X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Frelocation%2Ftrace_at.ma;h=b40b1c87b3be088ea46e2c08fea38a329e561cc9;hb=fb5c93c9812ea39fb78f1470da2095c80822e158;hp=f4b6f8d3ad033753006b53d8247855b8bb261ef8;hpb=1e94683f160df35b31f1eee8f4d99a6ec8008b36;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/trace_at.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/trace_at.ma
index f4b6f8d3a..b40b1c87b 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/trace_at.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/trace_at.ma
@@ -18,132 +18,184 @@ include "ground_2/relocation/trace.ma".
 (* RELOCATION TRACE *********************************************************)
 
 inductive at: trace → relation nat ≝
+   | at_empty: at (◊) 0 0
    | at_zero : ∀cs. at (Ⓣ @ cs) 0 0
-   | at_succ : ∀cs,i,j. at cs i j → at (Ⓣ @ cs) (⫯i) (⫯j)
-   | at_false: ∀cs,i,j. at cs i j → at (Ⓕ @ cs) i (⫯j).
+   | at_succ : ∀cs,i1,i2. at cs i1 i2 → at (Ⓣ @ cs) (⫯i1) (⫯i2)
+   | at_false: ∀cs,i1,i2. at cs i1 i2 → at (Ⓕ @ cs) i1 (⫯i2).
 
 interpretation "relocation (trace)"
    'RAt i1 cs i2 = (at cs i1 i2).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact at_inv_empty_aux: ∀cs,i,j. @⦃i, cs⦄ ≡ j → cs = ◊ → ⊥.
-#cs #i #j * -cs -i -j
-#cs [2,3: #i #j #_ ] #H destruct
+fact at_inv_empty_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → cs = ◊ → i1 = 0 ∧ i2 = 0.
+#cs #i1 #i2 * -cs -i1 -i2 /2 width=1 by conj/
+#cs #i1 #i2 #_ #H destruct
 qed-.
 
-lemma at_inv_empty: ∀i,j. @⦃i, ◊⦄ ≡ j → ⊥.
+lemma at_inv_empty: ∀i1,i2. @⦃i1, ◊⦄ ≡ i2 → i1 = 0 ∧ i2 = 0.
 /2 width=5 by at_inv_empty_aux/ qed-.
 
-fact at_inv_true_aux: ∀cs,i,j. @⦃i, cs⦄ ≡ j → ∀tl. cs = Ⓣ @ tl →
-                      (i = 0 ∧ j = 0) ∨
-                      ∃∃i0,j0. i = ⫯i0 & j = ⫯j0 & @⦃i0, tl⦄ ≡ j0.
-#cs #i #j * -cs -i -j
-#cs [2,3: #i #j #Hij ] #tl #H destruct
+lemma at_inv_empty_zero_sn: ∀i. @⦃0, ◊⦄ ≡ i → i = 0.
+#i #H elim (at_inv_empty … H) -H //
+qed-.
+
+lemma at_inv_empty_zero_dx: ∀i. @⦃i, ◊⦄ ≡ 0 → i = 0.
+#i #H elim (at_inv_empty … H) -H //
+qed-.
+
+lemma at_inv_empty_succ_sn: ∀i1,i2. @⦃⫯i1, ◊⦄ ≡ i2 → ⊥.
+#i1 #i2 #H elim (at_inv_empty … H) -H #H1 #H2 destruct
+qed-.
+
+lemma at_inv_empty_succ_dx: ∀i1,i2. @⦃i1, ◊⦄ ≡ ⫯i2 → ⊥.
+#i1 #i2 #H elim (at_inv_empty … H) -H #H1 #H2 destruct
+qed-.
+
+fact at_inv_true_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → ∀tl. cs = Ⓣ @ tl →
+                      (i1 = 0 ∧ i2 = 0) ∨
+                      ∃∃j1,j2. i1 = ⫯j1 & i2 = ⫯j2 & @⦃j1, tl⦄ ≡ j2.
+#cs #i1 #i2 * -cs -i1 -i2
+[2,3,4: #cs [2,3: #i1 #i2 #Hij ] ] #tl #H destruct
 /3 width=5 by ex3_2_intro, or_introl, or_intror, conj/
 qed-.
 
