(* 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. ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃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⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃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⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ≥[h] 𝐒⦃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) →
+ (∀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.
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃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) →
+ (∀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.
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.