lsubsx h G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})
| lsubsx_unit: ∀f,I,K1,K2. lsubsx h G f K1 K2 →
lsubsx h G (↑f) (K1.ⓤ{I}) (K2.ⓧ)
-| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ →
+| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ →
lsubsx h G f K1 K2 → lsubsx h G (↑f) (K1.ⓑ{I}V) (K2.ⓧ)
.
(* Basic inversion lemmas ***************************************************)
-fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
+fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
L1 = ⋆ → L2 = ⋆.
#h #g #G #L1 #L2 * -g -L1 -L2 //
[ #f #I #K1 #K2 #_ #H destruct
]
qed-.
-lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h, g] L2 → L2 = ⋆.
+lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h,g] L2 → L2 = ⋆.
/2 width=7 by lsubsx_inv_atom_sn_aux/ qed-.
-fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
+fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}.
+ ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}.
#h #g #G #L1 #L2 * -g -L1 -L2
[ #f #g #J #L1 #_ #H destruct
| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
]
qed-.
-lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h, ⫯f] L2 →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}.
+lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h,⫯f] L2 →
+ ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}.
/2 width=5 by lsubsx_inv_push_sn_aux/ qed-.
-fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
+fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+ ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
#h #g #G #L1 #L2 * -g -L1 -L2
[ #f #g #J #L1 #_ #H destruct
| #f #I #K1 #K2 #_ #g #J #L1 #H
]
qed-.
-lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h, ↑f] L2 →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h,↑f] L2 →
+ ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
/2 width=6 by lsubsx_inv_unit_sn_aux/ qed-.
-fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 →
+fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V →
- ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+ ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
+ G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
#h #g #G #L1 #L2 * -g -L1 -L2
[ #f #g #J #L1 #W #_ #H destruct
| #f #I #K1 #K2 #_ #g #J #L1 #W #H
qed-.
(* Basic_2A1: uses: lcosx_inv_pair *)
-lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, ↑f] L2 →
- ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ.
+lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,↑f] L2 →
+ ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
+ G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
/2 width=6 by lsubsx_inv_pair_sn_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, g] L2 →
- ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
- | ∃∃f,K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ↑f & L2 = K2.ⓧ.
+lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,g] L2 →
+ ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
+ | ∃∃f,K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
+ G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ↑f & L2 = K2.ⓧ.
#h #g #I #G #K1 #L2 #V #H
elim (pn_split g) * #f #Hf destruct
[ elim (lsubsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/
(* Advanced forward lemmas **************************************************)
-lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h, g] L2 →
- ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h, ⫱g] K2 & L2 = K2.ⓘ{I2}.
+lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h,g] L2 →
+ ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h,⫱g] K2 & L2 = K2.ⓘ{I2}.
#h #g #I1 #G #K1 #L2
elim (pn_split g) * #f #Hf destruct
[ #H elim (lsubsx_inv_push_sn … H) -H
(* Basic properties *********************************************************)
-lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2).
+lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2).
#h #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
[ #f #I #L1 #L2 #_ #IH #x #H
elim (eq_inv_px … H) -H /3 width=3 by lsubsx_push/
]
qed-.
-lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2).
+lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2).
#h #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/
qed-.