intros Q H' a b c nZb nZc p q r H'0 H'1 H'2 H'3 H'4 H'5 H'6 H'7 r1 s1 H'8 H'9.
cut (canonical A0 eqA ltM (pX b q)); [ intros Op0 | idtac ].
cut (canonical A0 eqA ltM (pX c r)); [ intros Op0bis | idtac ].
cut (Confluent Q (pX a p) (pX b q) (pX c r)); auto.
2: apply fconfl_top; auto.
2: inversion_clear H'; auto.
2: apply Reducep0 with (nZb := nZb) (nZc := nZc); auto.
2: apply inPolySet_imp_canonical with (L := Q); auto.
2: apply inPolySet_imp_canonical with (L := Q); auto.
intros H'10.
apply (Conf_inv1 Q a b c nZb nZc p q r r1 s1); auto.
Qed.

Theorem confl_mix :
forall (Q : list (poly A0 eqA ltM)) (a b : Term A n)
(nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b) (p q r : list (Term A n)),
canonical A0 eqA ltM (pX a p) ->
divP A A0 eqA multA divA n a b ->
inPolySet A A0 eqA n ltM (pX b r) Q ->
(forall q : list (Term A n),
canonical A0 eqA ltM q ->
ltP (A:=A) (n:=n) ltM q (pX a p) -> ReduStarConfluent Q q) ->
reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q
(pX a p) (pX a q) ->
reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q
(pX a p)
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b nZb
p r) ->
forall r1 s1 : list (Term A n),
reducestar A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q
(pX a q) r1 ->
reducestar A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b nZb
p r) s1 -> eqP A eqA n r1 s1.