alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con".
alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con".
alias id "ex_ind" = "cic:/Coq/Init/Logic/ex_ind.con".
+alias id "plus_Snm_nSm" = "cic:/Coq/Arith/Plus/plus_Snm_nSm.con".
+alias id "plus_lt_le_compat" = "cic:/Coq/Arith/Plus/plus_lt_le_compat.con".
+alias id "plus_lt_compat" = "cic:/Coq/Arith/Plus/plus_lt_compat.con".
theorem f_equal: \forall A,B:Type. \forall f:A \to B.
\forall x,y:A. x = y \to f x = f y.
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/cimp/props".
+
+include "cimp/defs.ma".
+
+include "getl/getl.ma".
+
+theorem cimp_flat_sx:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v)
+c)))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
+C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f)
+v) (CHead d1 (Bind b) w))).((match h in nat return (\lambda (n: nat).((getl n
+(CHead c (Flat f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl
+n c (CHead d2 (Bind b) w)))))) with [O \Rightarrow (\lambda (H0: (getl O
+(CHead c (Flat f) v) (CHead d1 (Bind b) w))).(ex_intro C (\lambda (d2:
+C).(getl O c (CHead d2 (Bind b) w))) d1 (getl_intro O c (CHead d1 (Bind b) w)
+c (drop_refl c) (clear_gen_flat f c (CHead d1 (Bind b) w) v (getl_gen_O
+(CHead c (Flat f) v) (CHead d1 (Bind b) w) H0))))) | (S n) \Rightarrow
+(\lambda (H0: (getl (S n) (CHead c (Flat f) v) (CHead d1 (Bind b)
+w))).(ex_intro C (\lambda (d2: C).(getl (S n) c (CHead d2 (Bind b) w))) d1
+(getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v n H0)))]) H)))))))).
+
+theorem cimp_flat_dx:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f)
+v))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
+C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind
+b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2
+(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))).
+
+theorem cimp_bind:
+ \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
+(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
+C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
+(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
+(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w:
+T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1
+(Bind b0) w))).((match h in nat return (\lambda (n: nat).((getl n (CHead c1
+(Bind b) v) (CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead
+c2 (Bind b) v) (CHead d2 (Bind b0) w)))))) with [O \Rightarrow (\lambda (H1:
+(getl O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).(let H2 \def (f_equal
+C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) w) (CHead
+c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O
+(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H3 \def (f_equal
+C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead d1
+(Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0)
+w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let
+H4 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1
+(Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0)
+w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in
+(\lambda (H5: (eq B b0 b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda
+(t: T).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind
+b0) t))))) (eq_ind_r B b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O
+(CHead c2 (Bind b) v) (CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2:
+C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2
+v)) b0 H5) w H4)))) H3)) H2))) | (S n) \Rightarrow (\lambda (H1: (getl (S n)
+(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r
+(Bind b) n) (getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v n H1)) in (let
+H2 \def H_x in (ex_ind C (\lambda (d2: C).(getl (r (Bind b) n) c2 (CHead d2
+(Bind b0) w))) (ex C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v)
+(CHead d2 (Bind b0) w)))) (\lambda (x: C).(\lambda (H3: (getl (r (Bind b) n)
+c2 (CHead x (Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S n) (CHead c2
+(Bind b) v) (CHead d2 (Bind b0) w))) x (getl_head (Bind b) n c2 (CHead x
+(Bind b0) w) H3 v)))) H2))))]) H0)))))))))).
+
+theorem cimp_getl_conf:
+ \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
+(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w))
+\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead
+d2 (Bind b) w)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
+C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
+(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
+(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def
+H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C
+(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall
+(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4:
+C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x
+(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3:
+C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0))
+\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0))))))))))
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0:
+B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h
+d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1
+(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0
+(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in
+(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (plus (S h) i) c2
+(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind
+b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (plus (S h) i) c2 (CHead x0
+(Bind b0) w0))).(let H_y0 \def (getl_conf_le (plus (S h) i) (CHead x0 (Bind
+b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (eq_ind nat (minus
+(plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind b) w) (CHead x0
+(Bind b0) w0))) (H_y0 (le_plus_r (S h) i)) (S h) (minus_plus_r (S h) i)) in
+(ex_intro C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0
+(getl_gen_S (Bind b) x (CHead x0 (Bind b0) w0) w h H6)))))) H4))))))))) H2)))
+H1)))))))))).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/getl".
+
+include "drop1/defs.ma".
+
+include "getl/defs.ma".
+
+axiom drop1_getl_trans:
+ \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
+\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
+i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2
+(CHead e2 (Bind b) (ctrans hds i v)))))))))))))
+.
+
include "drop/props.ma".
+include "getl/defs.ma".
+
theorem drop1_skip_bind:
\forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c:
C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b)
v))).(getl_flat c0 e2 (S n) (H e1 v (clear_gen_flat f c0 (CHead e1 (Bind b)
v) t H2) e2 n H1) f t))]) H0))))))))))) c)).
+theorem getl_clear_conf:
+ \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to
+(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3:
+C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2
+c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1
+c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c:
+C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in
+(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3
+(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3
+\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead
+x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c:
+C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3
+(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall
+(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2)
+\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall
+(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n)
+c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n)
+(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0)
+c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c:
+C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2:
+C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda
+(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).((match k in K return
+(\lambda (k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t)
+c2) \to (getl (S n) c2 c3)))) with [(Bind b) \Rightarrow (\lambda (H3: (getl
+(S n) (CHead c (Bind b) t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t)
+c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3))
+(getl_head (Bind b) n c c3 (getl_gen_S (Bind b) c c3 t n H3) t) c2
+(clear_gen_bind b c c2 t H4)))) | (Flat f) \Rightarrow (\lambda (H3: (getl (S
+n) (CHead c (Flat f) t) c3)).(\lambda (H4: (clear (CHead c (Flat f) t)
+c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n H3) c2 (clear_gen_flat f c c2 t
+H4))))]) H1 H2))))))))) c1)))) i).
+
set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/getl/drop".
-include "getl/fwd.ma".
+include "getl/props.ma".
include "clear/drop.ma".
nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t
n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
+theorem getl_drop_conf_lt:
+ \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c0 e0)))))))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1:
+C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d))
+c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i
+(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i
+(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda
+(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i:
+nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t)
+(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def
+(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C
+(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0
+(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x:
+C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead
+c1 (Bind b) u))).((match x in C return (\lambda (c2: C).((drop i O (CHead c0
+k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O (CHead c0
+k t) (CSort n))).(\lambda (H6: (clear (CSort n) (CHead c1 (Bind b)
+u))).(clear_gen_sort (CHead c1 (Bind b) u) n H6 (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0))))))) | (CHead c2 k0 t0) \Rightarrow (\lambda (H5: (drop i O
+(CHead c0 k t) (CHead c2 k0 t0))).(\lambda (H6: (clear (CHead c2 k0 t0)
+(CHead c1 (Bind b) u))).((match k0 in K return (\lambda (k1: K).((drop i O
+(CHead c0 k t) (CHead c2 k1 t0)) \to ((clear (CHead c2 k1 t0) (CHead c1 (Bind
+b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) with [(Bind b0) \Rightarrow
+(\lambda (H7: (drop i O (CHead c0 k t) (CHead c2 (Bind b0) t0))).(\lambda
+(H8: (clear (CHead c2 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind b)
+u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0
+H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat
+_) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0)
+(clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def
+(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind b)
+u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0
+H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 c2)).(let H14
+\def (eq_ind_r T t0 (\lambda (t0: T).(drop i O (CHead c0 k t) (CHead c2 (Bind
+b0) t0))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b: B).(drop i O
+(CHead c0 k t) (CHead c2 (Bind b) u))) H14 b H12) in (let H16 \def (eq_ind_r
+C c2 (\lambda (c: C).(drop i O (CHead c0 k t) (CHead c (Bind b) u))) H15 c1
+H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r
+(Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i O e (CHead e0
+(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r (Bind b) d) c1
+e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: T).(\lambda (x1:
+C).(\lambda (H17: (eq T u (lift h (r (Bind b) d) x0))).(\lambda (H18: (drop i
+O e (CHead x1 (Bind b) x0))).(\lambda (H19: (drop h (r (Bind b) d) c1
+x1)).(eq_ind_r T (lift h (r (Bind b) d) x0) (\lambda (t1: T).(ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x0) (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))) x0 x1 (refl_equal T (lift h d x0))
+(getl_intro i e (CHead x1 (Bind b) x0) (CHead x1 (Bind b) x0) H18 (clear_bind
+b x1 x0)) H19) u H17)))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H16 e
+h d H1)))))))) H10)) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (drop i O
+(CHead c0 k t) (CHead c2 (Flat f) t0))).(\lambda (H8: (clear (CHead c2 (Flat
+f) t0) (CHead c1 (Bind b) u))).((match i in nat return (\lambda (n:
+nat).((drop h (S (plus n d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t)
+(CHead c2 (Flat f) t0)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T
+u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind
+b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) with [O
+\Rightarrow (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) e)).(\lambda
+(H10: (drop O O (CHead c0 k t) (CHead c2 (Flat f) t0))).(let H11 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k t)
+(CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0)
+H10)) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow
+k])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t)
+(CHead c2 (Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t |
+(CHead _ _ t) \Rightarrow t])) (CHead c0 k t) (CHead c2 (Flat f) t0)
+(drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in (\lambda (H14:
+(eq K k (Flat f))).(\lambda (H15: (eq C c0 c2)).(let H16 \def (eq_ind_r C c2
+(\lambda (c: C).(clear c (CHead c1 (Bind b) u))) (clear_gen_flat f c2 (CHead
+c1 (Bind b) u) t0 H8) c0 H15) in (let H17 \def (eq_ind K k (\lambda (k:
+K).(drop h (S (plus O d)) (CHead c0 k t) e)) H9 (Flat f) H14) in (ex3_2_ind C
+T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda
+(_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda
+(e0: C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H18: (eq C e (CHead x0 (Flat f) x1))).(\lambda (H19: (eq T t
+(lift h (r (Flat f) (plus O d)) x1))).(\lambda (H20: (drop h (r (Flat f)
+(plus O d)) c0 x0)).(let H21 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
+(r (Flat f) (plus O d)) x1) H19) in (eq_ind_r C (CHead x0 (Flat f) x1)
+(\lambda (c3: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl O c3 (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H22 \def (H c1 u O
+(getl_intro O c0 (CHead c1 (Bind b) u) c0 (drop_refl c0) H16) x0 h d H20) in
+(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl O x0 (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: T).(\lambda (x3:
+C).(\lambda (H23: (eq T u (lift h d x2))).(\lambda (H24: (getl O x0 (CHead x3
+(Bind b) x2))).(\lambda (H25: (drop h d c1 x3)).(let H26 \def (eq_ind T u
+(\lambda (t: T).(clear c0 (CHead c1 (Bind b) t))) H16 (lift h d x2) H23) in
+(eq_ind_r T (lift h d x2) (\lambda (t1: T).(ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T (lift h d x2) (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift
+h d x2)) (getl_flat x0 (CHead x3 (Bind b) x2) O H24 f x1) H25) u H23)))))))
+H22)) e H18))))))) (drop_gen_skip_l c0 e t h (plus O d) (Flat f) H17)))))))
+H12)) H11)))) | (S n) \Rightarrow (\lambda (H9: (drop h (S (plus (S n) d))
+(CHead c0 k t) e)).(\lambda (H10: (drop (S n) O (CHead c0 k t) (CHead c2
+(Flat f) t0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead
+e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S n)
+d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S n) d)) c0
+e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl (S n) e (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H11: (eq C e (CHead x0 k x1))).(\lambda (H12:
+(eq T t (lift h (r k (plus (S n) d)) x1))).(\lambda (H13: (drop h (r k (plus
+(S n) d)) c0 x0)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
+(r k (plus (S n) d)) x1) H12) in (eq_ind_r C (CHead x0 k x1) (\lambda (c3:
+C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl (S n) c3 (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H15 \def (eq_ind
+nat (r k (plus (S n) d)) (\lambda (n: nat).(drop h n c0 x0)) H13 (plus (r k
+(S n)) d) (r_plus k (S n) d)) in (let H16 \def (eq_ind nat (r k (S n))
+(\lambda (n: nat).(drop h (plus n d) c0 x0)) H15 (S (r k n)) (r_S k n)) in
+(let H17 \def (H c1 u (r k n) (getl_intro (r k n) c0 (CHead c1 (Bind b) u)
+(CHead c2 (Flat f) t0) (drop_gen_drop k c0 (CHead c2 (Flat f) t0) t n H10)
+(clear_flat c2 (CHead c1 (Bind b) u) (clear_gen_flat f c2 (CHead c1 (Bind b)
+u) t0 H8) f t0)) x0 h d H16) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k n) x0
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2:
+T).(\lambda (x3: C).(\lambda (H18: (eq T u (lift h d x2))).(\lambda (H19:
+(getl (r k n) x0 (CHead x3 (Bind b) x2))).(\lambda (H20: (drop h d c1
+x3)).(let H21 \def (eq_ind T u (\lambda (t: T).(clear c2 (CHead c1 (Bind b)
+t))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) (lift h d x2) H18) in
+(eq_ind_r T (lift h d x2) (\lambda (t1: T).(ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T (lift h d x2) (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift
+h d x2)) (getl_head k n x0 (CHead x3 (Bind b) x2) H19 x1) H20) u H18)))))))
+H17)))) e H11))))))) (drop_gen_skip_l c0 e t h (plus (S n) d) k H9))))]) H1
+H7)))]) H5 H6)))]) H3 H4)))) H2)))))))))))))) c)).
+
+theorem getl_drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d
+h) i) \to (getl (minus i h) e a)))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
+d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H)
+in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0
+a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c
+x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i
+x c H3 e h d H0 H1) H4)))) H2)))))))))).
