X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground%2Frelocation%2Ftr_pat.ma;h=73b6dca66b6b85d3de7157b919295d62a3591d20;hb=ad6182251b8192ee7d25c53156afbce35e3715b6;hp=ec7b76a5b4771e88573e09e859fcc4a5034e63a4;hpb=33d0a7a9029859be79b25b5a495e0f30dab11f37;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/tr_pat.ma b/matita/matita/contribs/lambdadelta/ground/relocation/tr_pat.ma
index ec7b76a5b..73b6dca66 100644
--- a/matita/matita/contribs/lambdadelta/ground/relocation/tr_pat.ma
+++ b/matita/matita/contribs/lambdadelta/ground/relocation/tr_pat.ma
@@ -12,114 +12,68 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/notation/functions/apply_2.ma".
-include "ground/relocation/tr_map.ma".
-(*
 include "ground/arith/pnat_le_plus.ma".
-include "ground/relocation/pstream_eq.ma".
-include "ground/relocation/rtmap_istot.ma".
-*)
-(* POSITIVE APPLICATION FOR TOTAL RELOCATION MAPS ***************************)
-
-rec definition tr_pat (i: pnat) on i: tr_map → pnat.
-* #p #f cases i -i
-[ @p
-| #i lapply (tr_pat i f) -tr_pat -i -f
-  #i @(i+p)
-]
-defined.
-
-interpretation
-  "functional positive application (total relocation maps)"
-  'apply f i = (tr_pat i f).
+include "ground/relocation/pr_pat_lt.ma".
+include "ground/relocation/tr_map.ma".
 
-(* Properties on at (specific) ************************************************)
+(* TOTAL RELOCATION MAPS ****************************************************)
 
-lemma at_O1: ∀i2,f. @❪𝟏, i2⨮f❫ ≘ i2.
-#i2 elim i2 -i2 /2 width=5 by gr_pat_refl, gr_pat_next/
-qed.
+(* Constructions with pr_pat ***********************************************)
 
-lemma at_S1: ∀p,f,i1,i2. @❪i1, f❫ ≘ i2 → @❪↑i1, p⨮f❫ ≘ i2+p.
-#p elim p -p /3 width=7 by gr_pat_push, gr_pat_next/
+(*** at_O1 *)
+lemma tr_pat_unit_sn: ∀i2,f. ＠⧣❨𝟏,𝐭❨i2⨮f❩❩ ≘ i2.
+#i2 elim i2 -i2 /2 width=5 by pr_pat_refl, pr_pat_next/
 qed.
 
-lemma at_total: ∀i1,f. @❪i1, f❫ ≘ f@❨i1❩.
-#i1 elim i1 -i1
-[ * // | #i #IH * /3 width=1 by at_S1/ ]
+(*** at_S1 *)
+lemma tr_pat_succ_sn: ∀p,f,i1,i2. ＠⧣❨i1, 𝐭❨f❩❩ ≘ i2 → ＠⧣❨↑i1, 𝐭❨p⨮f❩❩ ≘ i2+p.
+#p elim p -p /3 width=7 by pr_pat_push, pr_pat_next/
 qed.
 
-lemma at_istot: ∀f. 𝐓❪f❫.
-/2 width=2 by ex_intro/ qed.
-
-lemma at_plus2: ∀f,i1,i,p,q. @❪i1, p⨮f❫ ≘ i → @❪i1, (p+q)⨮f❫ ≘ i+q.
+(*** at_plus2 *)
+lemma tr_pat_plus_dx (f):
+      ∀i1,i,p,q. ＠⧣❨i1, 𝐭❨p⨮f❩❩ ≘ i → ＠⧣❨i1, 𝐭❨(p+q)⨮f❩❩ ≘ i+q.
 #f #i1 #i #p #q #H elim q -q
-/2 width=5 by gr_pat_next/
+/2 width=5 by pr_pat_next/
 qed.
 
-(* Inversion lemmas on at (specific) ******************************************)
+(* Inversions with pr_pat ***************************************************)
 
-lemma at_inv_O1: ∀f,p,i2. @❪𝟏, p⨮f❫ ≘ i2 → p = i2.
-#f #p elim p -p /2 width=6 by gr_pat_inv_unit_push/
-#p #IH #i2 #H elim (gr_pat_inv_next … H) -H [|*: // ]
+(*** at_inv_O1 *)
+lemma tr_pat_inv_unit_sn (f):
+      ∀p,i2. ＠⧣❨𝟏, 𝐭❨p⨮f❩❩ ≘ i2 → p = i2.
+#f #p elim p -p /2 width=6 by pr_pat_inv_unit_push/
+#p #IH #i2 #H elim (pr_pat_inv_next … H) -H [|*: // ]
 #j2 #Hj * -i2 /3 width=1 by eq_f/
 qed-.
 
