--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst/defs.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))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/subst/fwd.ma".
+
+include "LambdaDelta-1/subst0/defs.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+theorem subst_lift_SO:
+ \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S
+O) d t)) t)))
+\def
+ \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq
+T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d:
+nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n)))
+(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T
+(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n))
+(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq
+T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n
+d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n)))
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T
+(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d
+(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T
+(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n)))
+(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0))
+(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T
+t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0)
+(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
+(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef
+(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O)
+(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d
+(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0))
+t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v
+(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)).
+
+theorem subst_subst0:
+ \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0
+d v t1 t2) \to (eq T (subst d v t1) (subst d v t2))))))
+\def
+ \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3))))))
+(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t:
+T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i)
+(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T
+(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0
+t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t))
+(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0)))
+(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0
+i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i)))
+(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1
+u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t:
+T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t))
+(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k
+(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst
+i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0:
+T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s
+k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t)))
+(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i))
+(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k:
+K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i:
+nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k
+i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst
+i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u
+t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda
+(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T
+(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead
+k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0
+u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u
+t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u
+t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0
+u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i)
+v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s
+k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r
+T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T
+(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0
+u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i
+v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda
+(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k
+i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4)))
+(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))))).
+
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