From 7fb60e1c2fcf644f97ec01162390ba99b75b7423 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <yliu8e5f7@protonmail.com>
Date: Thu, 19 Jun 2025 13:00:37 -0400
Subject: [PATCH] Add the coinductively defined safe predicate

---
 cosn.v | 44 ++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 44 insertions(+)
 create mode 100644 cosn.v

diff --git a/cosn.v b/cosn.v
new file mode 100644
index 0000000..4b75918
--- /dev/null
+++ b/cosn.v
@@ -0,0 +1,44 @@
+From Ltac2 Require Ltac2.
+Import Ltac2.Notations.
+
+Import Ltac2.Control.
+Require Import ssreflect ssrbool.
+Require Import FunInd.
+Require Import Arith.Wf_nat (well_founded_lt_compat).
+Require Import Psatz.
+From stdpp Require Import relations (rtc (..), rtc_once, rtc_r, sn).
+From Hammer Require Import Tactics.
+Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax common fp_red.
+
+Fixpoint nostuck (a : PTm) :=
+  match a with
+  | VarPTm i => true
+  | PAbs a => nostuck a
+  | PApp a b => (ishf a ==> isabs a) && nostuck a && nostuck b
+  | PBind _ A B => nostuck A && nostuck B
+  | PPair a b => nostuck a && nostuck b
+  | PProj _ a => (ishf a ==> ispair a) && nostuck a
+  | PZero => true
+  | PSuc a => nostuck a
+  | PInd P a b c => (ishf a ==> iszero a || issuc a) && nostuck b && nostuck c
+  | PNat => true
+  | PUniv _ => true
+  end.
+
+
+CoInductive safe a : Prop :=
+  safe_intro {safe_ok : nostuck a; safe_red : forall b,RRed.R a b -> safe b}.
+
+Arguments safe_intro {a}.
+Arguments safe_ok {a}.
+Arguments safe_red {a}.
+
+Lemma safe_coind P : (forall a, P a -> nostuck a /\  (forall b, RRed.R a b -> P b)) ->  forall a, P a -> safe a.
+  move => h.
+  cofix ih.
+  move => a ha. apply h in ha.
+  destruct ha as [ha0 ha1].
+  apply safe_intro.
+  apply ha0.
+  move => b hb. apply ha1 in hb. apply ih. apply hb.
+Qed.