(* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
(* Note: alternative definition of fsb *)
-inductive fsba (h) (g): relation3 genv lenv term ≝
+inductive fsba (h) (o): relation3 genv lenv term ≝
| fsba_intro: ∀G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → fsba h g G2 L2 T2
- ) → fsba h g G1 L1 T1.
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → fsba h o G2 L2 T2
+ ) → fsba h o G1 L1 T1.
interpretation
"'big tree' strong normalization (closure) alternative"
- 'BTSNAlt h g G L T = (fsba h g G L T).
+ 'BTSNAlt h o G L T = (fsba h o G L T).
(* Basic eliminators ********************************************************)
-lemma fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
- ∀G1,L1,T1. ⦥⦥[h,g] ⦃G1, L1, T1⦄ → (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2
+lemma fsba_ind_alt: ∀h,o. ∀R: relation3 …. (
+ ∀G1,L1,T1. ⦥⦥[h,o] ⦃G1, L1, T1⦄ → (
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2
) → R G1 L1 T1
) →
- ∀G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → R G L T.
-#h #g #R #IH #G #L #T #H elim H -G -L -T
+ ∀G,L,T. ⦥⦥[h, o] ⦃G, L, T⦄ → R G L T.
+#h #o #R #IH #G #L #T #H elim H -G -L -T
/4 width=1 by fsba_intro/
qed-.
(* Basic properties *********************************************************)
-lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦥⦥[h, g] ⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥⦥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
+lemma fsba_fpbs_trans: ∀h,o,G1,L1,T1. ⦥⦥[h, o] ⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦥⦥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
/4 width=5 by fsba_intro, fpbs_fpbg_trans/
qed-.
(* Main properties **********************************************************)
-theorem fsb_fsba: ∀h,g,G,L,T. ⦥[h, g] ⦃G, L, T⦄ → ⦥⦥[h, g] ⦃G, L, T⦄.
-#h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
+theorem fsb_fsba: ∀h,o,G,L,T. ⦥[h, o] ⦃G, L, T⦄ → ⦥⦥[h, o] ⦃G, L, T⦄.
+#h #o #G #L #T #H @(fsb_ind_alt … H) -G -L -T
#G1 #L1 #T1 #_ #IH @fsba_intro
#G2 #L2 #T2 * /3 width=5 by fsba_fpbs_trans/
qed.
(* Main inversion lemmas ****************************************************)
-theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → ⦥[h, g] ⦃G, L, T⦄.
-#h #g #G #L #T #H @(fsba_ind_alt … H) -G -L -T
+theorem fsba_inv_fsb: ∀h,o,G,L,T. ⦥⦥[h, o] ⦃G, L, T⦄ → ⦥[h, o] ⦃G, L, T⦄.
+#h #o #G #L #T #H @(fsba_ind_alt … H) -G -L -T
/4 width=1 by fsb_intro, fpb_fpbg/
qed-.
(* Advanced properties ******************************************************)
-lemma fsb_fpbs_trans: ∀h,g,G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥[h, g] ⦃G2, L2, T2⦄.
+lemma fsb_fpbs_trans: ∀h,o,G1,L1,T1. ⦥[h, o] ⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦥[h, o] ⦃G2, L2, T2⦄.
/4 width=5 by fsba_inv_fsb, fsb_fsba, fsba_fpbs_trans/ qed-.
(* Advanced eliminators *****************************************************)
-lemma fsb_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
- (∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+lemma fsb_ind_fpbg: ∀h,o. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦥[h, o] ⦃G1, L1, T1⦄ →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
R G1 L1 T1
) →
- ∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ → R G1 L1 T1.
-#h #g #R #IH #G1 #L1 #T1 #H @(fsba_ind_alt h g … G1 L1 T1)
+ ∀G1,L1,T1. ⦥[h, o] ⦃G1, L1, T1⦄ → R G1 L1 T1.
+#h #o #R #IH #G1 #L1 #T1 #H @(fsba_ind_alt h o … G1 L1 T1)
/3 width=1 by fsba_inv_fsb, fsb_fsba/
qed-.