(* *)
(**************************************************************************)
-include "basic_2/reducibility/cpr.ma".
+include "basic_2/reducibility/cnf.ma".
(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
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 @(TC_star_ind … HT1 IHT1 … HT12) //
+#L #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
qed-.
-axiom cprs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
+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
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
(* Basic properties *********************************************************)
(* Note: it does not hold replacing |L1| with |L2| *)
lemma cprs_lsubs_conf: ∀L1,T1,T2. L1 ⊢ T1 ➡* T2 →
- ∀L2. L1 [0, |L1|] ≼ L2 → L2 ⊢ T1 ➡* T2.
+ ∀L2. L1 ≼ [0, |L1|] L2 → L2 ⊢ T1 ➡* T2.
/3 width=3/
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/
qed-.
(* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1: â\88\80L,W1,T1,U2. L â\8a¢ â\93£W1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨
- â\88\83â\88\83W2,T2. L â\8a¢ W1 â\9e¡* W2 & L â\8a¢ T1 â\9e¡* T2 & U2 = â\93£W2.T2.
+lemma cprs_inv_cast1: â\88\80L,W1,T1,U2. L â\8a¢ â\93\9dW1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨
+ â\88\83â\88\83W2,T2. L â\8a¢ W1 â\9e¡* W2 & L â\8a¢ T1 â\9e¡* T2 & U2 = â\93\9dW2.T2.
#L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
#U2 #U #_ #HU2 * /3 width=3/ *
#W #T #HW1 #HT1 #H destruct
#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/
qed-.
-(* Basic_1: removed theorems 6:
+(* 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 //
+#T0 #T #H1T0 #_ #IHT #H2T0
+lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
+qed-.
+
+(* Basic_1: removed theorems 10:
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
*)