(* This file was automatically generated: do not edit *********************)
-include "Basic-1/T/defs.ma".
+include "basic_1/T/defs.ma".
-inductive C: Set \def
+inductive C: Type[0] \def
| CSort: nat \to C
| CHead: C \to (K \to (T \to C)).
-definition cweight:
- C \to nat
-\def
- let rec cweight (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O
-| (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight.
+let rec cweight (c: C) on c: nat \def match c with [(CSort _) \Rightarrow O |
+(CHead c0 _ t) \Rightarrow (let TMP_1 \def (cweight c0) in (let TMP_2 \def
+(tweight t) in (plus TMP_1 TMP_2)))].
definition clt:
C \to (C \to Prop)
\def
- \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))).
+ \lambda (c1: C).(\lambda (c2: C).(let TMP_1 \def (cweight c1) in (let TMP_2
+\def (cweight c2) in (lt TMP_1 TMP_2)))).
definition cle:
C \to (C \to Prop)
\def
- \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))).
+ \lambda (c1: C).(\lambda (c2: C).(let TMP_1 \def (cweight c1) in (let TMP_2
+\def (cweight c2) in (le TMP_1 TMP_2)))).
-definition CTail:
- K \to (T \to (C \to C))
-\def
- let rec CTail (k: K) (t: T) (c: C) on c: C \def (match c with [(CSort n)
-\Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k
-t d) h u)]) in CTail.
+let rec CTail (k: K) (t: T) (c: C) on c: C \def match c with [(CSort n)
+\Rightarrow (let TMP_2 \def (CSort n) in (CHead TMP_2 k t)) | (CHead d h u)
+\Rightarrow (let TMP_1 \def (CTail k t d) in (CHead TMP_1 h u))].
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1/C/defs.ma".
+
+let rec C_rect (P: (C \to Type[0])) (f: (\forall (n: nat).(P (CSort n))))
+(f0: (\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CHead c k
+t))))))) (c: C) on c: P c \def match c with [(CSort n) \Rightarrow (f n) |
+(CHead c0 k t) \Rightarrow (let TMP_1 \def ((C_rect P f f0) c0) in (f0 c0
+TMP_1 k t))].
+
+theorem C_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to
+(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CHead c k
+t))))))) \to (\forall (c: C).(P c))))
+\def
+ \lambda (P: ((C \to Prop))).(C_rect P).
+
+theorem clt_wf__q_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to
+Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0
+c))))) P n))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
+n) \to (P c)))))).(\lambda (c: C).(let TMP_1 \def (cweight c) in (let TMP_2
+\def (cweight c) in (let TMP_3 \def (refl_equal nat TMP_2) in (H TMP_1 c
+TMP_3))))))).
+
+theorem clt_wf_ind:
+ \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c)
+\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
+Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d)
+(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(let TMP_1 \def
+(\lambda (c0: C).(P c0)) in (let TMP_11 \def (\lambda (n: nat).(let TMP_2
+\def (\lambda (c0: C).(P c0)) in (let TMP_3 \def (Q TMP_2) in (let TMP_10
+\def (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) \to (Q
+(\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat
+(cweight c0) n0)).(let TMP_4 \def (\lambda (n1: nat).(\forall (m: nat).((lt m
+n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P c1)))))) in (let
+TMP_5 \def (cweight c0) in (let H2 \def (eq_ind_r nat n0 TMP_4 H0 TMP_5 H1)
+in (let TMP_9 \def (\lambda (d: C).(\lambda (H3: (lt (cweight d) (cweight
+c0))).(let TMP_6 \def (cweight d) in (let TMP_7 \def (cweight d) in (let
+TMP_8 \def (refl_equal nat TMP_7) in (H2 TMP_6 H3 d TMP_8)))))) in (H c0
+TMP_9))))))))) in (lt_wf_ind n TMP_3 TMP_10))))) in (clt_wf__q_ind TMP_1
+TMP_11 c)))))).
+
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/C/defs.ma".
+include "basic_1/C/fwd.ma".
-include "Basic-1/T/props.ma".
+include "basic_1/T/props.ma".
+
+theorem cle_r:
+ \forall (c: C).(cle c c)
+\def
+ \lambda (c: C).(let TMP_3 \def (\lambda (c0: C).(let TMP_1 \def (cweight c0)
+in (let TMP_2 \def (cweight c0) in (le TMP_1 TMP_2)))) in (let TMP_4 \def
+(\lambda (_: nat).(le_O_n O)) in (let TMP_8 \def (\lambda (c0: C).(\lambda
+(_: (le (cweight c0) (cweight c0))).(\lambda (_: K).(\lambda (t: T).(let
+TMP_5 \def (cweight c0) in (let TMP_6 \def (tweight t) in (let TMP_7 \def
+(plus TMP_5 TMP_6) in (le_n TMP_7)))))))) in (C_ind TMP_3 TMP_4 TMP_8 c)))).
+
+theorem cle_head:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (u1: T).(\forall
+(u2: T).((tle u1 u2) \to (\forall (k: K).(cle (CHead c1 k u1) (CHead c2 k
+u2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight
+c2))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (le (tweight u1)
+(tweight u2))).(\lambda (_: K).(let TMP_1 \def (cweight c1) in (let TMP_2
+\def (cweight c2) in (let TMP_3 \def (tweight u1) in (let TMP_4 \def (tweight
+u2) in (le_plus_plus TMP_1 TMP_2 TMP_3 TMP_4 H H0))))))))))).
+
+theorem cle_trans_head:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (k: K).(\forall
+(u: T).(cle c1 (CHead c2 k u))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight
+c2))).(\lambda (_: K).(\lambda (u: T).(let TMP_1 \def (cweight c1) in (let
+TMP_2 \def (cweight c2) in (let TMP_3 \def (tweight u) in (le_plus_trans
+TMP_1 TMP_2 TMP_3 H)))))))).
theorem clt_cong:
\forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t:
T).(clt (CHead c k t) (CHead d k t))))))
\def
\lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight
-d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d)
-(tweight t) H))))).
-(* COMMENTS
-Initial nodes: 33
-END *)
+d))).(\lambda (_: K).(\lambda (t: T).(let TMP_1 \def (cweight c) in (let
+TMP_2 \def (cweight d) in (let TMP_3 \def (tweight t) in (lt_reg_r TMP_1
+TMP_2 TMP_3 H)))))))).
theorem clt_head:
\forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight
-c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u))))
-(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c))
-(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
-(* COMMENTS
-Initial nodes: 69
-END *)
-
-theorem clt_wf__q_ind:
- \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to
-Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0
-c))))) P n))) \to (\forall (c: C).(P c)))
-\def
- let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
-C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
-n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight
-c)))))).
-(* COMMENTS
-Initial nodes: 61
-END *)
-
-theorem clt_wf_ind:
- \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c)
-\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
-\def
- let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
-C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
-Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d)
-(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind
-(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0:
-C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
-\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat
-(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
-(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P
-c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt
-(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight
-d))))))))))))) c)))).
-(* COMMENTS
-Initial nodes: 179
-END *)
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(let TMP_1 \def (cweight c)
+in (let TMP_2 \def (plus TMP_1 O) in (let TMP_6 \def (\lambda (n: nat).(let
+TMP_3 \def (cweight c) in (let TMP_4 \def (tweight u) in (let TMP_5 \def
+(plus TMP_3 TMP_4) in (lt n TMP_5))))) in (let TMP_7 \def (tweight u) in (let
+TMP_8 \def (cweight c) in (let TMP_9 \def (tweight_lt u) in (let TMP_10 \def
+(lt_reg_l O TMP_7 TMP_8 TMP_9) in (let TMP_11 \def (cweight c) in (let TMP_12
+\def (cweight c) in (let TMP_13 \def (plus_n_O TMP_12) in (eq_ind_r nat TMP_2
+TMP_6 TMP_10 TMP_11 TMP_13))))))))))))).
theorem chead_ctail:
\forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3
-K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t)
-(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k:
-K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
-(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead
-(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall
-(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
-(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def
-H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
-(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d:
-C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d))))))
-(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead
-c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1:
-C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead
-c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d:
-C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0
-(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0
-k t) H1))))) H0))))))))) c).
