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Generalize Pi to TBind so we have both sigma and pi

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Yiyun Liu 2024-12-27 12:12:19 -05:00
parent 368c83dd8e
commit 7f4c31b14e
3 changed files with 77 additions and 54 deletions
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5
syntax.sig
View file

@ -1,13 +1,16 @@
nat : Type
Tm(VarTm) : Type
PTag : Type
TTag : Type
TPi : TTag
TSig : TTag
PL : PTag
PR : PTag
Abs : (bind Tm in Tm) -> Tm
App : Tm -> Tm -> Tm
Pair : Tm -> Tm -> Tm
Proj : PTag -> Tm -> Tm
Pi : Tm -> (bind Tm in Tm) -> Tm
TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
Bot : Tm
Univ : nat -> Tm

88
theories/Autosubst2/syntax.v
View file

@ -19,13 +19,27 @@ Proof.
exact (eq_refl).
Qed.
Inductive TTag : Type :=
| TPi : TTag
| TSig : TTag.
Lemma congr_TPi : TPi = TPi.
Proof.
exact (eq_refl).
Qed.
Lemma congr_TSig : TSig = TSig.
Proof.
exact (eq_refl).
Qed.
Inductive Tm (n_Tm : nat) : Type :=
| VarTm : fin n_Tm -> Tm n_Tm
| Abs : Tm (S n_Tm) -> Tm n_Tm
| App : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
| Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
| Proj : PTag -> Tm n_Tm -> Tm n_Tm
| Pi : Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
| TBind : TTag -> Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
| Bot : Tm n_Tm
| Univ : nat -> Tm n_Tm.
@ -59,12 +73,14 @@ exact (eq_trans (eq_trans eq_refl (ap (fun x => Proj m_Tm x s1) H0))
(ap (fun x => Proj m_Tm t0 x) H1)).
Qed.
Lemma congr_Pi {m_Tm : nat} {s0 : Tm m_Tm} {s1 : Tm (S m_Tm)} {t0 : Tm m_Tm}
{t1 : Tm (S m_Tm)} (H0 : s0 = t0) (H1 : s1 = t1) :
Pi m_Tm s0 s1 = Pi m_Tm t0 t1.
Lemma congr_TBind {m_Tm : nat} {s0 : TTag} {s1 : Tm m_Tm} {s2 : Tm (S m_Tm)}
{t0 : TTag} {t1 : Tm m_Tm} {t2 : Tm (S m_Tm)} (H0 : s0 = t0) (H1 : s1 = t1)
(H2 : s2 = t2) : TBind m_Tm s0 s1 s2 = TBind m_Tm t0 t1 t2.
Proof.
exact (eq_trans (eq_trans eq_refl (ap (fun x => Pi m_Tm x s1) H0))
(ap (fun x => Pi m_Tm t0 x) H1)).
exact (eq_trans
(eq_trans (eq_trans eq_refl (ap (fun x => TBind m_Tm x s1 s2) H0))
(ap (fun x => TBind m_Tm t0 x s2) H1))
(ap (fun x => TBind m_Tm t0 t1 x) H2)).
Qed.
Lemma congr_Bot {m_Tm : nat} : Bot m_Tm = Bot m_Tm.
@ -98,7 +114,8 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
| App _ s0 s1 => App n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
| Pair _ s0 s1 => Pair n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
| Proj _ s0 s1 => Proj n_Tm s0 (ren_Tm xi_Tm s1)
| Pi _ s0 s1 => Pi n_Tm (ren_Tm xi_Tm s0) (ren_Tm (upRen_Tm_Tm xi_Tm) s1)
| TBind _ s0 s1 s2 =>
TBind n_Tm s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
| Bot _ => Bot n_Tm
| Univ _ s0 => Univ n_Tm s0
end.
@ -124,8 +141,8 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
| App _ s0 s1 => App n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
| Pair _ s0 s1 => Pair n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
| Proj _ s0 s1 => Proj n_Tm s0 (subst_Tm sigma_Tm s1)
| Pi _ s0 s1 =>
Pi n_Tm (subst_Tm sigma_Tm s0) (subst_Tm (up_Tm_Tm sigma_Tm) s1)
| TBind _ s0 s1 s2 =>
TBind n_Tm s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
| Bot _ => Bot n_Tm
| Univ _ s0 => Univ n_Tm s0
end.
@ -163,9 +180,9 @@ subst_Tm sigma_Tm s = s :=
congr_Pair (idSubst_Tm sigma_Tm Eq_Tm s0)
(idSubst_Tm sigma_Tm Eq_Tm s1)
| Proj _ s0 s1 => congr_Proj (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
| Pi _ s0 s1 =>
congr_Pi (idSubst_Tm sigma_Tm Eq_Tm s0)
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s1)
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@ -206,10 +223,10 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
| Pi _ s0 s1 =>
congr_Pi (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s1)
(upExtRen_Tm_Tm _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@ -251,10 +268,10 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
congr_Pair (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
| Proj _ s0 s1 => congr_Proj (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
| Pi _ s0 s1 =>
congr_Pi (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s1)
s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@ -296,10 +313,10 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
| Pi _ s0 s1 =>
congr_Pi (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s1)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@ -351,10 +368,11 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
| Pi _ s0 s1 =>
congr_Pi (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s1)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@ -427,10 +445,11 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
| Pi _ s0 s1 =>
congr_Pi (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s1)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@ -504,10 +523,11 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
| Pi _ s0 s1 =>
congr_Pi (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s1)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@ -620,10 +640,10 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
| Pi _ s0 s1 =>
congr_Pi (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s1)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@ -828,7 +848,7 @@ Arguments Univ {n_Tm}.
Arguments Bot {n_Tm}.
Arguments Pi {n_Tm}.
Arguments TBind {n_Tm}.
Arguments Proj {n_Tm}.

