inductive lsubv (a) (h) (G): relation lenv ≝
| lsubv_atom: lsubv a h G (⋆) (⋆)
| lsubv_bind: ∀I,L1,L2. lsubv a h G L1 L2 → lsubv a h G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubv_beta: ∀L1,L2,W,V. ⦃G, L1⦄ ⊢ ⓝW.V ![a,h] →
+| lsubv_beta: ∀L1,L2,W,V. ⦃G,L1⦄ ⊢ ⓝW.V ![a,h] →
lsubv a h G L1 L2 → lsubv a h G (L1.ⓓⓝW.V) (L2.ⓛW)
.
fact lsubv_inv_bind_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
∀I,K1. L1 = K1.ⓘ{I} →
∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
+ | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
G ⊢ K1 ⫃![a,h] K2 &
I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
#a #h #G #L1 #L2 * -L1 -L2
(* Basic_2A1: uses: lsubsv_inv_pair1 *)
lemma lsubv_inv_bind_sn (a) (h) (G): ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⫃![a,h] L2 →
∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
+ | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
G ⊢ K1 ⫃![a,h] K2 &
I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
/2 width=3 by lsubv_inv_bind_sn_aux/ qed-.
fact lsubv_inv_bind_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
∀I,K2. L2 = K2.ⓘ{I} →
∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
+ | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
#a #h #G #L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
(* Basic_2A1: uses: lsubsv_inv_pair2 *)
lemma lsubv_inv_bind_dx (a) (h) (G): ∀I,L1,K2. G ⊢ L1 ⫃![a,h] K2.ⓘ{I} →
∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
+ | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
/2 width=3 by lsubv_inv_bind_dx_aux/ qed-.
lemma lsubv_refl (a) (h) (G): reflexive … (lsubv a h G).
#a #h #G #L elim L -L /2 width=1 by lsubv_atom, lsubv_bind/
qed.
+
+(* Basic_2A1: removed theorems 3:
+ lsubsv_lstas_trans lsubsv_sta_trans
+ lsubsv_fwd_lsubd
+*)