-(* COMMENTS
-Initial nodes: 395
-END *)
+ \lambda (c: C).(let TMP_4 \def (\lambda (c0: C).(\forall (t: T).(\forall (k:
+K).(let TMP_3 \def (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(let TMP_1
+\def (CHead c0 k t) in (let TMP_2 \def (CTail h u d) in (eq C TMP_1
+TMP_2)))))) in (ex_3 K C T TMP_3))))) in (let TMP_13 \def (\lambda (n:
+nat).(\lambda (t: T).(\lambda (k: K).(let TMP_8 \def (\lambda (h: K).(\lambda
+(d: C).(\lambda (u: T).(let TMP_5 \def (CSort n) in (let TMP_6 \def (CHead
+TMP_5 k t) in (let TMP_7 \def (CTail h u d) in (eq C TMP_6 TMP_7))))))) in
+(let TMP_9 \def (CSort n) in (let TMP_10 \def (CSort n) in (let TMP_11 \def
+(CHead TMP_10 k t) in (let TMP_12 \def (refl_equal C TMP_11) in (ex_3_intro K
+C T TMP_8 k TMP_9 t TMP_12))))))))) in (let TMP_38 \def (\lambda (c0:
+C).(\lambda (H: ((\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
+K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t) (CTail h u
+d)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t0: T).(\lambda (k0:
+K).(let H_x \def (H t k) in (let H0 \def H_x in (let TMP_16 \def (\lambda (h:
+K).(\lambda (d: C).(\lambda (u: T).(let TMP_14 \def (CHead c0 k t) in (let
+TMP_15 \def (CTail h u d) in (eq C TMP_14 TMP_15)))))) in (let TMP_20 \def
+(\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(let TMP_17 \def (CHead c0 k
+t) in (let TMP_18 \def (CHead TMP_17 k0 t0) in (let TMP_19 \def (CTail h u d)
+in (eq C TMP_18 TMP_19))))))) in (let TMP_21 \def (ex_3 K C T TMP_20) in (let
+TMP_37 \def (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1:
+(eq C (CHead c0 k t) (CTail x0 x2 x1))).(let TMP_22 \def (CTail x0 x2 x1) in
+(let TMP_26 \def (\lambda (c1: C).(let TMP_25 \def (\lambda (h: K).(\lambda
+(d: C).(\lambda (u: T).(let TMP_23 \def (CHead c1 k0 t0) in (let TMP_24 \def
+(CTail h u d) in (eq C TMP_23 TMP_24)))))) in (ex_3 K C T TMP_25))) in (let
+TMP_30 \def (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(let TMP_27 \def
+(CTail x0 x2 x1) in (let TMP_28 \def (CHead TMP_27 k0 t0) in (let TMP_29 \def
+(CTail h u d) in (eq C TMP_28 TMP_29))))))) in (let TMP_31 \def (CHead x1 k0
+t0) in (let TMP_32 \def (CTail x0 x2 x1) in (let TMP_33 \def (CHead TMP_32 k0
+t0) in (let TMP_34 \def (refl_equal C TMP_33) in (let TMP_35 \def (ex_3_intro
+K C T TMP_30 x0 TMP_31 x2 TMP_34) in (let TMP_36 \def (CHead c0 k t) in
+(eq_ind_r C TMP_22 TMP_26 TMP_35 TMP_36 H1)))))))))))))) in (ex_3_ind K C T
+TMP_16 TMP_21 TMP_37 H0))))))))))))) in (C_ind TMP_4 TMP_13 TMP_38 c)))).
theorem clt_thead:
\forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
\def
- \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt
-c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0:
-C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t:
-T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
-(* COMMENTS
-Initial nodes: 71
-END *)
+ \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(let TMP_2 \def (\lambda (c0:
+C).(let TMP_1 \def (CTail k u c0) in (clt c0 TMP_1))) in (let TMP_4 \def
+(\lambda (n: nat).(let TMP_3 \def (CSort n) in (clt_head k TMP_3 u))) in (let
+TMP_6 \def (\lambda (c0: C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda
+(k0: K).(\lambda (t: T).(let TMP_5 \def (CTail k u c0) in (clt_cong c0 TMP_5
+H k0 t)))))) in (C_ind TMP_2 TMP_4 TMP_6 c)))))).
theorem c_tail_ind:
\forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to
\def
\lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort
n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t:
-T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P
-c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1)
-\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d:
-C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_:
-((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to
-(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind
-K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t)
-(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1:
-C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2
-x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def
-(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P
-d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2))
-(CHead c1 k t) H4))))) H3)))))))) c0)) c)))).
-(* COMMENTS
-Initial nodes: 295
-END *)
+T).(P (CTail k t c)))))))).(\lambda (c: C).(let TMP_1 \def (\lambda (c0:
+C).(P c0)) in (let TMP_20 \def (\lambda (c0: C).(let TMP_2 \def (\lambda (c1:
+C).(((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1))) in (let TMP_3 \def
+(\lambda (n: nat).(\lambda (_: ((\forall (d: C).((clt d (CSort n)) \to (P
+d))))).(H n))) in (let TMP_19 \def (\lambda (c1: C).(\lambda (_: ((((\forall
+(d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to (P d))))).(let
+H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (let TMP_6 \def (\lambda
+(h: K).(\lambda (d: C).(\lambda (u: T).(let TMP_4 \def (CHead c1 k t) in (let
+TMP_5 \def (CTail h u d) in (eq C TMP_4 TMP_5)))))) in (let TMP_7 \def (CHead
+c1 k t) in (let TMP_8 \def (P TMP_7) in (let TMP_18 \def (\lambda (x0:
+K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t)
+(CTail x0 x2 x1))).(let TMP_9 \def (CTail x0 x2 x1) in (let TMP_10 \def
+(\lambda (c2: C).(P c2)) in (let TMP_11 \def (CHead c1 k t) in (let TMP_12
+\def (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P d)))) in (let TMP_13
+\def (CTail x0 x2 x1) in (let H5 \def (eq_ind C TMP_11 TMP_12 H2 TMP_13 H4)
+in (let TMP_14 \def (clt_thead x0 x2 x1) in (let TMP_15 \def (H5 x1 TMP_14)
+in (let TMP_16 \def (H0 x1 TMP_15 x0 x2) in (let TMP_17 \def (CHead c1 k t)
+in (eq_ind_r C TMP_9 TMP_10 TMP_16 TMP_17 H4))))))))))))))) in (ex_3_ind K C
+T TMP_6 TMP_8 TMP_18 H3)))))))))))) in (C_ind TMP_2 TMP_3 TMP_19 c0))))) in
+(clt_wf_ind TMP_1 TMP_20 c)))))).
(tweight u) in (let TMP_2 \def (tweight t0) in (let TMP_3 \def (plus TMP_1
TMP_2) in (S TMP_3))))].
+definition tle:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(let TMP_1 \def (tweight t1) in (let TMP_2
+\def (tweight t2) in (le TMP_1 TMP_2)))).
+
in (let TMP_14 \def (le_plus_trans TMP_11 TMP_12 TMP_13 H) in (le_S TMP_7
TMP_10 TMP_14)))))))))))))) in (T_ind TMP_2 TMP_4 TMP_6 TMP_15 t))))).
+theorem tle_r:
+ \forall (t: T).(tle t t)
+\def
+ \lambda (t: T).(let TMP_3 \def (\lambda (t0: T).(let TMP_1 \def (tweight t0)
+in (let TMP_2 \def (tweight t0) in (le TMP_1 TMP_2)))) in (let TMP_5 \def
+(\lambda (_: nat).(let TMP_4 \def (S O) in (le_n TMP_4))) in (let TMP_7 \def
+(\lambda (_: nat).(let TMP_6 \def (S O) in (le_n TMP_6))) in (let TMP_12 \def
+(\lambda (_: K).(\lambda (t0: T).(\lambda (_: (le (tweight t0) (tweight
+t0))).(\lambda (t1: T).(\lambda (_: (le (tweight t1) (tweight t1))).(let
+TMP_8 \def (tweight t0) in (let TMP_9 \def (tweight t1) in (let TMP_10 \def
+(plus TMP_8 TMP_9) in (let TMP_11 \def (S TMP_10) in (le_n TMP_11))))))))))
+in (T_ind TMP_3 TMP_5 TMP_7 TMP_12 t))))).
+
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/C/defs.ma".
+include "basic_1/C/defs.ma".
-definition cbk:
- C \to nat
-\def
- let rec cbk (c: C) on c: nat \def (match c with [(CSort m) \Rightarrow m |
-(CHead c0 _ _) \Rightarrow (cbk c0)]) in cbk.
+let rec cbk (c: C) on c: nat \def match c with [(CSort m) \Rightarrow m |
+(CHead c0 _ _) \Rightarrow (cbk c0)].
-definition app1:
- C \to (T \to T)
-\def
- let rec app1 (c: C) on c: (T \to T) \def (\lambda (t: T).(match c with
-[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u
-t))])) in app1.
+let rec app1 (c: C) on c: T \to T \def \lambda (t: T).(match c with [(CSort
+_) \Rightarrow t | (CHead c0 k u) \Rightarrow (let TMP_1 \def (THead k u t)
+in (app1 c0 TMP_1))]).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/C/defs.ma".
+include "basic_1/C/defs.ma".
definition fweight:
C \to (T \to nat)
\def
- \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))).
+ \lambda (c: C).(\lambda (t: T).(let TMP_1 \def (cweight c) in (let TMP_2
+\def (tweight t) in (plus TMP_1 TMP_2)))).
definition flt:
C \to (T \to (C \to (T \to Prop)))
\def
- \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt
-(fweight c1 t1) (fweight c2 t2))))).
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(let
+TMP_1 \def (fweight c1 t1) in (let TMP_2 \def (fweight c2 t2) in (lt TMP_1
+TMP_2)))))).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1/flt/defs.ma".
+
+theorem flt_wf__q_ind:
+ \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C
+\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq
+nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall
+(t: T).(P c t))))
+\def
+ let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
+(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
+(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c:
+C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c:
+C).(\lambda (t: T).(let TMP_1 \def (fweight c t) in (let TMP_2 \def (fweight
+c t) in (let TMP_3 \def (refl_equal nat TMP_2) in (H TMP_1 c t TMP_3)))))))).
