(*** useful definitions and lemmas not really related to Fsub ***)
-let rec max m n \def
+definition max \def
+\lambda m,n.
match (leb m n) with
[true \Rightarrow n
|false \Rightarrow m].
+
+lemma le_n_max_m_n: \forall m,n:nat. n \le max m n.
+intros.
+unfold max.
+apply (leb_elim m n)
+ [simplify.intros.apply le_n
+ |simplify.intros.autobatch
+ ]
+qed.
+
+lemma le_m_max_m_n: \forall m,n:nat. m \le max m n.
+intros.
+unfold max.
+apply (leb_elim m n)
+ [simplify.intro.assumption
+ |simplify.intros.autobatch
+ ]
+qed.
+
+inductive in_list (A:Type): A → (list A) → Prop ≝
+| in_Base : ∀ x,l.(in_list A x (x::l))
+| in_Skip : ∀ x,y,l.(in_list A x l) → (in_list A x (y::l)).
-inductive in_list (A : Type) : A \to (list A) \to Prop \def
- | in_Base : \forall x:A.\forall l:(list A).
- (in_list A x (x :: l))
- | in_Skip : \forall x,y:A.\forall l:(list A).
- (in_list A x l) \to (in_list A x (y :: l)).
+notation > "hvbox(x ∈ l)"
+ non associative with precedence 30 for @{ 'inlist $x $l }.
+notation < "hvbox(x \nbsp ∈ \nbsp l)"
+ non associative with precedence 30 for @{ 'inlist $x $l }.
+interpretation "item in list" 'inlist x l =
+ (cic:/matita/Fsub/util/in_list.ind#xpointer(1/1) _ x l).
definition incl : \forall A.(list A) \to (list A) \to Prop \def
\lambda A,l,m.\forall x.(in_list A x l) \to (in_list A x m).
lemma in_list_nil : \forall A,x.\lnot (in_list A x []).
intros.unfold.intro.inversion H
- [intros;lapply (sym_eq ? ? ? H2);absurd (a::l = [])
- [assumption|apply nil_cons]
- |intros;lapply (sym_eq ? ? ? H4);absurd (a1::l = [])
- [assumption|apply nil_cons]]
+ [intros;lapply (sym_eq ? ? ? H2);destruct Hletin
+ |intros;destruct H4]
+qed.
+
+lemma in_cons_case : ∀A.∀x,h:A.∀t:list A.x ∈ h::t → x = h ∨ (x ∈ t).
+intros;inversion H;intros
+ [destruct H2;left;symmetry;assumption
+ |destruct H4;right;applyS H1]
+qed.
+
+lemma append_nil:\forall A:Type.\forall l:list A. l@[]=l.
+intros.
+elim l;intros;autobatch.
+qed.
+
+lemma append_cons:\forall A.\forall a:A.\forall l,l1.
+l@(a::l1)=(l@[a])@l1.
+intros.
+rewrite > associative_append.
+reflexivity.
+qed.
+
+lemma in_list_append1: \forall A.\forall x:A.
+\forall l1,l2. x \in l1 \to x \in l1@l2.
+intros.
+elim H
+ [simplify.apply in_Base
+ |simplify.apply in_Skip.assumption
+ ]
+qed.
+
+lemma in_list_append2: \forall A.\forall x:A.
+\forall l1,l2. x \in l2 \to x \in l1@l2.
+intros 3.
+elim l1
+ [simplify.assumption
+ |simplify.apply in_Skip.apply H.assumption
+ ]
+qed.
+
+theorem append_to_or_in_list: \forall A:Type.\forall x:A.
+\forall l,l1. x \in l@l1 \to (x \in l) \lor (x \in l1).
+intros 3.
+elim l
+ [right.apply H
+ |simplify in H1.inversion H1;intros
+ [destruct H3.left.rewrite < Hcut.
+ apply in_Base
+ |destruct H5.
+ elim (H l2)
+ [left.apply in_Skip.
+ rewrite < H4.assumption
+ |right.rewrite < H4.assumption
+ |rewrite > Hcut1.rewrite > H4.assumption
+ ]
+ ]
+ ]
qed.
lemma max_case : \forall m,n.(max m n) = match (leb m n) with
[ false \Rightarrow n
| true \Rightarrow m ].
intros;elim m;simplify;reflexivity;
+qed.
+
+lemma incl_A_A: ∀T,A.incl T A A.
+intros.unfold incl.intros.assumption.
qed.
\ No newline at end of file