(* *)
(**************************************************************************)
-include "basic_2/reducibility/cnf.ma".
-include "basic_2/computation/tprs.ma".
+include "basic_2/notation/relations/predstar_4.ma".
+include "basic_2/reduction/cnr.ma".
(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-definition cprs: lenv → relation term ≝
- λL. TC … (cpr L).
+(* Basic_1: includes: pr1_pr0 *)
+definition cprs: relation4 genv lenv term term ≝
+ λG. LTC … (cpr G).
interpretation "context-sensitive parallel computation (term)"
- 'PRedStar L T1 T2 = (cprs L T1 T2).
+ 'PRedStar G L T1 T2 = (cprs G L T1 T2).
(* Basic eliminators ********************************************************)
-lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 →
- (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) →
- ∀T2. L ⊢ T1 ➡* T2 → R T2.
-#L #T1 #R #HT1 #IHT1 #T2 #HT12
+lemma cprs_ind: ∀G,L,T1. ∀R:predicate term. R T1 →
+ (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → R T → R T2) →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T2.
+#G #L #T1 #R #HT1 #IHT1 #T2 #HT12
@(TC_star_ind … HT1 IHT1 … HT12) //
qed-.
-lemma cprs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → R T → R T1) →
- ∀T1. L ⊢ T1 ➡* T2 → R T1.
-#L #T2 #R #HT2 #IHT2 #T1 #HT12
+lemma cprs_ind_dx: ∀G,L,T2. ∀R:predicate term. R T2 →
+ (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → R T → R T1) →
+ ∀T1. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T1.
+#G #L #T2 #R #HT2 #IHT2 #T1 #HT12
@(TC_star_ind_dx … HT2 IHT2 … HT12) //
qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: pr3_pr2 *)
-lemma cpr_cprs: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ➡* T2.
-/2 width=1/ qed.
+lemma cpr_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+/2 width=1 by inj/ qed.
(* Basic_1: was: pr3_refl *)
-lemma cprs_refl: ∀L,T. L ⊢ T ➡* T.
-/2 width=1/ qed.
+lemma cprs_refl: ∀G,L,T. ⦃G, L⦄ ⊢ T ➡* T.
+/2 width=1 by cpr_cprs/ qed.
-lemma cprs_strap1: ∀L,T1,T,T2.
- L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → L ⊢ T1 ➡* T2.
-/2 width=3/ qed.
+lemma cprs_strap1: ∀G,L,T1,T,T2.
+ ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+normalize /2 width=3 by step/ qed.
(* Basic_1: was: pr3_step *)
-lemma cprs_strap2: ∀L,T1,T,T2.
- L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
-/2 width=3/ qed.
-
-(* Note: it does not hold replacing |L1| with |L2| *)
-lemma cprs_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➡* T2 →
- ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➡* T2.
-/3 width=3/
-qed.
+lemma cprs_strap2: ∀G,L,T1,T,T2.
+ ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+normalize /2 width=3 by TC_strap/ qed.
+
+lemma lsubr_cprs_trans: ∀G. lsub_trans … (cprs G) lsubr.
+/3 width=5 by lsubr_cpr_trans, LTC_lsub_trans/
+qed-.
+
+(* Basic_1: was: pr3_pr1 *)
+lemma tprs_cprs: ∀G,L,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+/2 width=3 by lsubr_cprs_trans/ qed.
+
+lemma cprs_bind_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+#G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
+/3 width=3 by cprs_strap2, cpr_cprs, cpr_pair_sn, cpr_bind/ qed.
(* Basic_1: was only: pr3_thin_dx *)
-lemma cprs_flat_dx: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L ⊢ T1 ➡* T2 →
- L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
-#I #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 /3 width=1/
-#T #T2 #_ #HT2 #IHT2
-@(cprs_strap1 … IHT2) -IHT2 /2 width=1/
+lemma cprs_flat_dx: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 →
+ ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+#I #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2
+/3 width=5 by cprs_strap1, cpr_flat, cpr_cprs, cpr_pair_sn/
qed.
-lemma tpss_cprs: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ T1 ➡* T2.
