(**************************************************************************)
include "basic_2/unfold/lsstas_lsstas.ma".
-include "basic_2/computation/llprs_cprs.ma".
+include "basic_2/computation/lprs_cprs.ma".
include "basic_2/computation/cpxs_cpxs.ma".
include "basic_2/computation/cpds.ma".
(* *)
(**************************************************************************)
-include "basic_2/reduction/llpr_llpr.ma".
+include "basic_2/reduction/lpr_lpr.ma".
include "basic_2/computation/cprs_lift.ma".
(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
(* Basic_1: was: pr3_t *)
(* Basic_1: includes: pr1_t *)
theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
-normalize /2 width=3 by trans_TC/ qed-.
+normalize /2 width=3 by trans_TC/ qed-.
(* Basic_1: was: pr3_confluence *)
(* Basic_1: includes: pr1_confluence *)
theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cprs_ind … H) -W2
/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
qed.
theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(cprs_ind_dx … H) -V1
/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
qed.
| #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
- /5 width=5 by cprs_flat_dx, cpr_cprs, cprs_bind, lsubr_abst, ex2_3_intro, or3_intro1/
+ /5 width=5 by cprs_bind, cprs_flat_dx, cpr_cprs, lsubr_abst, ex2_3_intro, or3_intro1/
| #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
- @or3_intro2 @(ex4_5_intro … HV2 HT10) /3 width=3 by cprs_flat_sn, cprs_strap1, cpr_cprs, cprs_bind/ (**) (* full auto is too slow 11s *)
+ /5 width=10 by cprs_flat_sn, cprs_bind_dx, cprs_strap1, ex4_5_intro, or3_intro2/
]
| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
(* Basic_1: was just: pr3_pr2_pr2_t *)
(* Basic_1: includes: pr3_pr0_pr2_t *)
-lemma llpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (llpr G 0).
+lemma lpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lpr G).
#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
[ /2 width=3 by/
| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
- elim (llpr_inv_lref_ge_dx … HL12 … HLK2) -L2
- /5 width=7 by cprs_delta, cprs_strap2, llpr_cpr_conf/
-| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (llpr_inv_bind_O … HL12) -HL12 /4 width=1 by cprs_bind/
-| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 /3 width=1 by cprs_flat/
-| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_zeta/
-| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) /3 width=1 by cprs_tau/
-| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_beta/
-| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_theta/
+ elim (lpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
+ /4 width=6 by cprs_strap2, cprs_delta/
+|3,7: /4 width=1 by lpr_pair, cprs_bind, cprs_beta/
+|4,6: /3 width=1 by cprs_flat, cprs_tau/
+|5,8: /4 width=3 by lpr_pair, cprs_zeta, cprs_theta, cprs_strap1/
]
qed-.
lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=9 by llpr_cpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
+/4 width=5 by lpr_cpr_trans, cprs_bind_dx, lpr_pair/ qed.
(* Advanced properties ******************************************************)
(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma cprs_llpr_trans: ∀G. s_rs_transitive … (cpr G) (llpr G 0).
-/3 width=6 by llpr_cpr_trans, llpr_cpr_conf, s_r_trans_LTC1/ qed-.
+lemma lpr_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lpr G).
+#G @s_r_trans_LTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
+qed-.
(* Basic_1: was: pr3_strip *)
(* Basic_1: includes: pr1_strip *)
lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
-lemma cprs_llpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #H @(cprs_ind_dx … T0 H) -T0 /2 width=3 by ex2_intro/
-#T0 #T #HT0 #_ #IHT1 #L1 #HL01
-elim (IHT1 … L1) /2 by llpr_cpr_conf/ -IHT1 #T2 #HT12 #HT2
-elim (llpr_cpr_conf_dx … HT0 … HL01) -L0 #T3 #HT03 #HT3
-elim (cprs_strip … HT2 … HT3) -T
+lemma cprs_lpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3 by ex2_intro/
+#T #T1 #_ #HT1 #IHT0 #L1 #HL01 elim (IHT0 … HL01)
+#T2 #HT2 #HT02 elim (lpr_cpr_conf_dx … HT1 … HL01) -L0
+#T3 #HT3 #HT13 elim (cprs_strip … HT2 … HT3) -T
/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
qed-.
-lemma cprs_llpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cprs_llpr_conf_dx … HT01 … HL01)
-/4 width=5 by cprs_llpr_trans, cprs_llpr_conf, ex2_intro/
+lemma cprs_lpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01
+/3 width=3 by lpr_cprs_trans, ex2_intro/
qed-.
lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=9 by cprs_llpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
+/4 width=5 by lpr_cprs_trans, cprs_bind_dx, lpr_pair/ qed.
(* *)
(**************************************************************************)
-include "basic_2/reduction/llpr_ldrop.ma".
+include "basic_2/reduction/cpr_lift.ma".
include "basic_2/computation/cprs.ma".
(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
elim (lift_total V1 0 (i+1)) /4 width=12 by cpr_lift, cprs_strap1/
qed.
-lemma cprs_llpr_conf: ∀G. s_r_confluent1 … (cprs G) (llpr G 0).
-/3 width=5 by llpr_cpr_conf, s_r_conf1_LTC1/ qed-.
-
(* Advanced inversion lemmas ************************************************)
(* Basic_1: was: pr3_gen_lref *)
| * #K #V1 #T1 #HLK #HVT1 #HT1
lapply (ldrop_fwd_drop2 … HLK) #H0LK
elim (cpr_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
- /4 width=6 by cprs_strap1, ex3_3_intro, or_intror/
+ /4 width=6 by cprs_strap1, ex3_3_intro, or_intror/
]
qed-.
lemma lleq_cpxs_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
∀L1. L1 ⋕[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2.
#h #g #G #L2 #T1 #T2 #H @(cpxs_ind_dx … H) -T1
-/4 width=6 by lleq_cpx_conf_dx, lleq_cpx_trans, cpxs_strap2/
+/4 width=6 by cpx_lleq_conf_dx, lleq_cpx_trans, cpxs_strap2/
qed-.
-lemma lleq_cpxs_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
+lemma cpxs_lleq_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
+ ∀L1. L2 ⋕[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2.
+/3 width=3 by lleq_cpxs_trans, lleq_sym/ qed-.
+
+lemma cpxs_lleq_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
∀L1. L1 ⋕[T1, 0] L2 → L1 ⋕[T2, 0] L2.
-#h #g #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by lleq_cpx_conf_dx/
+#h #g #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_lleq_conf_dx/
qed-.
-lemma lleq_cpxs_conf_sn: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 →
+lemma cpxs_lleq_conf_sn: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 →
∀L2. L1 ⋕[T1, 0] L2 → L1 ⋕[T2, 0] L2.
-/4 width=6 by lleq_cpxs_conf_dx, lleq_sym/ qed-.
+/4 width=6 by cpxs_lleq_conf_dx, lleq_sym/ 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 llprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+(*
+lamma llprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
/3 width=1 by llprs_llpxs, llpxs_fpbs/ qed.
-
-lemma cpr_llpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡[T2, 0] L2 →
- ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
-/4 width=5 by fpbs_strap1, llpr_fpb, cpr_fpb/ 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
theorem fpbs_trans: ∀h,g. tri_transitive … (fpbs h g).
/2 width=5 by tri_TC_transitive/ qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma cpr_llpr_ssta_fpbs: ∀h,g,G,L1,L2,T1,T2,U2,l2.
- ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡[T2, 0] L2 →
- ⦃G, L2⦄ ⊢ T2 ▪[h, g] l2+1 → ⦃G, L2⦄ ⊢ T2 •[h, g] U2 →
- ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, U2⦄.
-/3 width=5 by fpbs_trans, cpr_llpr_fpbs, ssta_fpbs/ qed.
∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
/3 width=5 by cpxs_fpbs, lsstas_cpxs/ qed.
+(* Note: this should be moved *)
lemma ssta_fpbs: ∀h,g,G,L,T,U,l.
⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •[h, g] U →
⦃G, L, T⦄ ≥[h, g] ⦃G, L, U⦄.
--- /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/llpx_lpr.ma".
+include "basic_2/computation/fpbs.ma".
+
+(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
+
+(* Properties on sn parallel reduction for local environments ***************)
+
+(* Note: this is used in the closure proof *)
+(* Note: original proof: /4 width=5 by fpbs_strap1, lpr_fpb, cpr_fpb/ *)
+(* Note: this should be moved *)
+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⦄.
+/5 width=5 by fpbs_strap1, cpr_fpb, fpb_llpx, lpr_llpx/ qed.
+
+(* Note: this is used in the closure proof *)
+(* Note: this should be moved *)
+lemma cpr_lpr_ssta_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, g] U2 →
+ ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, U2⦄.
+/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, ssta_cpx, fpb_cpx/ qed.
lemma cprs_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_cpxs, cprs_cpxs/ qed.
-
-lemma llprs_fpbu: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[T, 0] L2 → (L1 ⋕[T, 0] L2 → ⊥) →
+(*
+lamma llprs_fpbu: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[T, 0] L2 → (L1 ⋕[T, 0] L2 → ⊥) →
⦃G, L1, T⦄ ≻[h, g] ⦃G, L2, T⦄.
/3 width=1 by fpbu_llpxs, llprs_llpxs/ qed.
-
+*)
(* Basic forward lemmas *****************************************************)
lemma fpbu_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
#h #g #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
[ #G #L2 #U #H2 elim (lleq_fqup_trans … H2 … HT) -K2
/3 width=3 by fpbu_fqup, ex2_intro/
-| /4 width=10 by fpbu_cpxs, lleq_cpxs_conf_sn, lleq_cpxs_trans, ex2_intro/
+| /4 width=10 by fpbu_cpxs, cpxs_lleq_conf_sn, lleq_cpxs_trans, ex2_intro/
| /5 width=3 by fpbu_llpxs, lleq_llpxs_trans, lleq_canc_sn, ex2_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/lazypredsnstar_5.ma".
-include "basic_2/reduction/llpr.ma".
-
-(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
-
-definition llprs: genv → relation4 ynat term lenv lenv ≝
- λG,d. LTC … (llpr G d).
-
-interpretation "lazy parallel computation (local environment, sn variant)"
- 'LazyPRedSnStar G L1 L2 T d = (llprs G d T L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma llprs_ind: ∀G,L1,T,d. ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → R L → R L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L2.
-#G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma llprs_ind_dx: ∀G,L2,T,d. ∀R:predicate lenv. R L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → R L → R L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L1.
-#G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpr_llprs: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
-/2 width=1 by inj/ qed.
-
-lemma llprs_refl: ∀G,L,T,d. ⦃G, L⦄ ⊢ ➡*[T, d] L.
-/2 width=1 by lpr_llprs/ qed.
-
-lemma llprs_strap1: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
-normalize /2 width=3 by step/ qed-.
-
-lemma llprs_strap2: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
-normalize /2 width=3 by TC_strap/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llprs_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → |L1| = |L2|.
-#G #L1 #L2 #T #d #H @(llprs_ind … H) -L2
-/3 width=6 by llpr_fwd_length, trans_eq/
-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/relocation/llpx_sn_tc.ma".
-include "basic_2/computation/cprs_cprs.ma".
-include "basic_2/computation/llprs.ma".
-
-(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
-
-(* Advanced properties ******************************************************)
-
-lemma llprs_pair_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡*[T, 0] L.ⓑ{I}V2.
-/2 width=1 by llpx_sn_TC_pair_dx/ qed.
-
-(* Properties on context-sensitive parallel computation for terms ***********)
-
-lemma llprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (llprs G 0).
-/3 width=5 by cprs_llpr_trans, s_r_trans_LTC2/ qed-.
-
-(* Basic_1: was just: pr3_pr3_pr3_t *)
-lemma llprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (llprs G 0).
-#G @s_r_to_s_rs_trans @s_r_trans_LTC2
-/3 width=5 by cprs_llpr_trans, s_rs_trans_TC1/ (**) (* full auto too slow *)
-qed-.
-
-lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=3 by llprs_cprs_trans, llprs_pair_dx, cprs_bind/ qed.
-
-(* Inversion lemmas on context-sensitive parallel computation for terms *****)
-
-(* Basic_1: was: pr3_gen_abst *)
-lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
- U2 = ⓛ{a}W2.T2.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
-#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
-elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
-lapply (llprs_cpr_trans … HT02 (L.ⓛV1) ?)
-/3 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro/
-qed-.
-
-lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
- ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
-#a #G #L #W1 #W2 #T1 #T2 #H
-elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
-qed-.
-
-(* Basic_1: was pr3_gen_abbr *)
-lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
- U2 = ⓓ{a}V2.T2
- ) ∨
- ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
-#U0 #U2 #_ #HU02 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_abbr1 … HU02) -HU02 *
- [ #V2 #T2 #HV02 #HT02 #H destruct
- lapply (llprs_cpr_trans … HT02 (L.ⓓV1) ?)
- /4 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/
- | #T2 #HT02 #HUT2
- lapply (llprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02
- /4 width=3 by llprs_pair_dx, cprs_trans, ex3_intro, or_intror/
- ]
-| #U1 #HTU1 #HU01
- elim (lift_total U2 0 1) #U #HU2
- lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
- /4 width=3 by cprs_strap1, ldrop_drop, ex3_intro, 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/computation/llprs.ma".