+
+theorem getl_conf_ge_drop:
+ \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i:
+nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1
+c2) \to (drop i O c2 e))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda
+(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O))
+(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e
+u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S
+i))) (le_n (S i)) (plus i (S O)) (plus_comm i (S O)))) i (minus_Sx_SO i)) in
+H3)))))))).
+
+theorem getl_drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
+(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i
+c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2))
+(\lambda (c1: C).(drop (S i) j c1 e1)))))))))))
+\def
+ \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1
+e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2
+H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
+
+theorem drop_getl_trans_lt:
+ \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2:
+C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda
+(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda
+(e1: C).(drop h (minus d (S i)) e1 e2)))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
+c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i
+c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b)
+v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e:
+C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead
+e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d
+(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4:
+(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1
+e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1:
+C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1:
+C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O
+c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat
+(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i)))
+(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b
+e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h
+(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C
+(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v))))
+(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda
+(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda
+(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i
+c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h
+(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus
+d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S
+(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))).
+
+theorem drop_getl_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2
+e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 e2))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
+c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def
+(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
+(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_:
+C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i)
+e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x:
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def
+(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i
+O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda
+(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1:
+C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1
+e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h
+(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5)))))
+H2)))))))))).
+
+theorem drop_getl_trans_ge:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2)
+\to ((le d i) \to (getl (plus i h) c1 e2)))))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2:
+C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def
+(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
+(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x:
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro
+(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
+
+theorem getl_drop_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i
+h)) O c1 e2)))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h:
+nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2
+e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2:
+C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2
+H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda
+(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2:
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2
+e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall
+(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2:
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead
+c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3:
+C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b)
+t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop
+(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead
+c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S
+i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2))
+H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2
+(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead
+c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2
+t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_:
+(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i:
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t)
+e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S
+i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop
+(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2
+i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f:
+F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n:
+nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i:
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t)
+e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f)
+(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2)
+(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0)
+t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i
+n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2
+(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i)
+O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S
+(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/getl/getl".
+
+include "getl/drop.ma".
+
+include "getl/clear.ma".
+
+theorem getl_conf_le:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
+(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e
+a))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda
+(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda
+(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e
+a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x
+e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1)
+e H4)))) H2))))))))).
+
+theorem getl_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl
+h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1
+e2)))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
+nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2
+e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e:
+C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2)
+(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x
+e2)).((match i in nat return (\lambda (n: nat).((drop n O c2 x) \to (getl
+(plus n h) c1 e2))) with [O \Rightarrow (\lambda (H4: (drop O O c2 x)).(let
+H5 \def (eq_ind_r C x (\lambda (c: C).(clear c e2)) H3 c2 (drop_gen_refl c2 x
+H4)) in (getl_clear_trans (plus O h) c1 c2 H e2 H5))) | (S n) \Rightarrow
+(\lambda (H4: (drop (S n) O c2 x)).(let H_y \def (getl_drop_trans c1 c2 h H x
+n H4) in (getl_intro (plus (S n) h) c1 e2 x H_y H3)))]) H2)))) H1)))))))).
+
u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v
x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))).
+theorem getl_mono:
+ \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to
+(\forall (x2: C).((getl h c x2) \to (eq C x1 x2))))))
+\def
+ \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c
+x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all
+c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e:
+C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c
+x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in
+(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq
+C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear
+x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0
+(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0:
+C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def
+(eq_ind_r C x0 (\lambda (c: C).(clear c x1)) H6 x (drop_mono c x O h H2 x0
+H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/dec".
+
+include "subst0/defs.ma".
+
+include "lift/props.ma".
+
+theorem dnf_dec:
+ \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or
+(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
+\def
+ \lambda (w: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(ex
+T (\lambda (v: T).(or (subst0 d w t0 (lift (S O) d v)) (eq T t0 (lift (S O) d
+v))))))) (\lambda (n: nat).(\lambda (d: nat).(ex_intro T (\lambda (v: T).(or
+(subst0 d w (TSort n) (lift (S O) d v)) (eq T (TSort n) (lift (S O) d v))))
+(TSort n) (eq_ind_r T (TSort n) (\lambda (t0: T).(or (subst0 d w (TSort n)
+t0) (eq T (TSort n) t0))) (or_intror (subst0 d w (TSort n) (TSort n)) (eq T
+(TSort n) (TSort n)) (refl_equal T (TSort n))) (lift (S O) d (TSort n))
+(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d
+(ex T (\lambda (v: T).(or (subst0 d w (TLRef n) (lift (S O) d v)) (eq T
+(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(ex_intro T (\lambda
+(v: T).(or (subst0 d w (TLRef n) (lift (S O) d v)) (eq T (TLRef n) (lift (S
+O) d v)))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (subst0 d w
+(TLRef n) t0) (eq T (TLRef n) t0))) (or_intror (subst0 d w (TLRef n) (TLRef
+n)) (eq T (TLRef n) (TLRef n)) (refl_equal T (TLRef n))) (lift (S O) d (TLRef
+n)) (lift_lref_lt n (S O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n
+(\lambda (n0: nat).(ex T (\lambda (v: T).(or (subst0 n0 w (TLRef n) (lift (S
+O) n0 v)) (eq T (TLRef n) (lift (S O) n0 v)))))) (ex_intro T (\lambda (v:
+T).(or (subst0 n w (TLRef n) (lift (S O) n v)) (eq T (TLRef n) (lift (S O) n
+v)))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) (\lambda (t0: T).(or
+(subst0 n w (TLRef n) t0) (eq T (TLRef n) t0))) (or_introl (subst0 n w (TLRef
+n) (lift (S n) O w)) (eq T (TLRef n) (lift (S n) O w)) (subst0_lref w n))
+(lift (S O) n (lift n O w)) (lift_free w n (S O) O n (le_n (plus O n))
+(le_O_n n)))) d H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (v: T).(or
+(subst0 d w (TLRef n) (lift (S O) d v)) (eq T (TLRef n) (lift (S O) d v))))
+(TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (subst0 d w
+(TLRef n) t0) (eq T (TLRef n) t0))) (or_intror (subst0 d w (TLRef n) (TLRef
+n)) (eq T (TLRef n) (TLRef n)) (refl_equal T (TLRef n))) (lift (S O) d (TLRef
+(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (v: T).(or (subst0 d w t0
+(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))))).(\lambda (t1: T).(\lambda
+(H0: ((\forall (d: nat).(ex T (\lambda (v: T).(or (subst0 d w t1 (lift (S O)
+d v)) (eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d)
+in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 d w t0 (lift (S
+O) d v)) (eq T t0 (lift (S O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w
+(THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x: T).(\lambda (H2: (or (subst0 d w t0 (lift (S O) d x)) (eq T t0
+(lift (S O) d x)))).(or_ind (subst0 d w t0 (lift (S O) d x)) (eq T t0 (lift
+(S O) d x)) (ex T (\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O)
+d v)) (eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (H3: (subst0 d w t0
+(lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H4 \def H_x0 in
+(ex_ind T (\lambda (v: T).(or (subst0 (s k d) w t1 (lift (S O) (s k d) v))
+(eq T t1 (lift (S O) (s k d) v)))) (ex T (\lambda (v: T).(or (subst0 d w
+(THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x0: T).(\lambda (H5: (or (subst0 (s k d) w t1 (lift (S O) (s k d)
+x0)) (eq T t1 (lift (S O) (s k d) x0)))).(or_ind (subst0 (s k d) w t1 (lift
+(S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)) (ex T (\lambda (v:
+T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1)
+(lift (S O) d v))))) (\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s k d)
+x0))).(ex_intro T (\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O)
+d v)) (eq T (THead k t0 t1) (lift (S O) d v)))) (THead k x x0) (eq_ind_r T
+(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(or
+(subst0 d w (THead k t0 t1) t2) (eq T (THead k t0 t1) t2))) (or_introl
+(subst0 d w (THead k t0 t1) (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0))) (eq T (THead k t0 t1) (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0))) (subst0_both w t0 (lift (S O) d x) d H3 k t1 (lift (S O) (s k d) x0)
+H6)) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d)))) (\lambda
+(H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T (lift (S O) (s k d) x0)
+(\lambda (t2: T).(ex T (\lambda (v: T).(or (subst0 d w (THead k t0 t2) (lift
+(S O) d v)) (eq T (THead k t0 t2) (lift (S O) d v)))))) (ex_intro T (\lambda
+(v: T).(or (subst0 d w (THead k t0 (lift (S O) (s k d) x0)) (lift (S O) d v))
+(eq T (THead k t0 (lift (S O) (s k d) x0)) (lift (S O) d v)))) (THead k x x0)
+(eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (\lambda (t2:
+T).(or (subst0 d w (THead k t0 (lift (S O) (s k d) x0)) t2) (eq T (THead k t0
+(lift (S O) (s k d) x0)) t2))) (or_introl (subst0 d w (THead k t0 (lift (S O)
+(s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d) x0))) (eq T (THead
+k t0 (lift (S O) (s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0))) (subst0_fst w (lift (S O) d x) t0 d H3 (lift (S O) (s k d) x0) k))
+(lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d))) t1 H6)) H5)))
+H4)))) (\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d))
+in (let H4 \def H_x0 in (ex_ind T (\lambda (v: T).(or (subst0 (s k d) w t1
+(lift (S O) (s k d) v)) (eq T t1 (lift (S O) (s k d) v)))) (ex T (\lambda (v:
+T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1)
+(lift (S O) d v))))) (\lambda (x0: T).(\lambda (H5: (or (subst0 (s k d) w t1
+(lift (S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)))).(or_ind (subst0
+(s k d) w t1 (lift (S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)) (ex T
+(\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T
+(THead k t0 t1) (lift (S O) d v))))) (\lambda (H6: (subst0 (s k d) w t1 (lift
+(S O) (s k d) x0))).(eq_ind_r T (lift (S O) d x) (\lambda (t2: T).(ex T
+(\lambda (v: T).(or (subst0 d w (THead k t2 t1) (lift (S O) d v)) (eq T
+(THead k t2 t1) (lift (S O) d v)))))) (ex_intro T (\lambda (v: T).(or (subst0
+d w (THead k (lift (S O) d x) t1) (lift (S O) d v)) (eq T (THead k (lift (S
+O) d x) t1) (lift (S O) d v)))) (THead k x x0) (eq_ind_r T (THead k (lift (S
+O) d x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(or (subst0 d w (THead k
+(lift (S O) d x) t1) t2) (eq T (THead k (lift (S O) d x) t1) t2))) (or_introl
+(subst0 d w (THead k (lift (S O) d x) t1) (THead k (lift (S O) d x) (lift (S
+O) (s k d) x0))) (eq T (THead k (lift (S O) d x) t1) (THead k (lift (S O) d
+x) (lift (S O) (s k d) x0))) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H6
+(lift (S O) d x))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d)))
+t0 H3)) (\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T (lift (S
+O) (s k d) x0) (\lambda (t2: T).(ex T (\lambda (v: T).(or (subst0 d w (THead
+k t0 t2) (lift (S O) d v)) (eq T (THead k t0 t2) (lift (S O) d v))))))
+(eq_ind_r T (lift (S O) d x) (\lambda (t2: T).(ex T (\lambda (v: T).(or
+(subst0 d w (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)) (eq T
+(THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (ex_intro T
+(\lambda (v: T).(or (subst0 d w (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0)) (lift (S O) d v)) (eq T (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0)) (lift (S O) d v)))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0)) (\lambda (t2: T).(or (subst0 d w (THead k (lift (S
+O) d x) (lift (S O) (s k d) x0)) t2) (eq T (THead k (lift (S O) d x) (lift (S
+O) (s k d) x0)) t2))) (or_intror (subst0 d w (THead k (lift (S O) d x) (lift
+(S O) (s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d) x0))) (eq T
+(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0))) (refl_equal T (THead k (lift (S O) d x) (lift (S O)
+(s k d) x0)))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d))) t0
+H3) t1 H6)) H5))) H4)))) H2))) H1))))))))) 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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/defs".
+
+include "lift/defs.ma".
+
+inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
+| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift
+(S i) O v)))
+| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i:
+nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v
+(THead k u1 t) (THead k u2 t))))))))
+| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1:
+T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i
+v (THead k u t1) (THead k u t2))))))))
+| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i:
+nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2:
+T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2
+t2)))))))))).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd".
+
+include "subst0/defs.ma".
+
+include "lift/props.ma".
+
+theorem subst0_gen_sort:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
+i v (TSort n) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(let H0 \def (match H in
+subst0 return (\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (_: (subst0 n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to
+((eq T t0 (TSort n)) \to ((eq T t1 x) \to P))))))))) with [(subst0_lref v0
+i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0
+v)).(\lambda (H2: (eq T (TLRef i0) (TSort n))).(\lambda (H3: (eq T (lift (S
+i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef
+n0) (TSort n)) \to ((eq T (lift (S n0) O v0) x) \to P)))) (\lambda (H4: (eq T
+v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TSort n)) \to ((eq T
+(lift (S i) O t) x) \to P))) (\lambda (H5: (eq T (TLRef i) (TSort n))).(let
+H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (TSort n) H5) in (False_ind ((eq T (lift
+(S i) O v) x) \to P) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1
+H2 H3))))) | (subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq
+nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t)
+(TSort n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda
+(n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead
+k u2 t) x) \to ((subst0 n0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0
+v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k u1 t) (TSort n)) \to ((eq T
+(THead k u2 t) x) \to ((subst0 i t0 u1 u2) \to P)))) (\lambda (H6: (eq T
+(THead k u1 t) (TSort n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2)
+\to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4
+H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat
+i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TSort
+n))).(\lambda (H4: (eq T (THead k u t2) x)).(eq_ind nat i (\lambda (n0:
+nat).((eq T v0 v) \to ((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u
+t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to P))))) (\lambda (H5: (eq T v0
+v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TSort n)) \to ((eq T
+(THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to P)))) (\lambda (H6: (eq T
+(THead k u t1) (TSort n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H6) in (False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v
+t1 t2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4
+H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2:
+(eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1)
+(TSort n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda
+(n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TSort n)) \to ((eq T
+(THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2)
+\to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T
+(THead k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1
+u2) \to ((subst0 (s k i) t t1 t2) \to P))))) (\lambda (H7: (eq T (THead k u1
+t1) (TSort n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match
+e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
+H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to
+((subst0 (s k i) v t1 t2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq
+nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v)
+(refl_equal T (TSort n)) (refl_equal T x)))))))).