-#L #T1 #T2 #d #e #HT12
-lapply (cpr_intro … T1 … HT12) // /2 width=1/
+lemma cprs_flat_sn: ∀I,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+ ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+#I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind … H) -V2
+/3 width=3 by cprs_strap1, cpr_cprs, cpr_pair_sn, cpr_flat/
qed.
-(* Basic_1: was: pr3_pr1 *)
-lemma tprs_cprs: ∀T1,T2. T1 ➡* T2 → ∀L. L ⊢ T1 ➡* T2.
-#T1 #T2 #H @(tprs_ind … H) -T2 /2 width=1/ /3 width=3/
+lemma cprs_zeta: ∀G,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
+ ⦃G, L.ⓓV⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡* T2.
+#G #L #V #T1 #T #T2 #HT2 #H @(cprs_ind_dx … H) -T1
+/3 width=3 by cprs_strap2, cpr_cprs, cpr_bind, cpr_zeta/
+qed.
+
+lemma cprs_eps: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡* T2.
+#G #L #T1 #T2 #H @(cprs_ind … H) -T2
+/3 width=3 by cprs_strap1, cpr_cprs, cpr_eps/
+qed.
+
+lemma cprs_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+ ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 →
+ ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 * -T2
+/4 width=7 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_beta/
+qed.
+
+lemma cprs_theta_dx: ∀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 * -T2
+/4 width=9 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_theta/
qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1: ∀L,U2,k. L ⊢ ⋆k ➡* U2 → U2 = ⋆k.
-#L #U2 #k #H @(cprs_ind … H) -U2 //
+lemma cprs_inv_sort1: ∀G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡* U2 → U2 = ⋆k.
+#G #L #U2 #k #H @(cprs_ind … H) -U2 //
#U2 #U #_ #HU2 #IHU2 destruct
>(cpr_inv_sort1 … HU2) -HU2 //
qed-.
(* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1: ∀L,W1,T1,U2. L ⊢ ⓝW1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨
- ∃∃W2,T2. L ⊢ W1 ➡* W2 & L ⊢ T1 ➡* T2 & U2 = ⓝW2.T2.
-#L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
-#U2 #U #_ #HU2 * /3 width=3/ *
+lemma cprs_inv_cast1: ∀G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡* U2 → ⦃G, L⦄ ⊢ T1 ➡* U2 ∨
+ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L⦄ ⊢ T1 ➡* T2 & U2 = ⓝW2.T2.
+#G #L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+#U2 #U #_ #HU2 * /3 width=3 by cprs_strap1, or_introl/ *
#W #T #HW1 #HT1 #H destruct
-elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
-#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/
+elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3 by cprs_strap1, or_introl/ *
+#W2 #T2 #HW2 #HT2 #H destruct /4 width=5 by cprs_strap1, ex3_2_intro, or_intror/
qed-.
(* Basic_1: was: nf2_pr3_unfold *)
-lemma cprs_inv_cnf1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U.
-#L #T #U #H @(cprs_ind_dx … H) -T //
+lemma cprs_inv_cnr1: ∀G,L,T,U. ⦃G, L⦄ ⊢ T ➡* U → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → T = U.
+#G #L #T #U #H @(cprs_ind_dx … H) -T //
#T0 #T #H1T0 #_ #IHT #H2T0
-lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
+lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
qed-.
-lemma tprs_inv_cnf1: ∀T,U. T ➡* U → ⋆ ⊢ 𝐍⦃T⦄ → T = U.
-/3 width=3 by tprs_cprs, cprs_inv_cnf1/ qed-.
-
-(* Basic_1: removed theorems 10:
+(* Basic_1: removed theorems 13:
+ pr1_head_1 pr1_head_2 pr1_comp
clear_pr3_trans pr3_cflat pr3_gen_bind
pr3_head_1 pr3_head_2 pr3_head_21 pr3_head_12
- pr3_iso_appl_bind pr3_iso_appls_appl_bind pr3_iso_appls_bind
+ pr3_iso_appl_bind pr3_iso_applv_appl_bind pr3_iso_applv_bind
*)
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