-
-(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
-
-(* Main properties **********************************************************)
-
-theorem llprs_trans: ∀G,T,d. Transitive … (llprs G d T).
-normalize /2 width=3 by trans_TC/ qed-.
include "basic_2/notation/relations/lazypredsnstar_7.ma".
include "basic_2/reduction/llpx.ma".
-include "basic_2/computation/llprs.ma".
(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
(* Basic properties *********************************************************)
-lemma llprs_llpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
-normalize /3 width=3 by llpr_llpx, monotonic_TC/ qed.
-
lemma llpx_llpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
normalize /2 width=1 by inj/ qed.
#h #g #G #L1 #T #A #HT #L2 #HL12
@(TC_Conf3 … (λL,A. ⦃G, L⦄ ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by aaa_llpx_conf/
qed-.
-
-lemma aaa_llprs_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+(*
+lamma aaa_llprs_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
∀L2. ⦃G, L1⦄ ⊢ ➡*[T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
/3 width=5 by aaa_llpxs_conf, llprs_llpxs/ 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/llpx_lpr.ma".
+include "basic_2/computation/lprs.ma".
+include "basic_2/computation/llpxs.ma".
+
+(* LAZY SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS **************)
+
+(* Properties on sn parallel computation ************************************)
+
+(* Note: this should be moved *)
+lemma lprs_llpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
+normalize /3 width=3 by lpr_llpx, monotonic_TC/ 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/predsnstar_3.ma".
+include "basic_2/grammar/lpx_sn_tc.ma".
+include "basic_2/reduction/lpr.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+definition lprs: relation3 genv lenv lenv ≝
+ λG. TC … (lpr G).
+
+interpretation "parallel computation (local environment, sn variant)"
+ 'PRedSnStar G L1 L2 = (lprs G L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma lprs_ind: ∀G,L1. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L2.
+#G #L1 #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma lprs_ind_dx: ∀G,L2. ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → R L → R L1) →
+ ∀L1. ⦃G, L1⦄ ⊢ ➡* L2 → R L1.
+#G #L2 #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpr_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
+/2 width=1/ qed.
+
+lemma lprs_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡* L.
+/2 width=1/ qed.
+
+lemma lprs_strap1: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
+/2 width=3/ qed.
+
+lemma lprs_strap2: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
+/2 width=3/ qed.
+
+lemma lprs_pair_refl: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡* L2.ⓑ{I}V.
+/2 width=1 by TC_lpx_sn_pair_refl/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lprs_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡* L2 → L2 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
+
+lemma lprs_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡* ⋆ → L1 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lprs_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → |L1| = |L2|.
+/2 width=2 by TC_lpx_sn_fwd_length/ 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/predsnstaralt_3.ma".
+include "basic_2/computation/cprs_cprs.ma".
+include "basic_2/computation/lprs.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+(* alternative definition *)
+definition lprsa: relation3 genv lenv lenv ≝
+ λG. lpx_sn … (cprs G).
+
+interpretation "parallel computation (local environment, sn variant) alternative"
+ 'PRedSnStarAlt G L1 L2 = (lprsa G L1 L2).
+
+(* Main properties on the alternative definition ****************************)
+
+theorem lprsa_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
+/2 width=1 by lpx_sn_LTC_TC_lpx_sn/ qed-.
+
+(* Main inversion lemmas on the alternative definition **********************)
+
+theorem lprs_inv_lprsa: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡➡* L2.
+/3 width=3 by TC_lpx_sn_inv_lpx_sn_LTC, lpr_cprs_trans/ qed-.
+
+(* Alternative eliminators **************************************************)
+
+lemma lprs_ind_alt: ∀G. ∀R:relation lenv.
+ R (⋆) (⋆) → (
+ ∀I,K1,K2,V1,V2.
+ ⦃G, K1⦄ ⊢ ➡* K2 → ⦃G, K1⦄ ⊢ V1 ➡* V2 →
+ R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpr_cprs_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/cprs_cprs.ma".
+include "basic_2/computation/lprs.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lprs_pair: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+ ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
+/2 width=1 by TC_lpx_sn_pair/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lprs_inv_pair1: ∀I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡* L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
+ L2 = K2.ⓑ{I}V2.
+/3 width=3 by TC_lpx_sn_inv_pair1, lpr_cprs_trans/ qed-.
+
+lemma lprs_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡* K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
+ L1 = K1.ⓑ{I}V1.
+/3 width=3 by TC_lpx_sn_inv_pair2, lpr_cprs_trans/ qed-.
+
+(* Properties on context-sensitive parallel computation for terms ***********)
+
+lemma lprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lprs G).
+/3 width=5 by s_r_trans_LTC2, lpr_cprs_trans/ qed-.
+
+(* Basic_1: was just: pr3_pr3_pr3_t *)
+(* Note: alternative proof /3 width=5 by s_r_trans_LTC1, lprs_cpr_trans/ *)
+lemma lprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lprs G).
+#G @s_r_to_s_rs_trans @s_r_trans_LTC2
+@s_rs_trans_TC1 /2 width=3 by lpr_cprs_trans/ (**) (* full auto too slow *)
+qed-.
+
+lemma lprs_cprs_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #H @(lprs_ind … H) -L1 /2 width=3 by ex2_intro/
+#L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
+elim (cprs_lpr_conf_dx … HT1 … HL1) -HT1 #T2 #HT2
+elim (cprs_lpr_conf_dx … HT0 … HL1) -L #T3 #HT3
+elim (cprs_conf … HT2 … HT3) -T
+/3 width=5 by cprs_trans, ex2_intro/
+qed-.
+
+lemma lprs_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+/3 width=3 by lprs_cprs_conf_dx, cpr_cprs/ qed-.
+
+(* Note: this can be proved on its own using lprs_ind_dx *)
+lemma lprs_cprs_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01
+/3 width=3 by lprs_cprs_trans, ex2_intro/
+qed-.
+
+lemma lprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+/3 width=3 by lprs_cprs_conf_sn, cpr_cprs/ qed-.
+
+lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=5 by lprs_cprs_trans, lprs_pair, cprs_bind/ qed.
+
+(* Inversion lemmas on context-sensitive parallel computation for terms *****)
+
+(* Basic_1: was: pr3_gen_abst *)
+lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
+ U2 = ⓛ{a}W2.T2.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
+elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
+lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by lprs_pair, cprs_trans, cprs_strap1, ex3_2_intro/
+qed-.
+
+lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
+ ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
+#a #G #L #W1 #W2 #T1 #T2 #H elim (cprs_inv_abst1 … H) -H
+#W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
+qed-.
+
+(* Basic_1: was pr3_gen_abbr *)
+lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
+ U2 = ⓓ{a}V2.T2
+ ) ∨
+ ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
+#U0 #U2 #_ #HU02 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+ elim (cpr_inv_abbr1 … HU02) -HU02 *
+ [ #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?)
+ /4 width=5 by lprs_pair, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/
+ | #T2 #HT02 #HUT2
+ lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02
+ /4 width=3 by lprs_pair, cprs_trans, ex3_intro, or_intror/
+ ]
+| #U1 #HTU1 #HU01 elim (lift_total U2 0 1)
+ #U #HU2 lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
+ /4 width=3 by cprs_strap1, ldrop_drop, ex3_intro, or_intror/
+]
+qed-.
+
+(* More advanced properties *************************************************)
+
+lemma lprs_pair2: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+ ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
+/3 width=3 by lprs_pair, lprs_cprs_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/reduction/lpr_ldrop.ma".
+include "basic_2/computation/lprs.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+(* Properies on local environment slicing ***********************************)
+
+lemma lprs_ldrop_conf: ∀G. dropable_sn (lprs G).
+/3 width=3 by dropable_sn_TC, lpr_ldrop_conf/ qed-.
+
+lemma ldrop_lprs_trans: ∀G. dedropable_sn (lprs G).
+/3 width=3 by dedropable_sn_TC, ldrop_lpr_trans/ qed-.
+
+lemma lprs_ldrop_trans_O1: ∀G. dropable_dx (lprs G).
+/3 width=3 by dropable_dx_TC, lpr_ldrop_trans_O1/ 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/lpr_lpr.ma".
+include "basic_2/computation/lprs.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lprs_strip: ∀G. confluent2 … (lprs G) (lpr G).
+/3 width=3 by TC_strip1, lpr_conf/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem lprs_conf: ∀G. confluent2 … (lprs G) (lprs G).
+/3 width=3 by TC_confluent2, lpr_conf/ qed-.
+
+theorem lprs_trans: ∀G. Transitive … (lprs G).
+/2 width=3/ qed-.
(**************************************************************************)
include "basic_2/unfold/lsstas_lsstas.ma".
-include "basic_2/computation/fpbg_fpns.ma".
+include "basic_2/computation/fpbg_fleq.ma".
include "basic_2/equivalence/cpes_cpds.ma".
include "basic_2/dynamic/snv.ma".
(* *)
(**************************************************************************)
+include "basic_2/computation/fpbs_lpr.ma".
include "basic_2/dynamic/snv_lift.ma".
include "basic_2/dynamic/snv_cpcs.ma".
include "basic_2/dynamic/lsubsv_snv.ma".
(**************************************************************************)
include "basic_2/computation/cpds_cpds.ma".
+include "basic_2/computation/fpbs_lpr.ma".
include "basic_2/dynamic/snv_aaa.ma".
include "basic_2/dynamic/snv_cpcs.ma".
include "basic_2/dynamic/lsubsv_lsstas.ma".
lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
#G #L #T1 #T2 #H @(cpcs_ind … H) -T2
-[ /3 width=3/
+[ /3 width=3 by ex2_intro/
| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
[ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/
| /3 width=5 by cprs_strap2, ex2_intro/
(* Basic_1: was: pc3_wcpr0_t *)
(* Basic_1: note: pc3_wcpr0_t should be renamed *)
+(* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *)
lemma lpr_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
-/3 width=5 by lprs_cprs_conf, lpr_lprs/ qed-.
+#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1
+/2 width=3 by cprs_div/
+qed-.
(* Basic_1: was only: pc3_pr0_pr2_t *)
(* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
(* More inversion lemmas ****************************************************)
+(* Note: there must be a proof suitable for llpr *)
lemma cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
--- /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/llprs_cprs.ma".
+include "basic_2/conversion/cpc_cpc.ma".
+include "basic_2/equivalence/cpcs_cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
+#G #L #T1 #T2 #H @(cpcs_ind … H) -T2
+[ /3 width=3 by ex2_intro/
+| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
+ [ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/
+ | /3 width=5 by cprs_strap2, ex2_intro/
+ ]
+]
+qed-.
+
+(* Basic_1: was: pc3_gen_sort *)
+lemma cpcs_inv_sort: ∀G,L,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
+#G #L #k1 #k2 #H elim (cpcs_inv_cprs … H) -H
+#T #H1 >(cprs_inv_sort1 … H1) -T #H2
+lapply (cprs_inv_sort1 … H2) -L #H destruct //
+qed-.
+
+lemma cpcs_inv_abst1: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+#a #G #L #W1 #T1 #T #H
+elim (cpcs_inv_cprs … H) -H #X #H1 #H2
+elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
+/3 width=6 by cprs_bind, ex2_2_intro/
+qed-.
+
+lemma cpcs_inv_abst2: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+/3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
+
+(* Basic_1: was: pc3_gen_sort_abst *)
+lemma cpcs_inv_sort_abst: ∀a,G,L,W,T,k. ⦃G, L⦄ ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
+#a #G #L #W #T #k #H
+elim (cpcs_inv_cprs … H) -H #X #H1
+>(cprs_inv_sort1 … H1) -X #H2
+elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: pc3_gen_lift *)
+lemma cpcs_inv_lift: ∀G,L,K,s,d,e. ⇩[s, d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
+ ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2.
+#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
+elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
+elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
+elim (cprs_inv_lift1 … HU2 … HLK … HTU2) -L -U2 #X #HXU
+>(lift_inj … HXU … HTU) -X -U -d -e /2 width=3 by cprs_div/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpr_cpcs_trans: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡[T2, 0] L2 →
+ ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
+/4 width=5 by cprs_div, cprs_llpr_trans/
+qed-.
+
+lemma llprs_cpcs_trans: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡*[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡*[T2, 0] L2 →
+ ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
+/4 width=5 by cprs_div, llprs_cprs_trans/
+qed-.
+
+lemma cpr_cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cpr_cprs_div/
+qed-.
+
+lemma cprs_cpr_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cprs_cpr_div/
+qed-.
+
+lemma cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cprs_div/
+qed-.
+
+(* Basic_1: was: pc3_wcpr0_t *)
+(* Basic_1: note: pc3_wcpr0_t should be renamed *)
+lemma llpr_cprs_conf: ∀G,L1,L2,T1. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #T1 #HL12 #T2 #HT12 elim (cprs_llpr_conf_dx … HT12 … HL12) -L1
+/2 width=3 by cprs_div/
+qed-.