+
+theorem subst0_gen_lref:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
+i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst0 i v (TLRef n) x)).(let H0 \def (match H in subst0 return (\lambda
+(n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_:
+(subst0 n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TLRef n))
+\to ((eq T t1 x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))))))))
+with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda
+(H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TLRef n))).(\lambda (H3:
+(eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v)
+\to ((eq T (TLRef n0) (TLRef n)) \to ((eq T (lift (S n0) O v0) x) \to (land
+(eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H4: (eq T v0
+v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift
+(S i) O t) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) (\lambda
+(H5: (eq T (TLRef i) (TLRef n))).(let H6 \def (f_equal T nat (\lambda (e:
+T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i |
+(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n)
+H5) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) x) \to (land
+(eq nat n n0) (eq T x (lift (S n) O v))))) (\lambda (H7: (eq T (lift (S n) O
+v) x)).(eq_ind T (lift (S n) O v) (\lambda (t: T).(land (eq nat n n) (eq T t
+(lift (S n) O v)))) (conj (eq nat n n) (eq T (lift (S n) O v) (lift (S n) O
+v)) (refl_equal nat n) (refl_equal T (lift (S n) O v))) x H7)) i (sym_eq nat
+i n H6)))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) |
+(subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0
+i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TLRef
+n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0:
+nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2
+t) x) \to ((subst0 n0 v0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O
+v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T
+(THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1
+u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T
+(THead k u1 t) (TLRef n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2)
+\to (land (eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v
+H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0
+u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
+v)).(\lambda (H3: (eq T (THead k u t1) (TLRef n))).(\lambda (H4: (eq T (THead
+k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k
+u t1) (TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2)
+\to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0
+v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TLRef n)) \to ((eq T
+(THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T
+x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u t1) (TLRef n))).(let
+H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in
+(False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to (land
+(eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0
+(sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2
+H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0
+v)).(\lambda (H4: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H5: (eq T
+(THead k u2 t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T
+(THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 n0 v0
+u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to (land (eq nat n i) (eq T x (lift
+(S n) O v))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t:
+T).((eq T (THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to
+((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq
+T x (lift (S n) O v)))))))) (\lambda (H7: (eq T (THead k u1 t1) (TLRef
+n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in
+(False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s
+k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) H8))) v0
+(sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0
+(refl_equal nat i) (refl_equal T v) (refl_equal T (TLRef n)) (refl_equal T
+x))))))).
+
+theorem subst0_gen_head:
+ \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
+(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) x)).(let H0
+\def (match H in subst0 return (\lambda (n: nat).(\lambda (t: T).(\lambda
+(t0: T).(\lambda (t2: T).(\lambda (_: (subst0 n t t0 t2)).((eq nat n i) \to
+((eq T t v) \to ((eq T t0 (THead k u1 t1)) \to ((eq T t2 x) \to (or3 (ex2 T
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))))))
+with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda
+(H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (THead k u1 t1))).(\lambda
+(H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0
+v) \to ((eq T (TLRef n) (THead k u1 t1)) \to ((eq T (lift (S n) O v0) x) \to
+(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2)))
+(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
+t2))))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T
+(TLRef i) (THead k u1 t1)) \to ((eq T (lift (S i) O t) x) \to (or3 (ex2 T
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))
+(\lambda (H5: (eq T (TLRef i) (THead k u1 t1))).(let H6 \def (eq_ind T (TLRef
+i) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead k u1 t1) H5) in (False_ind ((eq T (lift (S i) O v) x) \to
+(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2)))
+(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
+t2)))))) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3)))))
+| (subst0_fst v0 u2 u0 i0 H0 t k0) \Rightarrow (\lambda (H1: (eq nat i0
+i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k
+u1 t1))).(\lambda (H4: (eq T (THead k0 u2 t) x)).(eq_ind nat i (\lambda (n:
+nat).((eq T v0 v) \to ((eq T (THead k0 u0 t) (THead k u1 t1)) \to ((eq T
+(THead k0 u2 t) x) \to ((subst0 n v0 u0 u2) \to (or3 (ex2 T (\lambda (u3:
+T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T
+(\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v
+t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3
+t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H5: (eq T v0
+v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k0 u0 t) (THead k u1 t1)) \to
+((eq T (THead k0 u2 t) x) \to ((subst0 i t0 u0 u2) \to (or3 (ex2 T (\lambda
+(u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2
+T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i)
+v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3
+t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H6: (eq T
+(THead k0 u0 t) (THead k u1 t1))).(let H7 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t |
+(TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t)
+(THead k u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t) (THead k u1
+t1) H6) in ((let H9 \def (f_equal T K (\lambda (e: T).(match e in T return
+(\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0
+| (THead k _ _) \Rightarrow k])) (THead k0 u0 t) (THead k u1 t1) H6) in
+(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t t1) \to ((eq T (THead
+k1 u2 t) x) \to ((subst0 i v u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x
+(THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2:
+T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H10: (eq T u0
+u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t t1) \to ((eq T (THead k u2 t) x)
+\to ((subst0 i v t0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3
+t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H11: (eq T t t1)).(eq_ind T t1
+(\lambda (t0: T).((eq T (THead k u2 t0) x) \to ((subst0 i v u1 u2) \to (or3
+(ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i
+v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1
+u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))
+(\lambda (H12: (eq T (THead k u2 t1) x)).(eq_ind T (THead k u2 t1) (\lambda
+(t0: T).((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t0 (THead
+k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq
+T t0 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i) v t1 t2))))))) (\lambda (H13: (subst0 i v u1
+u2)).(or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3
+t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T
+(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) (THead k
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (u3:
+T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1
+u3)) u2 (refl_equal T (THead k u2 t1)) H13))) x H12)) t (sym_eq T t t1 H11)))
+u0 (sym_eq T u0 u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0
+v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k0 v0 t2 t0 i0
+H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
+v)).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 t1))).(\lambda (H4: (eq T
+(THead k0 u t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T
+(THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0
+(s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1)))
+(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq
+T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0
+(s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1)))
+(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3))))))))) (\lambda (H6: (eq T (THead k0 u t0) (THead k u1 t1))).(let
+H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H8 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H9 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _)
+\Rightarrow k])) (THead k0 u t0) (THead k u1 t1) H6) in (eq_ind K k (\lambda
+(k1: K).((eq T u u1) \to ((eq T t0 t1) \to ((eq T (THead k1 u t2) x) \to
+((subst0 (s k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k
+u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T
+x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1
+(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead k t t2) x) \to ((subst0 (s k
+i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1)))
+(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3))))))))) (\lambda (H11: (eq T t0 t1)).(eq_ind T t1 (\lambda (t:
+T).((eq T (THead k u1 t2) x) \to ((subst0 (s k i) v t t2) \to (or3 (ex2 T
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))
+(\lambda (H12: (eq T (THead k u1 t2) x)).(eq_ind T (THead k u1 t2) (\lambda
+(t: T).((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T t
+(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3:
+T).(eq T t (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))) (\lambda (H13: (subst0 (s
+k i) v t1 t2)).(or3_intro1 (ex2 T (\lambda (u2: T).(eq T (THead k u1 t2)
+(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3:
+T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v
+t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2)
+(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex_intro2 T
+(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3)) t2 (refl_equal T (THead k u1 t2)) H13))) x H12))
+t0 (sym_eq T t0 t1 H11))) u (sym_eq T u u1 H10))) k0 (sym_eq K k0 k H9)))
+H8)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) |
+(subst0_both v0 u0 u2 i0 H0 k0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0
+i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k
+u1 t1))).(\lambda (H5: (eq T (THead k0 u2 t2) x)).(eq_ind nat i (\lambda (n:
+nat).((eq T v0 v) \to ((eq T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T
+(THead k0 u2 t2) x) \to ((subst0 n v0 u0 u2) \to ((subst0 (s k0 n) v0 t0 t2)
+\to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3)))
+(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
+t3))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T
+(THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to ((subst0
+i t u0 u2) \to ((subst0 (s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u3:
+T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T
+(\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v
+t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H7: (eq T
+(THead k0 u0 t0) (THead k u1 t1))).(let H8 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t0)
+(THead k u1 t1) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t0) (THead k u1
+t1) H7) in ((let H10 \def (f_equal T K (\lambda (e: T).(match e in T return
+(\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0
+| (THead k _ _) \Rightarrow k])) (THead k0 u0 t0) (THead k u1 t1) H7) in
+(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T
+(THead k1 u2 t2) x) \to ((subst0 i v u0 u2) \to ((subst0 (s k1 i) v t0 t2)
+\to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3)))
+(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
+t3))))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq
+T t0 t1) \to ((eq T (THead k u2 t2) x) \to ((subst0 i v t u2) \to ((subst0 (s
+k i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1)))
+(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3)))))))))) (\lambda (H12: (eq T t0 t1)).(eq_ind T t1 (\lambda (t:
+T).((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t
+t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3)))
+(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
+t3))))))))) (\lambda (H13: (eq T (THead k u2 t2) x)).(eq_ind T (THead k u2
+t2) (\lambda (t: T).((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to
+(or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T t (THead k u1 t3)))
+(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T t (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
+t3)))))))) (\lambda (H14: (subst0 i v u1 u2)).(\lambda (H15: (subst0 (s k i)
+v t1 t2)).(or3_intro2 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t2) (THead k
+u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T
+(THead k u2 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2_intro T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda
+(u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i) v t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H14
+H15)))) x H13)) t0 (sym_eq T t0 t1 H12))) u0 (sym_eq T u0 u1 H11))) k0
+(sym_eq K k0 k H10))) H9)) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i
+H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v)
+(refl_equal T (THead k u1 t1)) (refl_equal T x))))))))).
+
+theorem subst0_gen_lift_lt:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1)
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u t1 t2)))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d
+u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n:
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n))
+x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t:
+T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d))))
+in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n)
+t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S
+(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2:
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
+n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h
+(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H
+(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (and_ind (eq nat n i) (eq
+T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2:
+(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T
+(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))))
+(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0)
+O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T
+(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O
+(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3)))
+(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d))
+n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t:
+T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S
+(plus i d)) H0)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
+h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat
+(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d
+u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n))
+H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2:
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
+n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4))))
+(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t)
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
+(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift
+h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0
+t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t
+t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0))
+(\lambda (t: T).(subst0 i (lift h d u) t x)) H1 (THead k (lift h (S (plus i
+d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d))))
+in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus
+i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d))
+t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t)
+t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
+d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S
+(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h
+d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
+t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k
+(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S
+(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
+(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h
+(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
+T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d)))
+t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t)
+x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T
+x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T
+(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
+(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda
+(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d)))
+t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
+t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h
+(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k
+(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1
+t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3:
+(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2)))
+(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d)))
+t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i
+d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
+(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
+T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda
+(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0)
+x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2
+T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3:
+T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i
+d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S
+(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus
+i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0))
+H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
+(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
+(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d))
+x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s
+k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h
+(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i
+u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T
+(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1))
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
+t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda
+(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind
+T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
+(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i
+d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d))
+(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i
+d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d))))
+(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0
+H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
+(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d
+u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
+(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda
+(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x
+(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d))
+t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
+d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
+(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda
+(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i
+d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d))
+(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7
+(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
+(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda
+(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k
+i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T
+(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S
+(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S
+(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
+x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
+t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S
+(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind
+nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k
+(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus
+i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S
+(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2:
+T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead
+k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus
+i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T
+(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k
+x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u
+t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S
+(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d)))
+(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0
+i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k
+(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i
+H2))))))))))))) t1)).
