milestone update in ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / gr_nat_nat.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/gr_nat_nat.ma b/matita/matita/contribs/lambdadelta/ground/relocation/gr_nat_nat.ma
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambdadelta/ground/relocation/gr_nat_nat.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/nat_lt_pred.ma".
+include "ground/relocation/gr_nat.ma".
+
+(* NON-NEGATIVE APPLICATION FOR GENERIC RELOCATION MAPS *****************************)
+
+(* Main destructions ********************************************************)
+
+theorem gr_nat_monotonic (k2) (l2) (f):
+        @↑❪l2,f❫ ≘ k2 → ∀k1,l1. @↑❪l1,f❫ ≘ k1 → l1 < l2 → k1 < k2.
+#k2 @(nat_ind_succ … k2) -k2
+[ #l2 #f #H2f elim (gr_nat_inv_zero_dx … H2f) -H2f //
+  #g #H21 #_ #k1 #l1 #_ #Hi destruct
+  elim (nlt_inv_zero_dx … Hi)
+| #k2 #IH #l2 #f #H2f #k1 @(nat_ind_succ … k1) -k1 //
+  #k1 #_ #l1 #H1f #Hl elim (nlt_inv_gen … Hl)
+  #_ #Hl2 elim (gr_nat_inv_succ_bi … H2f (↓l2)) -H2f [1,3: * |*: // ]
+  #g #H2g #H
+  [ elim (gr_nat_inv_push_succ … H1f … H) -f
+    /4 width=8 by nlt_inv_succ_bi, nlt_succ_bi/
+  | /4 width=8 by gr_nat_inv_next_succ, nlt_succ_bi/
+  ]
+]
+qed-.
+
+theorem gr_nat_inv_monotonic (k1) (l1) (f):
+        @↑❪l1,f❫ ≘ k1 → ∀k2,l2. @↑❪l2,f❫ ≘ k2 → k1 < k2 → l1 < l2.
+#k1 @(nat_ind_succ … k1) -k1
+[ #l1 #f #H1f elim (gr_nat_inv_zero_dx … H1f) -H1f //
+  #g * -l1 #H #k2 #l2 #H2f #Hk
+  lapply (nlt_des_gen … Hk) -Hk #H22
+  elim (gr_nat_inv_push_succ … H2f … (↓k2) H) -f //
+| #k1 #IH #l1 @(nat_ind_succ … l1) -l1
+  [ #f #H1f elim (gr_nat_inv_zero_succ … H1f) -H1f [ |*: // ]
+    #g #H1g #H #k2 #l2 #H2f #Hj elim (nlt_inv_succ_sn … Hj) -Hj
+    /3 width=7 by gr_nat_inv_next_succ/
+  | #l1 #_ #f #H1f #k2 #l2 #H2f #Hj elim (nlt_inv_succ_sn … Hj) -Hj
+    #Hj #H22 elim (gr_nat_inv_succ_bi … H1f) -H1f [1,4: * |*: // ]
+    #g #Hg #H
+    [ elim (gr_nat_inv_push_succ … H2f … (↓k2) H) -f
+      /3 width=7 by nlt_succ_bi/
+    | /3 width=7 by gr_nat_inv_next_succ/
+    ]
+  ]
+]
+qed-.
+
+theorem gr_nat_mono (f) (l) (l1) (l2):
+        @↑❪l,f❫ ≘ l1 → @↑❪l,f❫ ≘ l2 → l2 = l1.
+#f #l #l1 #l2 #H1 #H2 elim (nat_split_lt_eq_gt l2 l1) //
+#Hi elim (nlt_ge_false l l)
+/2 width=6 by gr_nat_inv_monotonic/
+qed-.
+
+theorem gr_nat_inj (f) (l1) (l2) (l):
+        @↑❪l1,f❫ ≘ l → @↑❪l2,f❫ ≘ l → l1 = l2.
+#f #l1 #l2 #l #H1 #H2 elim (nat_split_lt_eq_gt l2 l1) //
+#Hi elim (nlt_ge_false l l)
+/2 width=6 by gr_nat_monotonic/
+qed-.