(* CONTEXT-SENSITIVE PARALLEL R-EQUIVALENCE FOR TERMS ***********************)
-definition cpcs: sh → relation4 genv lenv term term ≝
- λh,G. CTC … (cpc h G).
+definition cpcs (h) (G): relation3 lenv term term ≝
+ CTC … (cpc h G).
interpretation "context-sensitive parallel r-equivalence (term)"
'PConvStar h G L T1 T2 = (cpcs h G L T1 T2).
(* Basic eliminators ********************************************************)
(* Basic_2A1: was: cpcs_ind_dx *)
-lemma cpcs_ind_sn (h): ∀G,L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T → ⦃G, L⦄ ⊢ T ⬌*[h] T2 → R T → R T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → R T1.
+lemma cpcs_ind_sn (h) (G) (L) (T2):
+ ∀R:predicate term. R T2 →
+ (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T → ⦃G, L⦄ ⊢ T ⬌*[h] T2 → R T → R T1) →
+ ∀T1. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → R T1.
normalize /3 width=6 by TC_star_ind_dx/
qed-.
(* Basic_2A1: was: cpcs_ind *)
-lemma cpcs_ind_dx (h): ∀G,L,T1. ∀R:predicate term. R T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → ⦃G, L⦄ ⊢ T ⬌[h] T2 → R T → R T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → R T2.
+lemma cpcs_ind_dx (h) (G) (L) (T1):
+ ∀R:predicate term. R T1 →
+ (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → ⦃G, L⦄ ⊢ T ⬌[h] T2 → R T → R T2) →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → R T2.
normalize /3 width=6 by TC_star_ind/
qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: pc3_refl *)
-lemma cpcs_refl (h): ∀G. c_reflexive … (cpcs h G).
+lemma cpcs_refl (h) (G): c_reflexive … (cpcs h G).
/2 width=1 by inj/ qed.
(* Basic_1: was: pc3_s *)
-lemma cpcs_sym (h): ∀G,L. symmetric … (cpcs h G L).
+lemma cpcs_sym (h) (G) (L): symmetric … (cpcs h G L).
#h #G #L @TC_symmetric
/2 width=1 by cpc_sym/
qed-.
-lemma cpc_cpcs (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpc_cpcs (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
/2 width=1 by inj/ qed.
(* Basic_2A1: was: cpcs_strap2 *)
-lemma cpcs_step_sn (h): ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T → ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T →
+ ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
normalize /2 width=3 by TC_strap/
qed-.
(* Basic_2A1: was: cpcs_strap1 *)
-lemma cpcs_step_dx (h): ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → ⦃G, L⦄ ⊢ T ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G, L⦄ ⊢ T ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
normalize /2 width=3 by step/
qed-.
(* Basic_1: was: pc3_pr2_r *)
-lemma cpr_cpcs_dx (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cpcs_dx (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=1 by cpc_cpcs, or_introl/ qed.
(* Basic_1: was: pc3_pr2_x *)
-lemma cpr_cpcs_sn (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡[h] T1 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cpcs_sn (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T2 ➡[h] T1 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=1 by cpc_cpcs, or_intror/ qed.
(* Basic_1: was: pc3_pr2_u *)
(* Basic_2A1: was: cpcs_cpr_strap2 *)
-lemma cpcs_cpr_step_sn (h): ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_sn, or_introl/ qed-.
(* Basic_2A1: was: cpcs_cpr_strap1 *)
-lemma cpcs_cpr_step_dx (h): ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_dx, or_introl/ qed-.
-lemma cpcs_cpr_div (h): ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_dx, or_intror/ qed-.
-lemma cpr_div (h): ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
+ ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpr_cpcs_dx, cpcs_step_dx, or_intror/ qed-.
(* Basic_1: was: pc3_pr2_u2 *)
-lemma cpcs_cpr_conf (h): ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_conf (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T ➡[h] T1 →
+ ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_sn, or_intror/ qed-.
(* Basic_1: removed theorems 9: