--- /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/subst0/fwd.ma".
+
+theorem subst0_refl:
+ \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to
+(\forall (P: Prop).P))))
+\def
+ \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d:
+nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort
+n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda
+(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef
+n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O
+u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O
+u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1
+P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P:
+Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u
+t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0
+d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead
+k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T
+(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1)))
+(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T
+(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _)
+\Rightarrow t2])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def
+(eq_ind_r T x (\lambda (t2: T).(subst0 d u t0 t2)) H4 t0 H5) in (H d H6
+P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1)
+(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T
+(\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2:
+T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k
+t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2)
+\Rightarrow t2])) (THead k t0 t1) (THead k t0 x) H3) in (let H6 \def
+(eq_ind_r T x (\lambda (t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s
+k d) H6 P)))))) H2)) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0
+t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0
+x1))).(\lambda (H4: (subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1
+x1)).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead
+_ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in ((let H7
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2)
+\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T
+t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1
+t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u
+t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0
+t1 (THead k t0 t1) d H1)))))))))) t)).
+
+theorem subst0_lift_lt:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i
+(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall
+(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d
+t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda
+(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t:
+T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v))))
+(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0)))
+(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O
+v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t:
+T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S
+i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d
+v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r
+(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda
+(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_:
+(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall
+(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d
+u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt
+i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d)
+t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d
+(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t))
+(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d
+u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h
+d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k
+u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h
+d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1:
+((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0)
+(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda
+(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3
+\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s
+k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v)
+(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r
+T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0
+(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead
+k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h
+(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind
+nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v)
+(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s
+k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s
+k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d
+u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0))
+(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h
+d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d:
+nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v)
+(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda
+(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d:
+nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d
+(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda
+(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0))
+(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0:
+nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0
+t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1)
+(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t
+(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k
+d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift
+h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v)
+(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k
+d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s
+k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d)
+(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d
+(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0))
+(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))).
+
+theorem subst0_lift_ge:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
+(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0
+(plus i h) u (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n:
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le
+d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v:
+T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T
+(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift
+(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0
+(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda
+(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r
+nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0))
+O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S
+i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d
+(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d
+H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le
+d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t:
+T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T
+(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0
+h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift
+h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1)
+(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0
+h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2
+t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda
+(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d:
+nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d
+t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3
+\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0:
+nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1
+(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0)
+(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead
+k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda
+(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t))
+(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k
+d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k
+u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d)))))))))))))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda
+(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to
+(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k:
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0
+t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k
+i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le
+d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n:
+nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0)
+(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T
+(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0
+h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift
+h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1)
+(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0
+h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d
+i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead
+k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))).
+
+theorem subst0_lift_ge_S:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d
+t1) (lift (S O) d t2))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat
+(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d
+t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O))
+(plus_sym i (S O)))))))))).
+
+theorem subst0_lift_ge_s:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s
+(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda
+(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))).
+
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