-lemma at_inv_true: ∀cs,i,j. @⦃i, Ⓣ @ cs⦄ ≡ j →
-                   (i = 0 ∧ j = 0) ∨
-                   ∃∃i0,j0. i = ⫯i0 & j = ⫯j0 & @⦃i0, cs⦄ ≡ j0.
+lemma at_inv_true: ∀cs,i1,i2. @⦃i1, Ⓣ @ cs⦄ ≡ i2 →
+                   (i1 = 0 ∧ i2 = 0) ∨
+                   ∃∃j1,j2. i1 = ⫯j1 & i2 = ⫯j2 & @⦃j1, cs⦄ ≡ j2.
 /2 width=3 by at_inv_true_aux/ qed-.
 
-lemma at_inv_true_zero_sn: ∀cs,j. @⦃0, Ⓣ @ cs⦄ ≡ j → j = 0.
-#cs #j #H elim (at_inv_true … H) -H * //
-#i0 #j0 #H destruct
+lemma at_inv_true_zero_sn: ∀cs,i. @⦃0, Ⓣ @ cs⦄ ≡ i → i = 0.
+#cs #i #H elim (at_inv_true … H) -H * //
+#j1 #j2 #H destruct
 qed-.
 
 lemma at_inv_true_zero_dx: ∀cs,i. @⦃i, Ⓣ @ cs⦄ ≡ 0 → i = 0.
 #cs #i #H elim (at_inv_true … H) -H * //
-#i0 #j0 #_ #H destruct
+#j1 #j2 #_ #H destruct
 qed-.
 
-lemma at_inv_true_succ_sn: ∀cs,i,j. @⦃⫯i, Ⓣ @ cs⦄ ≡ j →
-                           ∃∃j0. j = ⫯j0 & @⦃i, cs⦄ ≡ j0.
-#cs #i #j #H elim (at_inv_true … H) -H *
+lemma at_inv_true_succ_sn: ∀cs,i1,i2. @⦃⫯i1, Ⓣ @ cs⦄ ≡ i2 →
+                           ∃∃j2. i2 = ⫯j2 & @⦃i1, cs⦄ ≡ j2.
+#cs #i1 #i2 #H elim (at_inv_true … H) -H *
 [ #H destruct
-| #i0 #j0 #H1 #H2 destruct /2 width=3 by ex2_intro/
+| #j1 #j2 #H1 #H2 destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma at_inv_true_succ_dx: ∀cs,i,j. @⦃i, Ⓣ @ cs⦄ ≡ ⫯j →
-                           ∃∃i0. i = ⫯i0 & @⦃i0, cs⦄ ≡ j.
-#cs #i #j #H elim (at_inv_true … H) -H *
+lemma at_inv_true_succ_dx: ∀cs,i1,i2. @⦃i1, Ⓣ @ cs⦄ ≡ ⫯i2 →
+                           ∃∃j1. i1 = ⫯j1 & @⦃j1, cs⦄ ≡ i2.
+#cs #i1 #i2 #H elim (at_inv_true … H) -H *
 [ #_ #H destruct
-| #i0 #j0 #H1 #H2 destruct /2 width=3 by ex2_intro/
+| #j1 #j2 #H1 #H2 destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma at_inv_true_succ: ∀cs,i,j. @⦃⫯i, Ⓣ @ cs⦄ ≡ ⫯j →
-                        @⦃i, cs⦄ ≡ j.
-#cs #i #j #H elim (at_inv_true … H) -H *
+lemma at_inv_true_succ: ∀cs,i1,i2. @⦃⫯i1, Ⓣ @ cs⦄ ≡ ⫯i2 →
+                        @⦃i1, cs⦄ ≡ i2.
+#cs #i1 #i2 #H elim (at_inv_true … H) -H *
 [ #H destruct
-| #i0 #j0 #H1 #H2 destruct //
+| #j1 #j2 #H1 #H2 destruct //
 ]
 qed-.
 
