(* Basic inversion lemmas ***************************************************)
+lemma isfin_inv_push: āg. š
ā¦gā¦ ā āf. āf = g ā š
ā¦fā¦.
+#g * /3 width=4 by fcla_inv_px, ex_intro/
+qed-.
+
lemma isfin_inv_next: āg. š
ā¦gā¦ ā āf. ā«Æf = g ā š
ā¦fā¦.
#g * #n #H #f #H0 elim (fcla_inv_nx ā¦ H ā¦ H0) -g
/2 width=2 by ex_intro/
qed-.
-(* Basic forward lemmas *****************************************************)
-
-lemma isfin_fwd_push: āg. š
ā¦gā¦ ā āf. āf = g ā š
ā¦fā¦.
-#g * /3 width=4 by fcla_inv_px, ex_intro/
-qed-.
-
(* Basic properties *********************************************************)
lemma isfin_eq_repl_back: eq_repl_back ā¦ isfin.
#f * /3 width=2 by fcla_next, ex_intro/
qed.
+(* Properties with iterated push ********************************************)
+
+lemma isfin_pushs: ān,f. š
ā¦fā¦ ā š
ā¦ā*[n]fā¦.
+#n elim n -n /3 width=3 by isfin_push/
+qed.
+
+(* Inversion lemmas with iterated push **************************************)
+
+lemma isfin_inv_pushs: ān,g. š
ā¦ā*[n]gā¦ ā š
ā¦gā¦.
+#n elim n -n /3 width=3 by isfin_inv_push/
+qed.
+
+(* Properties with tail *****************************************************)
+
lemma isfin_tl: āf. š
ā¦fā¦ ā š
ā¦ā«±fā¦.
#f elim (pn_split f) * #g #H #Hf destruct
-/3 width=3 by isfin_fwd_push, isfin_inv_next/
+/3 width=3 by isfin_inv_push, isfin_inv_next/
qed.
(* Inversion lemmas with tail ***********************************************)
#f elim (pn_split f) * /2 width=1 by isfin_next, isfin_push/
qed-.
-(* Inversion lemmas with tls ********************************************************)
+(* Inversion lemmas with iterated tail **************************************)
lemma isfin_inv_tls: ān,f. š
ā¦ā«±*[n]fā¦ ā š
ā¦fā¦.
#n elim n -n /3 width=1 by isfin_inv_tl/