Section Term.
Load "hCoefStructure".
Load "mCoefStructure".
Load "hOrderStructure".
Load "mOrderStructure".
Definition M1 := zero_mon n.
Definition Term := (A * mon n)%type.
Set Implicit Arguments.
Unset Strict Implicit.
Definition zeroP : Term -> Prop.
intros H'; case H'.
intros a H'1; exact (eqA a A0).
Defined.
Definition eqTerm : Term -> Term -> Prop.
intros H'; case H'.
intros a a' H'2; case H'2.
intros b b'; exact (eqA a b /\ a' = b').
Defined.
Set Strict Implicit.
Unset Implicit Arguments.
red in |- *.
intros x; case x; simpl in |- *; auto.
Qed.
red in |- *.
intros x y; case x; case y; simpl in |- *; intuition.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
Qed.
red in |- *.
intros x y z; case x; case y; case z; simpl in |- *; intuition.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := a0); auto.
rewrite H2; auto.
Qed.
Set Implicit Arguments.
Unset Strict Implicit.
Definition T2M : Term -> mon n.
intros t; case t; intros a m; exact m.
Defined.
Set Strict Implicit.
Unset Implicit Arguments.
Set Implicit Arguments.
Unset Strict Implicit.
Definition eqT (a b : Term) : Prop := T2M a = T2M b.
Hint Unfold eqT.
Set Strict Implicit.
Unset Implicit Arguments.
red in |- *; auto.
Qed.

Lemma eqT_sym : symmetric _ eqT.