+
+theorem flt_wf_ind:
+ \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2:
+T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
+\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
+\def
+ let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
+(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
+(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2:
+T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
+\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(let TMP_9 \def (\lambda
+(n: nat).(let TMP_1 \def (Q P) in (let TMP_8 \def (\lambda (n0: nat).(\lambda
+(H0: ((\forall (m: nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda
+(t0: T).(\lambda (H1: (eq nat (fweight c0 t0) n0)).(let TMP_2 \def (\lambda
+(n1: nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1:
+T).((eq nat (fweight c1 t1) m) \to (P c1 t1))))))) in (let TMP_3 \def
+(fweight c0 t0) in (let H2 \def (eq_ind_r nat n0 TMP_2 H0 TMP_3 H1) in (let
+TMP_7 \def (\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0
+t0)).(let TMP_4 \def (fweight c1 t1) in (let TMP_5 \def (fweight c1 t1) in
+(let TMP_6 \def (refl_equal nat TMP_5) in (H2 TMP_4 H3 c1 t1 TMP_6))))))) in
+(H c0 t0 TMP_7)))))))))) in (lt_wf_ind n TMP_1 TMP_8)))) in (flt_wf__q_ind P
+TMP_9 c t)))))).
+
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/flt/defs.ma".
+include "basic_1/flt/defs.ma".
-include "Basic-1/C/props.ma".
+include "basic_1/C/props.ma".
theorem flt_thead_sx:
\forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c
(THead k u t)))))
\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
-T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u)
-(tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u)
-(tweight t)) (le_plus_l (tweight u) (tweight t))))))).
-(* COMMENTS
-Initial nodes: 65
-END *)
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(let TMP_1
+\def (tweight u) in (let TMP_2 \def (tweight u) in (let TMP_3 \def (tweight
+t) in (let TMP_4 \def (plus TMP_2 TMP_3) in (let TMP_5 \def (S TMP_4) in (let
+TMP_6 \def (cweight c) in (let TMP_7 \def (tweight u) in (let TMP_8 \def
+(tweight u) in (let TMP_9 \def (tweight t) in (let TMP_10 \def (plus TMP_8
+TMP_9) in (let TMP_11 \def (tweight u) in (let TMP_12 \def (tweight t) in
+(let TMP_13 \def (le_plus_l TMP_11 TMP_12) in (let TMP_14 \def (le_n_S TMP_7
+TMP_10 TMP_13) in (lt_reg_l TMP_1 TMP_5 TMP_6 TMP_14)))))))))))))))))).
theorem flt_thead_dx:
\forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c
(THead k u t)))))
\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
-T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u)
-(tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u)
-(tweight t)) (le_plus_r (tweight u) (tweight t))))))).
-(* COMMENTS
-Initial nodes: 65
-END *)
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(let TMP_1
+\def (tweight t) in (let TMP_2 \def (tweight u) in (let TMP_3 \def (tweight
+t) in (let TMP_4 \def (plus TMP_2 TMP_3) in (let TMP_5 \def (S TMP_4) in (let
+TMP_6 \def (cweight c) in (let TMP_7 \def (tweight t) in (let TMP_8 \def
+(tweight u) in (let TMP_9 \def (tweight t) in (let TMP_10 \def (plus TMP_8
+TMP_9) in (let TMP_11 \def (tweight u) in (let TMP_12 \def (tweight t) in
+(let TMP_13 \def (le_plus_r TMP_11 TMP_12) in (let TMP_14 \def (le_n_S TMP_7
+TMP_10 TMP_13) in (lt_reg_l TMP_1 TMP_5 TMP_6 TMP_14)))))))))))))))))).
theorem flt_shift:
\forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c
k u) t c (THead k u t)))))
\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat
-(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt
-(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus
-(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus
-(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight
-c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight
-t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
-(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u)
-(tweight t))))))).
-(* COMMENTS
-Initial nodes: 179
-END *)
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(let TMP_1
+\def (cweight c) in (let TMP_2 \def (tweight u) in (let TMP_3 \def (tweight
+t) in (let TMP_4 \def (plus TMP_2 TMP_3) in (let TMP_5 \def (plus TMP_1
+TMP_4) in (let TMP_6 \def (S TMP_5) in (let TMP_12 \def (\lambda (n:
+nat).(let TMP_7 \def (cweight c) in (let TMP_8 \def (tweight u) in (let TMP_9
+\def (plus TMP_7 TMP_8) in (let TMP_10 \def (tweight t) in (let TMP_11 \def
+(plus TMP_9 TMP_10) in (lt TMP_11 n))))))) in (let TMP_13 \def (cweight c) in
+(let TMP_14 \def (tweight u) in (let TMP_15 \def (plus TMP_13 TMP_14) in (let
+TMP_16 \def (tweight t) in (let TMP_17 \def (plus TMP_15 TMP_16) in (let
+TMP_24 \def (\lambda (n: nat).(let TMP_18 \def (cweight c) in (let TMP_19
+\def (tweight u) in (let TMP_20 \def (plus TMP_18 TMP_19) in (let TMP_21 \def
+(tweight t) in (let TMP_22 \def (plus TMP_20 TMP_21) in (let TMP_23 \def (S
+n) in (lt TMP_22 TMP_23)))))))) in (let TMP_25 \def (cweight c) in (let
+TMP_26 \def (tweight u) in (let TMP_27 \def (plus TMP_25 TMP_26) in (let
+TMP_28 \def (tweight t) in (let TMP_29 \def (plus TMP_27 TMP_28) in (let
+TMP_30 \def (S TMP_29) in (let TMP_31 \def (le_n TMP_30) in (let TMP_32 \def
+(cweight c) in (let TMP_33 \def (tweight u) in (let TMP_34 \def (tweight t)
+in (let TMP_35 \def (plus TMP_33 TMP_34) in (let TMP_36 \def (plus TMP_32
+TMP_35) in (let TMP_37 \def (cweight c) in (let TMP_38 \def (tweight u) in
+(let TMP_39 \def (tweight t) in (let TMP_40 \def (plus_assoc_l TMP_37 TMP_38
+TMP_39) in (let TMP_41 \def (eq_ind_r nat TMP_17 TMP_24 TMP_31 TMP_36 TMP_40)
+in (let TMP_42 \def (cweight c) in (let TMP_43 \def (tweight u) in (let
+TMP_44 \def (tweight t) in (let TMP_45 \def (plus TMP_43 TMP_44) in (let
+TMP_46 \def (S TMP_45) in (let TMP_47 \def (plus TMP_42 TMP_46) in (let
+TMP_48 \def (cweight c) in (let TMP_49 \def (tweight u) in (let TMP_50 \def
+(tweight t) in (let TMP_51 \def (plus TMP_49 TMP_50) in (let TMP_52 \def
+(plus_n_Sm TMP_48 TMP_51) in (eq_ind nat TMP_6 TMP_12 TMP_41 TMP_47
+TMP_52))))))))))))))))))))))))))))))))))))))))))))).
theorem flt_arith0:
\forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t
(CHead c k t) (TLRef i)))))
\def
- \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_:
-nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))).
-(* COMMENTS
-Initial nodes: 21
-END *)
+ \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: nat).(let TMP_1
+\def (cweight c) in (let TMP_2 \def (tweight t) in (let TMP_3 \def (plus
+TMP_1 TMP_2) in (lt_x_plus_x_Sy TMP_3 O))))))).
theorem flt_arith1:
\forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle
\def
\lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda
(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_:
-K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1)
-(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H
-(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n:
-nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
-(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2)
-(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S
-O))))))))))).
-(* COMMENTS
-Initial nodes: 151
-END *)
+K).(\lambda (t2: T).(\lambda (_: nat).(let TMP_1 \def (cweight c1) in (let
+TMP_2 \def (tweight t1) in (let TMP_3 \def (plus TMP_1 TMP_2) in (let TMP_4
+\def (cweight c2) in (let TMP_5 \def (cweight c2) in (let TMP_6 \def (tweight
+t2) in (let TMP_7 \def (plus TMP_5 TMP_6) in (let TMP_8 \def (S O) in (let
+TMP_9 \def (plus TMP_7 TMP_8) in (let TMP_10 \def (S O) in (let TMP_11 \def
+(cweight c2) in (let TMP_12 \def (tweight t2) in (let TMP_13 \def (plus
+TMP_11 TMP_12) in (let TMP_14 \def (plus TMP_10 TMP_13) in (let TMP_16 \def
+(\lambda (n: nat).(let TMP_15 \def (cweight c2) in (lt TMP_15 n))) in (let
+TMP_17 \def (cweight c2) in (let TMP_18 \def (cweight c2) in (let TMP_19 \def
+(tweight t2) in (let TMP_20 \def (plus TMP_18 TMP_19) in (let TMP_21 \def
+(cweight c2) in (let TMP_22 \def (tweight t2) in (let TMP_23 \def (le_plus_l
+TMP_21 TMP_22) in (let TMP_24 \def (le_lt_n_Sm TMP_17 TMP_20 TMP_23) in (let
+TMP_25 \def (cweight c2) in (let TMP_26 \def (tweight t2) in (let TMP_27 \def
+(plus TMP_25 TMP_26) in (let TMP_28 \def (S O) in (let TMP_29 \def (plus
+TMP_27 TMP_28) in (let TMP_30 \def (cweight c2) in (let TMP_31 \def (tweight
+t2) in (let TMP_32 \def (plus TMP_30 TMP_31) in (let TMP_33 \def (S O) in
+(let TMP_34 \def (plus_sym TMP_32 TMP_33) in (let TMP_35 \def (eq_ind_r nat
+TMP_14 TMP_16 TMP_24 TMP_29 TMP_34) in (le_lt_trans TMP_3 TMP_4 TMP_9 H
+TMP_35)))))))))))))))))))))))))))))))))))))))))).
