record excedence : Type ≝ {
exc_carr:> Type;
- exc_relation: exc_carr → exc_carr → Prop;
+ exc_relation: exc_carr → exc_carr → Type;
exc_coreflexive: coreflexive ? exc_relation;
exc_cotransitive: cotransitive ? exc_relation
}.
ap_cotransitive: cotransitive ? ap_apart
}.
-notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
+notation "a break # b" non associative with precedence 50 for @{ 'apart $a $b}.
interpretation "axiomatic apartness" 'apart x y =
(cic:/matita/excedence/ap_apart.con _ x y).
intros (H1); apply (H x); cases H1; assumption;
|2: unfold; intros(x y H); cases H; clear H; [right|left] assumption;
|3: intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
- cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
+ cases Axy (H H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
[left; left|right; left|right; right|left; right] assumption]
qed.
definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
-notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.
+notation "a break ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.
interpretation "alikeness" 'napart a b =
(cic:/matita/excedence/eq.con _ a b).
intros (E); unfold; intros (x); apply ap_coreflexive;
qed.
-lemma eq_symmetric:∀E.symmetric ? (eq E).
+lemma eq_sym_:∀E.symmetric ? (eq E).
intros (E); unfold; intros (x y Exy); unfold; unfold; intros (Ayx); apply Exy;
apply ap_symmetric; assumption;
qed.
-lemma eq_transitive: ∀E.transitive ? (eq E).
+lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x := eq_sym_.
+
+coercion cic:/matita/excedence/eq_sym.con.
+
+lemma eq_trans_: ∀E.transitive ? (eq E).
(* bug. intros k deve fare whd quanto basta *)
intros 6 (E x y z Exy Eyz); intro Axy; cases (ap_cotransitive ???y Axy);
[apply Exy|apply Eyz] assumption.
qed.
+
+lemma eq_trans:∀E:apartness.∀x,y,z:E.x ≈ y → y ≈ z → x ≈ z ≝ eq_trans_.
+
(* BUG: vedere se ricapita *)
lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
intros 5 (E x y Lxy Lyx); intro H;
cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)
qed.
-theorem le_le_to_eq: ∀E:excedence.∀x,y:E. x ≤ y → y ≤ x → x ≈ y.
-intros 6 (E x y L1 L2 H); cases H; [apply (L1 H1)|apply (L2 H1)]
-qed.
-
lemma unfold_apart: ∀E:excedence. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
-unfold apart_of_excedence; unfold apart; simplify; intros; assumption;
+intros; assumption;
qed.
lemma le_rewl: ∀E:excedence.∀z,y,x:E. x ≈ y → x ≤ z → y ≤ z.
intros (E z y x Exy Lxz); apply (le_transitive ???? Lxz);
intro Xyz; apply Exy; apply unfold_apart; left; assumption;
qed.
+
+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]
+cases (Exy (ap_symmetric ??? a));
+qed.
+
+lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x.
+intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
+apply ap_symmetric; assumption;
+qed.
+
+lemma exc_rewl: ∀A:excedence.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z.
+intros (A x z y Exy Ayz); elim (exc_cotransitive ???x Ayz); [2:assumption]
+cases Exy; right; assumption;
+qed.
+
+lemma exc_rewr: ∀A:excedence.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x.
+intros (A x z y Exy Azy); elim (exc_cotransitive ???x Azy); [assumption]
+elim (Exy); left; assumption;
+qed.