+
+theorem subst0_gen_lift_false:
+ \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u
+(lift h d t) x) \to (\forall (P: Prop).P)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x:
+T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i
+(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P:
+Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda
+(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n))
+x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda
+(t: T).(subst0 i u t x)) H1 (TSort n) (lift_sort n h d)) in (subst0_gen_sort
+u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x:
+T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (le d
+i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d
+(TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P (\lambda (H2: (lt n d)).(let
+H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H1
+(TLRef n) (lift_lref_lt n h d H2)) in (and_ind (eq nat n i) (eq T x (lift (S
+n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O
+u))).(let H6 \def (eq_ind nat n (\lambda (n: nat).(lt n d)) H2 i H4) in
+(le_false d i P H H6)))) (subst0_gen_lref u x i n H3)))) (\lambda (H2: (le d
+n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t
+x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d H2)) in (and_ind (eq nat (plus
+n h) i) (eq T x (lift (S (plus n h)) O u)) P (\lambda (H4: (eq nat (plus n h)
+i)).(\lambda (_: (eq T x (lift (S (plus n h)) O u))).(let H6 \def (eq_ind_r
+nat i (\lambda (n: nat).(lt n (plus d h))) H0 (plus n h) H4) in (le_false d n
+P H2 (lt_le_S n d (simpl_lt_plus_r h n d H6)))))) (subst0_gen_lref u x i
+(plus n h) H3))))))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H:
+((\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).(\forall
+(i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u (lift h d t0) x)
+\to (\forall (P: Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall
+(u: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i:
+nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to
+(\forall (P: Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda
+(H2: (lt i (plus d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1))
+x)).(\lambda (P: Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1))
+(\lambda (t: T).(subst0 i u t x)) H3 (THead k (lift h d t0) (lift h (s k d)
+t1)) (lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x
+(THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0)
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
+(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
+(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2:
+T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u
+(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
+(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda
+(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7:
+(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda
+(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
+(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2:
+T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u
+(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k
+(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1)
+x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h))
+(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i
+(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda
+(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7:
+(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d)
+t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d
+t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
+
+theorem subst0_gen_lift_ge:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h)
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u t1 t2))))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h
+d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n:
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus
+d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0
+i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2
+T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i
+h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d
+(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef
+n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n))
+(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in
+(and_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq T
+x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
+(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5
+\def (eq_ind nat n (\lambda (n: nat).(lt n d)) H1 i H3) in (le_false (plus d
+h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0
+(minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5)))))
+(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind
+T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h))
+(lift_lref_ge n h d H1)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S
+(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
+(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n
+h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n
+h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S
+(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2))))
+(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S
+(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n)
+t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h
+d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u)
+(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n
+h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O
+u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
+nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift
+(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_comm n
+h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans d (S n)
+(plus O (S n)) (le_S d n H1) (le_n (plus O (S n)))) (le_O_n d))) (subst0_lref
+u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i H3))) (subst0_gen_lref
+u x i (plus n h) H2)))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H:
+((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d:
+nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t
+t2))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (i:
+nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h d t0) x) \to
+((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus i h) u t0 t2))))))))))).(\lambda (x:
+T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
+(subst0 i u (lift h d (THead k t t0)) x)).(\lambda (H2: (le (plus d h)
+i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) (\lambda (t: T).(subst0
+i u t x)) H1 (THead k (lift h d t) (lift h (s k d) t0)) (lift_head k t t0 h
+d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d)
+t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2))) (ex2 T (\lambda (t2:
+T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u
+(lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
+x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d
+t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d)
+t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (H4: (ex2 T (\lambda
+(u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i
+u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
+(s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2)) (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
+k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x (THead k x0 (lift h (s k
+d) t0)))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(eq_ind_r T (THead k x0
+(lift h (s k d) t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift
+h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
+(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0
+(minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 (lift h (s k
+d) t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0)
+t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h d x1))).(\lambda (H8:
+(subst0 (minus i h) u t x1)).(eq_ind_r T (lift h d x1) (\lambda (t2: T).(ex2
+T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k d) t0)) (lift h d t3)))
+(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift
+h d (THead k x1 t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift
+h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
+(ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k x1 t0)) (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x1
+t0) (refl_equal T (lift h d (THead k x1 t0))) (subst0_fst u x1 t (minus i h)
+H8 t0 k)) (THead k (lift h d x1) (lift h (s k d) t0)) (lift_head k x1 t0 h
+d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) (\lambda (H4: (ex2 T (\lambda
+(t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i)
+u (lift h (s k d) t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k
+(lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0)
+t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0
+(minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x
+(THead k (lift h d t) x0))).(\lambda (H6: (subst0 (s k i) u (lift h (s k d)
+t0) x0)).(eq_ind_r T (THead k (lift h d t) x0) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i
+h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (s k
+d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda
+(t2: T).(eq T (THead k (lift h d t) x0) (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7:
+(eq T x0 (lift h (s k d) x1))).(\lambda (H8: (subst0 (minus (s k i) h) u t0
+x1)).(eq_ind_r T (lift h (s k d) x1) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T (THead k (lift h d t) t2) (lift h d t3))) (\lambda (t3: T).(subst0
+(minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d (THead k t x1))
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda
+(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H9 \def (eq_ind_r
+nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x1)) H8 (s k (minus i
+h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2:
+T).(eq T (lift h d (THead k t x1)) (lift h d t2))) (\lambda (t2: T).(subst0
+(minus i h) u (THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h d
+(THead k t x1))) (subst0_snd k u x1 t0 (minus i h) H9 t))) (THead k (lift h d
+t) (lift h (s k d) x1)) (lift_head k t x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s
+k d) H6 (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le
+k (plus d h) i H2) (plus (s k d) h) (s_plus k d h)))) x H5)))) H4)) (\lambda
+(H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex2 T
+(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h)
+u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T
+x (THead k x0 x1))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(\lambda (H7:
+(subst0 (s k i) u (lift h (s k d) t0) x1)).(eq_ind_r T (THead k x0 x1)
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda
+(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2:
+T).(eq T x1 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h)
+u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x2:
+T).(\lambda (H8: (eq T x1 (lift h (s k d) x2))).(\lambda (H9: (subst0 (minus
+(s k i) h) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T
+(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
+k t t0) t2))) (\lambda (x3: T).(\lambda (H10: (eq T x0 (lift h d
+x3))).(\lambda (H11: (subst0 (minus i h) u t x3)).(eq_ind_r T (lift h d x3)
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 x1) (lift h d
+t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind_r
+T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k
+(lift h d x3) t2) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u
+(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x3 x2)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
+(minus i h) u (THead k t t0) t3)))) (let H12 \def (eq_ind_r nat (minus (s k
+i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 (s k (minus i h)) (s_minus k i
+h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h
+d (THead k x3 x2)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u
+(THead k t t0) t2)) (THead k x3 x2) (refl_equal T (lift h d (THead k x3 x2)))
+(subst0_both u t x3 (minus i h) H11 k t0 x2 H12))) (THead k (lift h d x3)
+(lift h (s k d) x2)) (lift_head k x3 x2 h d)) x1 H8) x0 H10)))) (H x0 i h d
+H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind nat (s k (plus d h)) (\lambda
+(n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) (s_plus k
+d h)))) x H5)))))) H4)) (subst0_gen_head k u (lift h d t) (lift h (s k d) t0)
+x i H3)))))))))))))) t1)).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/props".
+
+include "subst0/fwd.ma".
+
+include "lift/props.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).(and_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 _ t _)
+\Rightarrow t])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def (eq_ind_r
+T x (\lambda (t: T).(subst0 d u t0 t)) 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 _ _ t) \Rightarrow t]))
+(THead k t0 t1) (THead k t0 x) H3) in (let H6 \def (eq_ind_r T x (\lambda (t:
+T).(subst0 (s k d) u t1 t)) 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 _ t _)
+\Rightarrow t])) (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 _ _ t)
+\Rightarrow t])) (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 (t: T).(subst0 (s k d) u t1 t))
+H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t: T).(subst0 d u t0 t))
+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 (d: nat).((lt (s k
+i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d n) v) (lift h
+d t3) (lift h d 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 (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k
+i0) (lift h (minus d n) v) (lift h d t0) (lift h d 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) (lt_le_S (s k i0) (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_comm 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 (d: nat).((le
+d (s k i0)) \to (subst0 n v (lift h d t3) (lift h d 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 (d: nat).((le d (s k i0)) \to (subst0 n v (lift h d t0) (lift h
+d 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_comm 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)))))))).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0".
+
+include "subst0/props.ma".
+
+theorem subst0_subst0:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i
+u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t:
+T).(subst0 (S (plus i j)) u t t2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u:
+T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n
+u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3)))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i))
+(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda
+(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t:
+T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t
+(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v
+u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i))
+(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u0)).(\lambda (H1: ((\forall (u2: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda (t: T).(subst0 i u2 u1 t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t:
+T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0:
+nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i
+u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda
+(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0
+i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1
+x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0:
+T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i))
+u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst
+u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k:
+K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i:
+nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1:
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda
+(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k
+i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0:
+T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda
+(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k
+i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x:
+T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0
+(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n:
+nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in
+(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n
+u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T
+(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S
+(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3
+u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0
+H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda
+(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u2:
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda
+(t: T).(subst0 i u2 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t
+u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
+(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u1: T).(\forall (u:
+T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda (t: T).(subst0
+(s k i) u1 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t
+t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4:
+(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t:
+T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
+t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0
+x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda
+(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))
+(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t:
+T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0
+u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0
+(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def
+(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9
+(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t:
+T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
+t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5)
+(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4)))))
+(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))).
+
+theorem subst0_subst0_back:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i
+u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t:
+T).(subst0 (S (plus i j)) u t2 t)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u:
+T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n
+u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4)))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i))
+(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda
+(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t:
+T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift
+(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0
+(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0
+(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda
+(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1:
+((\forall (u2: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u2) \to
+(ex2 T (\lambda (t: T).(subst0 i u2 u1 t)) (\lambda (t: T).(subst0 (S (plus
+i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3:
+T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v
+u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0:
+T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3
+(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t)
+t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0
+(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k
+u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0))
+(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0
+i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v:
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0
+(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall
+(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1
+t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda
+(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2:
+(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t:
+T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0
+(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3
+x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def
+(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4
+(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k
+(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i)))
+(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u
+t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead
+k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6
+u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1:
+T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u0)).(\lambda (H1: ((\forall (u2: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u v u2) \to (ex2 T (\lambda (t: T).(subst0 i u2 u1 t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k:
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0
+t3)).(\lambda (H3: ((\forall (u1: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 t0 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3:
+T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v
+u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t:
+T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3
+(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3)
+t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6:
+(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i
+u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda
+(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0
+i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1
+x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r
+nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus
+i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0
+i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k
+(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0)
+t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0
+x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i))
+H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2
+t1 t2 H))))).
+
+theorem subst0_trans:
+ \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0
+i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1
+t3)))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to
+(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3:
+T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false
+v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0
+(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1:
+T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1:
+((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda
+(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k
+u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t)))
+(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3
+(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda
+(H3: (ex2 T (\lambda (u2: T).(eq T t3 (THead k u2 t))) (\lambda (u3:
+T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3
+t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3)
+(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0
+i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead
+k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda
+(H3: (ex2 T (\lambda (t2: T).(eq T t3 (THead k u2 t2))) (\lambda (t2:
+T).(subst0 (s k i0) v0 t t2)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k
+u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k
+u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda
+(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0:
+T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3
+H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
+t3 (THead k u2 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t t2))))).(ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5:
+(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T
+(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0))
+(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3))
+(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0:
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0
+(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0
+t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4:
+T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda
+(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))
+(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s
+k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2)))
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0
+(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2
+t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq
+T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0
+(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x
+t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda
+(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0)
+t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u
+t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t0 t2)))).(ex2_ind T (\lambda (t5:
+T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))
+(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4
+(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k
+u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3
+i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0
+t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3)
+t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0
+x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3)
+t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3))
+(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda
+(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1
+u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0
+u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2:
+(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0)
+v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4:
+(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda
+(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3
+t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1
+t0) t4) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t3)))
+(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead
+k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda
+(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0
+i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4
+H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u2 t2)))
+(\lambda (t2: T).(subst0 (s k i0) v0 t3 t2)))).(ex2_ind T (\lambda (t5:
+T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))
+(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4
+(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead
+k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1
+u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i0) v0 t3 t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda
+(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0
+i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))
+(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda
+(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t:
+T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0
+x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0
+H4))))))))))))))) i v t1 t2 H))))).
+
+theorem subst0_confluence_neq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
+nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall
+(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda
+(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t))))))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
+nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n:
+nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4:
+T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq
+nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5:
+T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda
+(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2
+(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(and_ind (eq nat i i2) (eq
+T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v)
+t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda
+(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n:
+nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda
+(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda
+(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in
+False return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift
+(S i) O v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with [])
+in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v:
+T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0
+i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2:
+nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t:
+T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda
+(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2:
+nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq
+nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t)))
+(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2
+(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T
+(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4:
+T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead
+k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead k u2 t))) (\lambda (u2: T).(subst0 i2 u3 u0
+u2)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4:
+T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t)
+t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq
+T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k
+x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t)
+t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3:
+T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda
+(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead
+k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda
+(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t)
+(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2
+H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead
+k u0 t3))) (\lambda (t2: T).(subst0 (s k i2) u3 t t2)))).(ex2_ind T (\lambda
+(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t
+t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0
+x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x)
+(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4))
+(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0
+i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3))
+(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k))
+t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i2) u3 t
+t2))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4
+t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3:
+T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0
+x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4:
+T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4))))
+(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i
+v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8:
+(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda
+(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead
+k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7)
+(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4))
+(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda
+(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0:
+(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2:
+T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2))
+\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k
+i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2:
+T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3)
+t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq
+T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda
+(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3
+t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t:
+T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (H4: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t3))) (\lambda
+(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k
+u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0
+i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u
+x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t:
+T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2
+k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T
+(\lambda (t2: T).(eq T t4 (THead k u t2))) (\lambda (t2: T).(subst0 (s k i2)
+u2 t3 t2)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda
+(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2
+(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2
+t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0
+(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t))
+(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7:
+(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2
+T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i
+v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u)
+(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (\lambda (H7: (eq nat
+(s k i) (s k i2))).(H3 (s_inj k i i2 H7))))) t4 H5)))) H4)) (\lambda (H4:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i2) u2 t3 t2))))).(ex3_2_ind T T (\lambda
+(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k
+u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u
+x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1)
+(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5))
+(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k
+i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t:
+T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0
+x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda
+(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2
+(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k
+x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0)))))
+(H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k i2))).(H3 (s_inj k i
+i2 H8))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 t4 i2
+H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda
+(i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: ((\forall (t2:
+T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) \to ((not (eq
+nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t:
+T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: ((\forall (t4:
+T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq
+nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t3 t)) (\lambda (t:
+T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: T).(\lambda (u3:
+T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k u0 t2)
+t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq
+T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T
+(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2)
+u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4
+t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t2))) (\lambda
+(u2: T).(subst0 i2 u3 u0 u2)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k
+u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0
+i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x
+t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2
+u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x
+t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0
+i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t2: T).(eq T t4 (THead k u0 t2))) (\lambda (t3: T).(subst0 (s k i2)
+u3 t2 t3)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda
+(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3
+t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0
+(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda
+(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x
+x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda
+(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3
+i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8
+(\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H9))))) t4 H7))))
+H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4
+(THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t2 t3))))).(ex3_2_ind T
+T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4:
+T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t4 (THead k x0 x1))).(\lambda (H8: (subst0 i2 u3 u0
+x0)).(\lambda (H9: (subst0 (s k i2) u3 t2 x1)).(eq_ind_r T (THead k x0 x1)
+(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u3 (THead k u2 t3) t5))
+(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 i2
+u3 u2 t)) (\lambda (t: T).(subst0 i v x0 t)) (ex2 T (\lambda (t: T).(subst0
+i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)))
+(\lambda (x: T).(\lambda (H10: (subst0 i2 u3 u2 x)).(\lambda (H11: (subst0 i
+v x0 x)).(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t:
+T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2
+t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t))) (\lambda (x2:
+T).(\lambda (H12: (subst0 (s k i2) u3 t3 x2)).(\lambda (H13: (subst0 (s k i)
+v x1 x2)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k x x2) (subst0_both
+u3 u2 x i2 H10 k t3 x2 H12) (subst0_both v x0 x i H11 k x1 x2 H13))))) (H3 x1
+u3 (s k i2) H9 (\lambda (H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2
+H12)))))))) (H1 x0 u3 i2 H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2
+t4 i2 H4)))))))))))))))))) i1 u1 t0 t1 H))))).