theorem flt_arith2:
\forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1
\def
\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda
(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda
-(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1)
-(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight
-t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S
-O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
-(* COMMENTS
-Initial nodes: 115
-END *)
+(_: K).(\lambda (t2: T).(\lambda (_: nat).(let TMP_1 \def (cweight c1) in
+(let TMP_2 \def (tweight t1) in (let TMP_3 \def (plus TMP_1 TMP_2) in (let
+TMP_4 \def (cweight c2) in (let TMP_5 \def (S O) in (let TMP_6 \def (plus
+TMP_4 TMP_5) in (let TMP_7 \def (cweight c2) in (let TMP_8 \def (tweight t2)
+in (let TMP_9 \def (plus TMP_7 TMP_8) in (let TMP_10 \def (S O) in (let
+TMP_11 \def (plus TMP_9 TMP_10) in (let TMP_12 \def (cweight c2) in (let
+TMP_13 \def (cweight c2) in (let TMP_14 \def (tweight t2) in (let TMP_15 \def
+(plus TMP_13 TMP_14) in (let TMP_16 \def (S O) in (let TMP_17 \def (S O) in
+(let TMP_18 \def (cweight c2) in (let TMP_19 \def (tweight t2) in (let TMP_20
+\def (le_plus_l TMP_18 TMP_19) in (let TMP_21 \def (S O) in (let TMP_22 \def
+(le_n TMP_21) in (let TMP_23 \def (le_plus_plus TMP_12 TMP_15 TMP_16 TMP_17
+TMP_20 TMP_22) in (lt_le_trans TMP_3 TMP_6 TMP_11 H
+TMP_23))))))))))))))))))))))))))))))).
+
+theorem cle_flt_trans:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (c3: C).(\forall
+(u2: T).(\forall (u3: T).((flt c2 u2 c3 u3) \to (flt c1 u2 c3 u3)))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight
+c2))).(\lambda (c3: C).(\lambda (u2: T).(\lambda (u3: T).(\lambda (H0: (lt
+(plus (cweight c2) (tweight u2)) (plus (cweight c3) (tweight u3)))).(let
+TMP_1 \def (cweight c1) in (let TMP_2 \def (tweight u2) in (let TMP_3 \def
+(plus TMP_1 TMP_2) in (let TMP_4 \def (cweight c2) in (let TMP_5 \def
+(tweight u2) in (let TMP_6 \def (plus TMP_4 TMP_5) in (let TMP_7 \def
+(cweight c3) in (let TMP_8 \def (tweight u3) in (let TMP_9 \def (plus TMP_7
+TMP_8) in (let TMP_10 \def (cweight c1) in (let TMP_11 \def (cweight c2) in
+(let TMP_12 \def (tweight u2) in (let TMP_13 \def (tweight u2) in (let TMP_14
+\def (tweight u2) in (let TMP_15 \def (le_n TMP_14) in (let TMP_16 \def
+(le_plus_plus TMP_10 TMP_11 TMP_12 TMP_13 H TMP_15) in (le_lt_trans TMP_3
+TMP_6 TMP_9 TMP_16 H0))))))))))))))))))))))).
theorem flt_trans:
\forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1
\def
\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3:
-T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1
-t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))).
-(* COMMENTS
-Initial nodes: 63
-END *)
-
-theorem flt_wf__q_ind:
- \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C
-\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq
-nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall
-(t: T).(P c t))))
-\def
- let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
-(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
-(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c:
-C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c:
-C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
-(* COMMENTS
-Initial nodes: 85
-END *)
-
-theorem flt_wf_ind:
- \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2:
-T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
-\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
-\def
- let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
-(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
-(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2:
-T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
-\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda
-(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m:
-nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda
-(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1:
-nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq
-nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0
-(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2
-(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c
-t))))).
-(* COMMENTS
-Initial nodes: 211
-END *)
+T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(let TMP_1 \def
+(fweight c1 t1) in (let TMP_2 \def (fweight c2 t2) in (let TMP_3 \def
+(fweight c3 t3) in (lt_trans TMP_1 TMP_2 TMP_3 H H0))))))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/tlist/defs.ma".
+include "basic_1/tlist/defs.ma".
-include "Basic-1/s/defs.ma".
+include "basic_1/s/defs.ma".
-definition lref_map:
- ((nat \to nat)) \to (nat \to (T \to T))
-\def
- let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t
-with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
-(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u
-t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in
-lref_map.
+let rec lref_map (f: (nat \to nat)) (d: nat) (t: T) on t: T \def match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (let TMP_4 \def
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]) in
+(TLRef TMP_4)) | (THead k u t0) \Rightarrow (let TMP_1 \def (lref_map f d u)
+in (let TMP_2 \def (s k d) in (let TMP_3 \def (lref_map f TMP_2 t0) in (THead
+k TMP_1 TMP_3))))].
definition lift:
nat \to (nat \to (T \to T))
\def
- \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x:
-nat).(plus x h)) i t))).
+ \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(let TMP_1 \def (\lambda
+(x: nat).(plus x h)) in (lref_map TMP_1 i t)))).
-definition lifts:
- nat \to (nat \to (TList \to TList))
-\def
- let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with
-[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts
-h d ts0))]) in lifts.
+let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def match ts with
+[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (let TMP_1 \def (lift h d
+t) in (let TMP_2 \def (lifts h d ts0) in (TCons TMP_1 TMP_2)))].
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/lift/defs.ma".
-
-theorem lift_sort:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
-n)) (TSort n))))
-\def
- \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort
-n)))).
-(* COMMENTS
-Initial nodes: 13
-END *)
-
-theorem lift_lref_lt:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T
-(lift h d (TLRef n)) (TLRef n)))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n
-d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true
-\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T
-(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
-(* COMMENTS
-Initial nodes: 72
-END *)
-
-theorem lift_lref_ge:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T
-(lift h d (TLRef n)) (TLRef (plus n h))))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d
-n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true
-\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h))))
-(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false
-(le_bge d n H)))))).
-(* COMMENTS
-Initial nodes: 80
-END *)
-
-theorem lift_head:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d)
-t)))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
-(* COMMENTS
-Initial nodes: 37
-END *)
-
-theorem lift_bind:
- \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u)
-(lift h (S d) t)))))))
-\def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
-(* COMMENTS
-Initial nodes: 37
-END *)
-
-theorem lift_flat:
- \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
-(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u)
-(lift h d t)))))))
-\def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
-(* COMMENTS
-Initial nodes: 35
-END *)
+include "basic_1/lift/props.ma".
theorem lift_gen_sort:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T
(TSort n) (lift h d t)) \to (eq T t (TSort n))))))
\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind
-(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))
-(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort
-n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H:
-(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort
-n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0))
-(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let
-H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
-(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0)
-H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) (\lambda (_: (le d
-n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T
-(TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let H1 \def
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
-(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef
-(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2))))))) (\lambda
-(k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq
-T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d
-t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d
-(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda
-(t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1))
-(lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TSort n) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k
-t0 t1) (TSort n)) H3))))))))) t)))).