+
+(* Basic_1: was only: pc3_pr0_pr2_t *)
+(* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
+lemma llpr_cpr_conf: ∀G,L1,L2,T1. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+/3 width=5 by llpr_cprs_conf, cpr_cprs/ qed-.
+
+(* Basic_1: was only: pc3_thin_dx *)
+lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
+#G #L #V1 #V2 #HV12 #T1 #T2 #HT12
+elim (cpcs_inv_cprs … HV12) -HV12
+elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_flat, cprs_div/
+qed.
+
+lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
+/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
+
+lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 →
+ ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2.
+#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, cprs_bind/
+qed.
+
+lemma cpcs_bind_sn: ∀a,I,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
+#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
+/3 width=5 by cprs_div, cprs_bind/
+qed.
+
+lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 →
+ ∀L2. L2 ⊑ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, lsubr_cprs_trans/
+qed-.
+
+(* Basic_1: was: pc3_lift *)
+lemma cpcs_lift: ∀G,L,K,s,d,e. ⇩[s, d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
+ ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
+#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
+elim (lift_total T d e) /3 width=12 by cprs_div, cprs_lift/
+qed.
+
+lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
+ ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
+#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-.
+
+(* More inversion lemmas ****************************************************)
+
+axiom cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+ ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
+(*
+#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
+elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
+lapply (llprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by llprs_pair/ -HT2 #HT2
+lapply (llprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by llprs_pair/ -HT2 #HT2
+/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
+qed-.
+*)
+lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+ ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2.
+#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
+#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was pc3_t *)
+theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
+
+theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
+
+theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
+
+lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
+ ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 →
+ ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
+
+lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
+ ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 →
+ ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
+
+(* Basic_1: was: pc3_wcpr0 *)
+lemma llpr_cpcs_conf: ∀G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ ➡[T1, 0] L2 → ⦃G, L1⦄ ⊢ ➡[T2, 0] L2 →
+ ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #T1 #T2 #HT1 #HT2 #H elim (cpcs_inv_cprs … H) -H
+/3 width=5 by cpcs_canc_dx, llpr_cprs_conf/
+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/llpr_llpr.ma".
+include "basic_2/computation/cprs_lift.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: pr3_t *)
+(* Basic_1: includes: pr1_t *)
+theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
+normalize /2 width=3 by trans_TC/ qed-.
+
+(* Basic_1: was: pr3_confluence *)
+(* Basic_1: includes: pr1_confluence *)
+theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L).
+normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-.
+
+theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
+/3 width=5 by cprs_trans, cprs_bind_dx/
+qed.
+
+(* Basic_1: was: pr3_flat *)
+theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2.
+#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
+/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/
+qed.
+
+theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/
+#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *)
+@(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/
+qed.
+
+theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/
+#V #V2 #_ #HV2 #IHV1
+@(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/
+qed.
+
+theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+ ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2
+/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
+qed.
+
+theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+ ⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+ ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
+/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was pr3_gen_appl *)
+lemma cprs_inv_appl1: ∀G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 &
+ U2 = ⓐV2. T2
+ | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
+ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
+ | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
+ ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
+ ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
+#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
+#U #U2 #_ #HU2 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+ elim (cpr_inv_appl1 … HU2) -HU2 *
+ [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/
+ | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
+ lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
+ lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
+ /5 width=5 by cprs_flat_dx, cpr_cprs, cprs_bind, lsubr_abst, ex2_3_intro, or3_intro1/
+ | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
+ @or3_intro2 @(ex4_5_intro … HV2 HT10) /3 width=3 by cprs_flat_sn, cprs_strap1, cpr_cprs, cprs_bind/ (**) (* full auto is too slow 11s *)
+ ]
+| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
+| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
+]
+qed-.
+
+(* Properties concerning sn parallel reduction on local environments ********)
+
+(* Basic_1: was just: pr3_pr2_pr2_t *)
+(* Basic_1: includes: pr3_pr0_pr2_t *)
+lemma llpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (llpr G 0).
+#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
+[ /2 width=3 by/
+| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
+ elim (llpr_inv_lref_ge_dx … HL12 … HLK2) -L2
+ /5 width=7 by cprs_delta, cprs_strap2, llpr_cpr_conf/
+| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (llpr_inv_bind_O … HL12) -HL12 /4 width=1 by cprs_bind/
+| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (llpr_inv_flat … HL12) -HL12 /3 width=1 by cprs_flat/
+| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
+ elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_zeta/
+| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12
+ elim (llpr_inv_flat … HL12) /3 width=1 by cprs_tau/
+| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
+ elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
+ elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_beta/
+| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
+ elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
+ elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_theta/
+]
+qed-.
+
+lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=9 by llpr_cpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
+lemma cprs_llpr_trans: ∀G. s_rs_transitive … (cpr G) (llpr G 0).
+/3 width=6 by llpr_cpr_trans, llpr_cpr_conf, s_r_trans_LTC1/ qed-.
+
+(* Basic_1: was: pr3_strip *)
+(* Basic_1: includes: pr1_strip *)
+lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
+normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
+
+lemma cprs_llpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #H @(cprs_ind_dx … T0 H) -T0 /2 width=3 by ex2_intro/
+#T0 #T #HT0 #_ #IHT1 #L1 #HL01
+elim (IHT1 … L1) /2 by llpr_cpr_conf/ -IHT1 #T2 #HT12 #HT2
+elim (llpr_cpr_conf_dx … HT0 … HL01) -L0 #T3 #HT03 #HT3
+elim (cprs_strip … HT2 … HT3) -T
+/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
+qed-.
+
+lemma cprs_llpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cprs_llpr_conf_dx … HT01 … HL01)
+/4 width=5 by cprs_llpr_trans, cprs_llpr_conf, ex2_intro/
+qed-.
+
+lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=9 by cprs_llpr_trans, cprs_bind_dx, llpr_bind_repl_O/ 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/llpr_ldrop.ma".
+include "basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Advanced properties ******************************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma cprs_delta: ∀G,L,K,V,V2,i.
+ ⇩[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡* V2 →
+ ∀W2. ⇧[0, i + 1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡* W2.
+#G #L #K #V #V2 #i #HLK #H elim H -V2 [ /3 width=6 by cpr_cprs, cpr_delta/ ]
+#V1 #V2 #_ #HV12 #IHV1 #W2 #HVW2
+lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
+elim (lift_total V1 0 (i+1)) /4 width=12 by cpr_lift, cprs_strap1/
+qed.
+
+lemma cprs_llpr_conf: ∀G. s_r_confluent1 … (cprs G) (llpr G 0).
+/3 width=5 by llpr_cpr_conf, s_r_conf1_LTC1/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: pr3_gen_lref *)
+lemma cprs_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡* T2 →
+ T2 = #i ∨
+ ∃∃K,V1,T1. ⇩[i] L ≡ K.ⓓV1 & ⦃G, K⦄ ⊢ V1 ➡* T1 &
+ ⇧[0, i + 1] T1 ≡ T2.
+#G #L #T2 #i #H @(cprs_ind … H) -T2 /2 width=1 by or_introl/
+#T #T2 #_ #HT2 *
+[ #H destruct
+ elim (cpr_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
+ * /4 width=6 by cpr_cprs, ex3_3_intro, or_intror/
+| * #K #V1 #T1 #HLK #HVT1 #HT1
+ lapply (ldrop_fwd_drop2 … HLK) #H0LK
+ elim (cpr_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
+ /4 width=6 by cprs_strap1, ex3_3_intro, or_intror/
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr3_lift *)
+lemma cprs_lift: ∀G. l_liftable (cprs G).
+/3 width=10 by l_liftable_LTC, cpr_lift/ qed.
+
+(* Basic_1: was: pr3_gen_lift *)
+lemma cprs_inv_lift1: ∀G. l_deliftable_sn (cprs G).
+/3 width=6 by l_deliftable_sn_LTC, cpr_inv_lift1/
+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 G , break term 46 L1 ⦄ ⊢ ➡ break [ term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyPRedSn $G $L1 $L2 $T $d }.
--- /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 G , break term 46 L1 ⦄ ⊢ ➡* break [ term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyPRedSnStar $G $L1 $L2 $T $d }.
--- /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/lazypredsn_5.ma".
+include "basic_2/relocation/llpx_sn.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
+
+definition llpr: genv → relation4 ynat term lenv lenv ≝ λG. llpx_sn (cpr G).
+
+interpretation "lazy parallel reduction (local environment, sn variant)"
+ 'LazyPRedSn G L1 L2 T d = (llpr G d T L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpr_inv_flat: ∀I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[ⓕ{I}V.T, d] L2 →
+ ⦃G, L1⦄ ⊢ ➡[V, d] L2 ∧ ⦃G, L1⦄ ⊢ ➡[T, d] L2.
+/2 width=2 by llpx_sn_inv_flat/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpr_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → |L1| = |L2|.
+/2 width=4 by llpx_sn_fwd_length/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpr_lref: ∀I,G,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 → ⦃G, L1⦄ ⊢ ➡[#i, d] L2.
+/2 width=9 by llpx_sn_lref/ qed.
+
+(* Note: lemma 250 *)
+lemma llpr_refl: ∀G,T,d. reflexive … (llpr G d T).
+/2 width=1 by llpx_sn_refl/ qed.
+
+(* Basic_1: removed theorems 5: wcpr0_gen_sort wcpr0_gen_head
+ wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+*)
--- /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/relocation/fquq_alt.ma".
+include "basic_2/static/ssta_llpx_sn.ma".
+include "basic_2/reduction/cpr_lift.ma".
+include "basic_2/reduction/cpr_llpx_sn.ma".
+include "basic_2/reduction/llpr.ma".
+
+(* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma llpr_inv_lref_ge_dx: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
+ ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
+ ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
+/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-.
+
+lemma llpr_inv_lref_ge_sn: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
+ ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
+ ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
+/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-.
+
+lemma llpr_inv_lref_ge_bi: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
+ ∀I1,I2,K1,K2,V1,V2.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
+/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
+
+lemma llpr_inv_bind_O: ∀a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡ [ⓑ{a,I}V.T, 0] L2 →
+ ⦃G, L1⦄ ⊢ ➡[V, 0] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind_O/ qed-.
+
+lemma llpr_bind_repl_O: ∀I,G,L1,L2,V1,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V2 →
+ ∀J,W1,W2. ⦃G, L1⦄ ⊢ ➡[W1, 0] L2 → ⦃G, L1⦄ ⊢ W1 ➡ W2 → ⦃G, L1.ⓑ{J}W1⦄ ⊢ ➡[T, 0] L2.ⓑ{J}W2.
+/2 width=4 by llpx_sn_bind_repl_O/ qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpr_ssta_conf: ∀h,g,G. s_r_confluent1 … (ssta h g G) (llpr G 0).
+/3 width=10 by ssta_llpx_sn_conf, cpr_lift/ qed-.
+
+lemma llpr_cpr_conf: ∀G. s_r_confluent1 … (cpr G) (llpr G 0).
+/3 width=10 by cpr_llpx_sn_conf, cpr_inv_lift1, cpr_lift/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=10 by llpr_lref, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
+#G #L #K #U #T #e #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (e+1)) #T2 #HUT2
+lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+qed-.
+
+lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=10 by llpr_lref, cpr_pair_sn, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
+#G #L #K #U #T #e #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (e+1)) #T2 #HUT2
+lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+qed-.
+
+lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_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/relocation/llpx_sn_llpx_sn.ma".
+include "basic_2/substitution/fqup.ma".
+include "basic_2/reduction/llpr_ldrop.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Main properties on context-sensitive parallel reduction for terms ********)
+
+fact cpr_conf_llpr_atom_atom:
+ ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
+/2 width=3 by cpr_atom, ex2_intro/ qed-.
+
+fact cpr_conf_llpr_atom_delta:
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
+ ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #V1 #HLK1 #HK01 #HV01
+elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_
+lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1))
+/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
+qed-.
+
+(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
+fact cpr_conf_llpr_delta_delta:
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
+ ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[i] L0 ≡ KX.ⓓVX →
+ ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #W1 #HLK1 #HK01 #_
+lapply (ldrop_fwd_drop2 … HLK1) -W1 #HLK1
+elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_
+lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_bind_bind:
+ ∀a,I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T.
+#a #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
+elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
+/3 width=5 by llpr_bind_repl_O, cpr_bind, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_bind_zeta:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
+elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_zeta_zeta:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
+elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2
+lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_flat_flat:
+ ∀I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T.
+#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) //
+elim (IH … HT01 … HT02 … H1T0 … H2T0) /3 width=5 by cpr_flat, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_flat_tau:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0
+elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_tau_tau:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0
+elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_llpr_flat_beta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ (**) (* full auto not tried *)
+/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
+qed-.
+
+(* Basic-1: includes:
+ pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
+ pr0_cong_upsilon_cong pr0_cong_upsilon_delta
+*)
+fact cpr_conf_llpr_flat_theta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2
+elim (cpr_inv_abbr1 … H) -H *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+ elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
+ elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0
+ /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/
+| #T1 #HT01 #HXT1 #H destruct
+ elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+ elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1
+ /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/
+]
+qed-.