+
+theorem subst0_confluence_eq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2)
+(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t)))
+(subst0 i u t1 t2) (subst0 i u t2 t1))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to
+(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5:
+T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda
+(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef
+i0) t2)).(and_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T (lift
+(S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t))
+(\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2)
+(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda
+(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2)
+(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t:
+T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2
+(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v
+t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2:
+T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0
+i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0
+i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda
+(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T
+t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t
+t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2)
+(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
+(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2
+t))) (\lambda (u2: T).(subst0 i0 v u1 u2)))).(ex2_ind T (\lambda (u3: T).(eq
+T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead
+k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
+(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x
+t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda
+(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v
+(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v
+(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x)
+(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x
+t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead
+k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k
+x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2
+x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t))
+(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4:
+T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x
+t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x
+t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t)
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead
+k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 v
+u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) (THead
+k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k
+x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x x0)).(or4_intro1 (eq T
+(THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v
+(THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))
+(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) (subst0_fst v x0 u2 i0 H7 t
+k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) (\lambda (H6: (subst0 i0 v u2
+x)).(or4_intro2 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x
+t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x
+t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t k))) (\lambda (H6: (subst0 i0 v
+x u2)).(or4_intro3 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x
+t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x
+t) (THead k u2 t)) (subst0_fst v u2 x i0 H6 t k))) (H1 x H5)) t2 H4)))) H3))
+(\lambda (H3: (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda
+(t2: T).(subst0 (s k i0) v t t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t t3)) (or4 (eq T
+(THead k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3))
+(\lambda (t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2)
+(subst0 i0 v t2 (THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2
+(THead k u1 x))).(\lambda (H5: (subst0 (s k i0) v t x)).(eq_ind_r T (THead k
+u1 x) (\lambda (t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4:
+T).(subst0 i0 v (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4)))
+(subst0 i0 v (THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind
+(eq T u2 u2) (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3:
+T).(subst0 i0 v u2 t3))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T
+(THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v
+(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)))
+(\lambda (_: (eq T u2 u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x))
+(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1
+x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1
+x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0
+H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda
+(t: T).(subst0 i0 v u2 t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 v u2 t3))
+(\lambda (t3: T).(subst0 i0 v u2 t3)) (or4 (eq T (THead k u2 t) (THead k u1
+x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1
+x)) (subst0 i0 v (THead k u1 x) (THead k u2 t))) (\lambda (x0: T).(\lambda
+(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T
+(THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v
+(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t))
+(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5
+u2) (subst0_fst v u2 u1 i0 H0 x k)))))) H6)) (\lambda (_: (subst0 i0 v u2
+u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1
+x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t)
+t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x)
+(subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_:
+(subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T
+(\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0
+v (THead k u1 x) t3))) (subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0
+v (THead k u1 x) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v
+(THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3)) (THead
+k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1
+u2 H0)) t2 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0
+i0 v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v t
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v t t3))) (or4 (eq T (THead k u2 t) t2)
+(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
+(THead k u2 t))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t2
+(THead k x0 x1))).(\lambda (H5: (subst0 i0 v u1 x0)).(\lambda (H6: (subst0 (s
+k i0) v t x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(or4 (eq T (THead
+k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v (THead k u2 t) t4)) (\lambda
+(t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v (THead k u2 t) t3) (subst0 i0 v t3
+(THead k u2 t)))) (or4_ind (eq T u2 x0) (ex2 T (\lambda (t3: T).(subst0 i0 v
+u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3))) (subst0 i0 v u2 x0) (subst0 i0
+v x0 u2) (or4 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0
+x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k u2 t))) (\lambda (H7: (eq T u2 x0)).(eq_ind_r T x0 (\lambda
+(t3: T).(or4 (eq T (THead k t3 t) (THead k x0 x1)) (ex2 T (\lambda (t4:
+T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: T).(subst0 i0 v (THead k x0
+x1) t4))) (subst0 i0 v (THead k t3 t) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k t3 t)))) (or4_intro2 (eq T (THead k x0 t) (THead k x0 x1))
+(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x0 t) t3)) (\lambda (t3:
+T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k x0 t) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t)) (subst0_snd k v x1 t i0 H6
+x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0
+v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t)
+(THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3))
+(\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2
+t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t))) (\lambda
+(x: T).(\lambda (H8: (subst0 i0 v u2 x)).(\lambda (H9: (subst0 i0 v x0
+x)).(or4_intro1 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0
+x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2
+t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3)) (THead k x x1)
+(subst0_both v u2 x i0 H8 k t x1 H6) (subst0_fst v x x0 i0 H9 x1 k)))))) H7))
+(\lambda (H7: (subst0 i0 v u2 x0)).(or4_intro2 (eq T (THead k u2 t) (THead k
+x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 t) (THead
+k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)) (subst0_both v u2 x0
+i0 H7 k t x1 H6))) (\lambda (H7: (subst0 i0 v x0 u2)).(or4_intro1 (eq T
+(THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v
+(THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t))
+(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v (THead k x0 x1) t3)) (THead k u2 x1) (subst0_snd k v x1 t i0
+H6 u2) (subst0_fst v u2 x0 i0 H7 x1 k)))) (H1 x0 H5)) t2 H4)))))) H3))
+(subst0_gen_head k v u1 t t2 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v:
+T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0
+(s k i0) v t3 t2)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v t3 t4)
+\to (or4 (eq T t2 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t))
+(\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t2 t4) (subst0
+(s k i0) v t4 t2)))))).(\lambda (u0: T).(\lambda (t4: T).(\lambda (H2:
+(subst0 i0 v (THead k u0 t3) t4)).(or3_ind (ex2 T (\lambda (u2: T).(eq T t4
+(THead k u2 t3))) (\lambda (u2: T).(subst0 i0 v u0 u2))) (ex2 T (\lambda (t5:
+T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5)))) (or4 (eq T (THead k u0 t2)
+t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t:
+T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) t4) (subst0 i0 v t4
+(THead k u0 t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2
+t3))) (\lambda (u2: T).(subst0 i0 v u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq
+T t4 (THead k u2 t3))) (\lambda (u2: T).(subst0 i0 v u0 u2)) (or4 (eq T
+(THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t))
+(\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) t4) (subst0
+i0 v t4 (THead k u0 t2))) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x
+t3))).(\lambda (H5: (subst0 i0 v u0 x)).(eq_ind_r T (THead k x t3) (\lambda
+(t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T (\lambda (t5: T).(subst0 i0 v
+(THead k u0 t2) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v
+(THead k u0 t2) t) (subst0 i0 v t (THead k u0 t2)))) (or4_ind (eq T t2 t2)
+(ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k
+i0) v t2 t))) (subst0 (s k i0) v t2 t2) (subst0 (s k i0) v t2 t2) (or4 (eq T
+(THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v
+(THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)))
+(\lambda (_: (eq T t2 t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x
+t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t:
+T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x
+t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3
+i0 H0 x)))) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t))
+(\lambda (t: T).(subst0 (s k i0) v t2 t)))).(ex2_ind T (\lambda (t:
+T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k i0) v t2 t)) (or4
+(eq T (THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v
+(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0
+i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0
+t2))) (\lambda (x0: T).(\lambda (_: (subst0 (s k i0) v t2 x0)).(\lambda (_:
+(subst0 (s k i0) v t2 x0)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x
+t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t:
+T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x
+t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3
+i0 H0 x)))))) H6)) (\lambda (_: (subst0 (s k i0) v t2 t2)).(or4_intro1 (eq T
+(THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t))) (subst0 i0 v
+(THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2))
+(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2
+k) (subst0_snd k v t2 t3 i0 H0 x)))) (\lambda (_: (subst0 (s k i0) v t2
+t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x t3)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x
+t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x
+t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0
+t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (THead k x t2)
+(subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 i0 H0 x)))) (H1 t2 H0))
+t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u0
+t2))) (\lambda (t2: T).(subst0 (s k i0) v t3 t2)))).(ex2_ind T (\lambda (t5:
+T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5))
+(or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2)
+t4) (subst0 i0 v t4 (THead k u0 t2))) (\lambda (x: T).(\lambda (H4: (eq T t4
+(THead k u0 x))).(\lambda (H5: (subst0 (s k i0) v t3 x)).(eq_ind_r T (THead k
+u0 x) (\lambda (t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T (\lambda (t5:
+T).(subst0 i0 v (THead k u0 t2) t5)) (\lambda (t5: T).(subst0 i0 v t t5)))
+(subst0 i0 v (THead k u0 t2) t) (subst0 i0 v t (THead k u0 t2)))) (or4_ind
+(eq T t2 x) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t:
+T).(subst0 (s k i0) v x t))) (subst0 (s k i0) v t2 x) (subst0 (s k i0) v x
+t2) (or4 (eq T (THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: T).(subst0
+i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t)))
+(subst0 i0 v (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x)
+(THead k u0 t2))) (\lambda (H6: (eq T t2 x)).(eq_ind_r T x (\lambda (t:
+T).(or4 (eq T (THead k u0 t) (THead k u0 x)) (ex2 T (\lambda (t5: T).(subst0
+i0 v (THead k u0 t) t5)) (\lambda (t5: T).(subst0 i0 v (THead k u0 x) t5)))
+(subst0 i0 v (THead k u0 t) (THead k u0 x)) (subst0 i0 v (THead k u0 x)
+(THead k u0 t)))) (or4_intro0 (eq T (THead k u0 x) (THead k u0 x)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u0 x) t)) (\lambda (t: T).(subst0 i0 v
+(THead k u0 x) t))) (subst0 i0 v (THead k u0 x) (THead k u0 x)) (subst0 i0 v
+(THead k u0 x) (THead k u0 x)) (refl_equal T (THead k u0 x))) t2 H6))
+(\lambda (H6: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t)) (\lambda (t:
+T).(subst0 (s k i0) v x t)))).(ex2_ind T (\lambda (t: T).(subst0 (s k i0) v
+t2 t)) (\lambda (t: T).(subst0 (s k i0) v x t)) (or4 (eq T (THead k u0 t2)
+(THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t))
+(\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2)
+(THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2))) (\lambda (x0:
+T).(\lambda (H7: (subst0 (s k i0) v t2 x0)).(\lambda (H8: (subst0 (s k i0) v
+x x0)).(or4_intro1 (eq T (THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0
+x) t))) (subst0 i0 v (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0
+x) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2)
+t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t)) (THead k u0 x0)
+(subst0_snd k v x0 t2 i0 H7 u0) (subst0_snd k v x0 x i0 H8 u0)))))) H6))
+(\lambda (H6: (subst0 (s k i0) v t2 x)).(or4_intro2 (eq T (THead k u0 t2)
+(THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t))
+(\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2)
+(THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) (subst0_snd k v
+x t2 i0 H6 u0))) (\lambda (H6: (subst0 (s k i0) v x t2)).(or4_intro3 (eq T
+(THead k u0 t2) (THead k u0 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v
+(THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2))
+(subst0_snd k v t2 x i0 H6 u0))) (H1 x H5)) t4 H4)))) H3)) (\lambda (H3:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i0) v t3 t2))))).(ex3_2_ind T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s k i0) v t3 t5))) (or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda
+(t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t)))
+(subst0 i0 v (THead k u0 t2) t4) (subst0 i0 v t4 (THead k u0 t2))) (\lambda
+(x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 x1))).(\lambda
+(H5: (subst0 i0 v u0 x0)).(\lambda (H6: (subst0 (s k i0) v t3 x1)).(eq_ind_r
+T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead k u0 t2) t) (ex2 T
+(\lambda (t5: T).(subst0 i0 v (THead k u0 t2) t5)) (\lambda (t5: T).(subst0
+i0 v t t5))) (subst0 i0 v (THead k u0 t2) t) (subst0 i0 v t (THead k u0
+t2)))) (or4_ind (eq T t2 x1) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t))
+(\lambda (t: T).(subst0 (s k i0) v x1 t))) (subst0 (s k i0) v t2 x1) (subst0
+(s k i0) v x1 t2) (or4 (eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda
+(t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k
+x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead
+k x0 x1) (THead k u0 t2))) (\lambda (H7: (eq T t2 x1)).(eq_ind_r T x1
+(\lambda (t: T).(or4 (eq T (THead k u0 t) (THead k x0 x1)) (ex2 T (\lambda
+(t5: T).(subst0 i0 v (THead k u0 t) t5)) (\lambda (t5: T).(subst0 i0 v (THead
+k x0 x1) t5))) (subst0 i0 v (THead k u0 t) (THead k x0 x1)) (subst0 i0 v
+(THead k x0 x1) (THead k u0 t)))) (or4_intro2 (eq T (THead k u0 x1) (THead k
+x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 x1) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u0 x1) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k u0 x1)) (subst0_fst v x0 u0 i0 H5
+x1 k)) t2 H7)) (\lambda (H7: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t2 t))
+(\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t:
+T).(subst0 (s k i0) v t2 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4
+(eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v
+(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0
+i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k
+u0 t2))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i0) v t2 x)).(\lambda
+(H9: (subst0 (s k i0) v x1 x)).(or4_intro1 (eq T (THead k u0 t2) (THead k x0
+x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k u0 t2)) (ex_intro2 T (\lambda (t:
+T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t)) (THead k x0 x) (subst0_both v u0 x0 i0 H5 k t2 x H8) (subst0_snd k v
+x x1 i0 H9 x0)))))) H7)) (\lambda (H7: (subst0 (s k i0) v t2 x1)).(or4_intro2
+(eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v
+(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0
+i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k
+u0 t2)) (subst0_both v u0 x0 i0 H5 k t2 x1 H7))) (\lambda (H7: (subst0 (s k
+i0) v x1 t2)).(or4_intro1 (eq T (THead k u0 t2) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k u0 t2) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v
+(THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k
+x0 t2) (subst0_fst v x0 u0 i0 H5 t2 k) (subst0_snd k v t2 x1 i0 H7 x0)))) (H1
+x1 H6)) t4 H4)))))) H3)) (subst0_gen_head k v u0 t3 t4 i0 H2))))))))))))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda
+(H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: T).((subst0 i0 v u1
+t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda
+(t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 i0 v t2
+u2)))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2:
+(subst0 (s k i0) v t2 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0)
+v t2 t4) \to (or4 (eq T t3 t4) (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3
+t)) (\lambda (t: T).(subst0 (s k i0) v t4 t))) (subst0 (s k i0) v t3 t4)
+(subst0 (s k i0) v t4 t3)))))).(\lambda (t4: T).(\lambda (H4: (subst0 i0 v
+(THead k u1 t2) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3
+t2))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda (t5: T).(eq T t4
+(THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s k i0) v t2 t5)))) (or4 (eq T (THead k u2 t3) t4) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 (THead k u2 t3)))
+(\lambda (H5: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t2))) (\lambda
+(u2: T).(subst0 i0 v u1 u2)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k
+u3 t2))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead k u2 t3) t4)
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4
