record excedence : Type ≝ {
exc_carr:> Type;
- exc_relation: exc_carr → exc_carr → Prop;
+ exc_relation: exc_carr → exc_carr → Type; (* Big bug: era in Prop!!! *)
exc_coreflexive: coreflexive ? exc_relation;
exc_cotransitive: cotransitive ? exc_relation
}.
cases (c x z y H2) (H4 H5); clear H2; [exact (Rxy H4)|exact (Ryz H5)]
qed.
-definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
+record apartness : Type ≝ {
+ ap_carr:> Type;
+ ap_apart: ap_carr → ap_carr → Type;
+ ap_coreflexive: coreflexive ? ap_apart;
+ ap_symmetric: symmetric ? ap_apart;
+ ap_cotransitive: cotransitive ? ap_apart
+}.
-notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart a b = (cic:/matita/excedence/apart.con _ 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).
-lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
-intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
-intros (H1); apply (H x); cases H1; assumption;
-qed.
+definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
-lemma apart_symmetric: ∀E.symmetric ? (apart E).
-intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption;
+definition apart_of_excedence: excedence → apartness.
+intros (E); apply (mk_apartness E (apart E));
+[1: unfold; cases E; simplify; clear E; intros (x); unfold;
+ 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 H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
+ [left; left|right; left|right; right|left; right] assumption]
qed.
-lemma apart_cotrans: ∀E. cotransitive ? (apart E).
-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;
-[left; left|right; left|right; right|left; right] assumption.
-qed.
+coercion cic:/matita/excedence/apart_of_excedence.con.
-definition eq ≝ λE:excedence.λa,b:E. ¬ (a # b).
+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).
lemma eq_reflexive:∀E. reflexive ? (eq E).
-intros (E); unfold; cases E (T f cRf _); simplify; unfold Not; intros (x H);
-apply (cRf x); cases H; assumption;
+intros (E); unfold; intros (x); apply ap_coreflexive;
qed.
lemma eq_symmetric:∀E.symmetric ? (eq E).
-intros (E); unfold; unfold eq; unfold Not;
-intros (x y H1 H2); apply H1; cases H2; [right|left] assumption;
+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).
-intros (E); unfold; cases E (T f _ cTf); simplify; unfold Not;
-intros (x y z H1 H2 H3); cases H3 (H4 H4); clear E H3; lapply (cTf ? ? y H4) as H5;
-cases H5; clear H5 H4 cTf; [1,4: apply H1|*:apply H2] clear H1 H2;
-[1,3:left|*:right] assumption;
+lemma eq_symmetric_:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x := eq_symmetric.
+
+coercion cic:/matita/excedence/eq_symmetric_.con.
+
+lemma eq_transitive_: ∀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 le_antisymmetric: ∀E.antisymmetric ? (le E) (eq E).
-intros (E); unfold; intros (x y Lxy Lyx); unfold; unfold; intros (H);
+lemma eq_transitive:∀E:apartness.∀x,y,z:E.x ≈ y → y ≈ z → x ≈ z ≝ eq_transitive_.
+
+(* BUG: vedere se ricapita *)
+lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
+intros 5 (E x y Lxy Lyx); intro H;
cases H; [apply Lxy;|apply Lyx] assumption;
qed.
(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);
-apply (apart_coreflexive ? x ABS);
+intros 2 (E x); intro H; cases H (_ ABS);
+apply (ap_coreflexive ? x ABS);
qed.
-
+
lemma lt_transitive: ∀E.transitive ? (lt E).
intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz);
split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
intros (E a b Lab); cases Lab (LEab Aab);
cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)
qed.
+
+(* CSC: lo avevi gia' dimostrato; ho messo apply! *)
+theorem le_le_to_eq: ∀E:excedence.∀x,y:E. x ≤ y → y ≤ x → x ≈ y.
+apply le_antisymmetric;
+qed.
+
+(* CSC: perche' quel casino: bastava intros; assumption! *)
+lemma unfold_apart: ∀E:excedence. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
+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; right; assumption;
+qed.
+
+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.
+
+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.
\ No newline at end of file
projects / helm.git / blobdiff