+
+fact cpr_conf_llpr_beta_beta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/
+/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+(* Basic_1: includes: pr0_upsilon_upsilon *)
+fact cpr_conf_llpr_theta_theta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
+ ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
+elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
+elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
+elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
+elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by ldrop_drop/
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/
+/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+theorem cpr_conf_llpr: ∀G. llpx_sn_confluent2 (cpr G) (cpr G).
+#G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
+[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_atom1 … H1) -H1
+ elim (cpr_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_conf_llpr_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_conf_llpr_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpr_conf_llpr_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpr_conf_llpr_delta_delta/
+ ]
+| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (cpr_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+ [ /3 width=10 by cpr_conf_llpr_bind_bind/
+ | /4 width=11 by ex2_commute, cpr_conf_llpr_bind_zeta/
+ | /3 width=11 by cpr_conf_llpr_bind_zeta/
+ | /3 width=12 by cpr_conf_llpr_zeta_zeta/
+ ]
+| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
+ | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+ ]
+ elim (cpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #HX2 #H2
+ |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
+ |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
+ ] destruct
+ [ /3 width=10 by cpr_conf_llpr_flat_flat/
+ | /4 width=8 by ex2_commute, cpr_conf_llpr_flat_tau/
+ | /4 width=12 by ex2_commute, cpr_conf_llpr_flat_beta/
+ | /4 width=14 by ex2_commute, cpr_conf_llpr_flat_theta/
+ | /3 width=8 by cpr_conf_llpr_flat_tau/
+ | /3 width=8 by cpr_conf_llpr_tau_tau/
+ | /3 width=12 by cpr_conf_llpr_flat_beta/
+ | /3 width=13 by cpr_conf_llpr_beta_beta/
+ | /3 width=14 by cpr_conf_llpr_flat_theta/
+ | /3 width=17 by cpr_conf_llpr_theta_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: includes: pr0_confluence pr2_confluence *)
+theorem cpr_conf: ∀G,L. confluent … (cpr G L).
+/2 width=6 by cpr_conf_llpr/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma llpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_llpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
+qed-.
+
+lemma llpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_llpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_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/lazypredsnstar_5.ma".
+include "basic_2/reduction/llpr.ma".
+
+(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
+
+definition llprs: genv → relation4 ynat term lenv lenv ≝
+ λG,d. LTC … (llpr G d).
+
+interpretation "lazy parallel computation (local environment, sn variant)"
+ 'LazyPRedSnStar G L1 L2 T d = (llprs G d T L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma llprs_ind: ∀G,L1,T,d. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L2.
+#G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma llprs_ind_dx: ∀G,L2,T,d. ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → R L → R L1) →
+ ∀L1. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L1.
+#G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpr_llprs: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+/2 width=1 by inj/ qed.
+
+lemma llprs_refl: ∀G,L,T,d. ⦃G, L⦄ ⊢ ➡*[T, d] L.
+/2 width=1 by lpr_llprs/ qed.
+
+lemma llprs_strap1: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+normalize /2 width=3 by step/ qed-.
+
+lemma llprs_strap2: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+normalize /2 width=3 by TC_strap/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llprs_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → |L1| = |L2|.
+#G #L1 #L2 #T #d #H @(llprs_ind … H) -L2
+/3 width=6 by llpr_fwd_length, trans_eq/
+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/relocation/llpx_sn_tc.ma".
+include "basic_2/computation/cprs_cprs.ma".
+include "basic_2/computation/llprs.ma".
+
+(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
+
+(* Advanced properties ******************************************************)
+
+lemma llprs_pair_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡*[T, 0] L.ⓑ{I}V2.
+/2 width=1 by llpx_sn_TC_pair_dx/ qed.
+
+(* Properties on context-sensitive parallel computation for terms ***********)
+
+lemma llprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (llprs G 0).
+/3 width=5 by cprs_llpr_trans, s_r_trans_LTC2/ qed-.
+
+(* Basic_1: was just: pr3_pr3_pr3_t *)
+lemma llprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (llprs G 0).
+#G @s_r_to_s_rs_trans @s_r_trans_LTC2
+/3 width=5 by cprs_llpr_trans, s_rs_trans_TC1/ (**) (* full auto too slow *)
+qed-.
+
+(* Note: this is an instance of a general theorem *)
+lemma llprs_cprs_conf_dx: ∀G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡* U2 →
+ ∀L0. ⦃G2, L0⦄ ⊢ ➡*[T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[U2, O] L2.
+#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx … H) -L0 //
+#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 … IHL2) -IHL2
+lapply (llprs_cprs_trans … HTU2 … HL2) -L2 #HTU2
+/3 width=3 by cprs_llpr_trans, cprs_llpr_conf/
+qed-.
+
+(* Note: this is an instance of a general theorem *)
+lemma llprs_cpr_conf_dx: ∀G2,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∀L0. ⦃G2, L0⦄ ⊢ ➡*[T2, O] L2 → ⦃G2, L0⦄ ⊢ ➡*[U2, O] L2.
+#G2 #L2 #T2 #U2 #HTU2 #L0 #H @(llprs_ind_dx … H) -L0 //
+#L0 #L #HL0 #HL2 #IHL2 @(llprs_strap2 … IHL2) -IHL2
+lapply (llprs_cpr_trans … HTU2 … HL2) -L2 #HTU2
+/3 width=3 by cprs_llpr_trans, cprs_llpr_conf/
+qed-.
+
+lemma llprs_cprs_conf_sn: ∀G,L0,L1,T0. ⦃G, L0⦄ ⊢ ➡*[T0, 0] L1 →
+ ∀T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #L1 #T0 #H @(llprs_ind_dx … H) -L0 /2 width=3 by ex2_intro/
+#L0 #L #HL0 #HL1 #IHL1 #T1 #HT01 elim (cprs_llpr_conf_sn … HT01 … HL0)
+#T2 #HT12 #HT02 elim (IHL1 … HT02) -IHL1 -HT02
+lapply (cprs_trans … HT01 … HT12) #HT02
+lapply (cprs_llpr_conf … HT02 … HL0) -HT02 -HL0
+/4 width=5 by cprs_llpr_trans, cprs_trans, ex2_intro/
+qed-.
+
+lemma llprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡*[T0, 0] L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+/3 width=3 by llprs_cprs_conf_sn, cpr_cprs/ qed-.
+
+lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=3 by llprs_cprs_trans, llprs_pair_dx, cprs_bind/ qed-.
+
+(* Inversion lemmas on context-sensitive parallel computation for terms *****)
+
+(* Basic_1: was: pr3_gen_abst *)
+lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
+ U2 = ⓛ{a}W2.T2.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
+elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
+lapply (llprs_cpr_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro/
+qed-.
+
+lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
+ ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
+#a #G #L #W1 #W2 #T1 #T2 #H
+elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
+qed-.
+
+(* Basic_1: was pr3_gen_abbr *)
+lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
+ U2 = ⓓ{a}V2.T2
+ ) ∨
+ ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
+#U0 #U2 #_ #HU02 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+ elim (cpr_inv_abbr1 … HU02) -HU02 *
+ [ #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (llprs_cpr_trans … HT02 (L.ⓓV1) ?)
+ /4 width=5 by llprs_pair_dx, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/
+ | #T2 #HT02 #HUT2
+ lapply (llprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02
+ /4 width=3 by llprs_pair_dx, cprs_trans, ex3_intro, or_intror/
+ ]
+| #U1 #HTU1 #HU01
+ elim (lift_total U2 0 1) #U #HU2
+ lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
+ /4 width=3 by cprs_strap1, ldrop_drop, ex3_intro, or_intror/
+]
+qed-.
+
+(* Note: we loose lprs_cprs_conf_dx and derivatives:
+ lprs_cpr_conf_dx lprs_cprs_conf
+*)
--- /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/llprs.ma".
+
+(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
+
+(* Main properties **********************************************************)
+
+theorem llprs_trans: ∀G,T,d. Transitive … (llprs G d T).
+normalize /2 width=3 by trans_TC/ 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/relocation/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+definition llpx_sn_confluent2: relation (lenv→relation term) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. llpx_sn R1 0 T0 L0 L1 → ∀L2. llpx_sn R2 0 T0 L0 L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+(* Note: we miss llpx_sn_conf and derivatives: lpr_conf lprs_conf *)
+++ /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/grammar/lpx_sn.ma".
-include "basic_2/relocation/ldrop_leq.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Properties on sn pointwise extension *************************************)
-
-lemma lpx_sn_deliftable_dropable: ∀R. l_deliftable_sn R → dropable_sn (lpx_sn R).
-#R #HR #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
- /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
-| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct
- /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 /3 width=3 by ldrop_drop, ex2_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
- elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (HR … HV12 … HLK1 … HWV1) -V1
- elim (IHLK1 … HL12) -L1 /3 width=5 by ldrop_skip, lpx_sn_pair, ex2_intro/
-]
-qed-.
-
-lemma lpx_sn_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) →
- l_liftable R → dedropable_sn (lpx_sn R).
-#R #H1R #H2R #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
- /4 width=4 by ldrop_atom, lpx_sn_atom, ex3_intro/
-| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
- #K2 #V2 #HK12 #HV12 #H destruct
- lapply (lpx_sn_fwd_length … HK12)
- #H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
- /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
- @leq_O2 normalize //
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
- /3 width=5 by ldrop_drop, leq_pair, lpx_sn_pair, ex3_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
- elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (lift_total W2 d e) #V2 #HWV2
- lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
- elim (IHLK1 … HK12) -K1
- /3 width=6 by ldrop_skip, leq_succ, lpx_sn_pair, ex3_intro/
-]
-qed-.
-
-fact lpx_sn_dropable_aux: ∀R,L2,K2,s,d,e. ⇩[s, d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
- d = 0 → ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & lpx_sn R K1 K2.
-#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom2 … H) -H
- /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
-| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
- #K1 #V1 #HK12 #HV12 #H destruct
- /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
-| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
- #L1 #V1 #HL12 #HV12 #H destruct
- elim (IHLK2 … HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/
-| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
- <plus_n_Sm #H destruct
-]
-qed-.
-
-lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
-/2 width=5 by lpx_sn_dropable_aux/ 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/predsn_3.ma".
-include "basic_2/grammar/lpx_sn.ma".
-include "basic_2/reduction/cpr.ma".
-
-(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
-
-definition lpr: relation3 genv lenv lenv ≝ λG. lpx_sn (cpr G).
-
-interpretation "parallel reduction (local environment, sn variant)"
- 'PRedSn G L1 L2 = (lpr G L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: includes: wcpr0_gen_sort *)
-lemma lpr_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡ L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-(* Basic_1: includes: wcpr0_gen_head *)
-lemma lpr_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L2 = K2.ⓑ{I}V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-lemma lpr_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡ ⋆ → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-lemma lpr_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-(* Note: lemma 250 *)
-lemma lpr_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡ L.
-/2 width=1 by lpx_sn_refl/ qed.
-
-lemma lpr_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡ K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 →
- ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ K2.ⓑ{I}V2.
-/2 width=1/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpr_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → |L1| = |L2|.
-/2 width=2 by lpx_sn_fwd_length/ qed-.
-
-(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
- pr0_subst1_back
-*)
+++ /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/relocation/fquq_alt.ma".
-include "basic_2/relocation/ldrop_lpx_sn.ma".
-include "basic_2/reduction/cpr_lift.ma".
-include "basic_2/reduction/lpr.ma".
-
-(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Properies on local environment slicing ***********************************)
-
-(* Basic_1: includes: wcpr0_drop *)
-lemma lpr_ldrop_conf: ∀G. dropable_sn (lpr G).
-/3 width=6 by lpx_sn_deliftable_dropable, cpr_inv_lift1/ qed-.
-
-(* Basic_1: includes: wcpr0_drop_back *)
-lemma ldrop_lpr_trans: ∀G. dedropable_sn (lpr G).
-/3 width=10 by lpx_sn_liftable_dedropable, cpr_lift/ qed-.
-
-lemma lpr_ldrop_trans_O1: ∀G. dropable_dx (lpr G).
-/2 width=3 by lpx_sn_dropable/ qed-.
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
-#G #L #K #U #T #e #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (e+1)) #T2 #HUT2
-lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
-qed-.
-
-lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
-#G #L #K #U #T #e #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (e+1)) #T2 #HUT2
-lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
-qed-.
-
-lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma fqu_lpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡ K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡ K1 & ⦃G1, L1⦄ ⊢ T1 ➡ T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpr_pair, ex3_2_intro/
-[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpr_inv_pair1 … H) -H
- #K2 #W2 #HLK2 #HVW2 #H destruct
- /3 width=5 by fqu_fquq, cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #K2 #HK12
- elim (ldrop_lpr_trans … HLK1 … HK12) -HK12
- /3 width=7 by fqu_drop, ex3_2_intro/
-]
-qed-.
-
-lemma fquq_lpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡ K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡ K1 & ⦃G1, L1⦄ ⊢ T1 ➡ T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_lpr_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_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/grammar/lpx_sn_lpx_sn.ma".
-include "basic_2/substitution/fqup.ma".
-include "basic_2/reduction/lpr_ldrop.ma".
-
-(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Main properties on context-sensitive parallel reduction for terms ********)
-
-fact cpr_conf_lpr_atom_atom:
- ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
-/2 width=3 by cpr_atom, ex2_intro/ qed-.
