--- /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/rat_3.ma".
+include "ground/xoa/ex_3_2.ma".
+include "ground/arith/pnat.ma".
+include "ground/relocation/gr_tl.ma".
+
+(* POSITIVE APPLICATION FOR GENERIC RELOCATION MAPS ***********************************************************)
+
+(*** at *)
+coinductive gr_pat: relation3 gr_map pnat pnat ≝
+(*** at_refl *)
+| gr_pat_refl (f) (g) (j1) (j2):
+ ⫯f = g → 𝟏 = j1 → 𝟏 = j2 → gr_pat g j1 j2
+(*** at_push *)
+| gr_pat_push (f) (i1) (i2):
+ gr_pat f i1 i2 → ∀g,j1,j2. ⫯f = g → ↑i1 = j1 → ↑i2 = j2 → gr_pat g j1 j2
+(*** at_next *)
+| gr_pat_next (f) (i1) (i2):
+ gr_pat f i1 i2 → ∀g,j2. ↑f = g → ↑i2 = j2 → gr_pat g i1 j2
+.
+
+interpretation
+ "relational positive application (generic relocation maps)"
+ 'RAt i1 f i2 = (gr_pat f i1 i2).
+
+(*** H_at_div *)
+definition H_gr_pat_div: relation4 gr_map gr_map gr_map gr_map ≝
+ λf2,g2,f1,g1.
+ ∀jf,jg,j. @❪jf,f2❫ ≘ j → @❪jg,g2❫ ≘ j →
+ ∃∃j0. @❪j0,f1❫ ≘ jf & @❪j0,g1❫ ≘ jg.
+
+(* Basic inversion lemmas ***************************************************)
+
+(*** at_inv_ppx *)
+lemma gr_pat_inv_unit_push (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g. 𝟏 = i1 → ⫯g = f → 𝟏 = i2.
+#f #i1 #i2 * -f -i1 -i2 //
+[ #f #i1 #i2 #_ #g #j1 #j2 #_ * #_ #x #H destruct
+| #f #i1 #i2 #_ #g #j2 * #_ #x #_ #H elim (eq_inv_gr_push_next … H)
+]
+qed-.
+
+(*** at_inv_npx *)
+lemma gr_pat_inv_succ_push (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f →
+ ∃∃j2. @❪j1,g❫ ≘ j2 & ↑j2 = i2.
+#f #i1 #i2 * -f -i1 -i2
+[ #f #g #j1 #j2 #_ * #_ #x #x1 #H destruct
+| #f #i1 #i2 #Hi #g #j1 #j2 * * * #x #x1 #H #Hf >(eq_inv_gr_push_bi … Hf) -g destruct /2 width=3 by ex2_intro/
+| #f #i1 #i2 #_ #g #j2 * #_ #x #x1 #_ #H elim (eq_inv_gr_push_next … H)
+]
+qed-.
+
+(*** at_inv_xnx *)
+lemma gr_pat_inv_next (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g. ↑g = f →
+ ∃∃j2. @❪i1,g❫ ≘ j2 & ↑j2 = i2.
+#f #i1 #i2 * -f -i1 -i2
+[ #f #g #j1 #j2 * #_ #_ #x #H elim (eq_inv_gr_next_push … H)
+| #f #i1 #i2 #_ #g #j1 #j2 * #_ #_ #x #H elim (eq_inv_gr_next_push … H)
+| #f #i1 #i2 #Hi #g #j2 * * #x #H >(eq_inv_gr_next_bi … H) -g /2 width=3 by ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(*** at_inv_ppn *)
+lemma gr_pat_inv_unit_push_succ (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g,j2. 𝟏 = i1 → ⫯g = f → ↑j2 = i2 → ⊥.
+#f #i1 #i2 #Hf #g #j2 #H1 #H <(gr_pat_inv_unit_push … Hf … H1 H) -f -g -i1 -i2
+#H destruct
+qed-.
+
+(*** at_inv_npp *)
+lemma gr_pat_inv_succ_push_unit (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f → 𝟏 = i2 → ⊥.
+#f #i1 #i2 #Hf #g #j1 #H1 #H elim (gr_pat_inv_succ_push … Hf … H1 H) -f -i1
+#x2 #Hg * -i2 #H destruct
+qed-.
+
+(*** at_inv_npn *)
+lemma gr_pat_inv_succ_push_succ (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g,j1,j2. ↑j1 = i1 → ⫯g = f → ↑j2 = i2 → @❪j1,g❫ ≘ j2.
+#f #i1 #i2 #Hf #g #j1 #j2 #H1 #H elim (gr_pat_inv_succ_push … Hf … H1 H) -f -i1
+#x2 #Hg * -i2 #H destruct //
+qed-.
+
+(*** at_inv_xnp *)
+lemma gr_pat_inv_next_unit (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g. ↑g = f → 𝟏 = i2 → ⊥.
+#f #i1 #i2 #Hf #g #H elim (gr_pat_inv_next … Hf … H) -f
+#x2 #Hg * -i2 #H destruct
+qed-.
+
+(*** at_inv_xnn *)
+lemma gr_pat_inv_next_succ (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g,j2. ↑g = f → ↑j2 = i2 → @❪i1,g❫ ≘ j2.
