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.
intros a b; case a; case b; simpl in |- *; auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
Qed.
intros a b c; case a; case b; case c; simpl in |- *; auto.
intuition.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs).
apply plusA_assoc with (1 := cs).
Qed.
intros a b c d; case a; case b; case c; case d; simpl in |- *; auto.
intuition.
Qed.
Hint Resolve eqTerm_plusTerm_comp.
Set Implicit Arguments.
Unset Strict Implicit.
Definition multTerm : Term -> Term -> Term.
intros H'; case H'; intros b2 c2 H1'; case H1'; intros b3 c3;
exact (multA b2 b3, mult_mon n c2 c3).
Defined.
Set Strict Implicit.
Unset Implicit Arguments.
intros a b; case a; case b; simpl in |- *; auto.
intros a1 H' b2 H'0 H'1;
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := multA A0 a1);
auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros a1 H' b2 H'0 H'1;
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := multA a1 A0);
auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := multA A0 a1); auto.
Qed.
intros a b c; case a; case b; case c; simpl in |- *; auto.
intros a1 m1 a2 m2 a3 m3; split; auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA (multA a1 a3) (multA a1 a2));
auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (multA a1 (plusA a3 a2)); auto.
Qed.
intros a b c; case a; case b; case c; simpl in |- *; auto.
Qed.
intros a b c d; case a; case b; case c; case d; unfold eqT in |- *;
simpl in |- *; auto.
intros H' c0 H'0 c2 H'1 c3 H'2 c4 H'3 H'4; rewrite H'3; rewrite H'4; auto.
Qed.
intros a b c; case a; case b; case c; simpl in |- *; auto.
intros a0 m a1 m0 a2 m1; split; auto.
apply multA_assoc with (1 := cs); auto.
apply mult_mon_assoc.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros a0 m a1 m0; split; auto.
apply mult_mon_com.
Qed.
intros a b c d; case a; case b; case c; case d; simpl in |- *; auto.
intros a1 m1 a2 m2 a3 m3 a4 m4; intros H1; case H1; intros H2 H3 H4; case H4;
intros H5 H6; split; auto.
rewrite H3; rewrite H6; auto.
Qed.
Hint Resolve eqTerm_multTerm_comp.
Set Implicit Arguments.
Unset Strict Implicit.
Definition invTerm : Term -> Term.
intros H; case H; intros b2 c2; exact (invA b2, c2).
Defined.
Set Strict Implicit.
Unset Implicit Arguments.
intros a; case a; simpl in |- *; auto.
intros; apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); apply invA_plusA with (1 := cs);
auto.
Qed.
intros a; case a; simpl in |- *; auto.
intros b H' H'0;
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := plusA A0 (invA A0));
auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := invA A0); auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := plusA (invA A0) A0); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
Qed.
intros a; case a; simpl in |- *; auto.
intros a0 m; split; auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA a0 A0); auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA a0 (plusA (invA a0) (invA (invA a0))));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (plusA a0 (invA a0)) (invA (invA a0)));
auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA A0 (invA (invA a0))); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA (invA (invA a0)) A0); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
Qed.
intros a H'; red in |- *; intros H'0; absurd (zeroP a); auto.
apply zeroP_comp_eqTerm with (a := invTerm (invTerm a)); auto.
apply zeroP_invTerm_zeroP; auto.
apply eqTerm_sym; apply invTerm_invol; auto.
Qed.
Hint Resolve nZero_invTerm_nZero.
Set Implicit Arguments.
Unset Strict Implicit.
Definition T1 : Term.
exact (A1, M1).
Defined.
Set Strict Implicit.
Unset Implicit Arguments.
simpl in |- *; auto.
apply A1_diff_A0 with (1 := cs).
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros a0 m a1 m0 H'; elim H'; intros H'0 H'1; rewrite H'1; clear H'; auto.
split; auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := multA A1 a0); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs).
apply multA_A1_l with (1 := cs); auto.
apply multA_eqA_comp with (1 := cs); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
apply sym_eq; unfold M1 in |- *; apply mult_mon_zero_l.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros a0 m a1 m0 H'; elim H'; intros H'0 H'1; rewrite H'1; clear H'.
split; auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := multA a0 A1); auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := multA A1 a0); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); apply multA_A1_l with (1 := cs); auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
apply sym_eq; unfold M1 in |- *; apply mult_mon_zero_r.
Qed.
apply nZero_invTerm_nZero; auto.
exact T1_nz.