-lemma at_inv_S1: ∀f,p,j1,i2. @❪↑j1, p⨮f❫ ≘ i2 →
-                 ∃∃j2. @❪j1, f❫ ≘ j2 & j2+p = i2.
-#f #p elim p -p /2 width=5 by gr_pat_inv_succ_push/
-#p #IH #j1 #i2 #H elim (gr_pat_inv_next … H) -H [|*: // ]
+(*** at_inv_S1 *)
+lemma tr_pat_inv_succ_sn (f):
+      ∀p,j1,i2. ＠⧣❨↑j1, 𝐭❨p⨮f❩❩ ≘ i2 →
+      ∃∃j2. ＠⧣❨j1, 𝐭❨f❩❩ ≘ j2 & j2+p = i2.
+#f #p elim p -p /2 width=5 by pr_pat_inv_succ_push/
+#p #IH #j1 #i2 #H elim (pr_pat_inv_next … H) -H [|*: // ]
 #j2 #Hj * -i2 elim (IH … Hj) -IH -Hj
 #i2 #Hi * -j2 /2 width=3 by ex2_intro/
 qed-.
 
-lemma at_inv_total: ∀f,i1,i2. @❪i1, f❫ ≘ i2 → f@❨i1❩ = i2.
-/2 width=6 by fr2_nat_mono/ qed-.
-
-(* Forward lemmas on at (specific) *******************************************)
+(* Destructions with pr_pat *************************************************)
 
-lemma at_increasing_plus: ∀f,p,i1,i2. @❪i1, p⨮f❫ ≘ i2 → i1 + p ≤ ↑i2.
+(* Note: a better conclusion would be: "i1 + ↓p ≤ i2" *)
+(*** at_increasing_plus *)
+lemma tr_pat_increasing_plus (f):
+      ∀p,i1,i2. ＠⧣❨i1, 𝐭❨p⨮f❩❩ ≘ i2 → i1 + p ≤ ↑i2.
 #f #p *
-[ #i2 #H <(at_inv_O1 … H) -i2 //
-| #i1 #i2 #H elim (at_inv_S1 … H) -H
+[ #i2 #H <(tr_pat_inv_unit_sn … H) -i2 //
+| #i1 #i2 #H elim (tr_pat_inv_succ_sn … H) -H
   #j1 #Ht * -i2 <pplus_succ_sn 
-  /4 width=2 by gr_pat_increasing, ple_plus_bi_dx, ple_succ_bi/
+  /4 width=2 by pr_pat_increasing, ple_plus_bi_dx, ple_succ_bi/
 ]
 qed-.
 
-lemma at_fwd_id: ∀f,p,i. @❪i, p⨮f❫ ≘ i → 𝟏 = p.
-#f #p #i #H elim (gr_pat_des_id … H) -H
-#g #H elim (push_inv_seq_dx … H) -H //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tr_pat_O1: ∀p,f. (p⨮f)@❨𝟏❩ = p.
-// qed.
-
-lemma tr_pat_S1: ∀p,f,i. (p⨮f)@❨↑i❩ = f@❨i❩+p.
-// qed.
-
-lemma tr_pat_eq_repl (i): gr_eq_repl … (λf1,f2. f1@❨i❩ = f2@❨i❩).
-#i elim i -i [2: #i #IH ] * #p1 #f1 * #p2 #f2 #H
-elim (eq_inv_seq_aux … H) -H #Hp #Hf //
->tr_pat_S1 >tr_pat_S1 /3 width=1 by eq_f2/
-qed.
-
-lemma tr_pat_S2: ∀f,i. (↑f)@❨i❩ = ↑(f@❨i❩).
-* #p #f * //
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem tr_pat_inj: ∀f,i1,i2,j. f@❨i1❩ = j → f@❨i2❩ = j → i1 = i2.
-/2 width=4 by gr_pat_inj/ qed-.
-
-corec theorem nstream_eq_inv_ext: ∀f1,f2. (∀i. f1@❨i❩ = f2@❨i❩) → f1 ≗ f2.
-* #p1 #f1 * #p2 #f2 #Hf @stream_eq_cons
-[ @(Hf (𝟏))
-| @nstream_eq_inv_ext -nstream_eq_inv_ext #i
-  ltr_pat (Hf (𝟏)) >tr_pat_O1 >tr_pat_O1 #H destruct
-  ltr_pat (Hf (↑i)) >tr_pat_S1 >tr_pat_S1 #H
-  /3 width=2 by eq_inv_pplus_bi_dx, eq_inv_psucc_bi/
-]
+(*** at_fwd_id *)
+lemma tr_pat_des_id (f):
+      ∀p,i. ＠⧣❨i, 𝐭❨p⨮f❩❩ ≘ i → 𝟏 = p.
+#f #p #i #H lapply (pr_pat_des_id … H) -H #H
+elim (eq_inv_pr_push_cons … H) -H //
 qed-.