+#f #i1 #i2 #Hf #g #j2 #H elim (gr_pat_inv_next … Hf … H) -f
+#x2 #Hg * -i2 #H destruct //
+qed-.
+
+(*** at_inv_pxp *)
+lemma gr_pat_inv_unit_bi (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → 𝟏 = i1 → 𝟏 = i2 → ∃g. ⫯g = f.
+#f elim (gr_map_split_tl … f) /2 width=2 by ex_intro/
+#H #i1 #i2 #Hf #H1 #H2 cases (gr_pat_inv_next_unit … Hf … H H2)
+qed-.
+
+(*** at_inv_pxn *)
+lemma gr_pat_inv_unit_succ (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀j2. 𝟏 = i1 → ↑j2 = i2 →
+ ∃∃g. @❪i1,g❫ ≘ j2 & ↑g = f.
+#f elim (gr_map_split_tl … f)
+#H #i1 #i2 #Hf #j2 #H1 #H2
+[ elim (gr_pat_inv_unit_push_succ … Hf … H1 H H2)
+| /3 width=5 by gr_pat_inv_next_succ, ex2_intro/
+]
+qed-.
+
+(*** at_inv_nxp *)
+lemma gr_pat_inv_succ_unit (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀j1. ↑j1 = i1 → 𝟏 = i2 → ⊥.
+#f elim (gr_map_split_tl f)
+#H #i1 #i2 #Hf #j1 #H1 #H2
+[ elim (gr_pat_inv_succ_push_unit … Hf … H1 H H2)
+| elim (gr_pat_inv_next_unit … Hf … H H2)
+]
+qed-.
+
+(*** at_inv_nxn *)
+lemma gr_pat_inv_succ_bi (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀j1,j2. ↑j1 = i1 → ↑j2 = i2 →
+ ∨∨ ∃∃g. @❪j1,g❫ ≘ j2 & ⫯g = f
+ | ∃∃g. @❪i1,g❫ ≘ j2 & ↑g = f.
+#f elim (gr_map_split_tl f) *
+/4 width=7 by gr_pat_inv_next_succ, gr_pat_inv_succ_push_succ, ex2_intro, or_intror, or_introl/
+qed-.
+
+(* Note: the following inversion lemmas must be checked *)
+(*** at_inv_xpx *)
+lemma gr_pat_inv_push (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g. ⫯g = f →
+ ∨∨ ∧∧ 𝟏 = i1 & 𝟏 = i2
+ | ∃∃j1,j2. @❪j1,g❫ ≘ j2 & ↑j1 = i1 & ↑j2 = i2.
+#f * [2: #i1 ] #i2 #Hf #g #H
+[ elim (gr_pat_inv_succ_push … Hf … H) -f /3 width=5 by or_intror, ex3_2_intro/
+| >(gr_pat_inv_unit_push … Hf … H) -f /3 width=1 by conj, or_introl/
+]
+qed-.
+
+(*** at_inv_xpp *)
+lemma gr_pat_inv_push_unit (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g. ⫯g = f → 𝟏 = i2 → 𝟏 = i1.
+#f #i1 #i2 #Hf #g #H elim (gr_pat_inv_push … Hf … H) -f * //
+#j1 #j2 #_ #_ * -i2 #H destruct
+qed-.
+
+(*** at_inv_xpn *)
+lemma gr_pat_inv_push_succ (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀g,j2. ⫯g = f → ↑j2 = i2 →
+ ∃∃j1. @❪j1,g❫ ≘ j2 & ↑j1 = i1.
+#f #i1 #i2 #Hf #g #j2 #H elim (gr_pat_inv_push … Hf … H) -f *
+[ #_ * -i2 #H destruct
+| #x1 #x2 #Hg #H1 * -i2 #H destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+(*** at_inv_xxp *)
+lemma gr_pat_inv_unit_dx (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → 𝟏 = i2 →
+ ∃∃g. 𝟏 = i1 & ⫯g = f.
+#f elim (gr_map_split_tl f)
+#H #i1 #i2 #Hf #H2
+[ /3 width=6 by gr_pat_inv_push_unit, ex2_intro/
+| elim (gr_pat_inv_next_unit … Hf … H H2)
+]
+qed-.
+
+(*** at_inv_xxn *)
+lemma gr_pat_inv_succ_dx (f) (i1) (i2):
+ @❪i1,f❫ ≘ i2 → ∀j2. ↑j2 = i2 →
+ ∨∨ ∃∃g,j1. @❪j1,g❫ ≘ j2 & ↑j1 = i1 & ⫯g = f
+ | ∃∃g. @❪i1,g❫ ≘ j2 & ↑g = f.
+#f elim (gr_map_split_tl f)
+#H #i1 #i2 #Hf #j2 #H2
+[ elim (gr_pat_inv_push_succ … Hf … H H2) -i2 /3 width=5 by or_introl, ex3_2_intro/
+| lapply (gr_pat_inv_next_succ … Hf … H H2) -i2 /3 width=3 by or_intror, ex2_intro/
+]
+qed-.
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