+(THead k u2 t3))) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x
+t2))).(\lambda (H7: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t2) (\lambda
+(t: T).(or4 (eq T (THead k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v
+(THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v
+(THead k u2 t3) t) (subst0 i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 t3)
+(ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k
+i0) v t3 t))) (subst0 (s k i0) v t3 t3) (subst0 (s k i0) v t3 t3) (or4 (eq T
+(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v
+(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)))
+(\lambda (_: (eq T t3 t3)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t:
+T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x)
+(subst0 i0 v x u2) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda
+(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k
+x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k
+x t2) (THead k u2 t3))) (\lambda (H9: (eq T u2 x)).(eq_ind_r T x (\lambda (t:
+T).(or4 (eq T (THead k t t3) (THead k x t2)) (ex2 T (\lambda (t5: T).(subst0
+i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x t2) t5)))
+(subst0 i0 v (THead k t t3) (THead k x t2)) (subst0 i0 v (THead k x t2)
+(THead k t t3)))) (or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v
+(THead k x t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H9))
+(\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x
+t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x0: T).(\lambda
+(H10: (subst0 i0 v u2 x0)).(\lambda (H11: (subst0 i0 v x x0)).(or4_intro1 (eq
+T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead
+k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v
+(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))
+(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t2) t)) (THead k x0 t3) (subst0_fst v x0 u2 i0 H10
+t3 k) (subst0_both v x x0 i0 H11 k t2 t3 H2)))))) H9)) (\lambda (H9: (subst0
+i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda
+(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k
+x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k
+x t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2
+t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x t3)
+(subst0_fst v x u2 i0 H9 t3 k) (subst0_snd k v t3 t2 i0 H2 x)))) (\lambda
+(H9: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k x t2))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x
+t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H9 k
+t2 t3 H2))) (H1 x H7))) (\lambda (H8: (ex2 T (\lambda (t: T).(subst0 (s k i0)
+v t3 t)) (\lambda (t: T).(subst0 (s k i0) v t3 t)))).(ex2_ind T (\lambda (t:
+T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v t3 t)) (or4
+(eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0
+i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2
+t3))) (\lambda (x0: T).(\lambda (_: (subst0 (s k i0) v t3 x0)).(\lambda (_:
+(subst0 (s k i0) v t3 x0)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t:
+T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x)
+(subst0 i0 v x u2) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda
+(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k
+x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k
+x t2) (THead k u2 t3))) (\lambda (H11: (eq T u2 x)).(eq_ind_r T x (\lambda
+(t: T).(or4 (eq T (THead k t t3) (THead k x t2)) (ex2 T (\lambda (t5:
+T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x
+t2) t5))) (subst0 i0 v (THead k t t3) (THead k x t2)) (subst0 i0 v (THead k x
+t2) (THead k t t3)))) (or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v
+(THead k x t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H11))
+(\lambda (H11: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x
+t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x1: T).(\lambda
+(H12: (subst0 i0 v u2 x1)).(\lambda (H13: (subst0 i0 v x x1)).(or4_intro1 (eq
+T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead
+k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v
+(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))
+(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t2) t)) (THead k x1 t3) (subst0_fst v x1 u2 i0 H12
+t3 k) (subst0_both v x x1 i0 H13 k t2 t3 H2)))))) H11)) (\lambda (H11:
+(subst0 i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v
+(THead k x t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k
+x t3) (subst0_fst v x u2 i0 H11 t3 k) (subst0_snd k v t3 t2 i0 H2 x))))
+(\lambda (H11: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k
+x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x
+t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H11
+k t2 t3 H2))) (H1 x H7))))) H8)) (\lambda (_: (subst0 (s k i0) v t3
+t3)).(or4_ind (eq T u2 x) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda
+(t: T).(subst0 i0 v x t))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T
+(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v
+(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)))
+(\lambda (H9: (eq T u2 x)).(eq_ind_r T x (\lambda (t: T).(or4 (eq T (THead k
+t t3) (THead k x t2)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3)
+t5)) (\lambda (t5: T).(subst0 i0 v (THead k x t2) t5))) (subst0 i0 v (THead k
+t t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k t t3))))
+(or4_intro3 (eq T (THead k x t3) (THead k x t2)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k x t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x
+t2) t))) (subst0 i0 v (THead k x t3) (THead k x t2)) (subst0 i0 v (THead k x
+t2) (THead k x t3)) (subst0_snd k v t3 t2 i0 H2 x)) u2 H9)) (\lambda (H9:
+(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x
+t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0
+i0 v x t)) (or4 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x
+t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x
+t2) (THead k u2 t3))) (\lambda (x0: T).(\lambda (H10: (subst0 i0 v u2
+x0)).(\lambda (H11: (subst0 i0 v x x0)).(or4_intro1 (eq T (THead k u2 t3)
+(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3)
+(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (ex_intro2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x t2) t)) (THead k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k)
+(subst0_both v x x0 i0 H11 k t2 t3 H2)))))) H9)) (\lambda (H9: (subst0 i0 v
+u2 x)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x
+t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x
+t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2
+t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x t3)
+(subst0_fst v x u2 i0 H9 t3 k) (subst0_snd k v t3 t2 i0 H2 x)))) (\lambda
+(H9: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k x t2))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x
+t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H9 k
+t2 t3 H2))) (H1 x H7))) (\lambda (_: (subst0 (s k i0) v t3 t3)).(or4_ind (eq
+T u2 x) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0
+v x t))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t3)
+(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3)
+(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (H9:
+(eq T u2 x)).(eq_ind_r T x (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k
+x t2)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5:
+T).(subst0 i0 v (THead k x t2) t5))) (subst0 i0 v (THead k t t3) (THead k x
+t2)) (subst0 i0 v (THead k x t2) (THead k t t3)))) (or4_intro3 (eq T (THead k
+x t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k x t3)
+(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k x t3)) (subst0_snd k v
+t3 t2 i0 H2 x)) u2 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2
+t)) (\lambda (t: T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t: T).(subst0
+i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x t)) (or4 (eq T (THead k u2 t3)
+(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3)
+(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))) (\lambda (x0:
+T).(\lambda (H10: (subst0 i0 v u2 x0)).(\lambda (H11: (subst0 i0 v x
+x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x
+t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x
+t2) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2
+t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t)) (THead k x0 t3)
+(subst0_fst v x0 u2 i0 H10 t3 k) (subst0_both v x x0 i0 H11 k t2 t3 H2))))))
+H9)) (\lambda (H9: (subst0 i0 v u2 x)).(or4_intro1 (eq T (THead k u2 t3)
+(THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3)
+(THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (ex_intro2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x t2) t)) (THead k x t3) (subst0_fst v x u2 i0 H9 t3 k) (subst0_snd
+k v t3 t2 i0 H2 x)))) (\lambda (H9: (subst0 i0 v x u2)).(or4_intro3 (eq T
+(THead k u2 t3) (THead k x t2)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v
+(THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3))
+(subst0_both v x u2 i0 H9 k t2 t3 H2))) (H1 x H7))) (H3 t3 H2)) t4 H6))))
+H5)) (\lambda (H5: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u1 t2)))
+(\lambda (t3: T).(subst0 (s k i0) v t2 t3)))).(ex2_ind T (\lambda (t5: T).(eq
+T t4 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5)) (or4 (eq T
+(THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0
+i0 v t4 (THead k u2 t3))) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k u1
+x))).(\lambda (H7: (subst0 (s k i0) v t2 x)).(eq_ind_r T (THead k u1 x)
+(\lambda (t: T).(or4 (eq T (THead k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0
+i0 v (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v
+(THead k u2 t3) t) (subst0 i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x)
+(ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k
+i0) v x t))) (subst0 (s k i0) v t3 x) (subst0 (s k i0) v x t3) (or4 (eq T
+(THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v
+(THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)))
+(\lambda (H8: (eq T t3 x)).(eq_ind_r T x (\lambda (t: T).(or4 (eq T (THead k
+u2 t) (THead k u1 x)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t)
+t5)) (\lambda (t5: T).(subst0 i0 v (THead k u1 x) t5))) (subst0 i0 v (THead k
+u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)))) (or4_ind
+(eq T u2 u2) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v u2 t))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T
+(THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v
+(THead k u2 x) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 x)))
+(\lambda (_: (eq T u2 u2)).(or4_intro3 (eq T (THead k u2 x) (THead k u1 x))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t:
+T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 x) (THead k u1
+x)) (subst0 i0 v (THead k u1 x) (THead k u2 x)) (subst0_fst v u2 u1 i0 H0 x
+k))) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v u2 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v u2 t)) (or4 (eq T (THead k u2 x) (THead k u1 x))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t:
+T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 x) (THead k u1
+x)) (subst0 i0 v (THead k u1 x) (THead k u2 x))) (\lambda (x0: T).(\lambda
+(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro3 (eq T
+(THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v
+(THead k u2 x) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 x))
+(subst0_fst v u2 u1 i0 H0 x k))))) H9)) (\lambda (_: (subst0 i0 v u2
+u2)).(or4_intro3 (eq T (THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k u1
+x) t))) (subst0 i0 v (THead k u2 x) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 x)) (subst0_fst v u2 u1 i0 H0 x k))) (\lambda (_: (subst0 i0 v
+u2 u2)).(or4_intro3 (eq T (THead k u2 x) (THead k u1 x)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 x) t)) (\lambda (t: T).(subst0 i0 v (THead k u1
+x) t))) (subst0 i0 v (THead k u2 x) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 x)) (subst0_fst v u2 u1 i0 H0 x k))) (H1 u2 H0)) t3 H8))
+(\lambda (H8: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 t)) (\lambda (t:
+T).(subst0 (s k i0) v x t)))).(ex2_ind T (\lambda (t: T).(subst0 (s k i0) v
+t3 t)) (\lambda (t: T).(subst0 (s k i0) v x t)) (or4 (eq T (THead k u2 t3)
+(THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3)
+(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (x0:
+T).(\lambda (H9: (subst0 (s k i0) v t3 x0)).(\lambda (H10: (subst0 (s k i0) v
+x x0)).(or4_ind (eq T u2 u2) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v u2 t))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2)
+(or4 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0
+v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)))
+(subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x)
+(THead k u2 t3))) (\lambda (_: (eq T u2 u2)).(or4_intro1 (eq T (THead k u2
+t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3)
+(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (ex_intro2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k u1 x) t)) (THead k u2 x0) (subst0_snd k v x0 t3 i0 H9 u2)
+(subst0_both v u1 u2 i0 H0 k x x0 H10)))) (\lambda (H11: (ex2 T (\lambda (t:
+T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t)))).(ex2_ind T
+(\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t)) (or4
+(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0
+i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2
+t3))) (\lambda (x1: T).(\lambda (_: (subst0 i0 v u2 x1)).(\lambda (_: (subst0
+i0 v u2 x1)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v
+(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k
+u2 x0) (subst0_snd k v x0 t3 i0 H9 u2) (subst0_both v u1 u2 i0 H0 k x x0
+H10)))))) H11)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k
+u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3)
+t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2
+t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (ex_intro2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k u1 x) t)) (THead k u2 x0) (subst0_snd k v x0 t3 i0 H9 u2)
+(subst0_both v u1 u2 i0 H0 k x x0 H10)))) (\lambda (_: (subst0 i0 v u2
+u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1
+x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3)
+t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x0)
+(subst0_snd k v x0 t3 i0 H9 u2) (subst0_both v u1 u2 i0 H0 k x x0 H10)))) (H1
+u2 H0))))) H8)) (\lambda (H8: (subst0 (s k i0) v t3 x)).(or4_ind (eq T u2 u2)
+(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2
+t))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t3)
+(THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3)
+(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (_:
+(eq T u2 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v
+(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k
+u2 x) (subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k))))
+(\lambda (H9: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v u2 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v u2 t)) (or4 (eq T (THead k u2 t3) (THead k u1
+x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1
+x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (x0: T).(\lambda
+(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T
+(THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v
+(THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))
+(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x) (subst0_snd k v x t3 i0 H8
+u2) (subst0_fst v u2 u1 i0 H0 x k)))))) H9)) (\lambda (_: (subst0 i0 v u2
+u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1
+x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3)
+t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k u2 x)
+(subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (_:
+(subst0 i0 v u2 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v
+(THead k u1 x) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t)) (THead k
+u2 x) (subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2
+H0))) (\lambda (H8: (subst0 (s k i0) v x t3)).(or4_ind (eq T u2 u2) (ex2 T
+(\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2 t)))
+(subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T (THead k u2 t3) (THead k
+u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1
+x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3))) (\lambda (_: (eq T u2
+u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1
+x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (\lambda (H9:
+(ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v u2
+t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0
+i0 v u2 t)) (or4 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1
+x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 t3))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2
+x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro3 (eq T (THead k u2 t3)
+(THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3)
+(THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (subst0_both v
+u1 u2 i0 H0 k x t3 H8))))) H9)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro3
+(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0
+i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2
+t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (\lambda (_: (subst0 i0 v u2
+u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1
+x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (H1 u2 H0))) (H3
+x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t2
+t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3)
+t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4
+(THead k u2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t4
+(THead k x0 x1))).(\lambda (H7: (subst0 i0 v u1 x0)).(\lambda (H8: (subst0 (s
+k i0) v t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead
+k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t3) t5))
+(\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v (THead k u2 t3) t) (subst0
+i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x1) (ex2 T (\lambda (t:
+T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))
+(subst0 (s k i0) v t3 x1) (subst0 (s k i0) v x1 t3) (or4 (eq T (THead k u2
+t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda
+(H9: (eq T t3 x1)).(eq_ind_r T x1 (\lambda (t: T).(or4 (eq T (THead k u2 t)
+(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t) t5))
+(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k u2
+t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)))) (or4_ind
+(eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T
+(THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v
+(THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2
+x1))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T
+(THead k t x1) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k
+t x1) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v
+(THead k t x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t
+x1)))) (or4_intro0 (eq T (THead k x0 x1) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k x0 x1) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k x0 x1)) (refl_equal T (THead k x0 x1))) u2 H10)) (\lambda
+(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v
+x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k u2 x1))) (\lambda (x: T).(\lambda (H11: (subst0 i0
+v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 x1)
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 x1)) (ex_intro2
+T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0
+v (THead k x0 x1) t)) (THead k x x1) (subst0_fst v x u2 i0 H11 x1 k)
+(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2
+x0)).(or4_intro2 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k u2 x1)) (subst0_fst v x0 u2 i0 H10 x1 k))) (\lambda (H10:
+(subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2
+T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0
