lstar removed from list.ma and placed in its own file

[helm.git] / matita / matita / contribs / lambda / preamble.ma

diff --git a/matita/matita/contribs/lambda/preamble.ma b/matita/matita/contribs/lambda/preamble.ma
index 53a433d404bb96b5c7435a02ce17a5b10d246ea8..c272595a2831f306ed7c12961cda21e2fa35bbf6 100644 (file)
--- a/matita/matita/contribs/lambda/preamble.ma
+++ b/matita/matita/contribs/lambda/preamble.ma
@@ -12,11 +12,15 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "arithmetics/nat.ma".
+include "basics/star.ma".
+include "basics/lists/list.ma".
+include "arithmetics/exp.ma".
 
 include "xoa_notation.ma".
 include "xoa.ma".
 
+(* logic *)
+
 (* Note: For some reason this cannot be in the standard library *) 
 interpretation "logical false" 'false = False.
 
@@ -24,6 +28,23 @@ notation "⊥"
   non associative with precedence 90
   for @{'false}.
 
+(* relations *)
+
+definition confluent1: ∀A. relation A → predicate A ≝ λA,R,a0.
+                       ∀a1. R a0 a1 → ∀a2. R a0 a2 →
+                       ∃∃a. R a1 a & R a2 a. 
+
+(* Note: confluent1 would be redundant if \Pi-reduction where enabled *)                       
+definition confluent: ∀A. predicate (relation A) ≝ λA,R.
+                      ∀a0. confluent1 … R a0.
+
+(* booleans *)
+
+definition is_false: predicate bool ≝ λb.
+                     false = b.
+
+(* arithmetics *)
+
 lemma lt_refl_false: ∀n. n < n → ⊥.
 #n #H elim (lt_to_not_eq … H) -H /2 width=1/
 qed-.
@@ -31,3 +52,47 @@ qed-.
 lemma lt_zero_false: ∀n. n < 0 → ⊥.
 #n #H elim (lt_to_not_le … H) -H /2 width=1/
 qed-.
+
+lemma plus_lt_false: ∀m,n. m + n < m → ⊥.
+#m #n #H elim (lt_to_not_le … H) -H /2 width=1/
+qed-.
+
+lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
+#m #n elim (lt_or_ge m n) /2 width=1/
+#H elim H -m /2 width=1/
+#m #Hm * #H /2 width=1/ /3 width=1/
+qed-.
+
+(* trichotomy operator *)
+
+(* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
+let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
+  match n1 with 
+  [ O    ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
+  | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]
+  ].
+
+lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1.
+#A #a1 #a2 #a3 #n2 elim n2 -n2
+[ #n1 #H elim (lt_zero_false … H)
+| #n2 #IH #n1 elim n1 -n1 // /3 width=1/
+]
+qed.
+
+lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2.
+#A #a1 #a2 #a3 #n elim n -n normalize //
+qed.
+
+lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
+#A #a1 #a2 #a3 #n1 elim n1 -n1
+[ #n2 #H elim (lt_zero_false … H)
+| #n1 #IH #n2 elim n2 -n2 // /3 width=1/
+]
+qed.
+
+(* lists *)
+
+(* Note: notation for nil not involving brackets *)
+notation > "◊"
+  non associative with precedence 90
+  for @{'nil}.