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.
red in |- *; auto.
Qed.
red in |- *; unfold eqT in |- *.
intros x y z H' H'0; rewrite H'; auto.
Qed.
intros a b; case a; case b; simpl in |- *; intuition.
Qed.
intros x y; case x; case y; simpl in |- *.
intros b2 c2 b3 c3.
case (eqA_dec b3 b2); intros eqAZ; auto.
case (eqmon_dec n c3 c2); intros eqAZ1; auto.
right; red in |- *; intros H'; elim H'; intros H'0 H'1; clear H'; auto.
right; red in |- *; intros H'; elim H'; intros H'0 H'1; clear H'; auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intuition.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := A0); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
Qed.
intros x; case x; simpl in |- *.
intros b H'.
apply eqA_dec; auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intuition.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := a1); auto;
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
Qed.
intros a b H' H'0; red in |- *; intros H'1.
apply H'.
apply zeroP_comp_eqTerm with (a := b); auto.
apply eqTerm_sym; auto.
Qed.
Set Implicit Arguments.
Unset Strict Implicit.
Definition plusTerm : Term -> Term -> Term.
intros x; case x; intros b2 c2; intros y; case y; intros b3 c3;
exact (plusA b2 b3, c2).
Defined.
Set Strict Implicit.
Unset Implicit Arguments.
intros a b; case a; case b; simpl in |- *; auto.
intros a1 m1 a2 m2 H1 H2; split; auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := plusA a2 A0); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros a1 m1 a2 m2 H1 H2; split; auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := plusA A0 a1); auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := plusA a1 A0); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
Qed.
intros a b c; case a; case b; case c; simpl in |- *; auto.
intuition.
Qed.
intros a b c; case a; case b; case c; simpl in |- *; auto.
intuition.
Qed.

Theorem plusTerm_com :
forall x y : Term, eqT x y -> eqTerm (plusTerm x y) (plusTerm y x).