partial commit in static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / syntax / sh_lt.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma
index d33515ef80dfb47e04a986bcdf781b1a77e1ebf3..bfeab590553cf8940ef01d581ae2515055da2e60 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground/arith/nat_lt_minus.ma".
 include "static_2/syntax/sh_props.ma".
 
 (* SORT HIERARCHY ***********************************************************)
@@ -19,38 +20,38 @@ include "static_2/syntax/sh_props.ma".
 (* strict monotonicity condition *)
 record sh_lt (h): Prop ≝
 {
-  next_lt: ∀s. s < ⫯[h]s
+  sh_next_lt: ∀s. s < ⇡[h]s
 }.
 
 (* Basic properties *********************************************************)
 
-lemma nexts_le (h): sh_lt h → ∀s,n. s ≤ (next h)^n s.
-#h #Hh #s #n elim n -n [ // ] normalize #n #IH
-lapply (next_lt … Hh ((next h)^n s)) #H
-lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
+lemma sh_nexts_le (h): sh_lt h → ∀s,n. s ≤ ⇡*[h,n]s.
+#h #Hh #s #n @(nat_ind_succ … n) -n [ // ] #n #IH <sh_nexts_succ
+lapply (sh_next_lt … Hh ((sh_next h)^n s)) #H
+lapply (nle_nlt_trans … IH H) -IH -H /2 width=2 by nlt_des_le/
 qed.
 
-lemma nexts_lt (h): sh_lt h → ∀s,n. s < (next h)^(↑n) s.
-#h #Hh #s #n normalize
-lapply (nexts_le … Hh s n) #H
-@(le_to_lt_to_lt … H) /2 width=1 by next_lt/
+lemma sh_nexts_lt (h): sh_lt h → ∀s,n. s < ⇡*[h,↑n]s.
+#h #Hh #s #n <sh_nexts_succ
+lapply (sh_nexts_le … Hh s n) #H
+@(nle_nlt_trans … H) /2 width=1 by sh_next_lt/
 qed.
 