-lemma at_inv_true_O_S: ∀cs,j. @⦃0, Ⓣ @ cs⦄ ≡ ⫯j → ⊥.
-#cs #j #H elim (at_inv_true … H) -H *
+lemma at_inv_true_O_S: ∀cs,i. @⦃0, Ⓣ @ cs⦄ ≡ ⫯i → ⊥.
+#cs #i #H elim (at_inv_true … H) -H *
 [ #_ #H destruct
-| #i0 #j0 #H1 destruct
+| #j1 #j2 #H destruct
 ]
 qed-.
 
 lemma at_inv_true_S_O: ∀cs,i. @⦃⫯i, Ⓣ @ cs⦄ ≡ 0 → ⊥.
 #cs #i #H elim (at_inv_true … H) -H *
 [ #H destruct
-| #i0 #j0 #_ #H1 destruct
+| #j1 #j2 #_ #H destruct
 ]
 qed-.
 
-fact at_inv_false_aux: ∀cs,i,j. @⦃i, cs⦄ ≡ j → ∀tl. cs = Ⓕ @ tl →
-                       ∃∃j0. j = ⫯j0 & @⦃i, tl⦄ ≡ j0.
-#cs #i #j * -cs -i -j
-#cs [2,3: #i #j #Hij ] #tl #H destruct
+fact at_inv_false_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → ∀tl. cs = Ⓕ @ tl →
+                       ∃∃j2. i2 = ⫯j2 & @⦃i1, tl⦄ ≡ j2.
+#cs #i1 #i2 * -cs -i1 -i2
+[2,3,4: #cs [2,3: #i1 #i2 #Hij ] ] #tl #H destruct
 /2 width=3 by ex2_intro/
 qed-.
 
-lemma at_inv_false: ∀cs,i,j. @⦃i, Ⓕ @ cs⦄ ≡ j →
-                    ∃∃j0. j = ⫯j0 & @⦃i, cs⦄ ≡ j0.
+lemma at_inv_false: ∀cs,i1,i2. @⦃i1, Ⓕ @ cs⦄ ≡ i2 →
+                    ∃∃j2. i2 = ⫯j2 & @⦃i1, cs⦄ ≡ j2.
 /2 width=3 by at_inv_false_aux/ qed-.
 
-lemma at_inv_false_S: ∀cs,i,j. @⦃i, Ⓕ @ cs⦄ ≡ ⫯j → @⦃i, cs⦄ ≡ j.
-#cs #i #j #H elim (at_inv_false … H) -H
-#j0 #H destruct //
+lemma at_inv_false_S: ∀cs,i1,i2. @⦃i1, Ⓕ @ cs⦄ ≡ ⫯i2 → @⦃i1, cs⦄ ≡ i2.
+#cs #i1 #i2 #H elim (at_inv_false … H) -H
+#j2 #H destruct //
 qed-.
 
