(* Inversion lemmas with context-sensitive stringly rt-normalizing terms ****)
-lemma fsb_inv_csx: ∀h,G,L,T. ≥[h] 𝐒❪G,L,T❫ → ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T❫.
+lemma fsb_inv_csx (h):
+ ∀G,L,T. ≥𝐒[h] ❪G,L,T❫ → ❪G,L❫ ⊢ ⬈*𝐒[h] T.
#h #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
qed-.
(* Propreties with context-sensitive stringly rt-normalizing terms **********)
-lemma csx_fsb_fpbs: ∀h,G1,L1,T1. ❪G1,L1❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- ∀G2,L2,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → ≥[h] 𝐒❪G2,L2,T2❫.
+lemma csx_fsb_fpbs (h):
+ ∀G1,L1,T1. ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
+ ∀G2,L2,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → ≥𝐒[h] ❪G2,L2,T2❫.
#h #G1 #L1 #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2
#G0 #L0 #T0 #IHu #H10
]
qed.
-lemma csx_fsb: ∀h,G,L,T. ❪G,L❫ ⊢ ⬈*[h] 𝐒❪T❫ → ≥[h] 𝐒❪G,L,T❫.
+lemma csx_fsb (h):
+ ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → ≥𝐒[h] ❪G,L,T❫.
/2 width=5 by csx_fsb_fpbs/ qed.
(* Advanced eliminators *****************************************************)
-lemma csx_ind_fpb: ∀h. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ❪G1,L1❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- (∀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❫ → Q G L T.
+lemma csx_ind_fpb (h) (Q:relation3 …):
+ (∀G1,L1,T1.
+ ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
+ (∀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 → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
-lemma csx_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ❪G1,L1❫ ⊢ ⬈*[h] 𝐒❪T1❫ →
- (∀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❫ → Q G L T.
+lemma csx_ind_fpbg (h) (Q:relation3 …):
+ (∀G1,L1,T1.
+ ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
+ (∀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 → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.