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
}.
}.
notation "a break # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "axiomatic apartness" 'apart x y =
+interpretation "apartness" 'apart x y =
(cic:/matita/excedence/ap_apart.con _ x y).
+definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
+
definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
definition apart_of_excedence: excedence → apartness.
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.
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,z,y:E.x ≈ y → y ≈ z → x ≈ z ≝
+ λE,x,y,z.eq_trans_ E x z y.
+
+notation > "'Eq'≈" non associative with precedence 50 for
+ @{'eqrewrite}.
+
+interpretation "eq_rew" 'eqrewrite =
+ (cic:/matita/excedence/eq_trans.con _ _ _).
+
(* BUG: vedere se ricapita *)
+alias id "antisymmetric" = "cic:/matita/constructive_higher_order_relations/antisymmetric.con".
lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
intros 5 (E x y Lxy Lyx); intro H;
cases H; [apply Lxy;|apply Lyx] assumption;
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.
intro Xyz; apply Exy; apply unfold_apart; right; assumption;
qed.
+notation > "'Ex'≪" non associative with precedence 50 for
+ @{'excedencerewritel}.
+
+interpretation "exc_rewl" 'excedencerewritel =
+ (cic:/matita/excedence/exc_rewl.con _ _ _).
+
lemma le_rewr: ∀E:excedence.∀z,y,x:E. x ≈ y → z ≤ x → z ≤ y.
intros (E z y x Exy Lxz); apply (le_transitive ???? Lxz);
intro Xyz; apply Exy; apply unfold_apart; left; assumption;
qed.
+
+notation > "'Ex'≫" non associative with precedence 50 for
+ @{'excedencerewriter}.
+
+interpretation "exc_rewr" 'excedencerewriter =
+ (cic:/matita/excedence/exc_rewr.con _ _ _).
+
+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.
+
+lemma lt_rewr: ∀A:excedence.∀x,z,y:A. x ≈ y → z < y → z < x.
+intros (A x y z E H); split; elim H;
+[apply (le_rewr ???? (eq_sym ??? E));|apply (ap_rewr ???? E)] assumption;
+qed.
+
+lemma lt_rewl: ∀A:excedence.∀x,z,y:A. x ≈ y → y < z → x < z.
+intros (A x y z E H); split; elim H;
+[apply (le_rewl ???? (eq_sym ??? E));| apply (ap_rewl ???? E);] assumption;
+qed.
+
+lemma lt_le_transitive: ∀A:excedence.∀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)]
+whd in LE LEx APx; cases APx (EXx EXx); [cases (LEx EXx)]
+cases (exc_cotransitive ??? z EXx) (EXz EXz); [cases (LE EXz)]
+right; assumption;
+qed.
+
+lemma le_lt_transitive: ∀A:excedence.∀x,y,z:A.x ≤ y → y < z → x < z.
+intros (A x y z LE LT); cases LT (LEx APx); split; [apply (le_transitive ???? LE LEx)]
+whd in LE LEx APx; cases APx (EXx EXx); [cases (LEx EXx)]
+cases (exc_cotransitive ??? x EXx) (EXz EXz); [right; assumption]
+cases LE; assumption;
+qed.
+
+lemma le_le_eq: ∀E:excedence.∀a,b:E. a ≤ b → b ≤ a → a ≈ b.
+intros (E x y L1 L2); intro H; cases H; [apply L1|apply L2] assumption;
+qed.
+
+lemma eq_le_le: ∀E:excedence.∀a,b:E. a ≈ b → a ≤ b ∧ b ≤ a.
+intros (E x y H); unfold apart_of_excedence in H; unfold apart in H;
+simplify in H; split; intro; apply H; [left|right] assumption.
+qed.
+
+lemma ap_le_to_lt: ∀E:excedence.∀a,c:E.c # a → c ≤ a → c < a.
+intros; split; assumption;
+qed.
+
+definition total_order_property : ∀E:excedence. Type ≝
+ λE:excedence. ∀a,b:E. a ≰ b → b < a.
+
projects / helm.git / blobdiff