 lemma at_inv_false_O: ∀cs,i. @⦃i, Ⓕ @ cs⦄ ≡ 0 → ⊥.
 #cs #i #H elim (at_inv_false … H) -H
-#j0 #H destruct
+#j2 #H destruct
+qed-.
+
+lemma at_inv_le: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → i1 ≤ ∥cs∥ ∧ i2 ≤ |cs|.
+#cs #i1 #i2 #H elim H -cs -i1 -i2 /2 width=1 by conj/
+#cs #i1 #i2 #_ * /3 width=1 by le_S_S, conj/ 
+qed-.
+
+lemma at_inv_gt1: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → ∥cs∥  < i1 → ⊥.
+#cs #i1 #i2 #H elim (at_inv_le … H) -H /2 width=4 by lt_le_false/
+qed-.
+
+lemma at_inv_gt2: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → |cs| < i2 → ⊥.
+#cs #i1 #i2 #H elim (at_inv_le … H) -H /2 width=4 by lt_le_false/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma at_monotonic: ∀cs,i2,j2. @⦃i2, cs⦄ ≡ j2 → ∀i1. i1 < i2 →
-                    ∃∃j1. @⦃i1, cs⦄ ≡ j1 & j1 < j2.
-#cs #i2 #j2 #H elim H -cs -i2 -j2
-[ #cs #i1 #H elim (lt_zero_false … H)
-| #cs #i #j #Hij #IH * /2 width=3 by ex2_intro, at_zero/
-  #i1 #H lapply (lt_S_S_to_lt … H) -H
-  #H elim (IH … H) -i
-  #j1 #Hij1 #H /3 width=3 by le_S_S, ex2_intro, at_succ/
-| #cs #i #j #_ #IH #i1 #H elim (IH … H) -i
+(* Note: lemma 250 *)
+lemma at_le: ∀cs,i1. i1 ≤ ∥cs∥ →
+             ∃∃i2. @⦃i1, cs⦄ ≡ i2 & i2 ≤ |cs|.
+#cs elim cs -cs
+[ #i1 #H <(le_n_O_to_eq … H) -i1 /2 width=3 by at_empty, ex2_intro/
+| * #cs #IH
+  [ * /2 width=3 by at_zero, ex2_intro/
+    #i1 #H lapply (le_S_S_to_le … H) -H
+    #H elim (IH … H) -IH -H /3 width=3 by at_succ, le_S_S, ex2_intro/
+  | #i1 #H elim (IH … H) -IH -H /3 width=3 by at_false, le_S_S, ex2_intro/
+  ]
+]
+qed-.
+
+lemma at_top: ∀cs. @⦃∥cs∥, cs⦄ ≡ |cs|.
+#cs elim cs -cs // * /2 width=1 by at_succ, at_false/
+qed. 
+
+lemma at_monotonic: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → ∀j1. j1 < i1 →
+                    ∃∃j2. @⦃j1, cs⦄ ≡ j2 & j2 < i2.
+#cs #i1 #i2 #H elim H -cs -i1 -i2
+[ #j1 #H elim (lt_zero_false … H)
+| #cs #j1 #H elim (lt_zero_false … H)
+| #cs #i1 #i2 #Hij #IH * /2 width=3 by ex2_intro, at_zero/
+  #j1 #H lapply (lt_S_S_to_lt … H) -H
+  #H elim (IH … H) -i1
+  #j2 #Hj12 #H /3 width=3 by le_S_S, ex2_intro, at_succ/
+| #cs #i1 #i2 #_ #IH #j1 #H elim (IH … H) -i1
   /3 width=3 by le_S_S, ex2_intro, at_false/
 ]
 qed-.
 
-lemma at_at_dec: ∀cs,i,j. Decidable (@⦃i, cs⦄ ≡ j).
+lemma at_at_dec: ∀cs,i1,i2. Decidable (@⦃i1, cs⦄ ≡ i2).
 #cs elim cs -cs [ | * #cs #IH ]
-[ /3 width=3 by at_inv_empty, or_intror/
-| * [2: #i ] * [2,4: #j ]
-  [ elim (IH i j) -IH
+[ * [2: #i1 ] * [2,4: #i2 ]
+  [4: /2 width=1 by at_empty, or_introl/
+  |*: @or_intror #H elim (at_inv_empty … H) #H1 #H2 destruct  
+  ]
+| * [2: #i1 ] * [2,4: #i2 ]
+  [ elim (IH i1 i2) -IH
     /4 width=1 by at_inv_true_succ, at_succ, or_introl, or_intror/
   | -IH /3 width=3 by at_inv_true_O_S, or_intror/
   | -IH /3 width=3 by at_inv_true_S_O, or_intror/
   | -IH /2 width=1 by or_introl, at_zero/
   ]
-| #i * [2: #j ]
-  [ elim (IH i j) -IH
+| #i1 * [2: #i2 ]
+  [ elim (IH i1 i2) -IH
     /4 width=1 by at_inv_false_S, at_false, or_introl, or_intror/
   | -IH /3 width=3 by at_inv_false_O, or_intror/
   ]
@@ -152,42 +204,40 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma at_increasing: ∀cs,i,j. @⦃i, cs⦄ ≡ j → i ≤ j.
-#cs #i elim i -i //
-#i0 #IHi #j #H elim (at_monotonic … H i0) -H
+lemma at_increasing: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → i1 ≤ i2.
+#cs #i1 elim i1 -i1 //
+#j1 #IHi #i2 #H elim (at_monotonic … H j1) -H
 /3 width=3 by le_to_lt_to_lt/
 qed-.
 
