exc_cotransitive: cotransitive ? exc_excess
}.
-interpretation "Excess base excess" 'nleq a b = (exc_excess _ a b).
+interpretation "Excess base excess" 'nleq a b = (exc_excess ? a b).
(* E(#,≰) → E(#,sym(≰)) *)
lemma make_dual_exc: excess_base → excess_base.
}.
notation "hvbox(a break # b)" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart x y = (ap_apart _ x y).
+interpretation "apartness" 'apart x y = (ap_apart ? x y).
definition apartness_of_excess_base: excess_base → apartness.
intros (E); apply (mk_apartness E (λa,b:E. a ≰ b ∨ b ≰ a));
definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
notation "hvbox(a break ≈ b)" non associative with precedence 50 for @{ 'napart $a $b}.
-interpretation "Apartness alikeness" 'napart a b = (eq _ a b).
-interpretation "Excess alikeness" 'napart a b = (eq (excess_base_OF_excess1 _) a b).
-interpretation "Excess (dual) alikeness" 'napart a b = (eq (excess_base_OF_excess _) a b).
+interpretation "Apartness alikeness" 'napart a b = (eq ? a b).
+interpretation "Excess alikeness" 'napart a b = (eq (excess_base_OF_excess1 ?) a b).
+interpretation "Excess (dual) alikeness" 'napart a b = (eq (excess_base_OF_excess ?) a b).
lemma eq_reflexive:∀E:apartness. reflexive ? (eq E).
intros (E); unfold; intros (x); apply ap_coreflexive;
λE,x,y,z.eq_trans_ E x z y.
notation > "'Eq'≈" non associative with precedence 50 for @{'eqrewrite}.
-interpretation "eq_rew" 'eqrewrite = (eq_trans _ _ _).
+interpretation "eq_rew" 'eqrewrite = (eq_trans ? ? ?).
alias id "antisymmetric" = "cic:/matita/constructive_higher_order_relations/antisymmetric.con".
lemma le_antisymmetric:
definition lt ≝ λE:excess.λa,b:E. a ≤ b ∧ a # b.
-interpretation "ordered sets less than" 'lt a b = (lt _ a b).
+interpretation "ordered sets less than" 'lt a b = (lt ? a b).
lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
intros 2 (E x); intro H; cases H (_ ABS);
qed.
notation > "'Le'≪" non associative with precedence 50 for @{'lerewritel}.
-interpretation "le_rewl" 'lerewritel = (le_rewl _ _ _).
+interpretation "le_rewl" 'lerewritel = (le_rewl ? ? ?).
notation > "'Le'≫" non associative with precedence 50 for @{'lerewriter}.
-interpretation "le_rewr" 'lerewriter = (le_rewr _ _ _).
+interpretation "le_rewr" 'lerewriter = (le_rewr ? ? ?).
lemma ap_rewl: ∀A:apartness.∀x,z,y:A. x ≈ y → y # z → x # z.
intros (A x z y Exy Ayz); cases (ap_cotransitive ???x Ayz); [2:assumption]
qed.
notation > "'Ap'≪" non associative with precedence 50 for @{'aprewritel}.
-interpretation "ap_rewl" 'aprewritel = (ap_rewl _ _ _).
+interpretation "ap_rewl" 'aprewritel = (ap_rewl ? ? ?).
notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}.
-interpretation "ap_rewr" 'aprewriter = (ap_rewr _ _ _).
+interpretation "ap_rewr" 'aprewriter = (ap_rewr ? ? ?).
alias symbol "napart" = "Apartness alikeness".
lemma exc_rewl: ∀A:excess.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z.
qed.
notation > "'Ex'≪" non associative with precedence 50 for @{'excessrewritel}.
-interpretation "exc_rewl" 'excessrewritel = (exc_rewl _ _ _).
+interpretation "exc_rewl" 'excessrewritel = (exc_rewl ? ? ?).
notation > "'Ex'≫" non associative with precedence 50 for @{'excessrewriter}.
-interpretation "exc_rewr" 'excessrewriter = (exc_rewr _ _ _).
+interpretation "exc_rewr" 'excessrewriter = (exc_rewr ? ? ?).
lemma lt_rewr: ∀A:excess.∀x,z,y:A. x ≈ y → z < y → z < x.
intros (A x y z E H); split; elim H;
qed.
notation > "'Lt'≪" non associative with precedence 50 for @{'ltrewritel}.
-interpretation "lt_rewl" 'ltrewritel = (lt_rewl _ _ _).
+interpretation "lt_rewl" 'ltrewritel = (lt_rewl ? ? ?).
notation > "'Lt'≫" non associative with precedence 50 for @{'ltrewriter}.
-interpretation "lt_rewr" 'ltrewriter = (lt_rewr _ _ _).
+interpretation "lt_rewr" 'ltrewriter = (lt_rewr ? ? ?).
lemma lt_le_transitive: ∀A:excess.∀x,y,z:A.x < y → y ≤ z → x < z.
intros (A x y z LT LE); cases LT (LEx APx); split; [apply (le_transitive ???? LEx LE)]