+v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0
+i0 v (THead k x0 x1) (THead k u2 x1)) (subst0_fst v u2 x0 i0 H10 x1 k))) (H1
+x0 H7)) t3 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3
+t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t:
+T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4
+(eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0
+i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k
+u2 t3))) (\lambda (x: T).(\lambda (H10: (subst0 (s k i0) v t3 x)).(\lambda
+(H11: (subst0 (s k i0) v x1 x)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t:
+T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2
+x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k u2 t3))) (\lambda (H12: (eq T u2 x0)).(eq_ind_r T
+x0 (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda
+(t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead
+k x0 x1) t5))) (subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v
+(THead k x0 x1) (THead k t t3)))) (or4_intro1 (eq T (THead k x0 t3) (THead k
+x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t3)) (ex_intro2 T (\lambda (t:
+T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t)) (THead k x0 x) (subst0_snd k v x t3 i0 H10 x0) (subst0_snd k v x x1
+i0 H11 x0))) u2 H12)) (\lambda (H12: (ex2 T (\lambda (t: T).(subst0 i0 v u2
+t)) (\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0
+i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3)
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda
+(x2: T).(\lambda (H13: (subst0 i0 v u2 x2)).(\lambda (H14: (subst0 i0 v x0
+x2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2
+t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x2 x)
+(subst0_both v u2 x2 i0 H13 k t3 x H10) (subst0_both v x0 x2 i0 H14 k x1 x
+H11)))))) H12)) (\lambda (H12: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead
+k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3)
+t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k
+u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))
+(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t)) (THead k x0 x) (subst0_both v u2 x0 i0
+H12 k t3 x H10) (subst0_snd k v x x1 i0 H11 x0)))) (\lambda (H12: (subst0 i0
+v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda
+(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k
+x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead
+k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k u2 x)
+(subst0_snd k v x t3 i0 H10 u2) (subst0_both v x0 u2 i0 H12 k x1 x H11))))
+(H1 x0 H7))))) H9)) (\lambda (H9: (subst0 (s k i0) v t3 x1)).(or4_ind (eq T
+u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0
+v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3)
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda
+(H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T (THead k t t3)
+(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5))
+(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k t
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t t3))))
+(or4_intro2 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k x0 t3)) (subst0_snd k v x1 t3 i0 H9 x0)) u2 H10)) (\lambda
+(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v
+x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda (H11: (subst0 i0
+v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 t3)
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2
+T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0
+v (THead k x0 x1) t)) (THead k x x1) (subst0_both v u2 x i0 H11 k t3 x1 H9)
+(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2
+x0)).(or4_intro2 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k u2 t3)) (subst0_both v u2 x0 i0 H10 k t3 x1 H9))) (\lambda
+(H10: (subst0 i0 v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t)) (THead k u2 x1) (subst0_snd k v x1 t3 i0 H9 u2) (subst0_fst v u2 x0
+i0 H10 x1 k)))) (H1 x0 H7))) (\lambda (H9: (subst0 (s k i0) v x1
+t3)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2)
+(or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0
+v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)))
+(subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1)
+(THead k u2 t3))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t:
+T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0
+i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5)))
+(subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1)
+(THead k t t3)))) (or4_intro3 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k x0 t3)) (subst0_snd k v t3 x1 i0 H9 x0)) u2 H10))
+(\lambda (H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0
+x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda
+(H11: (subst0 i0 v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq
+T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead
+k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v
+(THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2
+t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda
+(t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x t3) (subst0_fst v x u2 i0
+H11 t3 k) (subst0_both v x0 x i0 H12 k x1 t3 H9)))))) H10)) (\lambda (H10:
+(subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2
+T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0
+v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0
+i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0
+v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead
+k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0))))
+(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead
+k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda
+(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead
+k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2
+i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5))
+(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))).
+
+theorem subst0_confluence_lift:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i
+t2)) \to (eq T t1 t2)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0
+i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i
+t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t:
+T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S
+O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2)
+(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def
+(sym_equal T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O)
+i H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2)
+t)) (\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda
+(t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S
+O) i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S
+O) i t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1)
+x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S
+O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O))
+(plus_comm i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift
+(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1)
+(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n))
+(le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) H1 (eq T t1 t2)))
+(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i
+t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i)
+(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O)
+i)) (plus i (S O)) (plus_comm i (S O))) H1 (eq T t1 t2)))
+(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))).
+
--- /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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/tlt".
+
+include "subst0/defs.ma".
+
+include "lift/props.ma".
+
+include "lift/tlt.ma".
+
+theorem subst0_weight_le:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift
+(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O
+v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
+(plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
+g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S
+(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
+(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
+t0)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
+(wadd g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall
+(m: nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O
+v)) (g i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (plus_le_compat (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
+(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le
+f g H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(lt_le_S (plus (weight_map f0 u2) (weight_map f0 t0)) (S (plus
+(weight_map g u1) (weight_map g t0))) (le_lt_n_Sm (plus (weight_map f0 u2)
+(weight_map f0 t0)) (plus (weight_map g u1) (weight_map g t0))
+(plus_le_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t0)
+(weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g H2))))))))) k)))))))))
+(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: T).(\forall (t2:
+T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 t2) \to
+(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s
+k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
+(le (weight_map f (THead k0 u0 t2)) (weight_map g (THead k0 u0
+t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (v:
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s (Bind b0)
+i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2)
+(weight_map g t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
+(Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 t1)))))))))))))))
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
+t1)) (plus_le_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
+u0 f g H2)) m)) (lt_le_S (weight_map (wadd f (S (weight_map f u0))) (lift (S
+(S i)) O v)) (wadd g (S (weight_map g u0)) (S i)) (eq_ind nat (weight_map f
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f (S
+(weight_map f u0))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f
+u0)) v (S i) f))))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda
+(t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (u0:
+T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2:
+((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift
+(S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O)
+t2)) (plus (weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_compat
+(weight_map f u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map
+(wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m:
+nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O
+v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i))
+O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda
+(t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0)
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
+t1)) (plus_le_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd
+g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda
+(u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
+(H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0
+(lift (S i) O v)) (g i))).(lt_le_S (plus (weight_map f0 u0) (weight_map f0
+t2)) (S (plus (weight_map g u0) (weight_map g t1))) (le_lt_n_Sm (plus
+(weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g
+t1)) (plus_le_compat (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2)
+(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3)))))))))))))))) k))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
+(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt
+(weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f u2) (weight_map
+g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1:
+T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (le
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
+k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v
+t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2)
+(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
+(lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2))
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
+t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1
+f g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1)))
+(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1))
+(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (lt_le_S
+(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) (wadd g (S
+(weight_map g u1)) (S i)) (eq_ind nat (weight_map f (lift (S i) O v))
+(\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f u2)))
+(lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i)
+f)))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i)
+v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
+(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
+t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
+(weight_map g u1) (weight_map (wadd g O) t1)) (plus_le_compat (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O
+O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall
+(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f
+m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H5: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(lt_le_S (plus (weight_map f0 u2) (weight_map f0 t2)) (S (plus
+(weight_map g u1) (weight_map g t1))) (le_lt_n_Sm (plus (weight_map f0 u2)
+(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1))
+(plus_le_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2)
+(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5))))))))))))) k)))))))) d u t
+z H))))).
+
+theorem subst0_weight_lt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v:
+T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
+(plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
+g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S
+(weight_map g u1)) (le_S (S (weight_map f u2)) (weight_map g u1) (lt_le_S
+(weight_map f u2) (weight_map g u1) (H1 f g H2 H3))) n))))))))) (\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (plus_lt_le_compat (weight_map
+f u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
+(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n
+(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O
+O (le_n O) n))))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
+t0)) (plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd
+f O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
+(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd
+f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) b)) (\lambda
+(_: F).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
+(H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0
+(lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0
+t0)) (plus (weight_map g u1) (weight_map g t0)) (plus_lt_le_compat
+(weight_map f0 u2) (weight_map g u1) (weight_map f0 t0) (weight_map g t0) (H1
+f0 g H2 H3) (weight_le t0 f0 g H2)))))))) k))))))))) (\lambda (k: K).(K_ind
+(\lambda (k0: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall
+(i: nat).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: T).(\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map
+f (THead k0 u0 t2)) (weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b:
+B).(B_ind (\lambda (b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1:
+T).(\forall (i: nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind
+b0) i))) \to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
+(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
+t1)) (plus_le_lt_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd
+f (S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
+(S (weight_map g u0)) (lt_le_S (weight_map f u0) (S (weight_map g u0))
+(le_lt_n_Sm (weight_map f u0) (weight_map g u0) (weight_le u0 f g H2))) m))
+(lt_le_S (weight_map (wadd f (S (weight_map f u0))) (lift (S (S i)) O v))
+(wadd g (S (weight_map g u0)) (S i)) (eq_ind nat (weight_map f (lift (S i) O
+v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f
+u0))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i)
+f))))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
+(weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_lt_compat (weight_map
+f u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
+g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
+(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0)
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
+t1)) (plus_le_lt_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd
+f O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O)
+(wadd g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda
+(u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
+(H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0
+(lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f0 u0) (weight_map f0
+t2)) (plus (weight_map g u0) (weight_map g t1)) (plus_le_lt_compat
+(weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) (weight_map g t1)
+(weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) (\lambda (v:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (_: (subst0 i
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(k: K).(K_ind (\lambda (k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s
+k0 i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g
+t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f (THead k0 u2 t2)) (weight_map g (THead
+k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (t1:
+T).(\forall (t2: T).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind
+b0) i))) \to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map
+f (THead (Bind b0) u2 t2)) (weight_map g (THead (Bind b0) u1 t1))))))))))))
+(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (subst0 (S i) v t1
+t2)).(\lambda (_: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f (S (weight_map f u2))) t2)) (plus (weight_map g u1)
+(weight_map (wadd g (S (weight_map g u1))) t1)) (plus_lt_le_compat
+(weight_map f u2) (weight_map g u1) (weight_map (wadd f (S (weight_map f
+u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f g H4 H5)
+(subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) (wadd g (S
+(weight_map g u1))) (\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2))
+(S (weight_map g u1)) (le_S (S (weight_map f u2)) (weight_map g u1) (lt_le_S
+(weight_map f u2) (weight_map g u1) (H1 f g H4 H5))) m)) (lt_le_S (weight_map
+(wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) (wadd g (S (weight_map g
+u1)) (S i)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt
+n (g i))) H5 (weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v))
+(lift_weight_add_O (S (weight_map f u2)) v (S i) f)))))))))))))) (\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
+(weight_map g u1) (weight_map (wadd g O) t1)) (plus_lt_compat (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O
+m) (wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n
+O) m)))) (lt_le_S (weight_map (wadd f O) (lift (S (S i)) O v)) (wadd g O (S
+i)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g
+i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v
+(S i) f)))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0
+(S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
+t1)) (plus_lt_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m)
+(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (lt_le_S (weight_map (wadd f
+O) (lift (S (S i)) O v)) (wadd g O (S i)) (eq_ind nat (weight_map f (lift (S
+i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S
+(S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))) b)) (\lambda (_:
+F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda
+(f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall
+(m: nat).(le (f0 m) (g m))))).(\lambda (H5: (lt (weight_map f0 (lift (S i) O
+v)) (g i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t2)) (plus
+(weight_map g u1) (weight_map g t1)) (plus_lt_compat (weight_map f0 u2)
+(weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 f0 g H4 H5) (H3 f0
+g H4 H5)))))))))))) k)))))))) d u t z H))))).