-lemma at_length_lt: ∀cs,i,j. @⦃i, cs⦄ ≡ j → j < |cs|.
-#cs elim cs -cs [ | * #cs #IH ] #i #j #H normalize
-[ elim (at_inv_empty … H)
-| elim (at_inv_true … H) * -H //
-  #i0 #j0 #H1 #H2 #Hij0 destruct /3 width=2 by le_S_S/
-| elim (at_inv_false … H) -H
-  #j0 #H #H0 destruct /3 width=2 by le_S_S/
-]
+lemma at_increasing_strict: ∀cs,i1,i2. @⦃i1, Ⓕ @ cs⦄ ≡ i2 →
+                            i1 < i2 ∧ @⦃i1, cs⦄ ≡ ⫰i2.
+#cs #i1 #i2 #H elim (at_inv_false … H) -H
+#j2 #H #Hj2 destruct /4 width=2 by conj, at_increasing, le_S_S/
 qed-.
 
 (* Main properties **********************************************************)
 
-theorem at_mono: ∀cs,i,j1. @⦃i, cs⦄ ≡ j1 → ∀j2. @⦃i, cs⦄ ≡ j2 → j1 = j2.
-#cs #i #j1 #H elim H -cs -i -j1
-#cs [ |2,3: #i #j1 #_ #IH ] #j2 #H
-[ lapply (at_inv_true_zero_sn … H) -H //
-| elim (at_inv_true_succ_sn … H) -H
-  #j0 #H destruct #H >(IH … H) -cs -i -j1 //
+theorem at_mono: ∀cs,i,i1. @⦃i, cs⦄ ≡ i1 → ∀i2. @⦃i, cs⦄ ≡ i2 → i1 = i2.
+#cs #i #i1 #H elim H -cs -i -i1
+[2,3,4: #cs [2,3: #i #i1 #_ #IH ] ] #i2 #H
+[ elim (at_inv_true_succ_sn … H) -H
+  #j2 #H destruct #H >(IH … H) -cs -i -i1 //
 | elim (at_inv_false … H) -H
-  #j0 #H destruct #H >(IH … H) -cs -i -j1 //
+  #j2 #H destruct #H >(IH … H) -cs -i -i1 //
+| /2 width=2 by at_inv_true_zero_sn/
+| /2 width=1 by at_inv_empty_zero_sn/
 ]
 qed-.
 
-theorem at_inj: ∀cs,i1,j. @⦃i1, cs⦄ ≡ j → ∀i2. @⦃i2, cs⦄ ≡ j → i1 = i2.
-#cs #i1 #j #H elim H -cs -i1 -j
-#cs [ |2,3: #i1 #j #_ #IH ] #i2 #H
-[ lapply (at_inv_true_zero_dx … H) -H //
+theorem at_inj: ∀cs,i1,i. @⦃i1, cs⦄ ≡ i → ∀i2. @⦃i2, cs⦄ ≡ i → i1 = i2.
+#cs #i1 #i #H elim H -cs -i1 -i
+[2,3,4: #cs [ |2,3: #i1 #i #_ #IH ] ] #i2 #H
+[ /2 width=2 by at_inv_true_zero_dx/
 | elim (at_inv_true_succ_dx … H) -H
-  #i0 #H destruct #H >(IH … H) -cs -i1 -j //
+  #j2 #H destruct #H >(IH … H) -cs -i1 -i //
 | elim (at_inv_false … H) -H
-  #j0 #H destruct #H >(IH … H) -cs -i1 -j0 //
+  #j #H destruct #H >(IH … H) -cs -i1 -j //
+| /2 width=1 by at_inv_empty_zero_dx/
 ]
 qed-.