 lemma sh_lt_nexts_inv_lt (h): sh_lt h →
-      ∀s,n1,n2. (next h)^n1 s < (next h)^n2 s → n1 < n2.
+      ∀s,n1,n2. ⇡*[h,n1]s < ⇡*[h,n2]s → n1 < n2.
 #h #Hh
 @pull_2 #n1
-elim n1 -n1
-[ #s *
-  [ #H elim (lt_refl_false … H)
-  | #n2 //
+@(nat_ind_succ … n1) -n1
+[ #s #n2 @(nat_ind_succ … n2) -n2
+  [ #H elim (nlt_inv_refl … H)
+  | #n2 #_ //
   ]
-| #n1 #IH #s *
-  [ >iter_O #H
-    elim (lt_refl_false s)
-    /3 width=3 by nexts_lt, transitive_lt/
-  | #n2 >iter_S >iter_S <(iter_n_Sm … (next h)) <(iter_n_Sm … (next h)) #H
-    /3 width=2 by lt_S_S/
+| #n1 #IH #s #n2 @(nat_ind_succ … n2) -n2
+  [ <sh_nexts_zero #H
+    elim (nlt_inv_refl s)
+    /3 width=3 by sh_nexts_lt, nlt_trans/
+  | #n2 #_ <sh_nexts_succ <sh_nexts_succ >sh_nexts_swap >sh_nexts_swap #H
+    /3 width=2 by nlt_succ_bi/
   ]
 ]
 qed-.
@@ -59,16 +60,16 @@ lemma sh_lt_acyclic (h): sh_lt h → sh_acyclic h.
 #h #Hh
 @mk_sh_acyclic
 @pull_2 #n1
-elim n1 -n1
-[ #s * [ // ] #n2 >iter_O #H
-  elim (lt_refl_false s) >H in ⊢ (??%); -H
-  /2 width=1 by nexts_lt/
-| #n1 #IH #s *
-  [ >iter_O #H -IH
-    elim (lt_refl_false s) <H in ⊢ (??%); -H
-    /2 width=1 by nexts_lt/
-  | #n2 >iter_S >iter_S <(iter_n_Sm … (next h)) <(iter_n_Sm … (next h)) #H
-    /3 width=2 by eq_f/
+@(nat_ind_succ … n1) -n1
+[ #s #n2 @(nat_ind_succ … n2) -n2 [ // ] #n2 #_ <sh_nexts_zero #H
+  elim (nlt_inv_refl s) >H in ⊢ (??%); -H
+  /2 width=1 by sh_nexts_lt/
+| #n1 #IH #s #n2 @(nat_ind_succ … n2) -n2
+  [ <sh_nexts_zero #H -IH
+    elim (nlt_inv_refl s) <H in ⊢ (??%); -H
+    /2 width=1 by sh_nexts_lt/
+  | #n2 #_ <sh_nexts_succ <sh_nexts_succ >sh_nexts_swap >sh_nexts_swap #H
+    lapply (IH … H) -IH -H //
   ]
 ]
 qed.
@@ -76,33 +77,33 @@ qed.
 lemma sh_lt_dec (h): sh_lt h → sh_decidable h.
 #h #Hh
 @mk_sh_decidable #s1 #s2
-elim (lt_or_ge s2 s1) #Hs
+elim (nat_split_lt_ge s2 s1) #Hs
 [ @or_intror * #n #H destruct
-  @(lt_le_false … Hs) /2 width=1 by nexts_le/ (**) (* full auto too slow *)
-| @(nat_elim_le_sn … Hs) -s1 -s2 #s1 #s2 #IH #Hs12
-  elim (lt_or_eq_or_gt s2 (⫯[h]s1)) #Hs21 destruct
-  [ elim (le_to_or_lt_eq … Hs12) -Hs12 #Hs12 destruct
+  @(nlt_ge_false … Hs) /2 width=1 by sh_nexts_le/ (**) (* full auto too slow *)
+| @(nle_ind_sn … Hs) -s1 -s2 #s1 #s2 #IH #Hs12
+  elim (nat_split_lt_eq_gt s2 (⇡[h]s1)) #Hs21 destruct
+  [ elim (nle_split_lt_eq … Hs12) -Hs12 #Hs12 destruct
     [ -IH @or_intror * #n #H destruct
       generalize in match Hs21; -Hs21
-      <(iter_O … (next h) s1) in ⊢ (??%→?); <(iter_S … (next h)) #H
+      >sh_nexts_unit #H
       lapply (sh_lt_nexts_inv_lt … Hh … H) -H #H
-      <(le_n_O_to_eq n) in Hs12;
-      /2 width=2 by lt_refl_false, le_S_S_to_le/
+      <(nle_inv_zero_dx n) in Hs12;
+      /2 width=2 by nlt_inv_refl, nle_inv_succ_bi/
     | /3 width=2 by ex_intro, or_introl/
     ]
-  | -IH @or_introl @(ex_intro … 1) // (**) (* auto fails *)
-  | lapply (transitive_lt s1 ??? Hs21) [ /2 width=1 by next_lt/ ] -Hs12 #Hs12
-    elim (IH (s2-⫯[h]s1)) -IH
-    [3: /3 width=1 by next_lt, monotonic_lt_minus_r/ ]
-    >minus_minus_m_m [2,4: /2 width=1 by lt_to_le/ ] -Hs21
+  | -IH @or_introl @(ex_intro … 𝟏) // (**) (* auto fails *)
+  | lapply (nlt_trans s1 ??? Hs21) [ /2 width=1 by sh_next_lt/ ] -Hs12 #Hs12
+    elim (IH (s2-â\87¡[h]s1)) -IH
+    [3: /3 width=1 by sh_next_lt, nlt_minus_bi_sn/ ]
+    <nminus_minus_dx_refl_sn [2,4: /2 width=1 by nlt_des_le/ ] -Hs21
     [ * #n #H destruct
-      @or_introl @(ex_intro … (↑n)) >iter_S >iter_n_Sm //
+      @or_introl @(ex_intro … (↑n)) <sh_nexts_succ >sh_nexts_swap //
     | #H1 @or_intror * #n #H2 @H1 -H1 destruct
       generalize in match Hs12; -Hs12
-      <(iter_O … (next h) s1) in ⊢ (?%?→?); #H
+      >(sh_nexts_zero h s1) in ⊢ (?%?→?); #H
       lapply (sh_lt_nexts_inv_lt … Hh … H) -H #H
-      <(S_pred … H) -H
-      @(ex_intro … (↓n)) >(iter_n_Sm … (next h)) >iter_S //
+      >(nlt_des_gen … H) -H
+      @(ex_intro … (↓n)) <sh_nexts_succ >sh_nexts_swap //
     ]
   ]
 ]