(* Basic_2A1: uses: lsx_lpxs_trans *)
lemma lfsx_lfpxs_trans: āh,o,G,L1,T. G ā¢ ā¬*[h, o, T] šā¦L1ā¦ ā
āL2. ā¦G, L1ā¦ ā¢ ā¬*[h, T] L2 ā G ā¢ ā¬*[h, o, T] šā¦L2ā¦.
-#h #o #G #L1 #T #HL1 #L2 #H @(lfpxs_ind ā¦ H) -L2
+#h #o #G #L1 #T #HL1 #L2 #H @(lfpxs_ind_sn ā¦ H) -L2
/2 width=3 by lfsx_lfpx_trans/
qed-.
(* Advanced properties ******************************************************)
-fact lsx_bind_lpxs_aux: āh,o,p,I,G,L1,V. G ā¢ ā¬*[h, o, V] šā¦L1ā¦ ā
- āY,T. G ā¢ ā¬*[h, o, T] šā¦Yā¦ ā
- āL2. Y = L2.ā{I}V ā ā¦G, L1ā¦ ā¢ ā¬*[h, ā{p,I}V.T] L2 ā
- G ā¢ ā¬*[h, o, ā{p,I}V.T] šā¦L2ā¦.
+fact lfsx_bind_lfpxs_aux: āh,o,p,I,G,L1,V. G ā¢ ā¬*[h, o, V] šā¦L1ā¦ ā
+ āY,T. G ā¢ ā¬*[h, o, T] šā¦Yā¦ ā
+ āL2. Y = L2.ā{I}V ā ā¦G, L1ā¦ ā¢ ā¬*[h, ā{p,I}V.T] L2 ā
+ G ā¢ ā¬*[h, o, ā{p,I}V.T] šā¦L2ā¦.
#h #o #p #I #G #L1 #V #H @(lfsx_ind_lfpxs ā¦ H) -L1
-#L1 #HL1 #IHL1 #Y #T #H @(lfsx_ind_lfpxs ā¦ H) -Y
-#Y #HY #IHY #L2 #H #HL12 destruct @lfsx_intro_lfpxs
-#L0 #HL20 lapply (lfpxs_trans ā¦ HL12 ā¦ HL20)
-#HL10 #H elim (lfdneq_inv_bind ā¦ H) -H [ -HL1 -IHY | -HY -IHL1 ]
+#L1 #_ #IHL1 #Y #T #H @(lfsx_ind_lfpxs ā¦ H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct
+@lfsx_intro_lfpxs #L0 #HL20
+lapply (lfpxs_trans ā¦ HL12 ā¦ HL20) #HL10 #H
+elim (lfdneq_inv_bind ā¦ H) -H [ -IHY | -HY -IHL1 -HL12 ]
[ #HnV elim (lfdeq_dec h o L1 L2 V)
- [ #HV @(IHL1 ā¦ L0) -IHL1
- [2: /3 width=5 by lfdeq_canc_sn/
- |5: // |3: skip
- |6: //
-
- lfdeq_lfdneq_trans/
-
- /3 width=5 by lfsx_lfpxs_trans, lfpxs_pair, lfdeq_canc_sn/ (**) (* full auto too slow *)
- | -HnV -HL10 /4 width=5 by lsx_lpxs_trans, lpxs_pair/
+ [ #HV @(IHL1 ā¦ L0) -IHL1 -HL12
+ /3 width=4 by lfsx_lfpxs_trans, lfpxs_fwd_bind_dx, lfpxs_fwd_pair_sn, lfdeq_canc_sn/ (**) (* full auto too slow *)
+ | -HnV -HL10 /4 width=4 by lfsx_lfpxs_trans, lfpxs_fwd_pair_sn/
]
-| /3 width=4 by lpxs_pair/
+| /3 width=4 by lfpxs_fwd_bind_dx/
]
qed-.
-lemma lsx_bind: āh,o,a,I,G,L,V,l. G ā¢ ā¬*[h, o, V, l] L ā
- āT. G ā¢ ā¬*[h, o, T, ā«Æl] L.ā{I}V ā
- G ā¢ ā¬*[h, o, ā{a,I}V.T, l] L.
