+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground/notation/relations/isdivergent_1.ma".
-include "ground/relocation/rtmap_nexts.ma".
-include "ground/relocation/rtmap_tls.ma".
-
-(* RELOCATION MAP ***********************************************************)
-
-coinductive isdiv: predicate rtmap ≝
-| isdiv_next: ∀f,g. isdiv f → ↑f = g → isdiv g
-.
-
-interpretation "test for divergence (rtmap)"
- 'IsDivergent f = (isdiv f).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma isdiv_inv_gen: ∀g. 𝛀❪g❫ → ∃∃f. 𝛀❪f❫ & ↑f = g.
-#g * -g
-#f #g #Hf * /2 width=3 by ex2_intro/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma isdiv_inv_next: ∀g. 𝛀❪g❫ → ∀f. ↑f = g → 𝛀❪f❫.
-#g #H elim (isdiv_inv_gen … H) -H
-#f #Hf * -g #g #H >(injective_next … H) -H //
-qed-.
-
-lemma isdiv_inv_push: ∀g. 𝛀❪g❫ → ∀f. ⫯f = g → ⊥.
-#g #H elim (isdiv_inv_gen … H) -H
-#f #Hf * -g #g #H elim (discr_push_next … H)
-qed-.
-
-(* Main inversion lemmas ****************************************************)
-
-corec theorem isdiv_inv_eq_repl: ∀f1,f2. 𝛀❪f1❫ → 𝛀❪f2❫ → f1 ≡ f2.
-#f1 #f2 #H1 #H2
-cases (isdiv_inv_gen … H1) -H1
-cases (isdiv_inv_gen … H2) -H2
-/3 width=5 by eq_next/
-qed-.
-
-(* Basic properties *********************************************************)
-
-corec lemma isdiv_eq_repl_back: eq_repl_back … isdiv.
-#f1 #H cases (isdiv_inv_gen … H) -H
-#g1 #Hg1 #H1 #f2 #Hf cases (eq_inv_nx … Hf … H1) -f1
-/3 width=3 by isdiv_next/
-qed-.
-
-lemma isdiv_eq_repl_fwd: eq_repl_fwd … isdiv.
-/3 width=3 by isdiv_eq_repl_back, eq_repl_sym/ qed-.
-
-(* Alternative definition ***************************************************)
-
-corec lemma eq_next_isdiv: ∀f. ↑f ≡ f → 𝛀❪f❫.
-#f #H cases (eq_inv_nx … H) -H /4 width=3 by isdiv_next, eq_trans/
-qed.
-
-corec lemma eq_next_inv_isdiv: ∀f. 𝛀❪f❫ → ↑f ≡ f.
-#f * -f
-#f #g #Hf #Hg @(eq_next … Hg) [2: @eq_next_inv_isdiv // | skip ]
-@eq_f //
-qed-.
-
-(* Properties with iterated next ********************************************)
-
-lemma isdiv_nexts: ∀n,f. 𝛀❪f❫ → 𝛀❪↑*[n]f❫.
-#n @(nat_ind_succ … n) -n /3 width=3 by isdiv_next/
-qed.
-
-(* Inversion lemmas with iterated next **************************************)
-
-lemma isdiv_inv_nexts: ∀n,g. 𝛀❪↑*[n]g❫ → 𝛀❪g❫.
-#n @(nat_ind_succ … n) -n /3 width=3 by isdiv_inv_next/
-qed.
-
-(* Properties with tail *****************************************************)
-
-lemma isdiv_tl: ∀f. 𝛀❪f❫ → 𝛀❪⫱f❫.
-#f cases (pn_split f) * #g * -f #H
-[ elim (isdiv_inv_push … H) -H //
-| /2 width=3 by isdiv_inv_next/
-]
-qed.
-
-(* Properties with iterated tail ********************************************)
-
-lemma isdiv_tls: ∀n,g. 𝛀❪g❫ → 𝛀❪⫱*[n]g❫.
-#n @(nat_ind_succ … n) -n /3 width=1 by isdiv_tl/
-qed.