(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/ordered_sets/".
+set "baseuri" "cic:/matita/excedence/".
include "higher_order_defs/relations.ma".
include "nat/plus.ma".
}.
interpretation "excedence" 'nleq a b =
- (cic:/matita/ordered_sets/exc_relation.con _ a b).
+ (cic:/matita/excedence/exc_relation.con _ a b).
definition le ≝ λE:excedence.λa,b:E. ¬ (a ≰ b).
interpretation "ordered sets less or equal than" 'leq a b =
- (cic:/matita/ordered_sets/le.con _ a b).
+ (cic:/matita/excedence/le.con _ a b).
lemma le_reflexive: ∀E.reflexive ? (le E).
intros (E); unfold; cases E; simplify; intros (x); apply (H x);
definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart a b = (cic:/matita/ordered_sets/apart.con _ a b).
+interpretation "apartness" 'apart a b = (cic:/matita/excedence/apart.con _ a b).
lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.
interpretation "alikeness" 'napart a b =
- (cic:/matita/ordered_sets/eq.con _ a b).
+ (cic:/matita/excedence/eq.con _ a b).
lemma eq_reflexive:∀E. reflexive ? (eq E).
intros (E); unfold; cases E (T f cRf _); simplify; unfold Not; intros (x H);
definition lt ≝ λE:excedence.λa,b:E. a ≤ b ∧ a # b.
interpretation "ordered sets less than" 'lt a b =
- (cic:/matita/ordered_sets/lt.con _ a b).
+ (cic:/matita/excedence/lt.con _ a b).
lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
intros (E); unfold; unfold Not; intros (x H); cases H (_ ABS);