Rename samples subdirectories

samples/Coq/Rel.v

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+Require Export Logic.
+Import Playground1.
+
+Definition relation (X:Type) := X -> X -> Prop.
+
+Definition partial_function {X:Type} (R: relation X) :=
+	forall x y1 y2 : X, R x y1 -> R x y2 -> y1 = y2.
+
+Theorem next_nat_partial_function :
+	partial_function next_nat.
+Proof.
+	unfold partial_function.
+	intros x y1 y2 P Q.
+	inversion P.
+	inversion Q.
+	reflexivity.
+Qed.
+
+Theorem le_not_a_partial_function :
+	~ (partial_function le).
+Proof.
+	unfold not.
+	unfold partial_function.
+	intros H.
+	assert (O = S O) as Nonsense.
+		Case "Proof of assertion.".
+		apply H with O.
+		apply le_n.
+
+		apply le_S.
+		apply le_n.
+
+	inversion Nonsense.
+Qed.
+
+Theorem total_relation_not_partial_function :
+	~ (partial_function total_relation).
+Proof.
+	unfold not.
+	unfold partial_function.
+	intros H.
+	assert (O = S O) as Nonsense.
+	apply H with O.
+	apply total_relation1.
+
+	apply total_relation1.
+
+	inversion Nonsense.
+Qed.
+
+Theorem empty_relation_not_partial_funcion :
+	partial_function empty_relation.
+Proof.
+	unfold partial_function.
+	intros x y1 y2.
+	intros H.
+	inversion H.
+Qed.
+
+Definition reflexive {X:Type} (R: relation X) :=
+	forall a : X, R a a.
+
+Theorem le_reflexive :
+	reflexive le.
+Proof.
+	unfold reflexive.
+	intros n. apply le_n.
+Qed.
+
+Definition transitive {X:Type} (R: relation X) :=
+	forall a b c : X, (R a b) -> (R b c) -> (R a c).
+
+Theorem le_trans:
+	transitive le.
+Proof.
+	intros n m o Hnm Hmo.
+	induction Hmo.
+	Case "le_n". apply Hnm.
+	Case "le_S". apply le_S. apply IHHmo.
+Qed.
+
+Theorem lt_trans:
+	transitive lt.
+Proof.
+	unfold lt. unfold transitive.
+	intros n m o Hnm Hmo.
+	apply le_S in Hnm.
+	apply le_trans with (a := (S n)) (b := (S m)) (c := o).
+	apply Hnm.
+	apply Hmo.
+Qed.
+
+Theorem lt_trans' :
+	transitive lt.
+Proof.
+	unfold lt. unfold transitive.
+	intros n m o Hnm Hmo.
+	induction Hmo as [| m' Hm'o].
+	apply le_S.
+	apply Hnm.
+
+	apply le_S.
+	apply IHHm'o.
+Qed.
+
+Theorem le_Sn_le: forall n m, S n <= m -> n <= m.
+Proof.
+	intros n m H. apply le_trans with (S n).
+	apply le_S. apply le_n.
+	apply H. Qed.
+
+Theorem le_S_n : forall n m,
+	(S n <= S m) -> (n <= m).
+Proof.
+intros n m H.
+apply Sn_le_Sm__n_le_m.
+apply H.
+Qed.
+
+Theorem le_Sn_n : forall n,
+	~ (S n <= n).
+Proof.
+induction n.
+intros H.
+inversion H.
+
+unfold not in IHn.
+intros H.
+apply le_S_n in H.
+apply IHn.
+apply H.
+Qed.
+
+(*
+ TODO
+Theorem lt_trans'' :
+	transitive lt.
+Proof.
+	unfold lt. unfold transitive.
+	intros n m o Hnm Hmo.
+	induction o as [| o'].
+	*)
+
+Definition symmetric {X: Type} (R: relation X) :=
+	forall a b : X, (R a b) -> (R b a).
+
+Definition antisymmetric {X : Type} (R: relation X) :=
+	forall a b : X, (R a b) -> (R b a) -> a = b.
+
+Theorem le_antisymmetric :
+	antisymmetric le.
+Proof.
+intros a b.
+generalize dependent a.
+induction b.
+intros a.
+intros H.
+intros H1.
+inversion H.
