ ∃∃L1. ⦃F, K1, T⦄ ≻[h, g] ⦃G, L1, U⦄ & L1 ≡[U, 0] L2.
+#h #g #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
+[ #G #L2 #U #H2 elim (lleq_fqu_trans … H2 … HT) -K2
+ /3 width=3 by fpb_fqu, ex2_intro/
+| /4 width=10 by fpb_cpx, cpx_lleq_conf_sn, lleq_cpx_trans, ex2_intro/
+| #L2 #HKL2 #HnKL2 elim (lleq_lpx_trans … HKL2 … HT) -HKL2
+ /6 width=3 by fpb_lpx, lleq_canc_sn, ex2_intro/ (* 2 lleq_canc_sn *)
+]
+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 "basic_2/notation/relations/lazybtpredproper_8.ma".
-include "basic_2/multiple/fleq.ma".
-include "basic_2/reduction/fpbu.ma".
-
-(* "QRST" PROPER PARALLEL REDUCTION FOR CLOSURES ****************************)
-
-definition fpbc: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g,G1,L1,T1,G2,L2,T2.
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄.
-
-interpretation
- "'qrst' proper parallel reduction (closure)"
- 'LazyBTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbc h g G1 L1 T1 G2 L2 T2).
-
-(* Baic properties **********************************************************)
-
-lemma fpbu_fpbc: ∀h,g,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
-/2 width=5 by ex2_3_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 "basic_2/multiple/fleq_fleq.ma".
-include "basic_2/reduction/fpbu_fleq.ma".
-include "basic_2/reduction/fpbc.ma".
-
-(* "QRST" PROPER PARALLEL REDUCTION FOR CLOSURES ****************************)
-
-(* Properties on lazy equivalence on closures *******************************)
-
-lemma fpbc_fleq_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
-/3 width=9 by fleq_trans, ex2_3_intro/
-qed-.
-
-lemma fleq_fpbc_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 *
-#G0 #L0 #T0 #H0 #H02 elim (fleq_fpbu_trans … H1 … H0) -G -L -T
-/3 width=9 by fleq_trans, ex2_3_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 "basic_2/notation/relations/btpred_8.ma".
+include "basic_2/substitution/fquq.ma".
+include "basic_2/multiple/lleq.ma".
+include "basic_2/reduction/lpx.ma".
+
+(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+
+inductive fpbq (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h g G1 L1 T1 G2 L2 T2
+| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → fpbq h g G1 L1 T1 G1 L1 T2
+| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → fpbq h g G1 L1 T1 G1 L2 T1
+| fpbq_lleq: ∀L2. L1 ≡[T1, 0] L2 → fpbq h g G1 L1 T1 G1 L2 T1
+.
+
+interpretation
+ "'qrst' parallel reduction (closure)"
+ 'BTPRed h g G1 L1 T1 G2 L2 T2 = (fpbq h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpbq_refl: ∀h,g. tri_reflexive … (fpbq h g).
+/2 width=1 by fpbq_cpx/ qed.
+
+lemma cpr_fpbq: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
+/3 width=1 by fpbq_cpx, cpr_cpx/ qed.
+
+lemma lpr_fpbq: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1, T⦄ ≽[h, g] ⦃G, L2, T⦄.
+/3 width=1 by fpbq_lpx, lpr_lpx/ 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 "basic_2/static/aaa_fqus.ma".
+include "basic_2/static/aaa_lleq.ma".
+include "basic_2/reduction/lpx_aaa.ma".
+include "basic_2/reduction/fpbq.ma".
+
+(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma fpbq_aaa_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=6 by aaa_lleq_conf, lpx_aaa_conf, cpx_aaa_conf, aaa_fquq_conf, 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 "basic_2/reduction/cpx_lift.ma".
+include "basic_2/reduction/fpbq.ma".
+
+(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma sta_fpbq: ∀h,g,G,L,T1,T2,l.
+ ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 →
+ ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
+/3 width=4 by fpbq_cpx, sta_cpx/ 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 "basic_2/notation/relations/btpredproper_8.ma".
-include "basic_2/reduction/fpb.ma".
-
-(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
-
-inductive fpbu (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpbu_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpbu h g G1 L1 T1 G2 L2 T2
-| fpbu_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → fpbu h g G1 L1 T1 G1 L1 T2
-| fpbu_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T1, 0] L2 → ⊥) → fpbu h g G1 L1 T1 G1 L2 T1
-.
-
-interpretation
- "'rst' proper parallel reduction (closure)"
- 'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbu h g G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpr_fpbu: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) →
- ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
-/3 width=1 by fpbu_cpx, cpr_cpx/ qed.
