diff options
| author |  Hugo Herbelin <Hugo.Herbelin@inria.fr> | 2016-05-03 22:25:28 +0200 |
|---|---|---|
| committer | Hugo Herbelin <Hugo.Herbelin@inria.fr> | 2017-02-24 12:02:38 +0100 |
| commit | 4ccd6436a014d5d1b27d42e9bda40fb381d1bfce (patch) | |
| tree | 9af6a5ce6de2ba2671bf8e5ed96eb5ac97aec074 | |
| parent | 1682d4ed9df64937dfaa162e58233020036ff7b3 (diff) | |
Simplifying the proof of NoRetractToModalProposition.paradox in
Hurkens.v by dropping the artificial need for monad laws. Tactic eauto
decided to be dependent on the laws but there is a shorter proof
without them.
[Interestingly, eauto is not able to find the short proof. This is a
situation of the form
subgoal 1: H : ?A |- P x
subgoal 2: H : M ?A |- M (forall x, P x)
where addressing the second subgoal would find the right instance of
?A, but this instance is too hard to find by addressing first the
first subgoal.]
| -rw-r--r-- | theories/Logic/Hurkens.v | 13 |
1 files changed, 4 insertions, 9 deletions
diff --git a/theories/Logic/Hurkens.v b/theories/Logic/Hurkens.v index 56e03e965..a10c180cc 100644 --- a/theories/Logic/Hurkens.v +++ b/theories/Logic/Hurkens.v @@ -360,13 +360,12 @@ Module NoRetractToModalProposition. Section Paradox. Variable M : Prop -> Prop. -Hypothesis unit : forall A:Prop, A -> M A. -Hypothesis join : forall A:Prop, M (M A) -> M A. Hypothesis incr : forall A B:Prop, (A->B) -> M A -> M B. Lemma strength: forall A (P:A->Prop), M(forall x:A,P x) -> forall x:A,M(P x). Proof. - eauto. + intros A P h x. + eapply incr in h; eauto. Qed. (** ** The universe of modal propositions *) @@ -470,7 +469,7 @@ Hypothesis p2p2 : forall A:NProp, El A -> El (b2p (p2b A)). Theorem paradox : forall B:NProp, El B. Proof. intros B. - unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _ _ _))). + unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _))). + exact (fun P => ~~P). + exact bool. + exact p2b. @@ -480,8 +479,6 @@ Proof. + cbn. auto. + cbn. auto. + cbn. auto. - + auto. - + auto. Qed. End Paradox. @@ -516,7 +513,7 @@ Hypothesis p2p2 : forall A:NProp, El A -> El (b2p (p2b A)). Theorem mparadox : forall B:NProp, El B. Proof. intros B. - unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _ _ _))). + unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _))). + exact (fun P => P). + exact bool. + exact p2b. @@ -526,8 +523,6 @@ Proof. + cbn. auto. + cbn. auto. + cbn. auto. - + auto. - + auto. Qed. End MParadox. |