proposition: Lemma ax_sem_nax : forall m P i Q, ax_sem m P i Q -> nax m P i Q.
proof: Proof. induction 1. apply nax1; red; simpl; auto. apply nax2 with P; red; simpl; auto. apply nax3 with P Q R; try red; simpl; auto. apply nax4 with P; try red; simpl; auto. inversion_clear IHax_sem. apply nax1; unfold valid in *; simpl in *; auto. apply nax2 with P'0; unfold valid in *; simpl in *; auto. apply nax3 with P'0 Q'0 R'; unfold valid in *; simpl in *; auto. apply nax4 with P'0; unfold valid in *; simpl in *; auto. Qed.
imports: syntax little denot Bool List String ZArith