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Finish Pair EPar

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Yiyun Liu 2024-12-22 15:08:01 -05:00
parent 086e68f43e
commit fbe0bc4acc
1 changed files with 23 additions and 2 deletions
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25
theories/fp_red.v
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@ -393,8 +393,29 @@ Proof.
- move => n a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl). - move => n a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl).
split => [p|]. split => [p|].
+ move : iha => [/(_ p) [d [ih0 ih1]] _]. + move : iha => [/(_ p) [d [ih0 ih1]] _].
exists d. split=>//.
apply : rtc_l. apply RPar.ProjPair; eauto using RPar.refl.
set q := (X in rtc RPar.R X d).
by have -> : q = Proj p a1 by hauto lq:on.
+ move :iha => [iha _].
move : (iha PL) => [d0 [ih0 ih0']].
move : (iha PR) => [d1 [ih1 ih1']] {iha}.
exists d0, d1.
apply RPars.weakening in ih0, ih1.
repeat split => //=.
apply : rtc_l. apply RPar.AppPair; eauto using RPar.refl.
apply RPars.PairCong; apply RPars.AppCong; eauto using rtc_refl.
- move => n a0 a1 b0 b1 ha _ hb _ a b [*]. subst.
split.
+ move => p.
exists (if p is PL then a1 else b1).
split.
* apply rtc_once. apply : RPar.ProjPair'; eauto using RPar.refl.
* hauto lq:on rew:off.
+ exists a1, b1.
split. apply rtc_once. apply RPar.AppPair; eauto using RPar.refl.
split => //.
Qed.
Lemma commutativity n (a b0 b1 : Tm n) : Lemma commutativity n (a b0 b1 : Tm n) :
EPar.R a b0 -> RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c. EPar.R a b0 -> RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c.