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Add nat type definition

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Yiyun Liu 2025-02-21 13:23:38 -05:00
parent 0e0d9b20e5
commit fd0b48073d
3 changed files with 184 additions and 27 deletions
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165
theories/Autosubst2/syntax.v
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@ -41,7 +41,13 @@ Inductive PTm (n_PTm : nat) : Type :=
| PProj : PTag -> PTm n_PTm -> PTm n_PTm
| PBind : BTag -> PTm n_PTm -> PTm (S n_PTm) -> PTm n_PTm
| PUniv : nat -> PTm n_PTm
| PBot : PTm n_PTm.
| PBot : PTm n_PTm
| PNat : PTm n_PTm
| PZero : PTm n_PTm
| PSuc : PTm n_PTm -> PTm n_PTm
| PInd :
PTm (S n_PTm) ->
PTm n_PTm -> PTm n_PTm -> PTm (S (S n_PTm)) -> PTm n_PTm.
Lemma congr_PAbs {m_PTm : nat} {s0 : PTm (S m_PTm)} {t0 : PTm (S m_PTm)}
(H0 : s0 = t0) : PAbs m_PTm s0 = PAbs m_PTm t0.
@ -95,6 +101,37 @@ Proof.
exact (eq_refl).
Qed.
Lemma congr_PNat {m_PTm : nat} : PNat m_PTm = PNat m_PTm.
Proof.
exact (eq_refl).
Qed.
Lemma congr_PZero {m_PTm : nat} : PZero m_PTm = PZero m_PTm.
Proof.
exact (eq_refl).
Qed.
Lemma congr_PSuc {m_PTm : nat} {s0 : PTm m_PTm} {t0 : PTm m_PTm}
(H0 : s0 = t0) : PSuc m_PTm s0 = PSuc m_PTm t0.
Proof.
exact (eq_trans eq_refl (ap (fun x => PSuc m_PTm x) H0)).
Qed.
Lemma congr_PInd {m_PTm : nat} {s0 : PTm (S m_PTm)} {s1 : PTm m_PTm}
{s2 : PTm m_PTm} {s3 : PTm (S (S m_PTm))} {t0 : PTm (S m_PTm)}
{t1 : PTm m_PTm} {t2 : PTm m_PTm} {t3 : PTm (S (S m_PTm))} (H0 : s0 = t0)
(H1 : s1 = t1) (H2 : s2 = t2) (H3 : s3 = t3) :
PInd m_PTm s0 s1 s2 s3 = PInd m_PTm t0 t1 t2 t3.
Proof.
exact (eq_trans
(eq_trans
(eq_trans
(eq_trans eq_refl (ap (fun x => PInd m_PTm x s1 s2 s3) H0))
(ap (fun x => PInd m_PTm t0 x s2 s3) H1))
(ap (fun x => PInd m_PTm t0 t1 x s3) H2))
(ap (fun x => PInd m_PTm t0 t1 t2 x) H3)).
Qed.
Lemma upRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n) :
fin (S m) -> fin (S n).
Proof.
@ -119,6 +156,13 @@ Fixpoint ren_PTm {m_PTm : nat} {n_PTm : nat}
PBind n_PTm s0 (ren_PTm xi_PTm s1) (ren_PTm (upRen_PTm_PTm xi_PTm) s2)
| PUniv _ s0 => PUniv n_PTm s0
| PBot _ => PBot n_PTm
| PNat _ => PNat n_PTm
| PZero _ => PZero n_PTm
| PSuc _ s0 => PSuc n_PTm (ren_PTm xi_PTm s0)
| PInd _ s0 s1 s2 s3 =>
PInd n_PTm (ren_PTm (upRen_PTm_PTm xi_PTm) s0) (ren_PTm xi_PTm s1)
(ren_PTm xi_PTm s2)
(ren_PTm (upRen_PTm_PTm (upRen_PTm_PTm xi_PTm)) s3)
end.