-
-lemma lpr_fpbu: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → (L1 ≡[T, 0] L2 → ⊥) →
- ⦃G, L1, T⦄ ≻[h, g] ⦃G, L2, T⦄.
-/3 width=1 by fpbu_lpx, lpr_lpx/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fpbu_fwd_fpb: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=1 by fpb_fquq, fpb_cpx, fpb_lpx, fqu_fquq/
-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 "basic_2/multiple/fleq.ma".
-include "basic_2/reduction/fpbu_lleq.ma".
-
-(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
-
-(* Properties on lazy equivalence for closures ******************************)
-
-lemma fleq_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≡[0] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≡[0] ⦃G2, L2, U2⦄.
-#h #g #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
-#K2 #HK12 #G2 #L2 #U2 #H12 elim (lleq_fpbu_trans … HK12 … H12) -K2
-/3 width=5 by fleq_intro, ex2_3_intro/
-qed-.
-
-lemma fpb_fpbu: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
- ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-[ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H
- [ /3 width=1 by fpbu_fqu, or_intror/
- | * #H1 #H2 #H3 destruct /2 width=1 by or_introl/
- ]
-| #T2 #HT12 elim (eq_term_dec T1 T2)
- #HnT12 destruct /4 width=1 by fpbu_cpx, or_intror, or_introl/
-| #L2 #HL12 elim (lleq_dec … T1 L1 L2 0)
- /4 width=1 by fpbu_lpx, fleq_intro, or_intror, or_introl/
-| /3 width=1 by fleq_intro, or_introl/
-]
-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 "basic_2/unfold/lstas_da.ma".
-include "basic_2/reduction/cpx_lift.ma".
-include "basic_2/reduction/fpbu.ma".
-
-(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
-
-(* Advanced properties ******************************************************)
-
-lemma sta_fpbu: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
- ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #l #HT1 #HT12 elim (eq_term_dec T1 T2)
-/3 width=2 by fpbu_cpx, sta_cpx/ #H destruct
-elim (lstas_inv_refl_pos h G L T2 0) //
-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 "basic_2/multiple/lleq_fqus.ma".
-include "basic_2/multiple/lleq_lleq.ma".
-include "basic_2/reduction/lpx_lleq.ma".
-include "basic_2/reduction/fpbu.ma".
-
-(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
-
-(* Properties on lazy equivalence for local environments ********************)
-
-lemma lleq_fpbu_trans: ∀h,g,F,K1,K2,T. K1 ≡[T, 0] K2 →
- ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, g] ⦃G, L2, U⦄ →
- ∃∃L1. ⦃F, K1, T⦄ ≻[h, g] ⦃G, L1, U⦄ & L1 ≡[U, 0] L2.
-#h #g #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
-[ #G #L2 #U #H2 elim (lleq_fqu_trans … H2 … HT) -K2
- /3 width=3 by fpbu_fqu, ex2_intro/
-| /4 width=10 by fpbu_cpx, cpx_lleq_conf_sn, lleq_cpx_trans, ex2_intro/
-| #L2 #HKL2 #HnKL2 elim (lleq_lpx_trans … HKL2 … HT) -HKL2
- /6 width=3 by fpbu_lpx, lleq_canc_sn, ex2_intro/ (* 2 lleq_canc_sn *)
-]
-qed-.
/2 width=7 by fqu_fwd_length_lref1_aux/
qed-.
+(* Basic inversion lemmas ***************************************************)
+
+fact fqu_inv_eq_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ G1 = G2 → |L1| = |L2| → T1 = T2 → ⊥.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 normalize
+/2 width=4 by discr_tpair_xy_y, discr_tpair_xy_x, plus_xSy_x_false/
+#G #L #K #T #U #e #HLK #_ #_ #H #_ -G -T -U >(drop_fwd_length … HLK) in H; -L
+/2 width=4 by plus_xySz_x_false/
+qed-.
+
+lemma fqu_inv_eq: ∀G,L1,L2,T. ⦃G, L1, T⦄ ⊐ ⦃G, L2, T⦄ → |L1| = |L2| → ⊥.
+#G #L1 #L2 #T #H #H0 @(fqu_inv_eq_aux … H … H0) // (**) (* full auto fails *)
+qed-.
+
(* Advanced eliminators *****************************************************)
lemma fqu_wf_ind: ∀R:relation3 …. (
for context-sensitive computation on terms.
</news>
<news date="2014 June 9.">
- strong qrst-normalization
+ Strong qrst-normalization
for simply typed terms.
</news>
<news date="2014 April 16.">
- lazy equivalence on local environments
+ Lazy equivalence on local environments
addded as q-step to rst-computation on closures
(anniversary milestone).
</news>
projects / helm.git / commitdiff