include "basic_2/rt_computation/csx_aaa.ma".
include "basic_2/rt_computation/fpbs_aaa.ma".
-include "basic_2/rt_computation/fpbs_fpb.ma".
include "basic_2/rt_computation/fsb_csx.ma".
(* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
(* Main properties with atomic arity assignment for terms *******************)
-(* Note: this is the "big tree" theorem *)
-theorem aaa_fsb: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ≥[h, o] 𝐒⦃G, L, T⦄.
+theorem aaa_fsb (G) (L) (T) (A):
+ ❨G,L❩ ⊢ T ⁝ A → ≥𝐒 ❨G,L,T❩.
/3 width=2 by aaa_csx, csx_fsb/ qed.
(* Advanced eliminators with atomic arity assignment for terms **************)
-fact aaa_ind_fpb_aux: ∀h,o. ∀R:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
-#h #o #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
+fact aaa_ind_fpbc_aux (Q:relation3 …):
+ (∀G1,L1,T1,A.
+ ❨G1,L1❩ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ❨G1,L1,T1❩ ≻ ❨G2,L2,T2❩ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G,L,T. ❨G,L❩ ⊢ ⬈*𝐒 T → ∀A. ❨G,L❩ ⊢ T ⁝ A → Q G L T.
+#R #IH #G #L #T #H @(csx_ind_fpbc … H) -G -L -T
#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
-#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h o … G2 … L2 … T2 … HTA1) -A1
-/2 width=2 by fpb_fpbs/
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1
+/2 width=2 by fpbc_fpbs/
qed-.
-lemma aaa_ind_fpb: ∀h,o. ∀R:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃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-.
+lemma aaa_ind_fpbc (Q:relation3 …):
+ (∀G1,L1,T1,A.
+ ❨G1,L1❩ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ❨G1,L1,T1❩ ≻ ❨G2,L2,T2❩ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G,L,T,A. ❨G,L❩ ⊢ T ⁝ A → Q G L T.
+/4 width=4 by aaa_ind_fpbc_aux, aaa_csx/ qed-.
-fact aaa_ind_fpbg_aux: ∀h,o. ∀R:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
-#h #o #R #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
+fact aaa_ind_fpbg_aux (Q:relation3 …):
+ (∀G1,L1,T1,A.
+ ❨G1,L1❩ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ❨G1,L1,T1❩ > ❨G2,L2,T2❩ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G,L,T. ❨G,L❩ ⊢ ⬈*𝐒 T → ∀A. ❨G,L❩ ⊢ T ⁝ A → Q G L T.
+#Q #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 (fpbs_aaa_conf h o … G2 … L2 … T2 … HTA1) -A1
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1
/2 width=2 by fpbg_fwd_fpbs/
qed-.
-lemma aaa_ind_fpbg: ∀h,o. ∀R:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+lemma aaa_ind_fpbg (Q:relation3 …):
+ (∀G1,L1,T1,A.
+ ❨G1,L1❩ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ❨G1,L1,T1❩ > ❨G2,L2,T2❩ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G,L,T,A. ❨G,L❩ ⊢ T ⁝ A → Q G L T.
/4 width=4 by aaa_ind_fpbg_aux, aaa_csx/ qed-.