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Add injectivity lemma for abs

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Yiyun Liu 2025-02-14 21:55:44 -05:00
parent 300530a93f
commit 9bd554b513
2 changed files with 34 additions and 0 deletions
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1
theories/algorithmic.v
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@ -681,6 +681,7 @@ Proof.
suff [v0 [hv00 hv01]] : exists v0, rtc ERed.R va' v0 /\ rtc ERed.R vb' v0.
repeat split =>//=. sfirstorder.
simpl in *; by lia.
admit.
+ case : b => //=; try qauto depth:1 use:T_AbsPair_Imp, T_PairBind_Imp, T_PairUniv_Imp.
move => a1 b1 a0 b0 h _ _ Γ A hu0 hu1.

33
theories/fp_red.v
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@ -2319,6 +2319,14 @@ Module DJoin.
hauto lq:on rew:off ctrs:rtc unfold:R use:REReds.substing.
Qed.
Lemma renaming n m (a b : PTm n) (ρ : fin n -> fin m) :
R a b -> R (ren_PTm ρ a) (ren_PTm ρ b).
Proof. substify. apply substing. Qed.
Lemma weakening n (a b : PTm n) :
R a b -> R (ren_PTm shift a) (ren_PTm shift b).
Proof. apply renaming. Qed.
Lemma cong n m (a : PTm (S n)) c d (ρ : fin n -> PTm m) :
R c d -> R (subst_PTm (scons c ρ) a) (subst_PTm (scons d ρ) a).
Proof.
@ -2327,6 +2335,31 @@ Module DJoin.
hauto q:on ctrs:rtc inv:option use:REReds.cong.
Qed.
Lemma abs_inj n (a0 a1 : PTm (S n)) :
SN a0 -> SN a1 ->
R (PAbs a0) (PAbs a1) ->
R a0 a1.
Proof.
move => sn0 sn1.
move /weakening => /=.
move /AppCong. move /(_ (VarPTm var_zero) (VarPTm var_zero)).
move => /(_ ltac:(sfirstorder use:refl)).
move => h.
have /FromRRed1 h0 : RRed.R (PApp (PAbs (ren_PTm (upRen_PTm_PTm shift) a0)) (VarPTm var_zero)) a0 by apply RRed.AppAbs'; asimpl.
have /FromRRed0 h1 : RRed.R (PApp (PAbs (ren_PTm (upRen_PTm_PTm shift) a1)) (VarPTm var_zero)) a1 by apply RRed.AppAbs'; asimpl.
have sn0' := sn0.
have sn1' := sn1.
eapply sn_renaming with (ξ := (upRen_PTm_PTm shift)) in sn0.
eapply sn_renaming with (ξ := (upRen_PTm_PTm shift)) in sn1.
apply : transitive; try apply h0.
apply : N_Exp. apply N_β. sauto.
by asimpl.
apply : transitive; try apply h1.
apply : N_Exp. apply N_β. sauto.
by asimpl.
eauto.
Qed.
End DJoin.