38
theories/fp_red.v
View file

@ -68,10 +68,10 @@ Module Par.
| ProjCong p a0 a1 :
R a0 a1 ->
R (Proj p a0) (Proj p a1)
| PiCong A0 A1 B0 B1:
| BindCong p A0 A1 B0 B1:
R A0 A1 ->
R B0 B1 ->
R (Pi A0 B0) (Pi A1 B1)
R (TBind p A0 B0) (TBind p A1 B1)
(* Bot is useful for making the prov function computable *)
| BotCong :
R Bot Bot
@ -209,11 +209,11 @@ Module Par.
move : iha => [b0 [? ?]]. subst.
eexists. split. by apply ProjCong; eauto.
by asimpl.
- move => n A0 A1 B0 B1 ha iha hB ihB m ξ []//= t t0 [*]. subst.
- move => n p A0 A1 B0 B1 ha iha hB ihB m ξ []//= ? t t0 [*]. subst.
spec_refl.
move : iha => [b0 [? ?]].
move : ihB => [c0 [? ?]]. subst.
eexists. split. by apply PiCong; eauto.
eexists. split. by apply BindCong; eauto.
by asimpl.
- move => n n0 ξ []//=. hauto l:on.
- move => n i n0 ξ []//=. hauto l:on.
@ -286,7 +286,7 @@ Module RPar.
| ProjCong p a0 a1 :
R a0 a1 ->
R (Proj p a0) (Proj p a1)
| PiCong A0 A1 B0 B1:
| BindCong A0 A1 B0 B1:
R A0 A1 ->
R B0 B1 ->
R (Pi A0 B0) (Pi A1 B1)
@ -404,7 +404,7 @@ Module EPar.
| ProjCong p a0 a1 :
R a0 a1 ->
R (Proj p a0) (Proj p a1)
| PiCong A0 A1 B0 B1:
| BindCong A0 A1 B0 B1:
R A0 A1 ->
R B0 B1 ->
R (Pi A0 B0) (Pi A1 B1)
@ -533,7 +533,7 @@ Module RPars.
rtc RPar.R (App a0 b0) (App a1 b1).
Proof. solve_s. Qed.
Lemma PiCong n (a0 a1 : Tm n) b0 b1 :
Lemma BindCong n (a0 a1 : Tm n) b0 b1 :
rtc RPar.R a0 a1 ->
rtc RPar.R b0 b1 ->
rtc RPar.R (Pi a0 b0) (Pi a1 b1).
@ -754,7 +754,7 @@ Proof.
exists d. split => //.
hauto lq:on use:RPars.ProjCong, relations.rtc_transitive.
+ hauto lq:on ctrs:EPar.R use:RPars.ProjCong.
- hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.PiCong.
- hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.BindCong.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- hauto l:on ctrs:EPar.R inv:RPar.R.
Qed.
@ -927,7 +927,7 @@ Proof.
- move => n a0 a1 b0 b1 ha iha hb ihb c.
move /Pi_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
