diff options
Diffstat (limited to 'common/Smallstep.v')
-rw-r--r-- | common/Smallstep.v | 50 |
1 files changed, 50 insertions, 0 deletions
diff --git a/common/Smallstep.v b/common/Smallstep.v index cd61ec34..63426c17 100644 --- a/common/Smallstep.v +++ b/common/Smallstep.v @@ -66,6 +66,31 @@ Proof. intros. eapply star_step; eauto. apply star_refl. traceEq. Qed. +Lemma star_two: + forall ge s1 t1 s2 t2 s3 t, + step ge s1 t1 s2 -> step ge s2 t2 s3 -> t = t1 ** t2 -> + star ge s1 t s3. +Proof. + intros. eapply star_step; eauto. apply star_one; auto. +Qed. + +Lemma star_three: + forall ge s1 t1 s2 t2 s3 t3 s4 t, + step ge s1 t1 s2 -> step ge s2 t2 s3 -> step ge s3 t3 s4 -> t = t1 ** t2 ** t3 -> + star ge s1 t s4. +Proof. + intros. eapply star_step; eauto. eapply star_two; eauto. +Qed. + +Lemma star_four: + forall ge s1 t1 s2 t2 s3 t3 s4 t4 s5 t, + step ge s1 t1 s2 -> step ge s2 t2 s3 -> + step ge s3 t3 s4 -> step ge s4 t4 s5 -> t = t1 ** t2 ** t3 ** t4 -> + star ge s1 t s5. +Proof. + intros. eapply star_step; eauto. eapply star_three; eauto. +Qed. + Lemma star_trans: forall ge s1 t1 s2, star ge s1 t1 s2 -> forall t2 s3 t, star ge s2 t2 s3 -> t = t1 ** t2 -> star ge s1 t s3. @@ -103,6 +128,31 @@ Proof. intros. econstructor; eauto. apply star_refl. traceEq. Qed. +Lemma plus_two: + forall ge s1 t1 s2 t2 s3 t, + step ge s1 t1 s2 -> step ge s2 t2 s3 -> t = t1 ** t2 -> + plus ge s1 t s3. +Proof. + intros. eapply plus_left; eauto. apply star_one; auto. +Qed. + +Lemma plus_three: + forall ge s1 t1 s2 t2 s3 t3 s4 t, + step ge s1 t1 s2 -> step ge s2 t2 s3 -> step ge s3 t3 s4 -> t = t1 ** t2 ** t3 -> + plus ge s1 t s4. +Proof. + intros. eapply plus_left; eauto. eapply star_two; eauto. +Qed. + +Lemma plus_four: + forall ge s1 t1 s2 t2 s3 t3 s4 t4 s5 t, + step ge s1 t1 s2 -> step ge s2 t2 s3 -> + step ge s3 t3 s4 -> step ge s4 t4 s5 -> t = t1 ** t2 ** t3 ** t4 -> + plus ge s1 t s5. +Proof. + intros. eapply plus_left; eauto. eapply star_three; eauto. +Qed. + Lemma plus_star: forall ge s1 t s2, plus ge s1 t s2 -> star ge s1 t s2. Proof. |