-(* COMMENTS
-Initial nodes: 613
-END *)
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(let
+TMP_2 \def (\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (let TMP_1
+\def (TSort n) in (eq T t0 TMP_1)))) in (let TMP_5 \def (\lambda (n0:
+nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort n0)))).(let TMP_3 \def
+(TSort n) in (let TMP_4 \def (TSort n0) in (sym_eq T TMP_3 TMP_4 H))))) in
+(let TMP_49 \def (\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d
+(TLRef n0)))).(let TMP_6 \def (TLRef n0) in (let TMP_7 \def (TSort n) in (let
+TMP_8 \def (eq T TMP_6 TMP_7) in (let TMP_27 \def (\lambda (_: (lt n0
+d)).(let TMP_9 \def (TLRef n0) in (let TMP_10 \def (lift h d TMP_9) in (let
+TMP_12 \def (\lambda (t0: T).(let TMP_11 \def (TSort n) in (eq T TMP_11 t0)))
+in (let TMP_13 \def (TLRef n0) in (let TMP_14 \def (TSort n) in (let TMP_15
+\def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) in (let TMP_16 \def
+(TLRef n0) in (let TMP_17 \def (lift h d TMP_16) in (let H1 \def (eq_ind T
+TMP_14 TMP_15 I TMP_17 H) in (let TMP_18 \def (lt n0 d) in (let TMP_19 \def
+(False_ind TMP_18 H1) in (let TMP_20 \def (lift_lref_lt n0 h d TMP_19) in
+(let H1 \def (eq_ind T TMP_10 TMP_12 H TMP_13 TMP_20) in (let TMP_21 \def
+(TSort n) in (let TMP_22 \def (\lambda (ee: T).(match ee with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) in (let TMP_23 \def (TLRef n0) in (let H2 \def (eq_ind T TMP_21
+TMP_22 I TMP_23 H1) in (let TMP_24 \def (TLRef n0) in (let TMP_25 \def (TSort
+n) in (let TMP_26 \def (eq T TMP_24 TMP_25) in (False_ind TMP_26
+H2)))))))))))))))))))))) in (let TMP_48 \def (\lambda (_: (le d n0)).(let
+TMP_28 \def (TLRef n0) in (let TMP_29 \def (lift h d TMP_28) in (let TMP_31
+\def (\lambda (t0: T).(let TMP_30 \def (TSort n) in (eq T TMP_30 t0))) in
+(let TMP_32 \def (plus n0 h) in (let TMP_33 \def (TLRef TMP_32) in (let
+TMP_34 \def (TSort n) in (let TMP_35 \def (\lambda (ee: T).(match ee with
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow False])) in (let TMP_36 \def (TLRef n0) in (let TMP_37 \def (lift
+h d TMP_36) in (let H1 \def (eq_ind T TMP_34 TMP_35 I TMP_37 H) in (let
+TMP_38 \def (le d n0) in (let TMP_39 \def (False_ind TMP_38 H1) in (let
+TMP_40 \def (lift_lref_ge n0 h d TMP_39) in (let H1 \def (eq_ind T TMP_29
+TMP_31 H TMP_33 TMP_40) in (let TMP_41 \def (TSort n) in (let TMP_42 \def
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) in (let TMP_43 \def
+(plus n0 h) in (let TMP_44 \def (TLRef TMP_43) in (let H2 \def (eq_ind T
+TMP_41 TMP_42 I TMP_44 H1) in (let TMP_45 \def (TLRef n0) in (let TMP_46 \def
+(TSort n) in (let TMP_47 \def (eq T TMP_45 TMP_46) in (False_ind TMP_47
+H2)))))))))))))))))))))))) in (lt_le_e n0 d TMP_8 TMP_27 TMP_48)))))))) in
+(let TMP_68 \def (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort
+n) (lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_:
+(((eq T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq
+T (TSort n) (lift h d (THead k t0 t1)))).(let TMP_50 \def (THead k t0 t1) in
+(let TMP_51 \def (lift h d TMP_50) in (let TMP_53 \def (\lambda (t2: T).(let
+TMP_52 \def (TSort n) in (eq T TMP_52 t2))) in (let TMP_54 \def (lift h d t0)
+in (let TMP_55 \def (s k d) in (let TMP_56 \def (lift h TMP_55 t1) in (let
+TMP_57 \def (THead k TMP_54 TMP_56) in (let TMP_58 \def (lift_head k t0 t1 h
+d) in (let H2 \def (eq_ind T TMP_51 TMP_53 H1 TMP_57 TMP_58) in (let TMP_59
+\def (TSort n) in (let TMP_60 \def (\lambda (ee: T).(match ee with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) in (let TMP_61 \def (lift h d t0) in (let TMP_62 \def (s k d) in
+(let TMP_63 \def (lift h TMP_62 t1) in (let TMP_64 \def (THead k TMP_61
+TMP_63) in (let H3 \def (eq_ind T TMP_59 TMP_60 I TMP_64 H2) in (let TMP_65
+\def (THead k t0 t1) in (let TMP_66 \def (TSort n) in (let TMP_67 \def (eq T
+TMP_65 TMP_66) in (False_ind TMP_67 H3)))))))))))))))))))))))))) in (T_ind
+TMP_2 TMP_5 TMP_49 TMP_68 t)))))))).
theorem lift_gen_lref:
\forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T
(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le
(plus d h) i) (eq T t (TLRef (minus i h)))))))))
\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h:
+ \lambda (t: T).(let TMP_11 \def (\lambda (t0: T).(\forall (d: nat).(\forall
+(h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (let TMP_1
+\def (lt i d) in (let TMP_2 \def (TLRef i) in (let TMP_3 \def (eq T t0 TMP_2)
+in (let TMP_4 \def (land TMP_1 TMP_3) in (let TMP_5 \def (plus d h) in (let
+TMP_6 \def (le TMP_5 i) in (let TMP_7 \def (minus i h) in (let TMP_8 \def
+(TLRef TMP_7) in (let TMP_9 \def (eq T t0 TMP_8) in (let TMP_10 \def (land
+TMP_6 TMP_9) in (or TMP_4 TMP_10)))))))))))))))) in (let TMP_34 \def (\lambda
+(n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H:
+(eq T (TLRef i) (lift h d (TSort n)))).(let TMP_12 \def (TSort n) in (let
+TMP_13 \def (lift h d TMP_12) in (let TMP_15 \def (\lambda (t0: T).(let
+TMP_14 \def (TLRef i) in (eq T TMP_14 t0))) in (let TMP_16 \def (TSort n) in
+(let TMP_17 \def (lift_sort n h d) in (let H0 \def (eq_ind T TMP_13 TMP_15 H
+TMP_16 TMP_17) in (let TMP_18 \def (TLRef i) in (let TMP_19 \def (\lambda
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow
+True | (THead _ _ _) \Rightarrow False])) in (let TMP_20 \def (TSort n) in
+(let H1 \def (eq_ind T TMP_18 TMP_19 I TMP_20 H0) in (let TMP_21 \def (lt i
+d) in (let TMP_22 \def (TSort n) in (let TMP_23 \def (TLRef i) in (let TMP_24
+\def (eq T TMP_22 TMP_23) in (let TMP_25 \def (land TMP_21 TMP_24) in (let
+TMP_26 \def (plus d h) in (let TMP_27 \def (le TMP_26 i) in (let TMP_28 \def
+(TSort n) in (let TMP_29 \def (minus i h) in (let TMP_30 \def (TLRef TMP_29)
+in (let TMP_31 \def (eq T TMP_28 TMP_30) in (let TMP_32 \def (land TMP_27
+TMP_31) in (let TMP_33 \def (or TMP_25 TMP_32) in (False_ind TMP_33
+H1))))))))))))))))))))))))))))) in (let TMP_162 \def (\lambda (n:
+nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H: (eq T
+(TLRef i) (lift h d (TLRef n)))).(let TMP_35 \def (lt i d) in (let TMP_36
+\def (TLRef n) in (let TMP_37 \def (TLRef i) in (let TMP_38 \def (eq T TMP_36
+TMP_37) in (let TMP_39 \def (land TMP_35 TMP_38) in (let TMP_40 \def (plus d
+h) in (let TMP_41 \def (le TMP_40 i) in (let TMP_42 \def (TLRef n) in (let
+TMP_43 \def (minus i h) in (let TMP_44 \def (TLRef TMP_43) in (let TMP_45
+\def (eq T TMP_42 TMP_44) in (let TMP_46 \def (land TMP_41 TMP_45) in (let
+TMP_47 \def (or TMP_39 TMP_46) in (let TMP_90 \def (\lambda (H0: (lt n
+d)).(let TMP_48 \def (TLRef n) in (let TMP_49 \def (lift h d TMP_48) in (let
+TMP_51 \def (\lambda (t0: T).(let TMP_50 \def (TLRef i) in (eq T TMP_50 t0)))
+in (let TMP_52 \def (TLRef n) in (let TMP_53 \def (lift_lref_lt n h d H0) in
+(let H1 \def (eq_ind T TMP_49 TMP_51 H TMP_52 TMP_53) in (let TMP_54 \def
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow i | (TLRef n0)
+\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) in (let TMP_55 \def (TLRef i)
+in (let TMP_56 \def (TLRef n) in (let H2 \def (f_equal T nat TMP_54 TMP_55
+TMP_56 H1) in (let TMP_69 \def (\lambda (n0: nat).(let TMP_57 \def (lt n0 d)
+in (let TMP_58 \def (TLRef n) in (let TMP_59 \def (TLRef n0) in (let TMP_60
+\def (eq T TMP_58 TMP_59) in (let TMP_61 \def (land TMP_57 TMP_60) in (let
+TMP_62 \def (plus d h) in (let TMP_63 \def (le TMP_62 n0) in (let TMP_64 \def
+(TLRef n) in (let TMP_65 \def (minus n0 h) in (let TMP_66 \def (TLRef TMP_65)
+in (let TMP_67 \def (eq T TMP_64 TMP_66) in (let TMP_68 \def (land TMP_63
+TMP_67) in (or TMP_61 TMP_68)))))))))))))) in (let TMP_70 \def (lt n d) in
+(let TMP_71 \def (TLRef n) in (let TMP_72 \def (TLRef n) in (let TMP_73 \def
+(eq T TMP_71 TMP_72) in (let TMP_74 \def (land TMP_70 TMP_73) in (let TMP_75
+\def (plus d h) in (let TMP_76 \def (le TMP_75 n) in (let TMP_77 \def (TLRef
+n) in (let TMP_78 \def (minus n h) in (let TMP_79 \def (TLRef TMP_78) in (let
+TMP_80 \def (eq T TMP_77 TMP_79) in (let TMP_81 \def (land TMP_76 TMP_80) in
+(let TMP_82 \def (lt n d) in (let TMP_83 \def (TLRef n) in (let TMP_84 \def
+(TLRef n) in (let TMP_85 \def (eq T TMP_83 TMP_84) in (let TMP_86 \def (TLRef
+n) in (let TMP_87 \def (refl_equal T TMP_86) in (let TMP_88 \def (conj TMP_82
+TMP_85 H0 TMP_87) in (let TMP_89 \def (or_introl TMP_74 TMP_81 TMP_88) in
+(eq_ind_r nat n TMP_69 TMP_89 i H2))))))))))))))))))))))))))))))))) in (let
+TMP_161 \def (\lambda (H0: (le d n)).(let TMP_91 \def (TLRef n) in (let
+TMP_92 \def (lift h d TMP_91) in (let TMP_94 \def (\lambda (t0: T).(let
+TMP_93 \def (TLRef i) in (eq T TMP_93 t0))) in (let TMP_95 \def (plus n h) in
+(let TMP_96 \def (TLRef TMP_95) in (let TMP_97 \def (lift_lref_ge n h d H0)
+in (let H1 \def (eq_ind T TMP_92 TMP_94 H TMP_96 TMP_97) in (let TMP_98 \def
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow i | (TLRef n0)
+\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) in (let TMP_99 \def (TLRef i)
+in (let TMP_100 \def (plus n h) in (let TMP_101 \def (TLRef TMP_100) in (let
+H2 \def (f_equal T nat TMP_98 TMP_99 TMP_101 H1) in (let TMP_102 \def (plus n
+h) in (let TMP_115 \def (\lambda (n0: nat).(let TMP_103 \def (lt n0 d) in
+(let TMP_104 \def (TLRef n) in (let TMP_105 \def (TLRef n0) in (let TMP_106
+\def (eq T TMP_104 TMP_105) in (let TMP_107 \def (land TMP_103 TMP_106) in
+(let TMP_108 \def (plus d h) in (let TMP_109 \def (le TMP_108 n0) in (let
+TMP_110 \def (TLRef n) in (let TMP_111 \def (minus n0 h) in (let TMP_112 \def
+(TLRef TMP_111) in (let TMP_113 \def (eq T TMP_110 TMP_112) in (let TMP_114
+\def (land TMP_109 TMP_113) in (or TMP_107 TMP_114)))))))))))))) in (let