Lemma up_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm) :
@ -150,6 +194,13 @@ Fixpoint subst_PTm {m_PTm : nat} {n_PTm : nat}
(subst_PTm (up_PTm_PTm sigma_PTm) s2)
| PUniv _ s0 => PUniv n_PTm s0
| PBot _ => PBot n_PTm
| PNat _ => PNat n_PTm
| PZero _ => PZero n_PTm
| PSuc _ s0 => PSuc n_PTm (subst_PTm sigma_PTm s0)
| PInd _ s0 s1 s2 s3 =>
PInd n_PTm (subst_PTm (up_PTm_PTm sigma_PTm) s0)
(subst_PTm sigma_PTm s1) (subst_PTm sigma_PTm s2)
(subst_PTm (up_PTm_PTm (up_PTm_PTm sigma_PTm)) s3)
end.
Lemma upId_PTm_PTm {m_PTm : nat} (sigma : fin m_PTm -> PTm m_PTm)
@ -193,6 +244,15 @@ Fixpoint idSubst_PTm {m_PTm : nat} (sigma_PTm : fin m_PTm -> PTm m_PTm)
(idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s2)
| PUniv _ s0 => congr_PUniv (eq_refl s0)
| PBot _ => congr_PBot
| PNat _ => congr_PNat
| PZero _ => congr_PZero
| PSuc _ s0 => congr_PSuc (idSubst_PTm sigma_PTm Eq_PTm s0)
| PInd _ s0 s1 s2 s3 =>
congr_PInd
(idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s0)
(idSubst_PTm sigma_PTm Eq_PTm s1) (idSubst_PTm sigma_PTm Eq_PTm s2)
(idSubst_PTm (up_PTm_PTm (up_PTm_PTm sigma_PTm))
(upId_PTm_PTm _ (upId_PTm_PTm _ Eq_PTm)) s3)
end.
Lemma upExtRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n)
@ -239,6 +299,18 @@ ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
(upExtRen_PTm_PTm _ _ Eq_PTm) s2)
| PUniv _ s0 => congr_PUniv (eq_refl s0)
| PBot _ => congr_PBot
| PNat _ => congr_PNat
| PZero _ => congr_PZero
| PSuc _ s0 => congr_PSuc (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
| PInd _ s0 s1 s2 s3 =>
congr_PInd
(extRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
(upExtRen_PTm_PTm _ _ Eq_PTm) s0)
(extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
(extRen_PTm xi_PTm zeta_PTm Eq_PTm s2)
(extRen_PTm (upRen_PTm_PTm (upRen_PTm_PTm xi_PTm))
(upRen_PTm_PTm (upRen_PTm_PTm zeta_PTm))
(upExtRen_PTm_PTm _ _ (upExtRen_PTm_PTm _ _ Eq_PTm)) s3)
end.
Lemma upExt_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm)
@ -286,6 +358,18 @@ subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
(upExt_PTm_PTm _ _ Eq_PTm) s2)
| PUniv _ s0 => congr_PUniv (eq_refl s0)
| PBot _ => congr_PBot
| PNat _ => congr_PNat
| PZero _ => congr_PZero
| PSuc _ s0 => congr_PSuc (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
| PInd _ s0 s1 s2 s3 =>
congr_PInd
(ext_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
(upExt_PTm_PTm _ _ Eq_PTm) s0)
(ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
(ext_PTm sigma_PTm tau_PTm Eq_PTm s2)
(ext_PTm (up_PTm_PTm (up_PTm_PTm sigma_PTm))
(up_PTm_PTm (up_PTm_PTm tau_PTm))
(upExt_PTm_PTm _ _ (upExt_PTm_PTm _ _ Eq_PTm)) s3)
end.
Lemma up_ren_ren_PTm_PTm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@ -334,6 +418,20 @@ Fixpoint compRenRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
(upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s2)
| PUniv _ s0 => congr_PUniv (eq_refl s0)
| PBot _ => congr_PBot
| PNat _ => congr_PNat
| PZero _ => congr_PZero
| PSuc _ s0 =>
congr_PSuc (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
| PInd _ s0 s1 s2 s3 =>
congr_PInd
(compRenRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
(upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s0)
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s2)
(compRenRen_PTm (upRen_PTm_PTm (upRen_PTm_PTm xi_PTm))
(upRen_PTm_PTm (upRen_PTm_PTm zeta_PTm))
(upRen_PTm_PTm (upRen_PTm_PTm rho_PTm))
(up_ren_ren _ _ _ (up_ren_ren _ _ _ Eq_PTm)) s3)
end.
