(* *)
(**************************************************************************)
-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.
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