-/2 width=3 by lsx_bind_lpxs_aux/ qed.
+(* Basic_2A1: uses: lsx_bind *)
+lemma lfsx_bind: āh,o,p,I,G,L,V. G ā¢ ā¬*[h, o, V] šā¦Lā¦ ā
+ āT. G ā¢ ā¬*[h, o, T] šā¦L.ā{I}Vā¦ ā
+ G ā¢ ā¬*[h, o, ā{p,I}V.T] šā¦Lā¦.
+/2 width=3 by lfsx_bind_lfpxs_aux/ qed.
-lemma lsx_flat_lpxs: āh,o,I,G,L1,V,l. G ā¢ ā¬*[h, o, V, l] L1 ā
- āL2,T. G ā¢ ā¬*[h, o, T, l] L2 ā ā¦G, L1ā¦ ā¢ ā¬*[h, o] L2 ā
- G ā¢ ā¬*[h, o, ā{I}V.T, l] L2.
-#h #o #I #G #L1 #V #l #H @(lsx_ind_lfpxs ā¦ H) -L1
-#L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind_lfpxs ā¦ H) -L2
-#L2 #HL2 #IHL2 #HL12 @lsx_intro_lfpxs
-#L0 #HL20 lapply (lpxs_trans ā¦ HL12 ā¦ HL20)
-#HL10 #H elim (nlleq_inv_flat ā¦ H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 l)
- [ #HV @(IHL1 ā¦ L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
- | -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
+(* Basic_2A1: uses: lsx_flat_lpxs *)
+lemma lfsx_flat_lfpxs: āh,o,I,G,L1,V. G ā¢ ā¬*[h, o, V] šā¦L1ā¦ ā
+ āL2,T. G ā¢ ā¬*[h, o, T] šā¦L2ā¦ ā ā¦G, L1ā¦ ā¢ ā¬*[h, ā{I}V.T] L2 ā
+ G ā¢ ā¬*[h, o, ā{I}V.T] šā¦L2ā¦.
+#h #o #I #G #L1 #V #H @(lfsx_ind_lfpxs ā¦ H) -L1
+#L1 #HL1 #IHL1 #L2 #T #H @(lfsx_ind_lfpxs ā¦ H) -L2
+#L2 #HL2 #IHL2 #HL12 @lfsx_intro_lfpxs
+#L0 #HL20 lapply (lfpxs_trans ā¦ HL12 ā¦ HL20)
+#HL10 #H elim (lfdneq_inv_flat ā¦ H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (lfdeq_dec h o L1 L2 V)
+ [ #HV @(IHL1 ā¦ L0) -IHL1 -HL12
+ /3 width=5 by lfsx_lfpxs_trans, lfpxs_fwd_flat_dx, lfpxs_fwd_pair_sn, lfdeq_canc_sn/ (**) (* full auto too slow: 47s *)
+ | -HnV -HL10 /4 width=4 by lfsx_lfpxs_trans, lfpxs_fwd_pair_sn/
]
-| /3 width=1 by/
+| /3 width=3 by lfpxs_fwd_flat_dx/
]
qed-.
-lemma lsx_flat: āh,o,I,G,L,V,l. G ā¢ ā¬*[h, o, V, l] L ā
- āT. G ā¢ ā¬*[h, o, T, l] L ā G ā¢ ā¬*[h, o, ā{I}V.T, l] L.
-/2 width=3 by lsx_flat_lpxs/ qed.
+(* Basic_2A1: uses: lsx_flat *)
+lemma lfsx_flat: āh,o,I,G,L,V. G ā¢ ā¬*[h, o, V] šā¦Lā¦ ā
+ āT. G ā¢ ā¬*[h, o, T] šā¦Lā¦ ā G ā¢ ā¬*[h, o, ā{I}V.T] šā¦Lā¦.
+/2 width=3 by lfsx_flat_lfpxs/ qed.