(* This file was automatically generated: do not edit *********************)
-include "Basic-1/subst/fwd.ma".
+include "basic_1/subst/defs.ma".
-include "Basic-1/subst0/defs.ma".
+include "basic_1/subst0/fwd.ma".
-include "Basic-1/lift/props.ma".
+theorem 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)))).
+
+theorem 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))))).
+
+theorem 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)))).
+
+theorem 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)))))).
+
+theorem 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))))))).
theorem subst_lift_SO:
\forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S
(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O)
d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k
t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v
-(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1)))
-(f_equal3 K T T T THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v
-(lift (S O) (s k d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))) (subst d v
-(THead k (lift (S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S
-O) d t0) (lift (S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1))
-(lift_head k t0 t1 (S O) d)))))))) t)).
-(* COMMENTS
-Initial nodes: 879
-END *)
+(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq
+T (THead k t0 t1) (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v
+(lift (S O) (s k d) t1))) (sym_eq T (THead k (subst d v (lift (S O) d t0))
+(subst (s k d) v (lift (S O) (s k d) t1))) (THead k t0 t1) (f_equal3 K T T T
+THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v (lift (S O) (s k
+d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))))) (subst d v (THead k (lift
+(S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S O) d t0) (lift
+(S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1)) (lift_head k t0 t1 (S O)
+d)))))))) t)).
theorem subst_subst0:
\forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0
(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4))
(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1
t3 i))))))))))))) d v t1 t2 H))))).
-(* COMMENTS
-Initial nodes: 1363
-END *)