-(**************************************************************************)
-(* ___ *)
-(* ||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/arith/pnat_pred.ma".
-include "ground/arith/pnat_lt.ma".
-include "ground/relocation/gr_pat.ma".
-
-(* POSITIVE APPLICATION FOR GENERIC RELOCATION MAPS *************************)
-
-(* Destructions with plt and ple ********************************************)
-
-(*** at_increasing *)
-lemma gr_pat_increasing (i2) (i1) (f):
- @❪i1,f❫ ≘ i2 → i1 ≤ i2.
-#i2 elim i2 -i2
-[ #i1 #f #Hf elim (gr_pat_inv_unit_dx … Hf) -Hf //
-| #i2 #IH * //
- #i1 #f #Hf elim (gr_pat_inv_succ_bi … Hf) -Hf [1,4: * |*: // ]
- /3 width=2 by ple_succ_bi, ple_succ_dx/
-]
-qed-.
-
-(*** at_increasing_strict *)
-lemma gr_pat_increasing_strict (g) (i1) (i2):
- @❪i1,g❫ ≘ i2 → ∀f. ↑f = g →
- ∧∧ i1 < i2 & @❪i1,f❫ ≘ ↓i2.
-#g #i1 #i2 #Hg #f #H elim (gr_pat_inv_next … Hg … H) -Hg -H
-/4 width=2 by conj, gr_pat_increasing, ple_succ_bi/
-qed-.
-
-(*** at_fwd_id_ex *)
-lemma gr_pat_des_id (f) (i): @❪i,f❫ ≘ i → ⫯⫰f = f.
-#f elim (gr_map_split_tl f) //
-#H #i #Hf elim (gr_pat_inv_next … Hf … H) -Hf -H
-#j2 #Hg #H destruct lapply (gr_pat_increasing … Hg) -Hg
-#H elim (plt_ge_false … H) -H //
-qed-.
-
-(* Constructions with ple ***************************************************)
-
-(*** at_le_ex *)
-lemma gr_pat_le_ex (j2) (i2) (f):
- @❪i2,f❫ ≘ j2 → ∀i1. i1 ≤ i2 →
- ∃∃j1. @❪i1,f❫ ≘ j1 & j1 ≤ j2.
-#j2 elim j2 -j2 [2: #j2 #IH ] #i2 #f #Hf
-[ elim (gr_pat_inv_succ_dx … Hf) -Hf [1,3: * |*: // ]
- #g [ #x2 ] #Hg [ #H2 ] #H0
- [ * /3 width=3 by gr_pat_refl, ex2_intro/
- #i1 #Hi12 destruct lapply (ple_inv_succ_bi … Hi12) -Hi12
- #Hi12 elim (IH … Hg … Hi12) -x2 -IH
- /3 width=7 by gr_pat_push, ex2_intro, ple_succ_bi/
- | #i1 #Hi12 elim (IH … Hg … Hi12) -IH -i2
- /3 width=5 by gr_pat_next, ex2_intro, ple_succ_bi/
- ]
-| elim (gr_pat_inv_unit_dx … Hf) -Hf //
- #g * -i2 #H2 #i1 #Hi12 <(ple_inv_unit_dx … Hi12)
- /3 width=3 by gr_pat_refl, ex2_intro/
-]
-qed-.
-
-(*** at_id_le *)
-lemma gr_pat_id_le (i1) (i2):
- i1 ≤ i2 → ∀f. @❪i2,f❫ ≘ i2 → @❪i1,f❫ ≘ i1.
-#i1 #i2 #H
-@(ple_ind_alt … H) -i1 -i2 [ #i2 | #i1 #i2 #_ #IH ] #f #Hf
-lapply (gr_pat_des_id … Hf) #H <H in Hf; -H
-/4 width=7 by gr_pat_inv_succ_push_succ, gr_pat_push, gr_pat_refl/
-qed-.