+lemma subst_sort:
+ \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort
+k)) (TSort k))))
+\def
+ \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort
+k)))).
+
+lemma subst_lref_lt:
+ \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T
+(subst d v (TLRef i)) (TLRef i)))))
+\def
+ \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i
+d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true
+\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i)))
+(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))).
+
+lemma subst_lref_eq:
+ \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v)))
+\def
+ \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq
+T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with
+[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift
+i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))).
+
+lemma subst_lref_gt:
+ \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T
+(subst d v (TLRef i)) (TLRef (pred i))))))
+\def
+ \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d
+i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true
+\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef
+(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred
+i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d)
+(le_bge d i (lt_le_weak d i H)))))).
+
+lemma subst_head:
+ \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d:
+nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w
+t)))))))
+\def
+ \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d:
+nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))).