+TMP_130 \def (\lambda (n0: nat).(let TMP_116 \def (plus n h) in (let TMP_117
+\def (lt TMP_116 d) in (let TMP_118 \def (TLRef n) in (let TMP_119 \def (plus
+n h) in (let TMP_120 \def (TLRef TMP_119) in (let TMP_121 \def (eq T TMP_118
+TMP_120) in (let TMP_122 \def (land TMP_117 TMP_121) in (let TMP_123 \def
+(plus d h) in (let TMP_124 \def (plus n h) in (let TMP_125 \def (le TMP_123
+TMP_124) in (let TMP_126 \def (TLRef n) in (let TMP_127 \def (TLRef n0) in
+(let TMP_128 \def (eq T TMP_126 TMP_127) in (let TMP_129 \def (land TMP_125
+TMP_128) in (or TMP_122 TMP_129)))))))))))))))) in (let TMP_131 \def (plus n
+h) in (let TMP_132 \def (lt TMP_131 d) in (let TMP_133 \def (TLRef n) in (let
+TMP_134 \def (plus n h) in (let TMP_135 \def (TLRef TMP_134) in (let TMP_136
+\def (eq T TMP_133 TMP_135) in (let TMP_137 \def (land TMP_132 TMP_136) in
+(let TMP_138 \def (plus d h) in (let TMP_139 \def (plus n h) in (let TMP_140
+\def (le TMP_138 TMP_139) in (let TMP_141 \def (TLRef n) in (let TMP_142 \def
+(TLRef n) in (let TMP_143 \def (eq T TMP_141 TMP_142) in (let TMP_144 \def
+(land TMP_140 TMP_143) in (let TMP_145 \def (plus d h) in (let TMP_146 \def
+(plus n h) in (let TMP_147 \def (le TMP_145 TMP_146) in (let TMP_148 \def
+(TLRef n) in (let TMP_149 \def (TLRef n) in (let TMP_150 \def (eq T TMP_148
+TMP_149) in (let TMP_151 \def (le_n h) in (let TMP_152 \def (le_plus_plus d n
+h h H0 TMP_151) in (let TMP_153 \def (TLRef n) in (let TMP_154 \def
+(refl_equal T TMP_153) in (let TMP_155 \def (conj TMP_147 TMP_150 TMP_152
+TMP_154) in (let TMP_156 \def (or_intror TMP_137 TMP_144 TMP_155) in (let
+TMP_157 \def (plus n h) in (let TMP_158 \def (minus TMP_157 h) in (let
+TMP_159 \def (minus_plus_r n h) in (let TMP_160 \def (eq_ind_r nat n TMP_130
+TMP_156 TMP_158 TMP_159) in (eq_ind_r nat TMP_102 TMP_115 TMP_160 i
+H2))))))))))))))))))))))))))))))))))))))))))))))) in (lt_le_e n d TMP_47
+TMP_90 TMP_161))))))))))))))))))))) in (let TMP_191 \def (\lambda (k:
+K).(\lambda (t0: T).(\lambda (_: ((\forall (d: nat).(\forall (h:
nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i
-h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda
-(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def
-(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort
-n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee:
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
-(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i)))
-(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1))))))))
-(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
-nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or
-(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T
-(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind
-T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n)
-(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i |
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef
-n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef
-n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0
-h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus
-d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n)
-(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d
-n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef
-i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
-[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
-\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h)
-(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le
-(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n
-(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n
-h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0)))))
-(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land
-(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h)
-(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h))
-(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i
-H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d:
-nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to
-(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0
-(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d:
-nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to
-(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1
-(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
-nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def
-(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1
-(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let
-H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d)
-t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i)))
-(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h)))))
-H3)))))))))))) t).
-(* COMMENTS
-Initial nodes: 1221
-END *)
+h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: nat).(\forall (h:
+nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to (or (land (lt i d)
+(eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 (TLRef (minus i
+h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda
+(H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let TMP_163 \def (THead k
+t0 t1) in (let TMP_164 \def (lift h d TMP_163) in (let TMP_166 \def (\lambda
+(t2: T).(let TMP_165 \def (TLRef i) in (eq T TMP_165 t2))) in (let TMP_167
+\def (lift h d t0) in (let TMP_168 \def (s k d) in (let TMP_169 \def (lift h
+TMP_168 t1) in (let TMP_170 \def (THead k TMP_167 TMP_169) in (let TMP_171
+\def (lift_head k t0 t1 h d) in (let H2 \def (eq_ind T TMP_164 TMP_166 H1
+TMP_170 TMP_171) in (let TMP_172 \def (TLRef i) in (let TMP_173 \def (\lambda
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow
+True | (THead _ _ _) \Rightarrow False])) in (let TMP_174 \def (lift h d t0)
+in (let TMP_175 \def (s k d) in (let TMP_176 \def (lift h TMP_175 t1) in (let
+TMP_177 \def (THead k TMP_174 TMP_176) in (let H3 \def (eq_ind T TMP_172
+TMP_173 I TMP_177 H2) in (let TMP_178 \def (lt i d) in (let TMP_179 \def
+(THead k t0 t1) in (let TMP_180 \def (TLRef i) in (let TMP_181 \def (eq T
+TMP_179 TMP_180) in (let TMP_182 \def (land TMP_178 TMP_181) in (let TMP_183
+\def (plus d h) in (let TMP_184 \def (le TMP_183 i) in (let TMP_185 \def
+(THead k t0 t1) in (let TMP_186 \def (minus i h) in (let TMP_187 \def (TLRef
+TMP_186) in (let TMP_188 \def (eq T TMP_185 TMP_187) in (let TMP_189 \def
+(land TMP_184 TMP_188) in (let TMP_190 \def (or TMP_182 TMP_189) in
+(False_ind TMP_190 H3))))))))))))))))))))))))))))))))))))))) in (T_ind TMP_11
+TMP_34 TMP_162 TMP_191 t))))).
theorem lift_gen_lref_lt:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall
\def
\lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n
d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x
-\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d)
-(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq
-T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind
-(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n
-d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2:
-(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d
-h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus
-d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef
-(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq
-T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S
-n) d h H))) t H4))) H2)) H1)))))))).
-(* COMMENTS
-Initial nodes: 363
-END *)
+\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (let TMP_1 \def (lt n
+d) in (let TMP_2 \def (TLRef n) in (let TMP_3 \def (eq T t TMP_2) in (let
+TMP_4 \def (land TMP_1 TMP_3) in (let TMP_5 \def (plus d h) in (let TMP_6
+\def (le TMP_5 n) in (let TMP_7 \def (minus n h) in (let TMP_8 \def (TLRef
+TMP_7) in (let TMP_9 \def (eq T t TMP_8) in (let TMP_10 \def (land TMP_6
+TMP_9) in (let TMP_11 \def (TLRef n) in (let TMP_12 \def (eq T t TMP_11) in
+(let TMP_24 \def (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(let
+TMP_13 \def (lt n d) in (let TMP_14 \def (TLRef n) in (let TMP_15 \def (eq T
+t TMP_14) in (let TMP_16 \def (TLRef n) in (let TMP_17 \def (eq T t TMP_16)
+in (let TMP_23 \def (\lambda (_: (lt n d)).(\lambda (H4: (eq T t (TLRef
+n))).(let TMP_18 \def (TLRef n) in (let TMP_20 \def (\lambda (t0: T).(let
+TMP_19 \def (TLRef n) in (eq T t0 TMP_19))) in (let TMP_21 \def (TLRef n) in
+(let TMP_22 \def (refl_equal T TMP_21) in (eq_ind_r T TMP_18 TMP_20 TMP_22 t
+H4))))))) in (land_ind TMP_13 TMP_15 TMP_17 TMP_23 H2)))))))) in (let TMP_47
+\def (\lambda (H2: (land (le (plus d h) n) (eq T t (TLRef (minus n
+h))))).(let TMP_25 \def (plus d h) in (let TMP_26 \def (le TMP_25 n) in (let
+TMP_27 \def (minus n h) in (let TMP_28 \def (TLRef TMP_27) in (let TMP_29
+\def (eq T t TMP_28) in (let TMP_30 \def (TLRef n) in (let TMP_31 \def (eq T
+t TMP_30) in (let TMP_46 \def (\lambda (H3: (le (plus d h) n)).(\lambda (H4:
+(eq T t (TLRef (minus n h)))).(let TMP_32 \def (minus n h) in (let TMP_33
+\def (TLRef TMP_32) in (let TMP_35 \def (\lambda (t0: T).(let TMP_34 \def
+(TLRef n) in (eq T t0 TMP_34))) in (let TMP_36 \def (plus d h) in (let TMP_37
+\def (minus n h) in (let TMP_38 \def (TLRef TMP_37) in (let TMP_39 \def
+(TLRef n) in (let TMP_40 \def (eq T TMP_38 TMP_39) in (let TMP_41 \def (plus
+d h) in (let TMP_42 \def (S n) in (let TMP_43 \def (le_plus_trans TMP_42 d h
+H) in (let TMP_44 \def (lt_le_S n TMP_41 TMP_43) in (let TMP_45 \def
+(le_false TMP_36 n TMP_40 H3 TMP_44) in (eq_ind_r T TMP_33 TMP_35 TMP_45 t
+H4)))))))))))))))) in (land_ind TMP_26 TMP_29 TMP_31 TMP_46 H2)))))))))) in
+(or_ind TMP_4 TMP_10 TMP_12 TMP_24 TMP_47 H1)))))))))))))))))))))).