-
-fact cpr_conf_lpr_atom_delta:
- ∀G,L0,i. (
- ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
- ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
-elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
-lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1))
-/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
-qed-.
-
-(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
-fact cpr_conf_lpr_delta_delta:
- ∀G,L0,i. (
- ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
- ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀KX,VX. ⇩[i] L0 ≡ KX.ⓓVX →
- ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
-lapply (ldrop_fwd_drop2 … HLK1) -W1 #HLK1
-elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
-lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
-qed-.
-
-fact cpr_conf_lpr_bind_bind:
- ∀a,I,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T.
-#a #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
-/3 width=5 by lpr_pair, cpr_bind, ex2_intro/
-qed-.
-
-fact cpr_conf_lpr_bind_zeta:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 →
- ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
-#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/
-qed-.
-
-fact cpr_conf_lpr_zeta_zeta:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
- ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
-#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -T0 #T #HT1 #HT2
-elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1
-elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2
-lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_lpr_flat_flat:
- ∀I,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T.
-#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 … HL01 … HL02) /3 width=5 by cpr_flat, ex2_intro/
-qed-.
-
-fact cpr_conf_lpr_flat_tau:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/
-qed-.
-
-fact cpr_conf_lpr_tau_tau:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #V0 #T0 #IH #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_lpr_flat_beta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
-elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ (**) (* full auto not tried *)
-/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
-qed-.
-
-(* Basic-1: includes:
- pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
- pr0_cong_upsilon_cong pr0_cong_upsilon_delta
-*)
-fact cpr_conf_lpr_flat_theta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
- ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2
-elim (cpr_inv_abbr1 … H) -H *
-[ #W1 #T1 #HW01 #HT01 #H destruct
- elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
- elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
- /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/
-| #T1 #HT01 #HXT1 #H destruct
- elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
- elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1
- /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/
-]
-qed-.
-
-fact cpr_conf_lpr_beta_beta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ #W #HW1 #HW2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/
-lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/
-/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
-qed-.
-
-(* Basic_1: was: pr0_upsilon_upsilon *)
-fact cpr_conf_lpr_theta_theta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
- ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
- ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
-elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
-elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by ldrop_drop/
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/
-/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
-qed-.
-
-theorem cpr_conf_lpr: ∀G. lpx_sn_confluent (cpr G) (cpr G).
-#G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
-[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_atom1 … H1) -H1
- elim (cpr_inv_atom1 … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpr_conf_lpr_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpr_conf_lpr_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpr_conf_lpr_delta_delta/
- ]
-| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (cpr_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
- [ /3 width=10 by cpr_conf_lpr_bind_bind/
- | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/
- | /3 width=11 by cpr_conf_lpr_bind_zeta/
- | /3 width=12 by cpr_conf_lpr_zeta_zeta/
- ]
-| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #HX1 #H1
- | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
- | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
- ]
- elim (cpr_inv_flat1 … H2) -H2 *
- [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
- |2,6,10,14: #HX2 #H2
- |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
- |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
- ] destruct
- [ /3 width=10 by cpr_conf_lpr_flat_flat/
- | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/
- | /4 width=12 by ex2_commute, cpr_conf_lpr_flat_beta/
- | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/
- | /3 width=8 by cpr_conf_lpr_flat_tau/
- | /3 width=7 by cpr_conf_lpr_tau_tau/
- | /3 width=12 by cpr_conf_lpr_flat_beta/
- | /3 width=13 by cpr_conf_lpr_beta_beta/
- | /3 width=14 by cpr_conf_lpr_flat_theta/
- | /3 width=17 by cpr_conf_lpr_theta_theta/
- ]
-]
-qed-.
-
-(* Basic_1: includes: pr0_confluence pr2_confluence *)
-theorem cpr_conf: ∀G,L. confluent … (cpr G L).
-/2 width=6 by cpr_conf_lpr/ qed-.
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-lemma lpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) // -L0 /2 width=3 by ex2_intro/
-qed-.
-
-lemma lpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3 by ex2_intro/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lpr_conf: ∀G. confluent … (lpr G).
-/3 width=6 by lpx_sn_conf, cpr_conf_lpr/
-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/predsnstar_3.ma".
-include "basic_2/grammar/lpx_sn_tc.ma".
-include "basic_2/reduction/lpr.ma".
-
-(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
-
-definition lprs: relation3 genv lenv lenv ≝
- λG. TC … (lpr G).
-
-interpretation "parallel computation (local environment, sn variant)"
- 'PRedSnStar G L1 L2 = (lprs G L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma lprs_ind: ∀G,L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → R L → R L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L2.
-#G #L1 #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma lprs_ind_dx: ∀G,L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → R L → R L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡* L2 → R L1.
-#G #L2 #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpr_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
-/2 width=1/ qed.
-
-lemma lprs_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡* L.
-/2 width=1/ qed.
-
-lemma lprs_strap1: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
-/2 width=3/ qed.
-
-lemma lprs_strap2: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
-/2 width=3/ qed.
-
-lemma lprs_pair_refl: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡* L2.ⓑ{I}V.
-/2 width=1 by TC_lpx_sn_pair_refl/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lprs_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡* L2 → L2 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
-
-lemma lprs_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡* ⋆ → L1 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lprs_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → |L1| = |L2|.
-/2 width=2 by TC_lpx_sn_fwd_length/ 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/predsnstaralt_3.ma".
-include "basic_2/computation/cprs_cprs.ma".
-include "basic_2/computation/lprs.ma".
-
-(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
-
-(* alternative definition *)
-definition lprsa: relation3 genv lenv lenv ≝
- λG. lpx_sn … (cprs G).
-
-interpretation "parallel computation (local environment, sn variant) alternative"
- 'PRedSnStarAlt G L1 L2 = (lprsa G L1 L2).
-
-(* Main properties on the alternative definition ****************************)
-
-theorem lprsa_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
-/2 width=1 by lpx_sn_LTC_TC_lpx_sn/ qed-.
-
-(* Main inversion lemmas on the alternative definition **********************)
-
-theorem lprs_inv_lprsa: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡➡* L2.
-/3 width=3 by TC_lpx_sn_inv_lpx_sn_LTC, lpr_cprs_trans/ qed-.
-
-(* Alternative eliminators **************************************************)
-
-lemma lprs_ind_alt: ∀G. ∀R:relation lenv.
- R (⋆) (⋆) → (
- ∀I,K1,K2,V1,V2.
- ⦃G, K1⦄ ⊢ ➡* K2 → ⦃G, K1⦄ ⊢ V1 ➡* V2 →
- R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpr_cprs_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/cprs_cprs.ma".
-include "basic_2/computation/lprs.ma".
-
-(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma lprs_pair: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
- ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
-/2 width=1 by TC_lpx_sn_pair/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lprs_inv_pair1: ∀I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡* L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
- L2 = K2.ⓑ{I}V2.
-/3 width=3 by TC_lpx_sn_inv_pair1, lpr_cprs_trans/ qed-.
-
-lemma lprs_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡* K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
- L1 = K1.ⓑ{I}V1.
-/3 width=3 by TC_lpx_sn_inv_pair2, lpr_cprs_trans/ qed-.
-
-(* Properties on context-sensitive parallel computation for terms ***********)
-
-lemma lprs_cpr_trans: ∀G. s_r_trans … (cpr G) (lprs G).
-/3 width=5 by s_r_trans_TC2, lpr_cprs_trans/ qed-.
-
-(* Basic_1: was just: pr3_pr3_pr3_t *)
-lemma lprs_cprs_trans: ∀G. s_rs_trans … (cpr G) (lprs G).
-/3 width=5 by s_r_trans_TC1, lprs_cpr_trans/ qed-.
-
-lemma lprs_cprs_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #H elim H -L1
-[ #L1 #HL01
- elim (cprs_lpr_conf_dx … HT01 … HL01) -L0 /2 width=3 by ex2_intro/
-| #L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
- elim (cprs_lpr_conf_dx … HT1 … HL1) -HT1 #T2 #HT2 #HT12
- elim (cprs_lpr_conf_dx … HT0 … HL1) -L #T3 #HT3 #HT03
- elim (cprs_conf … HT2 … HT3) -T #T #HT2 #HT3
- lapply (cprs_trans … HT03 … HT3) -T3
- lapply (cprs_trans … HT12 … HT2) -T2 /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma lprs_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-/3 width=3 by lprs_cprs_conf_dx, cpr_cprs/ qed-.
-
-lemma lprs_cprs_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01 #T #HT1
-lapply (lprs_cprs_trans … HT1 … HL01) -HT1 /2 width=3 by ex2_intro/
-qed-.
-
-lemma lprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-/3 width=3 by lprs_cprs_conf_sn, cpr_cprs/ qed-.
-
-lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-#G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
-lapply (lprs_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1 by lprs_pair, cprs_bind/
-qed.
-
-(* Inversion lemmas on context-sensitive parallel computation for terms *****)
-
-(* Basic_1: was: pr3_gen_abst *)
-lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
- U2 = ⓛ{a}W2.T2.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
-#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
-elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
-lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?) /2 width=1 by lprs_pair/ -HT02 #HT02
-lapply (cprs_strap1 … HV10 … HV02) -V0
-lapply (cprs_trans … HT10 … HT02) -T0 /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
- ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
-#a #G #L #W1 #W2 #T1 #T2 #H
-elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
-qed-.
-
-(* Basic_1: was pr3_gen_abbr *)
-lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
- U2 = ⓓ{a}V2.T2
- ) ∨
- ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
-#U0 #U2 #_ #HU02 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_abbr1 … HU02) -HU02 *
- [ #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) /2 width=1 by lprs_pair/ -HT02 #HT02
- lapply (cprs_strap1 … HV10 … HV02) -V0
- lapply (cprs_trans … HT10 … HT02) -T0 /3 width=5 by ex3_2_intro, or_introl/
- | #T2 #HT02 #HUT2
- lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02 /2 width=1 by lprs_pair/ -V0 #HT02
- lapply (cprs_trans … HT10 … HT02) -T0 /3 width=3 by ex3_intro, or_intror/
- ]
-| #U1 #HTU1 #HU01
- elim (lift_total U2 0 1) #U #HU2
- lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
- /4 width=3 by cprs_strap1, ldrop_drop, ex3_intro, or_intror/
-]
-qed-.
-
-(* More advanced properties *************************************************)
-
-lemma lprs_pair2: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
- ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
-/3 width=3 by lprs_pair, lprs_cprs_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/reduction/lpr_ldrop.ma".
-include "basic_2/computation/lprs.ma".
-
-(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma lprs_ldrop_conf: ∀G. dropable_sn (lprs G).
-/3 width=3 by dropable_sn_TC, lpr_ldrop_conf/ qed-.
-
-lemma ldrop_lprs_trans: ∀G. dedropable_sn (lprs G).
-/3 width=3 by dedropable_sn_TC, ldrop_lpr_trans/ qed-.
-
-lemma lprs_ldrop_trans_O1: ∀G. dropable_dx (lprs G).
-/3 width=3 by dropable_dx_TC, lpr_ldrop_trans_O1/ 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/lpr_lpr.ma".
-include "basic_2/computation/lprs.ma".
-
-(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma lprs_strip: ∀G. confluent2 … (lprs G) (lpr G).
-/3 width=3 by TC_strip1, lpr_conf/ qed-.
-
-(* Main properties **********************************************************)
-
-theorem lprs_conf: ∀G. confluent2 … (lprs G) (lprs G).
-/3 width=3 by TC_confluent2, lpr_conf/ qed-.
-
-theorem lprs_trans: ∀G. Transitive … (lprs G).
-/2 width=3/ 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/grammar/lenv_length.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-inductive lpx_sn (R:lenv→relation term): relation lenv ≝
-| lpx_sn_atom: lpx_sn R (⋆) (⋆)
-| lpx_sn_pair: ∀I,K1,K2,V1,V2.
- lpx_sn R K1 K2 → R K1 V1 V2 →
- lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_sn_refl: ∀R. (∀L. reflexive ? (R L)) → reflexive … (lpx_sn R).
-#R #HR #L elim L -L /2 width=1 by lpx_sn_atom, lpx_sn_pair/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #L1 #L2 * -L1 -L2
-[ #J #K1 #V1 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 →
- ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
-#R #L1 #L2 * -L1 -L2
-[ //
-| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-#R #L1 #L2 * -L1 -L2
-[ #J #K2 #V2 #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) →
- ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-lemma lpx_sn_inv_pair: ∀R,I1,I2,L1,L2,V1,V2.
- lpx_sn R (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
- ∧∧ lpx_sn R L1 L2 & R L1 V1 V2 & I1 = I2.
-#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lpx_sn_inv_pair1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L1 -L2 normalize //
-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/grammar/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-definition lpx_sn_confluent: relation (lenv→relation term) ≝ λR1,R2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-definition lpx_sn_transitive: relation (lenv→relation term) ≝ λR1,R2.
- ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
- ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
-
-(* Main properties **********************************************************)
-
-theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
-#R #HR #L1 #L #H elim H -L1 -L //
-#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
-elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5/
-qed-.
-
-theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
- confluent2 … (lpx_sn R1) (lpx_sn R2).
