(* RELOCATION MAP ***********************************************************)
definition isfin: predicate rtmap ≝
- λf. â\88\83n. ð\9d\90\82â¦\83fâ¦\84 â\89¡ n.
+ λf. â\88\83n. ð\9d\90\82â¦\83fâ¦\84 â\89\98 n.
interpretation "test for finite colength (rtmap)"
'IsFinite f = (isfin f).
+(* Basic eliminators ********************************************************)
+
+lemma isfin_ind (R:predicate rtmap): (∀f. 𝐈⦃f⦄ → R f) →
+ (∀f. 𝐅⦃f⦄ → R f → R (⫯f)) →
+ (∀f. 𝐅⦃f⦄ → R f → R (↑f)) →
+ ∀f. 𝐅⦃f⦄ → R f.
+#R #IH1 #IH2 #IH3 #f #H elim H -H
+#n #H elim H -f -n /3 width=2 by ex_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma isfin_inv_push: ∀g. 𝐅⦃g⦄ → ∀f. ⫯f = g → 𝐅⦃f⦄.
+#g * /3 width=4 by fcla_inv_px, ex_intro/
+qed-.
+
+lemma isfin_inv_next: ∀g. 𝐅⦃g⦄ → ∀f. ↑f = g → 𝐅⦃f⦄.
+#g * #n #H #f #H0 elim (fcla_inv_nx … H … H0) -g
+/2 width=2 by ex_intro/
+qed-.
+
(* Basic properties *********************************************************)
+lemma isfin_eq_repl_back: eq_repl_back … isfin.
+#f1 * /3 width=4 by fcla_eq_repl_back, ex_intro/
+qed-.
+
+lemma isfin_eq_repl_fwd: eq_repl_fwd … isfin.
+/3 width=3 by isfin_eq_repl_back, eq_repl_sym/ qed-.
+
lemma isfin_isid: ∀f. 𝐈⦃f⦄ → 𝐅⦃f⦄.
/3 width=2 by fcla_isid, ex_intro/ qed.
-lemma isfin_push: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 ð\9d\90\85â¦\83â\86\91f⦄.
+lemma isfin_push: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 ð\9d\90\85â¦\83⫯f⦄.
#f * /3 width=2 by fcla_push, ex_intro/
qed.
-lemma isfin_next: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 ð\9d\90\85â¦\83⫯f⦄.
+lemma isfin_next: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 ð\9d\90\85â¦\83â\86\91f⦄.
#f * /3 width=2 by fcla_next, ex_intro/
qed.
-lemma isfin_eq_repl_back: eq_repl_back … isfin.
-#f1 * /3 width=4 by fcla_eq_repl_back, ex_intro/
-qed-.
+(* Properties with iterated push ********************************************)
-lemma isfin_eq_repl_fwd: eq_repl_fwd … isfin.
-/3 width=3 by isfin_eq_repl_back, eq_repl_sym/ qed-.
+lemma isfin_pushs: ∀n,f. 𝐅⦃f⦄ → 𝐅⦃⫯*[n]f⦄.
+#n elim n -n /3 width=3 by isfin_push/
+qed.
-(* Basic eliminators ********************************************************)
+(* Inversion lemmas with iterated push **************************************)
-lemma isfin_ind (R:predicate rtmap): (∀f. 𝐈⦃f⦄ → R f) →
- (∀f. 𝐅⦃f⦄ → R f → R (↑f)) →
- (∀f. 𝐅⦃f⦄ → R f → R (⫯f)) →
- ∀f. 𝐅⦃f⦄ → R f.
-#R #IH1 #IH2 #IH3 #f #H elim H -H
-#n #H elim H -f -n /3 width=2 by ex_intro/
-qed-.
+lemma isfin_inv_pushs: ∀n,g. 𝐅⦃⫯*[n]g⦄ → 𝐅⦃g⦄.
+#n elim n -n /3 width=3 by isfin_inv_push/
+qed.
-(* Basic inversion lemmas ***************************************************)
+(* Properties with tail *****************************************************)
-lemma isfin_inv_next: ∀g. 𝐅⦃g⦄ → ∀f. ⫯f = g → 𝐅⦃f⦄.
-#g * #n #H #f #H0 elim (fcla_inv_nx … H … H0) -g
-/2 width=2 by ex_intro/
+lemma isfin_tl: ∀f. 𝐅⦃f⦄ → 𝐅⦃⫱f⦄.
+#f elim (pn_split f) * #g #H #Hf destruct
+/3 width=3 by isfin_inv_push, isfin_inv_next/
+qed.
+
+(* Inversion lemmas with tail ***********************************************)
+
+lemma isfin_inv_tl: ∀f. 𝐅⦃⫱f⦄ → 𝐅⦃f⦄.
+#f elim (pn_split f) * /2 width=1 by isfin_next, isfin_push/
qed-.
-(* Basic forward lemmas *****************************************************)
+(* Inversion lemmas with iterated tail **************************************)
-lemma isfin_fwd_push: ∀g. 𝐅⦃g⦄ → ∀f. ↑f = g → 𝐅⦃f⦄.
-#g * /3 width=4 by fcla_inv_px, ex_intro/
+lemma isfin_inv_tls: ∀n,f. 𝐅⦃⫱*[n]f⦄ → 𝐅⦃f⦄.
+#n elim n -n /3 width=1 by isfin_inv_tl/
qed-.