- more results on relocation

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_isfin.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isfin.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isfin.ma
index 45c8ea854656683341e2106b4668ae3e741bcf3b..875378b987c00d56b339ed8ffeb79163b6a15165 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isfin.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isfin.ma
@@ -23,8 +23,36 @@ definition isfin: predicate rtmap ≝
 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.
 
@@ -36,32 +64,33 @@ lemma isfin_next: ∀f. 𝐅⦃f⦄ → 𝐅⦃⫯f⦄.
 #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-.