-#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #n #IH *
-[ #_ #X1 #H1 #X2 #H2 -n
- >(lpx_sn_inv_atom1 … H1) -X1
- >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3/
-| #L0 #I #V0 #Hn #X1 #H1 #X2 #H2 destruct
- elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
- elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
- elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5/
-]
-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/grammar/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* Properties on transitive_closure *****************************************)
-
-lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
-#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
-[ /2 width=1 by lpx_sn_refl/
-| /3 width=1 by TC_reflexive, lpx_sn_refl/
-| /3 width=5 by lpx_sn_pair, step/
-]
-qed-.
-
-lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀V1,V2. LTC … R L1 V1 V2 →
- TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
-#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
-[ /2 width=1 by TC_lpx_sn_pair_refl/
-| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
-]
-qed-.
-
-lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
- TC … (lpx_sn R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
-qed-.
-
-(* Inversion lemmas on transitive closure ***********************************)
-
-lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
-#R #L1 #H @(TC_ind_dx … L1 H) -L1
-[ /2 width=2 by lpx_sn_inv_atom2/
-| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
-[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
-| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
- elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
- lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_ind: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀S:relation lenv.
- S (⋆) (⋆) → (
- ∀I,K1,K2,V1,V2.
- TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
- S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
-#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
-[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
-| #L2 #I #V2 #IHL2 #X #H
- elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
- #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
-#R #L2 #H elim H -L2
-[ /2 width=2 by lpx_sn_inv_atom1/
-| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
-]
-qed-.
-
-fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
-[ #J #K #W #H destruct
-| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
-
-lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- lpx_sn (LTC … R) L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
-
-(* Forward lemmas on transitive closure *************************************)
-
-lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L2
-[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
-| #L #L2 #_ #HL2 #IHL1
- >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
-]
-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 G , break term 46 L1 ⦄ ⊢ ➡ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'PRedSn $G $L1 $L2 }.
+++ /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 G , break term 46 L1 ⦄ ⊢ ➡* break term 46 L2 )"
- non associative with precedence 45
- for @{ 'PRedSnStar $G $L1 $L2 }.
+++ /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 G , break term 46 L1 ⦄ ⊢ ➡ ➡ * break term 46 L2 )"
- non associative with precedence 45
- for @{ 'PRedSnStarAlt $G $L1 $L2 }.
--- /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/grammar/lenv_length.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+inductive lpx_sn (R:lenv→relation term): relation lenv ≝
+| lpx_sn_atom: lpx_sn R (⋆) (⋆)
+| lpx_sn_pair: ∀I,K1,K2,V1,V2.
+ lpx_sn R K1 K2 → R K1 V1 V2 →
+ lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
+.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_refl: ∀R. (∀L. reflexive ? (R L)) → reflexive … (lpx_sn R).
+#R #HR #L elim L -L /2 width=1 by lpx_sn_atom, lpx_sn_pair/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #L1 #L2 * -L1 -L2
+[ #J #K1 #V1 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 →
+ ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #L1 #L2 * -L1 -L2
+[ #J #K2 #V2 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) →
+ ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+lemma lpx_sn_inv_pair: ∀R,I1,I2,L1,L2,V1,V2.
+ lpx_sn R (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
+ ∧∧ lpx_sn R L1 L2 & R L1 V1 V2 & I1 = I2.
+#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lpx_sn_inv_pair1 … H) -H
+#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L1 -L2 normalize //
+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/grammar/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+definition lpx_sn_confluent: relation (lenv→relation term) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+definition lpx_sn_transitive: relation (lenv→relation term) ≝ λR1,R2.
+ ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
+ ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
+#R #HR #L1 #L #H elim H -L1 -L //
+#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
+elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5/
+qed-.
+
+theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
+ confluent2 … (lpx_sn R1) (lpx_sn R2).
+#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #n #IH *
+[ #_ #X1 #H1 #X2 #H2 -n
+ >(lpx_sn_inv_atom1 … H1) -X1
+ >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3/
+| #L0 #I #V0 #Hn #X1 #H1 #X2 #H2 destruct
+ elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
+ elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
+ elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5/
+]
+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/grammar/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* Properties on transitive_closure *****************************************)
+
+lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
+#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
+[ /2 width=1 by lpx_sn_refl/
+| /3 width=1 by TC_reflexive, lpx_sn_refl/
+| /3 width=5 by lpx_sn_pair, step/
+]
+qed-.
+
+lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀V1,V2. LTC … R L1 V1 V2 →
+ TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
+#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
+[ /2 width=1 by TC_lpx_sn_pair_refl/
+| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
+]
+qed-.
+
+lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
+ TC … (lpx_sn R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
+qed-.
+
+(* Inversion lemmas on transitive closure ***********************************)
+
+lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
+#R #L1 #H @(TC_ind_dx … L1 H) -L1
+[ /2 width=2 by lpx_sn_inv_atom2/
+| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
+[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
+| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
+ elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
+ lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_ind: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀S:relation lenv.
+ S (⋆) (⋆) → (
+ ∀I,K1,K2,V1,V2.
+ TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
+ S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
+#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
+[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
+| #L2 #I #V2 #IHL2 #X #H
+ elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
+ #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
+#R #L2 #H elim H -L2
+[ /2 width=2 by lpx_sn_inv_atom1/
+| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
+]
+qed-.
+
+fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
+[ #J #K #W #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
+
+lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ lpx_sn (LTC … R) L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
+
+(* Forward lemmas on transitive closure *************************************)
+
+lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L2
+[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+ >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
+]
+qed-.
c: conversion
d: decomposed extended reduction
e: decomposed extended conversion
-n: reduction for "big tree" normal forms
q: restricted reduction
r: reduction
s: substitution
+++ /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 G , break term 46 L1 ⦄ ⊢ ➡ break [ term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyPRedSn $G $L1 $L2 $T $d }.
+++ /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 G , break term 46 L1 ⦄ ⊢ ➡* break [ term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyPRedSnStar $G $L1 $L2 $T $d }.
--- /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 G , break term 46 L1 ⦄ ⊢ ➡ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedSn $G $L1 $L2 }.
--- /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 G , break term 46 L1 ⦄ ⊢ ➡* break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedSnStar $G $L1 $L2 }.
--- /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 G , break term 46 L1 ⦄ ⊢ ➡ ➡ * break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedSnStarAlt $G $L1 $L2 }.
(* Properties on lazy sn pointwise extensions *******************************)
-lemma llpx_sn_cpr_conf: ∀R. (∀L. reflexive … (R L)) → l_liftable R → l_deliftable_sn R →
+lemma cpr_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → l_liftable R → l_deliftable_sn R →
∀G. s_r_confluent1 … (cpr G) (llpx_sn R 0).
#R #H1R #H2R #H3R #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
[ //
]
qed-.
-lemma lleq_cpx_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+lemma cpx_lleq_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
∀L1. L2 ⋕[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
-lemma lleq_cpx_conf_sn: ∀h,g,G. s_r_confluent1 … (cpx h g G) (lleq 0).
-/3 width=6 by llpx_sn_cpx_conf, lift_mono, ex2_intro/ qed-.
+lemma cpx_lleq_conf_sn: ∀h,g,G. s_r_confluent1 … (cpx h g G) (lleq 0).
+/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
-lemma lleq_cpx_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+lemma cpx_lleq_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
∀L1. L1 ⋕[T1, 0] L2 → L1 ⋕[T2, 0] L2.
-/4 width=6 by lleq_cpx_conf_sn, lleq_sym/ qed-.
+/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.
(* Properties on lazy sn pointwise extensions *******************************)
(* Note: lemma 1000 *)
-lemma llpx_sn_cpx_conf: ∀R. (∀L. reflexive … (R L)) → l_liftable R → l_deliftable_sn R →
+lemma cpx_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → l_liftable R → l_deliftable_sn R →
∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx_sn R 0).
#R #H1R #H2R #H3R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
[ //
lemma cpr_fpb: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
/3 width=1 by fpb_cpx, cpr_cpx/ qed.
-
-lemma llpr_fpb: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡[T, 0] L2 → ⦃G, L1, T⦄ ≽[h, g] ⦃G, L2, T⦄.
+(*
+lamma llpr_fpb: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡[T, 0] L2 → ⦃G, L1, T⦄ ≽[h, g] ⦃G, L2, T⦄.
/3 width=1 by fpb_llpx, llpr_llpx/ 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/lazypredsn_5.ma".
-include "basic_2/relocation/llpx_sn.ma".
-include "basic_2/reduction/cpr.ma".
-
-(* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
-
-definition llpr: genv → relation4 ynat term lenv lenv ≝ λG. llpx_sn (cpr G).
-
-interpretation "lazy parallel reduction (local environment, sn variant)"
- 'LazyPRedSn G L1 L2 T d = (llpr G d T L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma llpr_inv_flat: ∀I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[ⓕ{I}V.T, d] L2 →
- ⦃G, L1⦄ ⊢ ➡[V, d] L2 ∧ ⦃G, L1⦄ ⊢ ➡[T, d] L2.
-/2 width=2 by llpx_sn_inv_flat/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma llpr_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → |L1| = |L2|.
-/2 width=4 by llpx_sn_fwd_length/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llpr_lref: ∀I,G,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 → ⦃G, L1⦄ ⊢ ➡[#i, d] L2.
-/2 width=9 by llpx_sn_lref/ qed.
-
-(* Note: lemma 250 *)
-lemma llpr_refl: ∀G,T,d. reflexive … (llpr G d T).
-/2 width=1 by llpx_sn_refl/ qed.
-
-(* Basic_1: removed theorems 5: wcpr0_gen_sort wcpr0_gen_head
- wcpr0_getl wcpr0_getl_back
- pr0_subst1_back
-*)
+++ /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/relocation/fquq_alt.ma".
-include "basic_2/reduction/cpr_lift.ma".
-include "basic_2/reduction/cpr_llpx_sn.ma".
-include "basic_2/reduction/llpr.ma".
-
-(* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma llpr_inv_lref_ge_dx: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
- ∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
-/2 width=5 by llpx_sn_inv_lref_ge_dx/ qed-.
-
-lemma llpr_inv_lref_ge_sn: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
- ∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
-/2 width=5 by llpx_sn_inv_lref_ge_sn/ qed-.
-
-lemma llpr_inv_lref_ge_bi: ∀G,L1,L2,d,i. ⦃G, L1⦄ ⊢ ➡[#i, d] L2 → d ≤ i →
- ∀I1,I2,K1,K2,V1,V2.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2.
-/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
-
-lemma llpr_inv_bind_O: ∀a,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡ [ⓑ{a,I}V.T, 0] L2 →
- ⦃G, L1⦄ ⊢ ➡[V, 0] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_inv_bind_O/ qed-.
-
-lemma llpr_bind_repl_O: ∀I,G,L1,L2,V1,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[T, 0] L2.ⓑ{I}V2 →
- ∀J,W1,W2. ⦃G, L1⦄ ⊢ ➡[W1, 0] L2 → ⦃G, L1⦄ ⊢ W1 ➡ W2 → ⦃G, L1.ⓑ{J}W1⦄ ⊢ ➡[T, 0] L2.ⓑ{J}W2.
-/2 width=4 by llpx_sn_bind_repl_O/ qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma llpr_cpr_conf: ∀G. s_r_confluent1 … (cpr G) (llpr G 0).
-/3 width=10 by llpx_sn_cpr_conf, cpr_inv_lift1, cpr_lift/ qed-.
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=10 by llpr_lref, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
-#G #L #K #U #T #e #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (e+1)) #T2 #HUT2
-lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
-qed-.
-
-lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=10 by llpr_lref, cpr_pair_sn, cpr_bind, cpr_flat, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ldrop_pair, ex3_2_intro/
-#G #L #K #U #T #e #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (e+1)) #T2 #HUT2
-lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
-qed-.
-
-lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[T1, 0] L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_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/relocation/llpx_sn_llpx_sn.ma".
-include "basic_2/substitution/fqup.ma".
-include "basic_2/reduction/llpr_ldrop.ma".
-
-(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Main properties on context-sensitive parallel reduction for terms ********)
-
-fact cpr_conf_llpr_atom_atom:
- ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
-/2 width=3 by cpr_atom, ex2_intro/ qed-.
-
-fact cpr_conf_llpr_atom_delta:
- ∀G,L0,i. (
- ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
- ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #V1 #HLK1 #HK01 #HV01
-elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_
-lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
-lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1))
-/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
-qed-.
-
-(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
-fact cpr_conf_llpr_delta_delta:
- ∀G,L0,i. (
- ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
- ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀KX,VX. ⇩[i] L0 ≡ KX.ⓓVX →
- ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[#i, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[#i, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (llpr_inv_lref_ge_sn … HL01 … HLK0) -HL01 // #K1 #W1 #HLK1 #HK01 #_
-lapply (ldrop_fwd_drop2 … HLK1) -W1 #HLK1
-elim (llpr_inv_lref_ge_sn … HL02 … HLK0) -HL02 // #K2 #W2 #HLK2 #HK02 #_
-lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
-lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_bind_bind:
- ∀a,I,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓑ{a,I}V0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T.
