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rename lemmas for classical reasoning to avoid clash with fintype #219

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Merged
CohenCyril merged 1 commit into from
Jun 5, 2020
Merged

rename lemmas for classical reasoning to avoid clash with fintype #219

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4 changes: 2 additions & 2 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -4,8 +4,8 @@

### Added

- in `boolp.v`, lemmas for classical reasoning `existsNP`, `existsP`,
`forallNP`, `forallP`, `imply_classic`, `orC`.
- in `boolp.v`, lemmas for classical reasoning `existsNP`, `existsPN`,
`forallNP`, `forallPN`, `Nimply`, `orC`.
- in `classical_sets.v`, definitions for supremums: `ul`, `lb`,
`supremum`
- in `ereal.v`:
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9 changes: 4 additions & 5 deletions theories/boolp.v
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Expand Up @@ -140,7 +140,6 @@ Qed.
Lemma gen_choiceMixin {T : Type} : Choice.mixin_of T.
Proof. by case: classic. Qed.


Lemma pdegen (P : Prop): P = True \/ P = False.
Proof. by have [p|Np] := pselect P; [left|right]; rewrite propeqE. Qed.

Expand Down Expand Up @@ -561,7 +560,7 @@ split => [[x Px h]|/asboolP]; [exact: Px|].
by rewrite asbool_neg => /existsp_asboolPn.
Qed.

Lemma existsP (T : Type) (P : T -> Prop) :
Lemma existsPN (T : Type) (P : T -> Prop) :
(exists x : T, P x) <-> (~ forall x : T, ~ P x).
Proof.
split => [[x Px h]|/asboolP]; [exact: (h x)|].
Expand All @@ -575,16 +574,16 @@ split => [h [x Px]|/asboolP]; [exact: (h x)|].
by rewrite asbool_neg => /forallp_asboolPn.
Qed.

Lemma forallP (T : Type) (P : T -> Prop) :
Lemma forallPN (T : Type) (P : T -> Prop) :
(forall x : T, P x) <-> (~ exists x : T, ~ P x).
Proof.
split => [h [x px]|]; [exact/px/h|move=> /forallNP h x].
by move/contrapT : (h x).
Qed.

Lemma imply_classic (P Q : Prop) : (P /\ ~ Q) <-> ~ (P -> Q).
Lemma Nimply (P Q : Prop) : ~ (P -> Q) <-> (P /\ ~ Q).
Proof.
split => [[p nq pq]|/asboolP]; [exact/nq/pq|].
split=> [/asboolP|[p nq pq]]; [|exact/nq/pq].
by rewrite asbool_neg => /imply_asboolPn.
Qed.

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4 changes: 2 additions & 2 deletions theories/ereal.v
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Expand Up @@ -476,7 +476,7 @@ have [Spoo|Spoo] := pselect (S +oo).
by exists +oo; split; [apply/ereal_ub_pinfty | apply/lbP => /= y /ubP; apply].
have [r Sr] : exists r, S r%:E.
move: S0 Snoo Spoo => [[r Sr _ _|//|Snoo Snoo1 Spoo]]; first by exists r.
apply/existsP => nS; move: Snoo1; apply; apply/eqP; rewrite predeqE.
apply/existsPN => nS; move: Snoo1; apply; apply/eqP; rewrite predeqE.
by case=> // r; split => // /nS.
set U := [set x | (real_of_er_def r @` S) x ].
have [ubU|/set0P/negP] := pselect (ub U !=set0); last first.
Expand Down Expand Up @@ -545,7 +545,7 @@ have : ~ ub S (ereal_sup S - e%:num%:E)%E.
move/ub_ereal_sup; apply/negP.
by rewrite -ltNge Sr lte_subl_addr lte_fin ltr_addl.
move/asboolP; rewrite asbool_neg; case/existsp_asboolPn => /= x.
case/imply_classic => ? ?; exists x; split => //.
case/Nimply => ? ?; exists x; split => //.
by rewrite ltNge; apply/negP.
Qed.

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2 changes: 1 addition & 1 deletion theories/reals.v
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Expand Up @@ -327,7 +327,7 @@ split; last first.
move=> h [x] /ubP hle; case/(_ x): h => y /hle.
by rewrite leNgt => /negbTE ->.
move/forallNP => h x; have {h} := h x.
move=> /ubP /existsNP => -[y /imply_classic[Ey /negP]].
move=> /ubP /existsNP => -[y /Nimply[Ey /negP]].
by rewrite -ltNge => ltx; exists y.
Qed.

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