theorem lift_gen_lref_false:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
\lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T
(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d
-h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n)))
-(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land
-(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P
-(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H
-H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n
-h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda
-(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false
-(plus d h) n P H4 H0))) H3)) H2)))))))))).
-(* COMMENTS
-Initial nodes: 269
-END *)
+h n H1) in (let H2 \def H_x in (let TMP_1 \def (lt n d) in (let TMP_2 \def
+(TLRef n) in (let TMP_3 \def (eq T t TMP_2) in (let TMP_4 \def (land TMP_1
+TMP_3) in (let TMP_5 \def (plus d h) in (let TMP_6 \def (le TMP_5 n) in (let
+TMP_7 \def (minus n h) in (let TMP_8 \def (TLRef TMP_7) in (let TMP_9 \def
+(eq T t TMP_8) in (let TMP_10 \def (land TMP_6 TMP_9) in (let TMP_15 \def
+(\lambda (H3: (land (lt n d) (eq T t (TLRef n)))).(let TMP_11 \def (lt n d)
+in (let TMP_12 \def (TLRef n) in (let TMP_13 \def (eq T t TMP_12) in (let
+TMP_14 \def (\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef
+n))).(le_false d n P H H4))) in (land_ind TMP_11 TMP_13 P TMP_14 H3)))))) in
+(let TMP_23 \def (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus
+n h))))).(let TMP_16 \def (plus d h) in (let TMP_17 \def (le TMP_16 n) in
+(let TMP_18 \def (minus n h) in (let TMP_19 \def (TLRef TMP_18) in (let
+TMP_20 \def (eq T t TMP_19) in (let TMP_22 \def (\lambda (H4: (le (plus d h)
+n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(let TMP_21 \def (plus d h) in
+(le_false TMP_21 n P H4 H0)))) in (land_ind TMP_17 TMP_20 P TMP_22 H3))))))))
+in (or_ind TMP_4 TMP_10 P TMP_15 TMP_23 H2)))))))))))))))))))))).
theorem lift_gen_lref_ge:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
\def
\lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d
-t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in
-(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus
-d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n))
-(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind
-(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda
-(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r
-T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq
-T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d
-(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2))
-(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
-h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
-h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda
-(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n
-h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus
-(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
-(* COMMENTS
-Initial nodes: 473
-END *)
+t))).(let TMP_1 \def (plus n h) in (let H_x \def (lift_gen_lref t d h TMP_1
+H0) in (let H1 \def H_x in (let TMP_2 \def (plus n h) in (let TMP_3 \def (lt
+TMP_2 d) in (let TMP_4 \def (plus n h) in (let TMP_5 \def (TLRef TMP_4) in
+(let TMP_6 \def (eq T t TMP_5) in (let TMP_7 \def (land TMP_3 TMP_6) in (let
+TMP_8 \def (plus d h) in (let TMP_9 \def (plus n h) in (let TMP_10 \def (le
+TMP_8 TMP_9) in (let TMP_11 \def (plus n h) in (let TMP_12 \def (minus TMP_11
+h) in (let TMP_13 \def (TLRef TMP_12) in (let TMP_14 \def (eq T t TMP_13) in
+(let TMP_15 \def (land TMP_10 TMP_14) in (let TMP_16 \def (TLRef n) in (let
+TMP_17 \def (eq T t TMP_16) in (let TMP_41 \def (\lambda (H2: (land (lt (plus
+n h) d) (eq T t (TLRef (plus n h))))).(let TMP_18 \def (plus n h) in (let
+TMP_19 \def (lt TMP_18 d) in (let TMP_20 \def (plus n h) in (let TMP_21 \def
+(TLRef TMP_20) in (let TMP_22 \def (eq T t TMP_21) in (let TMP_23 \def (TLRef
+n) in (let TMP_24 \def (eq T t TMP_23) in (let TMP_40 \def (\lambda (H3: (lt
+(plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(let TMP_25 \def
+(plus n h) in (let TMP_26 \def (TLRef TMP_25) in (let TMP_28 \def (\lambda
+(t0: T).(let TMP_27 \def (TLRef n) in (eq T t0 TMP_27))) in (let TMP_29 \def
+(plus n h) in (let TMP_30 \def (TLRef TMP_29) in (let TMP_31 \def (TLRef n)
+in (let TMP_32 \def (eq T TMP_30 TMP_31) in (let TMP_33 \def (plus n h) in
+(let TMP_34 \def (plus d h) in (let TMP_35 \def (le_plus_l d h) in (let
+TMP_36 \def (lt_le_trans TMP_33 d TMP_34 H3 TMP_35) in (let TMP_37 \def
+(simpl_lt_plus_r h n d TMP_36) in (let TMP_38 \def (lt_le_S n d TMP_37) in
+(let TMP_39 \def (le_false d n TMP_32 H TMP_38) in (eq_ind_r T TMP_26 TMP_28
+TMP_39 t H4))))))))))))))))) in (land_ind TMP_19 TMP_22 TMP_24 TMP_40
+H2)))))))))) in (let TMP_61 \def (\lambda (H2: (land (le (plus d h) (plus n
+h)) (eq T t (TLRef (minus (plus n h) h))))).(let TMP_42 \def (plus d h) in
+(let TMP_43 \def (plus n h) in (let TMP_44 \def (le TMP_42 TMP_43) in (let
+TMP_45 \def (plus n h) in (let TMP_46 \def (minus TMP_45 h) in (let TMP_47
+\def (TLRef TMP_46) in (let TMP_48 \def (eq T t TMP_47) in (let TMP_49 \def
+(TLRef n) in (let TMP_50 \def (eq T t TMP_49) in (let TMP_60 \def (\lambda
+(_: (le (plus d h) (plus n h))).(\lambda (H4: (eq T t (TLRef (minus (plus n
+h) h)))).(let TMP_51 \def (plus n h) in (let TMP_52 \def (minus TMP_51 h) in
+(let TMP_53 \def (TLRef TMP_52) in (let TMP_55 \def (\lambda (t0: T).(let
+TMP_54 \def (TLRef n) in (eq T t0 TMP_54))) in (let TMP_56 \def (plus n h) in
+(let TMP_57 \def (minus TMP_56 h) in (let TMP_58 \def (minus_plus_r n h) in
+(let TMP_59 \def (f_equal nat T TLRef TMP_57 n TMP_58) in (eq_ind_r T TMP_53
+TMP_55 TMP_59 t H4))))))))))) in (land_ind TMP_44 TMP_48 TMP_50 TMP_60
+H2)))))))))))) in (or_ind TMP_7 TMP_15 TMP_17 TMP_41 TMP_61
+H1))))))))))))))))))))))))))).