+
+theorem subst0_tlt_head:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt
+(THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
+z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
+(weight_map (\lambda (_: nat).O) u))) t)) (plus_le_lt_compat (weight_map
+(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n
+(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m)))
+(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n:
+nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda
+(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_:
+nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u)
+(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda
+(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda
+(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
+
+theorem subst0_tlt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z
+(THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
+z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx
+(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
+
include "drop1/props.ma".
+include "drop1/getl.ma".
+
include "clear/defs.ma".
include "clear/fwd.ma".
include "getl/drop.ma".
+include "getl/getl.ma".
+
include "getl/dec.ma".
include "getl/flt.ma".
include "cimp/defs.ma".
+include "cimp/props.ma".
+
+include "subst0/defs.ma".
+
+include "subst0/fwd.ma".
+
+include "subst0/props.ma".
+
+include "subst0/subst0.ma".
+
+include "subst0/tlt.ma".
+
+include "subst0/dec.ma".
+
include "G/defs.ma".
include "next_plus/defs.ma".
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* Problematic objects for disambiguation/typechecking ********************)
+(* FG: PLEASE COMMENT THE NON WORKING OBJECTS *****************************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
+
+include "LambdaDelta/theory.ma".
+
+theorem iso_trans:
+ \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
+t3) \to (iso t1 t3)))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
+(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
+(TSort n2) t3)).(let H1 \def (match H0 in iso return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (iso t t0)).((eq T t (TSort n2)) \to ((eq T t0 t3) \to
+(iso (TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H0: (eq
+T (TSort n0) (TSort n2))).(\lambda (H1: (eq T (TSort n3) t3)).((let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow n0])) (TSort n0) (TSort n2) H0) in (eq_ind nat n2 (\lambda (_:
+nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H3: (eq T
+(TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t))
+(iso_sort n1 n3) t3 H3)) n0 (sym_eq nat n0 n2 H2))) H1))) | (iso_lref i1 i2)
+\Rightarrow (\lambda (H0: (eq T (TLRef i1) (TSort n2))).(\lambda (H1: (eq T
+(TLRef i2) t3)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H0) in
+(False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H2)) H1))) |
+(iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1)
+(TSort n2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T
+(THead k v1 t1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TSort n2) H0) in (False_ind ((eq T (THead k v2 t2)
+t3) \to (iso (TSort n1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TSort n2))
+(refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3:
+T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 in iso return
+(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (TLRef
+i2)) \to ((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2)
+\Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef i2))).(\lambda (H1: (eq T
+(TSort n2) t3)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef
+_) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H0) in
+(False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H2)) H1))) |
+(iso_lref i0 i3) \Rightarrow (\lambda (H0: (eq T (TLRef i0) (TLRef
+i2))).(\lambda (H1: (eq T (TLRef i3) t3)).((let H2 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i2) H0) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef
+i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H3: (eq T (TLRef i3) t3)).(eq_ind
+T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H3)) i0
+(sym_eq nat i0 i2 H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda
+(H0: (eq T (THead k v1 t1) (TLRef i2))).(\lambda (H1: (eq T (THead k v2 t2)
+t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H0) in
+(False_ind ((eq T (THead k v2 t2) t3) \to (iso (TLRef i1) t3)) H2)) H1)))])
+in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k:
+K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def
+(match H0 in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t
+t0)).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3)
+t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1)
+(THead k v2 t4))).(\lambda (H1: (eq T (TSort n2) t5)).((let H2 \def (eq_ind T
+(TSort n1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TSort n2)
+t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_lref i1 i2) \Rightarrow
+(\lambda (H0: (eq T (TLRef i1) (THead k v2 t4))).(\lambda (H1: (eq T (TLRef
+i2) t5)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0)
+in (False_ind ((eq T (TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1)))
+| (iso_head k0 v0 v3 t0 t4) \Rightarrow (\lambda (H0: (eq T (THead k0 v0 t0)
+(THead k v2 t4))).(\lambda (H1: (eq T (THead k0 v3 t4) t5)).((let H2 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H3 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _)
+\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H4 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _)
+\Rightarrow k])) (THead k0 v0 t0) (THead k v2 t4) H0) in (eq_ind K k (\lambda
+(k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to ((eq T (THead k1 v3 t4) t5) \to
+(iso (THead k v1 t3) t5))))) (\lambda (H5: (eq T v0 v2)).(eq_ind T v2
+(\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k v3 t4) t5) \to (iso (THead
+k v1 t3) t5)))) (\lambda (H6: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).((eq
+T (THead k v3 t4) t5) \to (iso (THead k v1 t3) t5))) (\lambda (H7: (eq T
+(THead k v3 t4) t5)).(eq_ind T (THead k v3 t4) (\lambda (t: T).(iso (THead k
+v1 t3) t)) (iso_head k v1 v3 t3 t4) t5 H7)) t0 (sym_eq T t0 t4 H6))) v0
+(sym_eq T v0 v2 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H1
+(refl_equal T (THead k v2 t4)) (refl_equal T t5)))))))))) t1 t2 H))).
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* Problematic objects for disambiguation/typechecking ********************)
+(* FG: PLEASE COMMENT THE NON WORKING OBJECTS *****************************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
+
+include "LambdaDelta/theory.ma".
+
+theorem drop1_getl_trans:
+ \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
+\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
+i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2
+(CHead e2 (Bind b) (ctrans hds i v)))))))))))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
+C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1:
+C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to
+(ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i
+v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H
+in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
+(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to
+(ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with
+[(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2:
+(eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C
+c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))
+(\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2:
+C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i
+c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2
+H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3:
+(eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5:
+(eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
+(False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1
+hds c3 c4) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))
+H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2)
+(refl_equal C c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda
+(hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2
+c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i:
+nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl
+(trans hds0 i) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0
+in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
+(_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq
+C c0 c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
+h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
+\Rightarrow (ctrans hds0 i v)])))))))))))) with [(drop1_nil c) \Rightarrow
+(\lambda (H2: (eq PList PNil (PCons h d hds0))).(\lambda (H3: (eq C c
+c2)).(\lambda (H4: (eq C c c1)).((let H5 \def (eq_ind PList PNil (\lambda (e:
+PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
+\Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds0) H2)
+in (False_ind ((eq C c c2) \to ((eq C c c1) \to (ex C (\lambda (e2: C).(getl
+(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) H5)) H3
+H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds0 H3) \Rightarrow (\lambda (H4: (eq
+PList (PCons h0 d0 hds0) (PCons h d hds0))).(\lambda (H5: (eq C c0
+c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda
+(e: PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
+\Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h
+d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e
+in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d0 | (PCons _
+n _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H9
+\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
+(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
+(PCons h0 d0 hds0) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n:
+nat).((eq nat d0 d) \to ((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4
+c1) \to ((drop n d0 c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2:
+C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i)
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match
+(blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))))
+(\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds0
+hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1
+hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d)
+with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0
+i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
+\Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda (H11: (eq PList hds0
+hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C c4 c1)
+\to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C (\lambda (e2: C).(getl
+(match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))) (\lambda
+(H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d
+c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt
+(trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow
+(plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d)
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) (\lambda (H13: (eq C c4
+c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) \to ((drop1 hds0 c3 c)
+\to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
+(CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift
+h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans
+hds0 i v)])))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1
+hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex
+C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) |
+false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)])))))) (\lambda (x_x:
+bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to
+(ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i)
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0
+with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)) | false \Rightarrow (ctrans hds0 i v)]))))))) (\lambda (H0: (eq bool (blt
+(trans hds0 i) d) true)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16
+\def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2
+(Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2
+(CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v)))))) (\lambda (x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind
+b) (ctrans hds0 i v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 i) d
+(le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt (S d)
+(S (trans hds0 i)) H0))) c2 c3 h H14 b x (ctrans hds0 i v) H17) in (let H
+\def H_x0 in (ex2_ind C (\lambda (e1: C).(getl (trans hds0 i) c2 (CHead e1
+(Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) (\lambda
+(e1: C).(drop h (minus d (S (trans hds0 i))) e1 x)) (ex C (\lambda (e2:
+C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i v)))))) (\lambda (x0: C).(\lambda (H1: (getl (trans hds0
+i) c2 (CHead x0 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i
+v))))).(\lambda (_: (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro C
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d
+(S (trans hds0 i))) (ctrans hds0 i v))))) x0 H1)))) H))))) H16)))) (\lambda
+(H0: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def (H c1 c3 H15 b e1
+v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0
+i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl
+(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))) (\lambda
+(x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0
+i v)))).(let H \def (drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x
+(Bind b) (ctrans hds0 i v)) H17) in (ex_intro C (\lambda (e2: C).(getl (plus
+(trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x (H (bge_le d
+(trans hds0 i) H0)))))) H16)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq
+C c0 c2 H12))) hds0 (sym_eq PList hds0 hds0 H11))) d0 (sym_eq nat d0 d H10)))
+h0 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal
+PList (PCons h d hds0)) (refl_equal C c2) (refl_equal C c1)))))))))))))))
+hds).
+
(**************************************************************************)
(* Problematic objects for disambiguation/typechecking ********************)
-(* FG: PLEASE COMMENT THE NON WORKING OBJECTS *****************************)
set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems".
include "LambdaDelta/theory.ma".
-(*
+(* Problem 1: disambiguation errors with these objects *)
-(* Problem 2: disambiguation errors with these objects *)
+(* iso_trans (in problems-1)
+ * drop1_getl_trans (in problems-2)
+ *)
-iso_trans (in iso/props)
-
-theorem iso_trans:
- \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
-t3) \to (iso t1 t3)))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
-(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
-(TSort n2) t3)).(let H1 \def (match H0 in iso return (\lambda (t: T).(\lambda
-(t0: T).(\lambda (_: (iso t t0)).((eq T t (TSort n2)) \to ((eq T t0 t3) \to
-(iso (TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H0: (eq
-T (TSort n0) (TSort n2))).(\lambda (H1: (eq T (TSort n3) t3)).((let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow n0])) (TSort n0) (TSort n2) H0) in (eq_ind nat n2 (\lambda (_:
-nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H3: (eq T
-(TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t))
-(iso_sort n1 n3) t3 H3)) n0 (sym_eq nat n0 n2 H2))) H1))) | (iso_lref i1 i2)
-\Rightarrow (\lambda (H0: (eq T (TLRef i1) (TSort n2))).(\lambda (H1: (eq T
-(TLRef i2) t3)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H0) in
-(False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H2)) H1))) |
-(iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1)
-(TSort n2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T
-(THead k v1 t1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TSort n2) H0) in (False_ind ((eq T (THead k v2 t2)
-t3) \to (iso (TSort n1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TSort n2))
-(refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3:
-T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 in iso return
-(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (TLRef
-i2)) \to ((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2)
-\Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef i2))).(\lambda (H1: (eq T
-(TSort n2) t3)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef
-_) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H0) in
-(False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H2)) H1))) |
-(iso_lref i0 i3) \Rightarrow (\lambda (H0: (eq T (TLRef i0) (TLRef
-i2))).(\lambda (H1: (eq T (TLRef i3) t3)).((let H2 \def (f_equal T nat
-(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
-(TLRef i0) (TLRef i2) H0) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef
-i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H3: (eq T (TLRef i3) t3)).(eq_ind
-T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H3)) i0
-(sym_eq nat i0 i2 H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda
-(H0: (eq T (THead k v1 t1) (TLRef i2))).(\lambda (H1: (eq T (THead k v2 t2)
-t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H0) in
-(False_ind ((eq T (THead k v2 t2) t3) \to (iso (TLRef i1) t3)) H2)) H1)))])
-in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k:
-K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def
-(match H0 in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t
-t0)).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3)
-t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1)
-(THead k v2 t4))).(\lambda (H1: (eq T (TSort n2) t5)).((let H2 \def (eq_ind T
-(TSort n1) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TSort n2)
-t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_lref i1 i2) \Rightarrow
-(\lambda (H0: (eq T (TLRef i1) (THead k v2 t4))).(\lambda (H1: (eq T (TLRef
-i2) t5)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0)
-in (False_ind ((eq T (TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1)))
-| (iso_head k0 v0 v3 t0 t4) \Rightarrow (\lambda (H0: (eq T (THead k0 v0 t0)
-(THead k v2 t4))).(\lambda (H1: (eq T (THead k0 v3 t4) t5)).((let H2 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H3 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _)
-\Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H4 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _)
-\Rightarrow k])) (THead k0 v0 t0) (THead k v2 t4) H0) in (eq_ind K k (\lambda
-(k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to ((eq T (THead k1 v3 t4) t5) \to
-(iso (THead k v1 t3) t5))))) (\lambda (H5: (eq T v0 v2)).(eq_ind T v2
-(\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k v3 t4) t5) \to (iso (THead
-k v1 t3) t5)))) (\lambda (H6: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).((eq
-T (THead k v3 t4) t5) \to (iso (THead k v1 t3) t5))) (\lambda (H7: (eq T
-(THead k v3 t4) t5)).(eq_ind T (THead k v3 t4) (\lambda (t: T).(iso (THead k
-v1 t3) t)) (iso_head k v1 v3 t3 t4) t5 H7)) t0 (sym_eq T t0 t4 H6))) v0
-(sym_eq T v0 v2 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H1
-(refl_equal T (THead k v2 t4)) (refl_equal T t5)))))))))) t1 t2 H))).
-
-
-drop1_getl_trans
-
-(* Problem 1: does not typecheck a match on an empty type *)
+(* Problem 2: does not typecheck a match on an empty type *)
theorem subst0_confluence_neq:
\forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t))))))))))))
(* Problem 3: assertion failure raised by type checker on this object *)
-tau1
-
-*)
+(* tau1
+ *)
projects / helm.git / commitdiff