Qed.
Hint Resolve nZero_invTerm_T1.
intros a b; case a; case b; simpl in |- *; auto.
intros a0 m a1 m0; split; auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA (multA (invA a1) a0) A0);
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with
(plusA (multA (invA a1) a0) (plusA (multA a1 a0) (invA (multA a1 a0))));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with
(plusA (plusA (multA (invA a1) a0) (multA a1 a0)) (invA (multA a1 a0)));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (multA (plusA (invA a1) a1) a0) (invA (multA a1 a0)));
auto.
apply plusA_eqA_comp with (1 := cs); auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (multA a0 (invA a1)) (multA a0 a1));
auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (multA a0 (plusA (invA a1) a1));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (multA (plusA a1 (invA a1)) a0) (invA (multA a1 a0)));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (multA A0 a0) (invA (multA a1 a0)));
auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA A0 (invA (multA a1 a0)));
auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs);
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA (invA (multA a1 a0)) A0);
auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros a0 m a1 m0; split; auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA (multA a1 (invA a0)) A0);
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with
(plusA (multA a1 (invA a0)) (plusA (multA a1 a0) (invA (multA a1 a0))));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with
(plusA (plusA (multA a1 (invA a0)) (multA a1 a0)) (invA (multA a1 a0)));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (multA a1 (plusA (invA a0) a0)) (invA (multA a1 a0)));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (multA a1 (plusA a0 (invA a0))) (invA (multA a1 a0)));
auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (multA a1 A0) (invA (multA a1 a0)));
auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs); auto.
apply
(eqA_trans _ _ _ _ _ _ _ _ _ cs)
with (plusA (multA A0 a1) (invA (multA a1 a0)));
auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA A0 (invA (multA a1 a0)));
auto.
apply (eqA_sym _ _ _ _ _ _ _ _ _ cs);
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (plusA (invA (multA a1 a0)) A0);
auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intuition.
Qed.
intros a; case a; simpl in |- *; auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros a1 m1 a2 m2 H1; case H1; intros H2 H3; auto.
rewrite H3; auto.
unfold eqT in |- *; simpl in |- *; apply sym_eq; unfold M1 in |- *;
apply mult_mon_zero_l.
Qed.
intros a; case a; simpl in |- *; auto.
intuition.
Qed.
Set Implicit Arguments.
Unset Strict Implicit.
Definition minusTerm : Term -> Term -> Term.
intros H; case H; intros b2 c2 H'; case H'; intros b3 c3;
exact (minusA b2 b3, c2).
Defined.
Set Strict Implicit.
Unset Implicit Arguments.
intros a b; case a; case b; simpl in |- *; auto.
split; auto.
apply minusA_def with (1 := cs); auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
Qed.
intros a; apply zeroP_comp_eqTerm with (a := plusTerm a (invTerm a)); auto.
apply plusTerm_invTerm_zeroP; auto.
apply eqTerm_sym; apply eqTerm_minusTerm_plusTerm_invTerm; auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros a0 m a1 m0 H.
apply A_sep with (2 := eqA_dec) (1 := cs); auto.
Qed.
intros a b c eqT1.
apply eqTerm_trans with (y := multTerm (plusTerm a (invTerm b)) c); auto.
apply
eqTerm_trans with (y := plusTerm (multTerm a c) (multTerm (invTerm b) c));
auto.
apply
eqTerm_trans with (y := plusTerm (multTerm a c) (invTerm (multTerm b c)));
auto.
apply eqTerm_minusTerm_plusTerm_invTerm; auto.
apply eqTerm_plusTerm_comp; auto.
apply eqT_trans with (y := multTerm b c); auto.
apply multTerm_eqT; auto.
apply invTerm_eqT; auto.
apply multTerm_eqT; auto.
apply eqT_trans with (y := b); auto.
apply invTerm_eqT; auto.
apply eqTerm_refl; auto.
apply eqTerm_sym; apply mult_invTerm_com; auto.
apply multTerm_plusTerm_dist_l; auto.
apply eqTerm_multTerm_comp; auto.
apply eqTerm_sym; apply eqTerm_minusTerm_plusTerm_invTerm; auto.
apply eqTerm_refl; auto.
Qed.
Hint Resolve multTerm_minusTerm_dist_l.

Theorem nzeroP_multTerm :
forall a b : Term, ~ zeroP a -> ~ zeroP b -> ~ zeroP (multTerm a b).