Lemma up_ren_subst_PTm_PTm {k : nat} {l : nat} {m_PTm : nat}
@ -391,6 +489,21 @@ Fixpoint compRenSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
(up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s2)
| PUniv _ s0 => congr_PUniv (eq_refl s0)
| PBot _ => congr_PBot
| PNat _ => congr_PNat
| PZero _ => congr_PZero
| PSuc _ s0 =>
congr_PSuc (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
| PInd _ s0 s1 s2 s3 =>
congr_PInd
(compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
(up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s0)
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s2)
(compRenSubst_PTm (upRen_PTm_PTm (upRen_PTm_PTm xi_PTm))
(up_PTm_PTm (up_PTm_PTm tau_PTm))
(up_PTm_PTm (up_PTm_PTm theta_PTm))
(up_ren_subst_PTm_PTm _ _ _ (up_ren_subst_PTm_PTm _ _ _ Eq_PTm))
s3)
end.
Lemma up_subst_ren_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
@ -468,6 +581,21 @@ ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
(up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s2)
| PUniv _ s0 => congr_PUniv (eq_refl s0)
| PBot _ => congr_PBot
| PNat _ => congr_PNat
| PZero _ => congr_PZero
| PSuc _ s0 =>
congr_PSuc (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
| PInd _ s0 s1 s2 s3 =>
congr_PInd
(compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
(up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s0)
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s2)
(compSubstRen_PTm (up_PTm_PTm (up_PTm_PTm sigma_PTm))
(upRen_PTm_PTm (upRen_PTm_PTm zeta_PTm))
(up_PTm_PTm (up_PTm_PTm theta_PTm))
(up_subst_ren_PTm_PTm _ _ _ (up_subst_ren_PTm_PTm _ _ _ Eq_PTm))
s3)
end.
Lemma up_subst_subst_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
@ -547,6 +675,21 @@ subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
(up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s2)
| PUniv _ s0 => congr_PUniv (eq_refl s0)
| PBot _ => congr_PBot
| PNat _ => congr_PNat
| PZero _ => congr_PZero
| PSuc _ s0 =>
congr_PSuc (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
| PInd _ s0 s1 s2 s3 =>
congr_PInd
(compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
(up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s0)
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s2)
(compSubstSubst_PTm (up_PTm_PTm (up_PTm_PTm sigma_PTm))
(up_PTm_PTm (up_PTm_PTm tau_PTm))
(up_PTm_PTm (up_PTm_PTm theta_PTm))
(up_subst_subst_PTm_PTm _ _ _
(up_subst_subst_PTm_PTm _ _ _ Eq_PTm)) s3)
end.
Lemma renRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
@ -665,6 +808,18 @@ Fixpoint rinst_inst_PTm {m_PTm : nat} {n_PTm : nat}
(rinstInst_up_PTm_PTm _ _ Eq_PTm) s2)
| PUniv _ s0 => congr_PUniv (eq_refl s0)
| PBot _ => congr_PBot
| PNat _ => congr_PNat
| PZero _ => congr_PZero
| PSuc _ s0 => congr_PSuc (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
| PInd _ s0 s1 s2 s3 =>
congr_PInd
(rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
(rinstInst_up_PTm_PTm _ _ Eq_PTm) s0)
(rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
(rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s2)
(rinst_inst_PTm (upRen_PTm_PTm (upRen_PTm_PTm xi_PTm))
(up_PTm_PTm (up_PTm_PTm sigma_PTm))
(rinstInst_up_PTm_PTm _ _ (rinstInst_up_PTm_PTm _ _ Eq_PTm)) s3)
end.
Lemma rinstInst'_PTm {m_PTm : nat} {n_PTm : nat}
@ -871,6 +1026,14 @@ Core.
Arguments VarPTm {n_PTm}.
Arguments PInd {n_PTm}.
Arguments PSuc {n_PTm}.
Arguments PZero {n_PTm}.
Arguments PNat {n_PTm}.
Arguments PBot {n_PTm}.
Arguments PUniv {n_PTm}.