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Make the internal language even smaller

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This commit is contained in:
Yiyun Liu 2025-04-03 21:18:17 -04:00
parent f402f4e528
commit 3b8fe388dc
4 changed files with 133 additions and 504 deletions
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64
theories/Autosubst2/syntax.v
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@ -19,29 +19,13 @@ 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 PTm : Type :=
| VarPTm : nat -> PTm
| PAbs : PTm -> PTm
| PApp : PTm -> PTm -> PTm
| PPair : PTm -> PTm -> PTm
| PProj : PTag -> PTm -> PTm
| PConst : TTag -> PTm
| PUniv : nat -> PTm
| PBot : PTm.
| PConst : nat -> PTm.
Lemma congr_PAbs {s0 : PTm} {t0 : PTm} (H0 : s0 = t0) : PAbs s0 = PAbs t0.
Proof.
@ -69,22 +53,12 @@ exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj x s1) H0))
(ap (fun x => PProj t0 x) H1)).
Qed.
Lemma congr_PConst {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
Lemma congr_PConst {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
PConst s0 = PConst t0.
Proof.
exact (eq_trans eq_refl (ap (fun x => PConst x) H0)).
Qed.
Lemma congr_PUniv {s0 : nat} {t0 : nat} (H0 : s0 = t0) : PUniv s0 = PUniv t0.
Proof.
exact (eq_trans eq_refl (ap (fun x => PUniv x) H0)).
Qed.
Lemma congr_PBot : PBot = PBot.
Proof.
exact (eq_refl).
Qed.
Lemma upRen_PTm_PTm (xi : nat -> nat) : nat -> nat.
Proof.
exact (up_ren xi).
@ -98,8 +72,6 @@ Fixpoint ren_PTm (xi_PTm : nat -> nat) (s : PTm) {struct s} : PTm :=
| PPair s0 s1 => PPair (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
| PProj s0 s1 => PProj s0 (ren_PTm xi_PTm s1)
| PConst s0 => PConst s0
| PUniv s0 => PUniv s0
| PBot => PBot
end.
Lemma up_PTm_PTm (sigma : nat -> PTm) : nat -> PTm.
@ -115,8 +87,6 @@ Fixpoint subst_PTm (sigma_PTm : nat -> PTm) (s : PTm) {struct s} : PTm :=
| PPair s0 s1 => PPair (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
| PProj s0 s1 => PProj s0 (subst_PTm sigma_PTm s1)
| PConst s0 => PConst s0
| PUniv s0 => PUniv s0
| PBot => PBot
end.
Lemma upId_PTm_PTm (sigma : nat -> PTm) (Eq : forall x, sigma x = VarPTm x) :
@ -145,8 +115,6 @@ subst_PTm sigma_PTm s = s :=
(idSubst_PTm sigma_PTm Eq_PTm s1)
| PProj s0 s1 => congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
| PConst s0 => congr_PConst (eq_refl s0)
| PUniv s0 => congr_PUniv (eq_refl s0)
| PBot => congr_PBot
end.
Lemma upExtRen_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat)
@ -177,8 +145,6 @@ ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
| PProj s0 s1 =>
congr_PProj (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
| PConst s0 => congr_PConst (eq_refl s0)
| PUniv s0 => congr_PUniv (eq_refl s0)
| PBot => congr_PBot
end.
Lemma upExt_PTm_PTm (sigma : nat -> PTm) (tau : nat -> PTm)
@ -210,8 +176,6 @@ subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
| PProj s0 s1 =>
congr_PProj (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
| PConst s0 => congr_PConst (eq_refl s0)
| PUniv s0 => congr_PUniv (eq_refl s0)
| PBot => congr_PBot
end.
Lemma up_ren_ren_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat)
@ -242,8 +206,6 @@ Fixpoint compRenRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat)
congr_PProj (eq_refl s0)
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
| PConst s0 => congr_PConst (eq_refl s0)
| PUniv s0 => congr_PUniv (eq_refl s0)
| PBot => congr_PBot
end.
Lemma up_ren_subst_PTm_PTm (xi : nat -> nat) (tau : nat -> PTm)
@ -278,8 +240,6 @@ Fixpoint compRenSubst_PTm (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm)
congr_PProj (eq_refl s0)
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
| PConst s0 => congr_PConst (eq_refl s0)
| PUniv s0 => congr_PUniv (eq_refl s0)
| PBot => congr_PBot
end.
Lemma up_subst_ren_PTm_PTm (sigma : nat -> PTm) (zeta_PTm : nat -> nat)
@ -325,8 +285,6 @@ ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
congr_PProj (eq_refl s0)
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
| PConst s0 => congr_PConst (eq_refl s0)
| PUniv s0 => congr_PUniv (eq_refl s0)
| PBot => congr_PBot
end.
Lemma up_subst_subst_PTm_PTm (sigma : nat -> PTm) (tau_PTm : nat -> PTm)
@ -373,8 +331,6 @@ subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
congr_PProj (eq_refl s0)
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
| PConst s0 => congr_PConst (eq_refl s0)
| PUniv s0 => congr_PUniv (eq_refl s0)
| PBot => congr_PBot
end.
Lemma renRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) (s : PTm) :
@ -464,8 +420,6 @@ Fixpoint rinst_inst_PTm (xi_PTm : nat -> nat) (sigma_PTm : nat -> PTm)
| PProj s0 s1 =>
congr_PProj (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
| PConst s0 => congr_PConst (eq_refl s0)
| PUniv s0 => congr_PUniv (eq_refl s0)
| PBot => congr_PBot
end.
Lemma rinstInst'_PTm (xi_PTm : nat -> nat) (s : PTm) :
@ -527,6 +481,20 @@ Proof.
exact (fun x => 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 : Type :=
| VarTm : nat -> Tm
| Abs : Tm -> Tm