update for the article

[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 459d4d16bce3ae285c1cff8564bda5a4a9ca1fad..b7e10f851883233aae42c87faa1966256653c025 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma
@@ -12,25 +12,98 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "static_2/syntax/sort.ma".
+include "static_2/syntax/sh_props.ma".
 
 (* SORT HIERARCHY ***********************************************************)
 
-record is_lt (h): Prop ≝
+(* strict monotonicity condition *)
+record sh_lt (h): Prop ≝
 {
-  next_lt: ∀s. s < ⫯[h]s (* strict monotonicity condition *)
+  next_lt: ∀s. s < ⫯[h]s
 }.
 
 (* Basic properties *********************************************************)
 
-lemma nexts_le (h): is_lt h → ∀s,n. s ≤ (next h)^n s.
+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/
 qed.
 
-lemma nexts_lt (h): is_lt h → ∀s,n. s < (next h)^(↑n) s.
+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/
 qed.
+
+lemma sh_lt_nexts_inv_lt (h): sh_lt h →
+      ∀s,n1,n2. (next h)^n1 s < (next h)^n2 s → n1 < n2.
+#h #Hh
+@pull_2 #n1
+elim n1 -n1
+[ #s *
+  [ #H elim (lt_refl_false … 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/
+  ]
+]
+qed-.
+
+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/
+  ]
+]
+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
+[ @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
+    [ -IH @or_intror * #n #H destruct
+      generalize in match Hs21; -Hs21
+      <(iter_O … (next h) s1) in ⊢ (??%→?); <(iter_S … (next h)) #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/
+    | /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
+    [ * #n #H destruct
+      @or_introl @(ex_intro … (↑n)) >iter_S >iter_n_Sm //
+    | #H1 @or_intror * #n #H2 @H1 -H1 destruct
+      generalize in match Hs12; -Hs12
+      <(iter_O … (next 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 //
+    ]
+  ]
+]
+qed-.