+reflexivity.
+
+intros a H1 H2.
+destruct a.
+inversion H2.
+
+apply Sn_le_Sm__n_le_m in H1.
+apply Sn_le_Sm__n_le_m in H2.
+apply IHb in H1.
+rewrite H1 in |- *.
+reflexivity.
+
+apply H2.
+Qed.
+
+(*
+ TODO
+Theorem le_step : forall n m p,
+	n < m ->
+	n <= S p ->
+	n <= p.
+Proof.
+*)
+
+Definition equivalence {X:Type} (R: relation X) :=
+	(reflexive R) /\ (symmetric R) /\ (transitive R).
+
+Definition order {X:Type} (R: relation X) :=
+	(reflexive R) /\ (antisymmetric R) /\ (transitive R).
+
+Definition preorder {X:Type} (R: relation X) :=
+	(reflexive R) /\ (transitive R).
+
+Theorem le_order :
+	order le.
+Proof.
+	unfold order. split.
+	Case "refl". apply le_reflexive.
+	split.
+		Case "antisym". apply le_antisymmetric.
+		Case "transitive". apply le_trans. Qed.
+
+Inductive clos_refl_trans {A:Type} (R: relation A) : relation A :=
+	| rt_step : forall x y, R x y -> clos_refl_trans R x y
+	| rt_refl : forall x, clos_refl_trans R x x
+	| rt_trans : forall x y z,
+		clos_refl_trans R x y -> clos_refl_trans R y z -> clos_refl_trans R x z.
+
+Theorem next_nat_closure_is_le : forall n m,
+	(n <= m) <-> ((clos_refl_trans next_nat) n m).
+Proof.
+intros n m.
+split.
+intro H.
+induction H.
+apply rt_refl.
+
+apply rt_trans with m.
+apply IHle.
+
+apply rt_step.
+apply nn.
+
+intro H.
+induction H.
+inversion H.
+apply le_S.
+apply le_n.
+
+apply le_n.
+
+apply le_trans with y.
+apply IHclos_refl_trans1.
+
+apply IHclos_refl_trans2.
+Qed.																    
+
+Inductive refl_step_closure {X : Type} (R: relation X)
+											: X -> X -> Prop :=
+	| rsc_refl : forall (x : X), refl_step_closure R x x
+	| rsc_step : forall (x y z : X), R x y ->
+								refl_step_closure R y z ->
+								refl_step_closure R x z.
+
+Tactic Notation "rt_cases" tactic(first) ident(c) :=
+	first;
+	[ Case_aux c "rt_step" | Case_aux c "rt_refl" | Case_aux c "rt_trans" ].
+
+Tactic Notation "rsc_cases" tactic(first) ident(c) :=
+	  first;
+		  [ Case_aux c "rsc_refl" | Case_aux c "rsc_step" ].
+
+Theorem rsc_R : forall (X:Type) (R:relation X) (x y:X),
+				R x y -> refl_step_closure R x y.
+Proof.
+intros X R x y r.
+apply rsc_step with y.
+ apply r.
+  
+  apply rsc_refl.
+Qed.
+
+Theorem rsc_trans :
+	forall (X : Type) (R : relation X) (x y z : X),
+		refl_step_closure R x y ->
+			refl_step_closure R y z ->
+				refl_step_closure R x z.
+Proof.
+intros X.
+intros R x y z.
+intros H.
+induction H.
+intros H1.
+apply H1.
+
+intros H1.
+apply IHrefl_step_closure in H1.
+apply rsc_step with y.
+apply H.
+
+apply H1.
+Qed.
+
+Theorem rtc_rsc_coincide:
+	forall (X:Type) (R: relation X) (x y : X),
+		clos_refl_trans R x y <-> refl_step_closure R x y.
+Proof.
+intros X R x y.
+split.
+intros H.
+induction H.
+apply rsc_step with y.
+apply H.
+
+apply rsc_refl.
+
+apply rsc_refl.
+
+apply rsc_trans with y.
+apply IHclos_refl_trans1.
+
+apply IHclos_refl_trans2.
+
+intros H1.
+induction H1.
+apply rt_refl.
+
+apply rt_trans with y.
+apply rt_step.
+apply H.
+
+apply IHrefl_step_closure.
+Qed.