have : EPar.R (Pi a2 b2)(Pi a3 b3)
by hauto l:on use:EPar.PiCong.
by hauto l:on use:EPar.BindCong.
move : OExp.commutativity0 h2; repeat move/[apply].
move => [d h].
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
@ -1375,24 +1375,24 @@ Module ERPar.
- sfirstorder use:EPar.AppCong, @rtc_once.
Qed.
Lemma PiCong n (a0 a1 : Tm n) b0 b1:
Lemma BindCong n (a0 a1 : Tm n) b0 b1:
R a0 a1 ->
R b0 b1 ->
rtc R (Pi a0 b0) (Pi a1 b1).
Proof.
move => [] + [].
- sfirstorder use:RPar.PiCong, @rtc_once.
- sfirstorder use:RPar.BindCong, @rtc_once.
- move => h0 h1.
apply : rtc_l.
left. apply RPar.PiCong; eauto; apply RPar.refl.
left. apply RPar.BindCong; eauto; apply RPar.refl.
apply rtc_once.
hauto l:on use:EPar.PiCong, EPar.refl.
hauto l:on use:EPar.BindCong, EPar.refl.
- move => h0 h1.
apply : rtc_l.
left. apply RPar.PiCong; eauto; apply RPar.refl.
left. apply RPar.BindCong; eauto; apply RPar.refl.
apply rtc_once.
hauto l:on use:EPar.PiCong, EPar.refl.
- sfirstorder use:EPar.PiCong, @rtc_once.
hauto l:on use:EPar.BindCong, EPar.refl.
- sfirstorder use:EPar.BindCong, @rtc_once.
Qed.
Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
@ -1423,7 +1423,7 @@ Module ERPar.
End ERPar.
Hint Resolve ERPar.AppCong ERPar.refl ERPar.AbsCong ERPar.PairCong ERPar.ProjCong ERPar.PiCong : erpar.
Hint Resolve ERPar.AppCong ERPar.refl ERPar.AbsCong ERPar.PairCong ERPar.ProjCong ERPar.BindCong : erpar.
Module ERPars.
#[local]Ltac solve_s_rec :=
@ -1454,7 +1454,7 @@ Module ERPars.
rtc ERPar.R (Proj p a0) (Proj p a1).
Proof. solve_s. Qed.
Lemma PiCong n (a0 a1 : Tm n) b0 b1:
Lemma BindCong n (a0 a1 : Tm n) b0 b1:
rtc ERPar.R a0 a1 ->
rtc ERPar.R b0 b1 ->
rtc ERPar.R (Pi a0 b0) (Pi a1 b1).
@ -1512,7 +1512,7 @@ Proof.
- sfirstorder use:ERPars.AppCong.
- sfirstorder use:ERPars.PairCong.
- sfirstorder use:ERPars.ProjCong.
- sfirstorder use:ERPars.PiCong.
- sfirstorder use:ERPars.BindCong.
- sfirstorder.
- sfirstorder.
Qed.