diff options
Diffstat (limited to 'mathcomp/algebra/poly.v')
| -rw-r--r-- | mathcomp/algebra/poly.v | 15 |
1 files changed, 8 insertions, 7 deletions
diff --git a/mathcomp/algebra/poly.v b/mathcomp/algebra/poly.v index e0a8e5d..d898774 100644 --- a/mathcomp/algebra/poly.v +++ b/mathcomp/algebra/poly.v @@ -1298,13 +1298,14 @@ Qed. (* Lifting a ring predicate to polynomials. *) -Definition polyOver (S : pred_class) := - [qualify a p : {poly R} | all (mem S) p]. +Implicit Type S : {pred R}. + +Definition polyOver S := [qualify a p : {poly R} | all (mem S) p]. Fact polyOver_key S : pred_key (polyOver S). Proof. by []. Qed. Canonical polyOver_keyed S := KeyedQualifier (polyOver_key S). -Lemma polyOverS (S1 S2 : pred_class) : +Lemma polyOverS (S1 S2 : {pred R}) : {subset S1 <= S2} -> {subset polyOver S1 <= polyOver S2}. Proof. by move=> sS12 p /(all_nthP 0)S1p; apply/(all_nthP 0)=> i /S1p; apply: sS12. @@ -1313,7 +1314,7 @@ Qed. Lemma polyOver0 S : 0 \is a polyOver S. Proof. by rewrite qualifE polyseq0. Qed. -Lemma polyOver_poly (S : pred_class) n E : +Lemma polyOver_poly S n E : (forall i, i < n -> E i \in S) -> \poly_(i < n) E i \is a polyOver S. Proof. move=> S_E; apply/(all_nthP 0)=> i lt_i_p /=; rewrite coef_poly. @@ -1322,7 +1323,7 @@ Qed. Section PolyOverAdd. -Variables (S : predPredType R) (addS : addrPred S) (kS : keyed_pred addS). +Variables (S : {pred R}) (addS : addrPred S) (kS : keyed_pred addS). Lemma polyOverP {p} : reflect (forall i, p`_i \in kS) (p \in polyOver kS). Proof. @@ -1354,7 +1355,7 @@ Canonical polyOver_zmodPred S addS kS := ZmodPred (@polyOverNr S addS kS). Section PolyOverSemiring. -Context (S : pred_class) (ringS : @semiringPred R S) (kS : keyed_pred ringS). +Variables (S : {pred R}) (ringS : semiringPred S) (kS : keyed_pred ringS). Fact polyOver_mulr_closed : mulr_closed (polyOver kS). Proof. @@ -1382,7 +1383,7 @@ End PolyOverSemiring. Section PolyOverRing. -Context (S : pred_class) (ringS : @subringPred R S) (kS : keyed_pred ringS). +Variables (S : {pred R}) (ringS : subringPred S) (kS : keyed_pred ringS). Canonical polyOver_smulrPred := SmulrPred (polyOver_mulr_closed kS). Canonical polyOver_subringPred := SubringPred (polyOver_mulr_closed kS). |