"context-sensitive parallel r-computation (term)"
'PRedStar h G L T1 T2 = (cpms h G L O T1 T2).
+(* Basic eliminators ********************************************************)
+
+lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
+ Q 0 T2 →
+ (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
+ ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1.
+#h #G #L #T2 #R @ltc_ind_sn_refl //
+qed-.
+
+lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
+ Q 0 T1 →
+ (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) →
+ ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2.
+#h #G #L #T1 #R @ltc_ind_dx_refl //
+qed-.
+
(* Basic properties *********************************************************)
+(* Basic_1: includes: pr1_pr0 *)
+(* Basic_1: uses: pr3_pr2 *)
+(* Basic_2A1: includes: cpr_cprs *)
lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2.
/2 width=1 by ltc_rc/ qed.
∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
/2 width=3 by ltc_dx/ qed-.
+(* Basic_2A1: uses: cprs_bind_dx *)
+lemma cpms_bind_dx (n) (h) (G) (L):
+ ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 →
+ ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+#n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
+/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
+
(* Basic properties with r-transition ***************************************)
+(* Basic_1: was: pr3_refl *)
lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0).
/2 width=1 by cpm_cpms/ qed.
-(* Basic eliminators ********************************************************)
-
-lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
- Q 0 T2 →
- (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
- ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1.
-#h #G #L #T2 #R @ltc_ind_sn_refl //
-qed-.
-
-lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
- Q 0 T1 →
- (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) →
- ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2.
-#h #G #L #T1 #R @ltc_ind_dx_refl //
-qed-.
-
(* Basic_2A1: removed theorems 4:
sta_cprs_scpds lstas_scpds scpds_strap1 scpds_fwd_cprs
*)