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Finish the soundness proof completely

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Yiyun Liu 2025-02-16 22:43:56 -05:00
parent d24991e994
commit bdba6f50e5
3 changed files with 119 additions and 5 deletions
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28
theories/fp_red.v
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@ -1900,6 +1900,22 @@ Module LoRed.
RRed.R a b.
Proof. induction 1; hauto lq:on ctrs:RRed.R. Qed.
Lemma AppAbs' n a (b : PTm n) u :
u = (subst_PTm (scons b VarPTm) a) ->
R (PApp (PAbs a) b) u.
Proof. move => ->. apply AppAbs. Qed.
Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
move => n a b m ξ /=.
apply AppAbs'. by asimpl.
all : try qauto ctrs:R use:ne_nf_ren, ishf_ren.
Qed.
End LoRed.
Module LoReds.
@ -2467,6 +2483,18 @@ Module DJoin.
eauto.
Qed.
Lemma ejoin_abs_inj n (a0 a1 : PTm (S n)) :
nf a0 -> nf a1 ->
EJoin.R (PAbs a0) (PAbs a1) ->
EJoin.R a0 a1.
Proof.
move => h0 h1.
have [? [? [? ?]]] : SN a0 /\ SN a1 /\ SN (PAbs a0) /\ SN (PAbs a1) by sauto lqb:on rew:off use:ne_nf_embed.
move /FromEJoin.
move /abs_inj.
hauto l:on use:ToEJoin.
Qed.
Lemma standardization n (a b : PTm n) :
SN a -> SN b -> R a b ->
exists va vb, rtc RRed.R a va /\ rtc RRed.R b vb /\ nf va /\ nf vb /\ EJoin.R va vb.