-#a #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
-elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
-/3 width=5 by llpr_bind_repl_O, cpr_bind, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_bind_zeta:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 →
- ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
-#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
-elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_zeta_zeta:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
- ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[+ⓓV0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
-#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_bind_O … HL01) -HL01 #H1V0 #H1T0
-elim (llpr_inv_bind_O … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=4 by llpr_bind_repl_O/ -L0 -T0 #T #HT1 #HT2
-elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1
-elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2
-lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_flat_flat:
- ∀I,G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓕ{I}V0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T.
-#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) //
-elim (IH … HT01 … HT02 … H1T0 … H2T0) /3 width=5 by cpr_flat, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_flat_tau:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0
-elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_tau_tau:
- ∀G,L0,V0,T0. (
- ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓝV0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
-#G #L0 #V0 #T0 #IH #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #_ #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #_ #H2T0
-elim (IH … HT01 … HT02 … H1T0 … H2T0) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
-qed-.
-
-fact cpr_conf_llpr_flat_beta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/ #W #HW1 #HW2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ (**) (* full auto not tried *)
-/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
-qed-.
-
-(* Basic-1: includes:
- pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
- pr0_cong_upsilon_cong pr0_cong_upsilon_delta
-*)
-fact cpr_conf_llpr_flat_theta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
- ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2
-elim (cpr_inv_abbr1 … H) -H *
-[ #W1 #T1 #HW01 #HT01 #H destruct
- elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
- elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0
- /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/
-| #T1 #HT01 #HXT1 #H destruct
- elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
- elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1
- /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/
-]
-qed-.
-
-fact cpr_conf_llpr_beta_beta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓛ{a}W0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1/ #W #HW1 #HW2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/
-lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/
-/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
-qed-.
-
-(* Basic_1: includes: pr0_upsilon_upsilon *)
-fact cpr_conf_llpr_theta_theta:
- ∀a,G,L0,V0,W0,T0. (
- ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
- ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[T, 0] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[T, 0] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
- ) →
- ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
- ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 →
- ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
- ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[ⓐV0.ⓓ{a}W0.T0, 0] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
-#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
-#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (llpr_inv_flat … HL01) -HL01 #H1V0 #HL01
-elim (llpr_inv_bind_O … HL01) -HL01 #H1W0 #H1T0
-elim (llpr_inv_flat … HL02) -HL02 #H2V0 #HL02
-elim (llpr_inv_bind_O … HL02) -HL02 #H2W0 #H2T0
-elim (IH … HV01 … HV02 … H1V0 … H2V0) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … H1W0 … H2W0) /2 width=1 by/
-elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=4 by llpr_bind_repl_O/ -L0 -V0 -W0 -T0
-elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by ldrop_drop/
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/
-/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
-qed-.
-
-theorem cpr_conf_llpr: ∀G. llpx_sn_confluent2 (cpr G) (cpr G).
-#G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
-[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_atom1 … H1) -H1
- elim (cpr_inv_atom1 … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpr_conf_llpr_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpr_conf_llpr_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /4 width=10 by ex2_commute, cpr_conf_llpr_atom_delta/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpr_conf_llpr_delta_delta/
- ]
-| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (cpr_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
- [ /3 width=10 by cpr_conf_llpr_bind_bind/
- | /4 width=11 by ex2_commute, cpr_conf_llpr_bind_zeta/
- | /3 width=11 by cpr_conf_llpr_bind_zeta/
- | /3 width=12 by cpr_conf_llpr_zeta_zeta/
- ]
-| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpr_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #HX1 #H1
- | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
- | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
- ]
- elim (cpr_inv_flat1 … H2) -H2 *
- [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
- |2,6,10,14: #HX2 #H2
- |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
- |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
- ] destruct
- [ /3 width=10 by cpr_conf_llpr_flat_flat/
- | /4 width=8 by ex2_commute, cpr_conf_llpr_flat_tau/
- | /4 width=12 by ex2_commute, cpr_conf_llpr_flat_beta/
- | /4 width=14 by ex2_commute, cpr_conf_llpr_flat_theta/
- | /3 width=8 by cpr_conf_llpr_flat_tau/
- | /3 width=8 by cpr_conf_llpr_tau_tau/
- | /3 width=12 by cpr_conf_llpr_flat_beta/
- | /3 width=13 by cpr_conf_llpr_beta_beta/
- | /3 width=14 by cpr_conf_llpr_flat_theta/
- | /3 width=17 by cpr_conf_llpr_theta_theta/
- ]
-]
-qed-.
-
-(* Basic_1: includes: pr0_confluence pr2_confluence *)
-theorem cpr_conf: ∀G,L. confluent … (cpr G L).
-/2 width=6 by cpr_conf_llpr/ qed-.
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-lemma llpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_llpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
-qed-.
-
-lemma llpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_llpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/
-qed-.
(**************************************************************************)
include "basic_2/notation/relations/lazypredsn_7.ma".
-include "basic_2/reduction/llpr.ma".
+include "basic_2/relocation/llpx_sn.ma".
include "basic_2/reduction/cpx.ma".
(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
lemma llpx_refl: ∀h,g,G,T,d. reflexive … (llpx h g G d T).
/2 width=1 by llpx_sn_refl/ qed.
-
-lemma llpr_llpx: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
-/3 width=3 by llpx_sn_co, cpr_cpx/ qed.
lemma aaa_cpr_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
/3 width=5 by aaa_cpx_conf, cpr_cpx/ qed-.
-
-lemma aaa_llpr_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+(*
+lamma aaa_llpr_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[T, 0] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
/3 width=5 by aaa_llpx_conf, llpr_llpx/ qed-.
+*)
(* Advanced properties ******************************************************)
lemma llpx_cpx_conf: ∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx h g G 0).
-/3 width=10 by llpx_sn_cpx_conf, cpx_inv_lift1, cpx_lift/ qed-.
+/3 width=10 by cpx_llpx_sn_conf, cpx_inv_lift1, cpx_lift/ qed-.
(* Inversion lemmas on relocation *******************************************)
--- /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/relocation/llpx_sn_lpx_sn.ma".
+include "basic_2/reduction/lpr.ma".
+include "basic_2/reduction/llpx.ma".
+
+(* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
+
+(* Properties on sn parallel reduction **************************************)
+
+(* Note: this should be moved *)
+lemma lpr_llpx: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
+/3 width=4 by cpr_cpx, lpx_sn_llpx_sn, llpx_sn_co/ 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/predsn_3.ma".
+include "basic_2/grammar/lpx_sn.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+definition lpr: relation3 genv lenv lenv ≝ λG. lpx_sn (cpr G).
+
+interpretation "parallel reduction (local environment, sn variant)"
+ 'PRedSn G L1 L2 = (lpr G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: includes: wcpr0_gen_sort *)
+lemma lpr_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡ L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+(* Basic_1: includes: wcpr0_gen_head *)
+lemma lpr_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L2 = K2.ⓑ{I}V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpr_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡ ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpr_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Note: lemma 250 *)
+lemma lpr_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡ L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpr_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡ K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 →
+ ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ K2.ⓑ{I}V2.
+/2 width=1 by lpx_sn_pair/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpr_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
+
+(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
+ pr0_subst1_back
+*)
--- /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/relocation/fquq_alt.ma".
+include "basic_2/relocation/ldrop_lpx_sn.ma".
+include "basic_2/reduction/cpr_lift.ma".
+include "basic_2/reduction/lpr.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Properies on local environment slicing ***********************************)
+
+(* Basic_1: includes: wcpr0_drop *)
+lemma lpr_ldrop_conf: ∀G. dropable_sn (lpr G).
+/3 width=6 by lpx_sn_deliftable_dropable, cpr_inv_lift1/ qed-.
+
+(* Basic_1: includes: wcpr0_drop_back *)
+lemma ldrop_lpr_trans: ∀G. dedropable_sn (lpr G).
+/3 width=10 by lpx_sn_liftable_dedropable, cpr_lift/ qed-.
+
+lemma lpr_ldrop_trans_O1: ∀G. dropable_dx (lpr G).
+/2 width=3 by lpx_sn_dropable/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
+#G #L #K #U #T #e #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (e+1)) #T2 #HUT2
+lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+qed-.
+
+lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
+#G #L #K #U #T #e #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (e+1)) #T2 #HUT2
+lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+qed-.
+
+lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+ ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fqu_lpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ➡ K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡ K1 & ⦃G1, L1⦄ ⊢ T1 ➡ T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpr_pair, ex3_2_intro/
+[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpr_inv_pair1 … H) -H
+ #K2 #W2 #HLK2 #HVW2 #H destruct
+ /3 width=5 by fqu_fquq, cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
+| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #K2 #HK12
+ elim (ldrop_lpr_trans … HLK1 … HK12) -HK12
+ /3 width=7 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fquq_lpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ➡ K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡ K1 & ⦃G1, L1⦄ ⊢ T1 ➡ T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_lpr_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_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/grammar/lpx_sn_lpx_sn.ma".
+include "basic_2/substitution/fqup.ma".
+include "basic_2/reduction/lpr_ldrop.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Main properties on context-sensitive parallel reduction for terms ********)
+
+fact cpr_conf_lpr_atom_atom:
+ ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
+/2 width=3 by cpr_atom, ex2_intro/ qed-.
+
+fact cpr_conf_lpr_atom_delta:
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
+ ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1))
+/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
+qed-.
+
+(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
+fact cpr_conf_lpr_delta_delta:
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
+ ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[i] L0 ≡ KX.ⓓVX →
+ ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_drop2 … HLK1) -W1 #HLK1
+elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_bind_bind:
+ ∀a,I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T.
+#a #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
+/3 width=5 by lpr_pair, cpr_bind, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_bind_zeta:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, ldrop_drop, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_zeta_zeta:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by ldrop_drop/ #T1 #HT1 #HXT1
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by ldrop_drop/ #T2 #HT2 #HXT2
+lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_flat:
+ ∀I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T.
+#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) /3 width=5 by cpr_flat, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_tau:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3 by cpr_tau, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_tau_tau:
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_beta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ (**) (* full auto not tried *)
+/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
+qed-.
+
+(* Basic-1: includes:
+ pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
+ pr0_cong_upsilon_cong pr0_cong_upsilon_delta
+*)
+fact cpr_conf_lpr_flat_theta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/ #HU2
+elim (cpr_inv_abbr1 … H) -H *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+ elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
+ elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
+ /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/
+| #T1 #HT01 #HXT1 #H destruct
+ elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+ elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1
+ /4 width=9 by cpr_flat, cpr_zeta, ldrop_drop, lift_flat, ex2_intro/
+]
+qed-.
+
+fact cpr_conf_lpr_beta_beta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/
+/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+(* Basic_1: was: pr0_upsilon_upsilon *)
+fact cpr_conf_lpr_theta_theta:
+ ∀a,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃G, L, T⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+ ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+ ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
+ ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by ldrop_drop/
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by ldrop_drop/
+/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+theorem cpr_conf_lpr: ∀G. lpx_sn_confluent (cpr G) (cpr G).
+#G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
+[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_atom1 … H1) -H1
+ elim (cpr_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_conf_lpr_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_conf_lpr_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpr_conf_lpr_delta_delta/
+ ]
+| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_bind1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #T1 #HT01 #HXT1 #H11 #H12
+ ]
+ elim (cpr_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #T2 #HT02 #HXT2 #H21 #H22
+ ] destruct
+ [ /3 width=10 by cpr_conf_lpr_bind_bind/
+ | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/
+ | /3 width=11 by cpr_conf_lpr_bind_zeta/
+ | /3 width=12 by cpr_conf_lpr_zeta_zeta/
+ ]
+| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpr_inv_flat1 … H1) -H1 *
+ [ #V1 #T1 #HV01 #HT01 #H1
+ | #HX1 #H1
+ | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
+ | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+ ]
+ elim (cpr_inv_flat1 … H2) -H2 *
+ [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+ |2,6,10,14: #HX2 #H2
+ |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
+ |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
+ ] destruct
+ [ /3 width=10 by cpr_conf_lpr_flat_flat/
+ | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/
+ | /4 width=12 by ex2_commute, cpr_conf_lpr_flat_beta/
+ | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/
+ | /3 width=8 by cpr_conf_lpr_flat_tau/
+ | /3 width=7 by cpr_conf_lpr_tau_tau/
+ | /3 width=12 by cpr_conf_lpr_flat_beta/
+ | /3 width=13 by cpr_conf_lpr_beta_beta/
+ | /3 width=14 by cpr_conf_lpr_flat_theta/
+ | /3 width=17 by cpr_conf_lpr_theta_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: includes: pr0_confluence pr2_confluence *)
+theorem cpr_conf: ∀G,L. confluent … (cpr G L).
+/2 width=6 by cpr_conf_lpr/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma lpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
+qed-.
+
+lemma lpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lpr_conf: ∀G. confluent … (lpr G).
+/3 width=6 by lpx_sn_conf, cpr_conf_lpr/
+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/grammar/lpx_sn.ma".
+include "basic_2/relocation/ldrop_leq.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Properties on sn pointwise extension *************************************)
+
+lemma lpx_sn_ldrop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
+ ∀I,K1,V1,i. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
+ [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
+ | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+ /3 width=5 by ldrop_drop_lt, ex3_2_intro/
+ ]
+]
+qed-.
