(* Main properties with atomic arity assignment for terms *******************)
-theorem aaa_fsb: ∀h,G,L,T,A. ❪G,L❫ ⊢ T ⁝ A → ≥[h] 𝐒❪G,L,T❫.
+theorem aaa_fsb (h):
+ ∀G,L,T,A. ❪G,L❫ ⊢ T ⁝ A → ≥𝐒[h] ❪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. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ❪G1,L1❫ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) →
- ∀G,L,T. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T❫ → ∀A. ❪G,L❫ ⊢ T ⁝ A → Q G L T.
+fact aaa_ind_fpb_aux (h) (Q:relation3 …):
+ (∀G1,L1,T1,A.
+ ❪G1,L1❫ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → ∀A. ❪G,L❫ ⊢ T ⁝ A → Q G L T.
#h #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 (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1
/2 width=2 by fpb_fpbs/
qed-.
-lemma aaa_ind_fpb: ∀h. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ❪G1,L1❫ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) →
- ∀G,L,T,A. ❪G,L❫ ⊢ T ⁝ A → Q G L T.
+lemma aaa_ind_fpb (h) (Q:relation3 …):
+ (∀G1,L1,T1,A.
+ ❪G1,L1❫ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ ≻[h] ❪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_fpb_aux, aaa_csx/ qed-.
-fact aaa_ind_fpbg_aux: ∀h. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ❪G1,L1❫ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) →
- ∀G,L,T. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T❫ → ∀A. ❪G,L❫ ⊢ T ⁝ A → Q G L T.
+fact aaa_ind_fpbg_aux (h) (Q:relation3 …):
+ (∀G1,L1,T1,A.
+ ❪G1,L1❫ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → ∀A. ❪G,L❫ ⊢ T ⁝ A → Q G L T.
#h #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 … G2 … L2 … T2 … HTA1) -A1
/2 width=2 by fpbg_fwd_fpbs/
qed-.
-lemma aaa_ind_fpbg: ∀h. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ❪G1,L1❫ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) →
- ∀G,L,T,A. ❪G,L❫ ⊢ T ⁝ A → Q G L T.
+lemma aaa_ind_fpbg (h) (Q:relation3 …):
+ (∀G1,L1,T1,A.
+ ❪G1,L1❫ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪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-.