(* *)
(**************************************************************************)
+include "basic_2/notation/relations/pconvstar_4.ma".
include "basic_2/conversion/cpc.ma".
(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-definition cpcs: lenv → relation term ≝
- λL. TC … (cpc L).
+definition cpcs: relation4 genv lenv term term ≝
+ λG. LTC … (cpc G).
interpretation "context-sensitive parallel equivalence (term)"
- 'PConvStar L T1 T2 = (cpcs L T1 T2).
+ 'PConvStar G L T1 T2 = (cpcs G L T1 T2).
(* Basic eliminators ********************************************************)
-lemma cpcs_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) //
+lemma cpcs_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 cpcs_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 cpcs_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: pc3_refl *)
-lemma cpcs_refl: ∀L. reflexive … (cpcs L).
+lemma cpcs_refl: ∀G,L. reflexive … (cpcs G L).
/2 width=1/ qed.
(* Basic_1: was: pc3_s *)
-lemma cpcs_sym: ∀L. symmetric … (cpcs L).
-/3 width=1/ qed.
+lemma cpcs_sym: ∀G,L. symmetric … (cpcs G L).
+#G #L @TC_symmetric // qed.
-lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
-/2 width=3/ qed.
+lemma cpc_cpcs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T2.
+/2 width=1/ qed.
-lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
-/2 width=3/ qed.
+lemma cpcs_strap1: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L @step qed.
-(* Basic_1: was: pc3_pr2_r *)
-lemma cpcs_cpr_dx: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ⬌* T2.
-/3 width=1/ qed.
+lemma cpcs_strap2: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L @TC_strap qed.
-lemma cpcs_tpr_dx: ∀L,T1,T2. T1 ➡ T2 → L ⊢ T1 ⬌* T2.
+(* Basic_1: was: pc3_pr2_r *)
+lemma cpr_cpcs_dx: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
/3 width=1/ qed.
(* Basic_1: was: pc3_pr2_x *)
-lemma cpcs_cpr_sn: ∀L,T1,T2. L ⊢ T2 ➡ T1 → L ⊢ T1 ⬌* T2.
-/3 width=1/ qed.
-
-lemma cpcs_tpr_sn: ∀L,T1,T2. T2 ➡ T1 → L ⊢ T1 ⬌* T2.
+lemma cpr_cpcs_sn: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡ T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
/3 width=1/ qed.
-lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cpr_strap1: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
/3 width=3/ qed.
(* Basic_1: was: pc3_pr2_u *)
-lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cpr_strap2: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
/3 width=3/ qed.
-lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
+lemma cpcs_cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
/3 width=3/ qed.
-lemma cpr_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
+lemma cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
/3 width=3/ qed-.
(* Basic_1: was: pc3_pr2_u2 *)
-lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cpr_conf: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
/3 width=3/ qed.
(* Basic_1: removed theorems 9:
clear_pc3_trans pc3_ind_left
pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
- pc3_pr2_fsubst0 pc3_pr2_fsubst0_back pc3_fsubst0
- Basic_1: removed local theorems 6:
+ pc3_pr2_fqubst0 pc3_pr2_fqubst0_back pc3_fqubst0
+ pc3_gen_abst pc3_gen_abst_shift
+*)
+(* Basic_1: removed local theorems 6:
pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left
pc3_wcpr0_t_aux
*)