+
+lemma lpx_sn_ldrop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
+ ∀I,K2,V2,i. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (ldrop_inv_O1_pair1 … H) * -H
+ [ -IHK12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ex3_2_intro/
+ | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+ /3 width=5 by ldrop_drop_lt, ex3_2_intro/
+ ]
+]
+qed-.
+
+lemma lpx_sn_deliftable_dropable: ∀R. l_deliftable_sn R → dropable_sn (lpx_sn R).
+#R #HR #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
+ /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
+| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct
+ /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
+| #I #L1 #K1 #V1 #e #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 /3 width=3 by ldrop_drop, ex2_intro/
+| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (HR … HV12 … HLK1 … HWV1) -V1
+ elim (IHLK1 … HL12) -L1 /3 width=5 by ldrop_skip, lpx_sn_pair, ex2_intro/
+]
+qed-.
+
+lemma lpx_sn_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) →
+ l_liftable R → dedropable_sn (lpx_sn R).
+#R #H1R #H2R #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
+ /4 width=4 by ldrop_atom, lpx_sn_atom, ex3_intro/
+| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+ #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (lpx_sn_fwd_length … HK12)
+ #H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
+ /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
+ @leq_O2 normalize //
+| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
+ /3 width=5 by ldrop_drop, leq_pair, lpx_sn_pair, ex3_intro/
+| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (lift_total W2 d e) #V2 #HWV2
+ lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
+ elim (IHLK1 … HK12) -K1
+ /3 width=6 by ldrop_skip, leq_succ, lpx_sn_pair, ex3_intro/
+]
+qed-.
+
+fact lpx_sn_dropable_aux: ∀R,L2,K2,s,d,e. ⇩[s, d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
+ d = 0 → ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & lpx_sn R K1 K2.
+#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
+[ #d #e #He #X #H >(lpx_sn_inv_atom2 … H) -H
+ /4 width=3 by ldrop_atom, lpx_sn_atom, ex2_intro/
+| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
+ #K1 #V1 #HK12 #HV12 #H destruct
+ /3 width=5 by ldrop_pair, lpx_sn_pair, ex2_intro/
+| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
+ #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12) -L2 /3 width=3 by ldrop_drop, ex2_intro/
+| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
+ <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
+/2 width=5 by lpx_sn_dropable_aux/ 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/relocation/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-definition llpx_sn_confluent2: relation (lenv→relation term) ≝ λR1,R2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. llpx_sn R1 0 T0 L0 L1 → ∀L2. llpx_sn R2 0 T0 L0 L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-(* Note: we miss llpx_sn_conf and derivatives: lpr_conf lprs_conf *)
--- /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/relocation/ldrop_lpx_sn.ma".
+include "basic_2/relocation/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on pointwise extensions ***************************************)
+
+lemma lpx_sn_llpx_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀T,L1,L2,d. lpx_sn R L1 L2 → llpx_sn R d T L1 L2.
+#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#n #IH #L1 * *
+[ -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_sort/
+| -HR #i elim (lt_or_ge i (|L1|))
+ [2: -IH /4 width=4 by lpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux/ ]
+ #Hi #Hn #L2 #d elim (ylt_split i d)
+ [ -n /3 width=2 by llpx_sn_skip, lpx_sn_fwd_length/ ]
+ #Hdi #HL12 elim (ldrop_O1_lt L1 i) //
+ #I #K1 #V1 #HLK1 elim (lpx_sn_ldrop_conf … HL12 … HLK1) -HL12
+ /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
+| -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_gref/
+| /4 width=1 by llpx_sn_bind, lpx_sn_pair/
+| -HR /3 width=1 by llpx_sn_flat/
+]
+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/relocation/llpx_sn_ldrop.ma".
+include "basic_2/static/ssta.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ******************************)
+
+(* Properties on lazy sn pointwise extensions *******************************)
+
+lemma ssta_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → l_liftable R →
+ ∀h,g,G. s_r_confluent1 … (ssta h g G) (llpx_sn R 0).
+#R #H1R #H2R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
+[ /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
+| #G #Ls #Ks #V1s #W2s #V2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1d #HLKd #HV1s #HV1sd
+ lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
+ lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
+ @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| #G #Ls #Ks #V1s #W1s #l #i #HLKs #Hl #HVW1s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1d #HLKd #HV1s #HV1sd
+ lapply (ldrop_fwd_drop2 … HLKs) -HLKs #HLKs
+ lapply (ldrop_fwd_drop2 … HLKd) -HLKd #HLKd
+ @(llpx_sn_lift_le … HLKs HLKd … HVW1s) -HLKs -HLKd -HVW1s /2 width=1 by/ (**) (* full auto too slow *)
+| #a #I #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+ /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
+]
+qed-.
}
]
[ { "strongly normalizing extended computation" * } {
- [ "lcosx ( ? ⊢ ⧤⬊*[?,?,?] ? )" "lcosx_cpxs" * ]
- [ "lsx ( ? ⊢ ⋕⬊*[?,?,?,?] ? )" "lsx_ldrop" + "lsx_cpxs" + "lsx_csx" * ]
+(* [ "lcosx ( ? ⊢ ⧤⬊*[?,?,?] ? )" "lcosx_cpxs" * ] *)
+ [ "llsx ( ? ⊢ ⋕⬊*[?,?,?,?] ? )" "llsx_alt ( ? ⊢ ⋕⬊⬊*[?,?,?,?] ? )" "llsx_ldrop" + "llsx_llpx" + "llsx_llpxs" + "llsx_csx" * ]
[ "csx_vector ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csx_tstc_vector" + "csx_aaa" * ]
[ "csx ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csx_alt ( ⦃?,?⦄ ⊢ ⬊⬊*[?,?] ? )" "csx_lift" + "csx_lpx" + "csx_lpxs" + "csx_fpbs" * ]
}
]
[ { "\"big tree\" parallel computation" * } {
- [ "fpbg ( ⦃?,?,?⦄ >⋕[?,?] ⦃?,?,?⦄ )" "fpbg_lift" + "fpbg_fpns" + "fpbg_fpbg" * ]
- [ "fpbc ( ⦃?,?,?⦄ ≻⋕[?,?] ⦃?,?,?⦄ )" "fpbc_fpns" + "fpbc_fpbs" * ]
- [ "fpbu ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpbu_lift" + "fpbu_fpns" * ]
- [ "fpbs ( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_alt ( ⦃?,?,?⦄ ≥≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_aaa" + "fpbs_fpns" + "fpbs_fpbs" + "fpbs_ext" * ]
- }
- ]
- [ { "parallel computation for \"big tree\" normal forms" * } {
- [ "fpns ( ⦃?,?,?⦄ ⊢ ⋕➡*[?,?] ⦃?,?,?⦄ )" "fpns_fpns" * ]
+ [ "fpbg ( ⦃?,?,?⦄ >⋕[?,?] ⦃?,?,?⦄ )" "fpbg_lift" + "fpbg_fleq" + "fpbg_fpbg" * ]
+ [ "fpbc ( ⦃?,?,?⦄ ≻⋕[?,?] ⦃?,?,?⦄ )" "fpbc_fleq" + "fpbc_fpbs" * ]
+ [ "fpbu ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpbu_lift" + "fpbu_lleq" "fpbu_fleq" * ]
+ [ "fpbs ( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_alt ( ⦃?,?,?⦄ ≥≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_fleq" + "fpbs_aaa" + "fpbs_lpr" + "fpbs_fpbs" + "fpbs_ext" * ]
}
]
[ { "decomposed extended computation" * } {
}
]
[ { "context-sensitive extended computation" * } {
- [ "lpxs ( ⦃?,?⦄ ⊢ ➡*[?,?] ? )" "lpxs_alt ( ⦃?,?⦄ ⊢ ➡➡*[?,?] ? )" "lpxs_ldrop" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ]
+ [ "llpxs ( ⦃?,?⦄ ⊢ ➡*[?,?,?,?] ? )" "llpxs_lleq" + "llpxs_aaa" + "llpxs_lprs" "llpxs_cpxs" + "llpxs_llpxs" * ]
+(* [ "lpxs ( ⦃?,?⦄ ⊢ ➡*[?,?] ? )" "lpxs_alt ( ⦃?,?⦄ ⊢ ➡➡*[?,?] ? )" "lpxs_ldrop" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ] *)
[ "cpxs ( ⦃?,?⦄ ⊢ ? ➡*[?,?] ? )" "cpxs_tstc" + "cpxs_tstc_vector" + "cpxs_leq" + "cpxs_lift" + "cpxs_lleq" + "cpxs_aaa" + "cpxs_cpxs" * ]
}
]
}
]
[ { "context-sensitive extended reduction" * } {
- [ "lpx ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lpx_ldrop" + "lpx_lleq" + "lpx_aaa" * ]
+ [ "llpx ( ⦃?,?⦄ ⊢ ➡[?,?,?,?] ? )" "llpx_ldrop" + "llpx_lleq" + "llpx_aaa" + "llpx_lpr" * ]
+(* [ "lpx ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lpx_ldrop" + "lpx_lleq" + "lpx_aaa" * ] *)
[ "cpx ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpx_leq" + "cpx_lift" + "cpx_llpx_sn" + "cpx_lleq" + "cpx_cix" * ]
}
]
]
[ { "context-sensitive reduction" * } {
[ "lpr ( ⦃?,?⦄ ⊢ ➡ ? )" "lpr_ldrop" + "lpr_lpr" * ]
- [ "cpr ( ⦃?,?⦄ ⊢ ? ➡ ? )" "cpr_lift" + "cpr_cir" * ]
+ [ "cpr ( ⦃?,?⦄ ⊢ ? ➡ ? )" "cpr_lift" + "cpr_llpx_sn" + "cpr_cir" * ]
}
]
[ { "irreducible forms for context-sensitive reduction" * } {
}
]
[ { "stratified static type assignment" * } {
- [ "ssta ( ⦃?,?⦄ ⊢ ? •[?,?] ? )" "ssta_lift" + "ssta_ssta" * ]
+ [ "ssta ( ⦃?,?⦄ ⊢ ? •[?,?] ? )" "ssta_lift" + "ssta_lpx_sn" + "ssta_ssta" * ]
}
]
[ { "local env. ref. for degree assignment" * } {
]
class "yellow"
[ { "substitution" * } {
+ [ { "lazy equivalence" * } {
+ [ "fleq ( ⦃?,?,?⦄ ⋕[?] ⦃?,?,?⦄ )" "fleq_fleq" * ]
+ [ "lleq ( ? ⋕[?,?] ? )" "lleq_leq" + "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" * ]
+ }
+ ]
[ { "iterated structural successor for closures" * } {
- [ "fqus ( ⦃?,?,?⦄ ⊃* ⦃?,?,?⦄ )" "fqus_alt" + "fqus_lleq" + "fqus_fqus" * ]
- [ "fqup ( ⦃?,?,?⦄ ⊃+ ⦃?,?,?⦄ )" "fqup_lleq" + "fqup_fqup" * ]
+ [ "fqus ( ⦃?,?,?⦄ ⊃* ⦃?,?,?⦄ )" "fqus_alt" + "fqus_fqus" * ]
+ [ "fqup ( ⦃?,?,?⦄ ⊃+ ⦃?,?,?⦄ )" "fqup_fqup" * ]
}
]
[ { "iterated local env. slicing" * } {
class "orange"
[ { "relocation" * } {
[ { "structural successor for closures" * } {
- [ "fquq ( ⦃?,?,?⦄ ⊃⸮ ⦃?,?,?⦄ )" "fquq_alt ( ⦃?,?,?⦄ ⊃⊃⸮ ⦃?,?,?⦄ )" "fquq_lleq" * ]
- [ "fqu ( ⦃?,?,?⦄ ⊃ ⦃?,?,?⦄ )" "fqu_lleq" * ]
+ [ "fquq ( ⦃?,?,?⦄ ⊃⸮ ⦃?,?,?⦄ )" "fquq_alt ( ⦃?,?,?⦄ ⊃⊃⸮ ⦃?,?,?⦄ )" * ]
+ [ "fqu ( ⦃?,?,?⦄ ⊃ ⦃?,?,?⦄ )" * ]
}
]
[ { "global env. slicing" * } {
[ "gdrop ( ⇩[?] ? ≡ ? )" "gdrop_gdrop" * ]
}
]
- [ { "lazy equivalence for local environments" * } {
- [ "lleq ( ? ⋕[?,?] ? )" "lleq_leq" + "lleq_ldrop" + "lleq_lleq" * ]
- }
- ]
[ { "lazy pointwise extension of a relation" * } {
- [ "llpx_sn" "llpx_sn_leq" + "llpx_sn_ldrop" + "llpx_sn_llpx_sn" * ]
+ [ "llpx_sn" "llpx_sn_tc" + "llpx_sn_leq" + "llpx_sn_ldrop" + "llpx_sn_lpx_sn" * ]
}
- ]
- [ { "basic local env. slicing" * } {
+ ]
+ [ { "basic local env. slicing" * } {
[ "ldrop ( ⇩[?,?,?] ? ≡ ? )" "ldrop_lpx_sn" + "ldrop_leq" + "ldrop_ldrop" * ]
}
]
projects / helm.git / commitdiff