include "basic_2/computation/csx_aaa.ma".
include "basic_2/computation/fsb_csx.ma".
-(* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
+(* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
(* Main properties **********************************************************)
-(* Note: this is the "big tree" theorem ("small step" version) *)
-theorem aaa_fsb: â\88\80h,g,G,L,T,A. â¦\83G, Lâ¦\84 â\8a¢ T â\81\9d A â\86\92 â¦\83G, Lâ¦\84 â\8a¢ ⦥[h, g] T.
+(* Note: this is the "big tree" theorem ("RST" version) *)
+theorem aaa_fsb: â\88\80h,g,G,L,T,A. â¦\83G, Lâ¦\84 â\8a¢ T â\81\9d A â\86\92 ⦥[h, g] â¦\83G, L, Tâ¦\84.
/3 width=2 by aaa_csx, csx_fsb/ qed.
-(* Note: this is the "big tree" theorem ("big step" version) *)
-theorem aaa_fsba: â\88\80h,g,G,L,T,A. â¦\83G, Lâ¦\84 â\8a¢ T â\81\9d A â\86\92 â¦\83G, Lâ¦\84 â\8a¢ ⦥⦥[h, g] T.
+(* Note: this is the "big tree" theorem ("QRST" version) *)
+theorem aaa_fsba: â\88\80h,g,G,L,T,A. â¦\83G, Lâ¦\84 â\8a¢ T â\81\9d A â\86\92 ⦥⦥[h, g] â¦\83G, L, Tâ¦\84.
/3 width=2 by fsb_fsba, aaa_fsb/ qed.
(* Advanced eliminators on atomica arity assignment for terms ***************)
-fact aaa_ind_fpbu_aux: ∀h,g. ∀R:relation3 genv lenv term.
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
-#h #g #R #IH #G #L #T #H @(csx_ind_fpbu … H) -G -L -T
+fact aaa_ind_fpb_aux: ∀h,g. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) →
+ ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+#h #g #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
-#G2 #L2 #T2 #H12 elim (aaa_fpbs_conf h g … G2 … L2 … T2 … HTA1) -A1
-/2 width=2 by fpbu_fwd_fpbs/
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h g … G2 … L2 … T2 … HTA1) -A1
+/2 width=2 by fpb_fpbs/
qed-.
-lemma aaa_ind_fpbu: ∀h,g. ∀R:relation3 genv lenv term.
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
-/4 width=4 by aaa_ind_fpbu_aux, aaa_csx/ qed-.
+lemma aaa_ind_fpb: ∀h,g. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+/4 width=4 by aaa_ind_fpb_aux, aaa_csx/ qed-.
fact aaa_ind_fpbg_aux: ∀h,g. ∀R:relation3 genv lenv term.
(∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (â\88\80G2,L2,T2. â¦\83G1, L1, T1â¦\84 >â\8b\95[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ (â\88\80G2,L2,T2. â¦\83G1, L1, T1â¦\84 >â\89¡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
R G1 L1 T1
) →
∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
#h #g #R #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
-#G2 #L2 #T2 #H12 elim (aaa_fpbs_conf h g … G2 … L2 … T2 … HTA1) -A1
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h g … G2 … L2 … T2 … HTA1) -A1
/2 width=2 by fpbg_fwd_fpbs/
qed-.
lemma aaa_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
(∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (â\88\80G2,L2,T2. â¦\83G1, L1, T1â¦\84 >â\8b\95[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ (â\88\80G2,L2,T2. â¦\83G1, L1, T1â¦\84 >â\89¡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
R G1 L1 T1
) →
∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.