theorem lift_gen_head:
\forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
T).(eq T t (lift h (s k d) z)))))))))))
\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
-(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
-(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n:
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(let TMP_8
+\def (\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u
+t) (lift h d t0)) \to (let TMP_2 \def (\lambda (y: T).(\lambda (z: T).(let
+TMP_1 \def (THead k y z) in (eq T t0 TMP_1)))) in (let TMP_4 \def (\lambda
+(y: T).(\lambda (_: T).(let TMP_3 \def (lift h d y) in (eq T u TMP_3)))) in
+(let TMP_7 \def (\lambda (_: T).(\lambda (z: T).(let TMP_5 \def (s k d) in
+(let TMP_6 \def (lift h TMP_5 z) in (eq T t TMP_6))))) in (ex3_2 T T TMP_2
+TMP_4 TMP_7)))))))) in (let TMP_27 \def (\lambda (n: nat).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TSort
+n)))).(let TMP_9 \def (TSort n) in (let TMP_10 \def (lift h d TMP_9) in (let
+TMP_12 \def (\lambda (t0: T).(let TMP_11 \def (THead k u t) in (eq T TMP_11
+t0))) in (let TMP_13 \def (TSort n) in (let TMP_14 \def (lift_sort n h d) in
+(let H0 \def (eq_ind T TMP_10 TMP_12 H TMP_13 TMP_14) in (let TMP_15 \def
+(THead k u t) in (let TMP_16 \def (\lambda (ee: T).(match ee with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) in (let TMP_17 \def (TSort n) in (let H1 \def (eq_ind T TMP_15 TMP_16
+I TMP_17 H0) in (let TMP_20 \def (\lambda (y: T).(\lambda (z: T).(let TMP_18
+\def (TSort n) in (let TMP_19 \def (THead k y z) in (eq T TMP_18 TMP_19)))))
+in (let TMP_22 \def (\lambda (y: T).(\lambda (_: T).(let TMP_21 \def (lift h
+d y) in (eq T u TMP_21)))) in (let TMP_25 \def (\lambda (_: T).(\lambda (z:
+T).(let TMP_23 \def (s k d) in (let TMP_24 \def (lift h TMP_23 z) in (eq T t
+TMP_24))))) in (let TMP_26 \def (ex3_2 T T TMP_20 TMP_22 TMP_25) in
+(False_ind TMP_26 H1))))))))))))))))))) in (let TMP_77 \def (\lambda (n:
nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t)
-(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda
-(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1
-\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef
-n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
-(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
-(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0:
-(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T
-(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def
-(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H2)))) (\lambda (H0: (le d n)).(let H1 \def
-(eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H
-(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (eq_ind T (THead
-k u t) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0:
+(lift h d (TLRef n)))).(let TMP_30 \def (\lambda (y: T).(\lambda (z: T).(let
+TMP_28 \def (TLRef n) in (let TMP_29 \def (THead k y z) in (eq T TMP_28
+TMP_29))))) in (let TMP_32 \def (\lambda (y: T).(\lambda (_: T).(let TMP_31
+\def (lift h d y) in (eq T u TMP_31)))) in (let TMP_35 \def (\lambda (_:
+T).(\lambda (z: T).(let TMP_33 \def (s k d) in (let TMP_34 \def (lift h
+TMP_33 z) in (eq T t TMP_34))))) in (let TMP_36 \def (ex3_2 T T TMP_30 TMP_32
+TMP_35) in (let TMP_55 \def (\lambda (H0: (lt n d)).(let TMP_37 \def (TLRef
+n) in (let TMP_38 \def (lift h d TMP_37) in (let TMP_40 \def (\lambda (t0:
+T).(let TMP_39 \def (THead k u t) in (eq T TMP_39 t0))) in (let TMP_41 \def
+(TLRef n) in (let TMP_42 \def (lift_lref_lt n h d H0) in (let H1 \def (eq_ind
+T TMP_38 TMP_40 H TMP_41 TMP_42) in (let TMP_43 \def (THead k u t) in (let
+TMP_44 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in (let
+TMP_45 \def (TLRef n) in (let H2 \def (eq_ind T TMP_43 TMP_44 I TMP_45 H1) in
+(let TMP_48 \def (\lambda (y: T).(\lambda (z: T).(let TMP_46 \def (TLRef n)
+in (let TMP_47 \def (THead k y z) in (eq T TMP_46 TMP_47))))) in (let TMP_50
+\def (\lambda (y: T).(\lambda (_: T).(let TMP_49 \def (lift h d y) in (eq T u
+TMP_49)))) in (let TMP_53 \def (\lambda (_: T).(\lambda (z: T).(let TMP_51
+\def (s k d) in (let TMP_52 \def (lift h TMP_51 z) in (eq T t TMP_52))))) in
+(let TMP_54 \def (ex3_2 T T TMP_48 TMP_50 TMP_53) in (False_ind TMP_54
+H2)))))))))))))))) in (let TMP_76 \def (\lambda (H0: (le d n)).(let TMP_56
+\def (TLRef n) in (let TMP_57 \def (lift h d TMP_56) in (let TMP_59 \def
+(\lambda (t0: T).(let TMP_58 \def (THead k u t) in (eq T TMP_58 t0))) in (let
+TMP_60 \def (plus n h) in (let TMP_61 \def (TLRef TMP_60) in (let TMP_62 \def
+(lift_lref_ge n h d H0) in (let H1 \def (eq_ind T TMP_57 TMP_59 H TMP_61
+TMP_62) in (let TMP_63 \def (THead k u t) in (let TMP_64 \def (\lambda (ee:
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) in (let TMP_65 \def (plus n h) in (let
+TMP_66 \def (TLRef TMP_65) in (let H2 \def (eq_ind T TMP_63 TMP_64 I TMP_66
+H1) in (let TMP_69 \def (\lambda (y: T).(\lambda (z: T).(let TMP_67 \def
+(TLRef n) in (let TMP_68 \def (THead k y z) in (eq T TMP_67 TMP_68))))) in
+(let TMP_71 \def (\lambda (y: T).(\lambda (_: T).(let TMP_70 \def (lift h d
+y) in (eq T u TMP_70)))) in (let TMP_74 \def (\lambda (_: T).(\lambda (z:
+T).(let TMP_72 \def (s k d) in (let TMP_73 \def (lift h TMP_72 z) in (eq T t
+TMP_73))))) in (let TMP_75 \def (ex3_2 T T TMP_69 TMP_71 TMP_74) in
+(False_ind TMP_75 H2)))))))))))))))))) in (lt_le_e n d TMP_36 TMP_55
+TMP_76))))))))))) in (let TMP_205 \def (\lambda (k0: K).(\lambda (t0:
T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h:
nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0
-t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2:
-T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1))
-(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0
-(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2]))
-(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2)
-\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1))
-H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let
-H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k
-H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z:
-T).(eq T (THead k1 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s
-k d) z)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0:
-nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let
-H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0:
-nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift
-h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k
-d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
-(THead k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k d)
-z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
-nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
-t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10
-(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
+t0 t1)))).(let TMP_78 \def (THead k0 t0 t1) in (let TMP_79 \def (lift h d
+TMP_78) in (let TMP_81 \def (\lambda (t2: T).(let TMP_80 \def (THead k u t)
+in (eq T TMP_80 t2))) in (let TMP_82 \def (lift h d t0) in (let TMP_83 \def
+(s k0 d) in (let TMP_84 \def (lift h TMP_83 t1) in (let TMP_85 \def (THead k0
+TMP_82 TMP_84) in (let TMP_86 \def (lift_head k0 t0 t1 h d) in (let H2 \def
+(eq_ind T TMP_79 TMP_81 H1 TMP_85 TMP_86) in (let TMP_87 \def (\lambda (e:
+T).(match e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead
+k1 _ _) \Rightarrow k1])) in (let TMP_88 \def (THead k u t) in (let TMP_89
+\def (lift h d t0) in (let TMP_90 \def (s k0 d) in (let TMP_91 \def (lift h
+TMP_90 t1) in (let TMP_92 \def (THead k0 TMP_89 TMP_91) in (let H3 \def
+(f_equal T K TMP_87 TMP_88 TMP_92 H2) in (let TMP_93 \def (\lambda (e:
+T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead
+_ t2 _) \Rightarrow t2])) in (let TMP_94 \def (THead k u t) in (let TMP_95
+\def (lift h d t0) in (let TMP_96 \def (s k0 d) in (let TMP_97 \def (lift h
+TMP_96 t1) in (let TMP_98 \def (THead k0 TMP_95 TMP_97) in (let H4 \def
+(f_equal T T TMP_93 TMP_94 TMP_98 H2) in (let TMP_99 \def (\lambda (e:
+T).(match e with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead
+_ _ t2) \Rightarrow t2])) in (let TMP_100 \def (THead k u t) in (let TMP_101
+\def (lift h d t0) in (let TMP_102 \def (s k0 d) in (let TMP_103 \def (lift h
+TMP_102 t1) in (let TMP_104 \def (THead k0 TMP_101 TMP_103) in (let H5 \def
+(f_equal T T TMP_99 TMP_100 TMP_104 H2) in (let TMP_203 \def (\lambda (H6:
+(eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let TMP_107 \def (\lambda
+(k1: K).(let TMP_105 \def (s k1 d) in (let TMP_106 \def (lift h TMP_105 t1)
+in (eq T t TMP_106)))) in (let H8 \def (eq_ind_r K k0 TMP_107 H5 k H7) in
+(let TMP_116 \def (\lambda (k1: K).(let TMP_110 \def (\lambda (y: T).(\lambda
+(z: T).(let TMP_108 \def (THead k1 t0 t1) in (let TMP_109 \def (THead k y z)
+in (eq T TMP_108 TMP_109))))) in (let TMP_112 \def (\lambda (y: T).(\lambda
+(_: T).(let TMP_111 \def (lift h d y) in (eq T u TMP_111)))) in (let TMP_115
+\def (\lambda (_: T).(\lambda (z: T).(let TMP_113 \def (s k d) in (let
+TMP_114 \def (lift h TMP_113 z) in (eq T t TMP_114))))) in (ex3_2 T T TMP_110
+TMP_112 TMP_115))))) in (let TMP_124 \def (\lambda (t2: T).(\forall (h0:
+nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (let
+TMP_118 \def (\lambda (y: T).(\lambda (z: T).(let TMP_117 \def (THead k y z)
+in (eq T t1 TMP_117)))) in (let TMP_120 \def (\lambda (y: T).(\lambda (_:
+T).(let TMP_119 \def (lift h0 d0 y) in (eq T u TMP_119)))) in (let TMP_123
+\def (\lambda (_: T).(\lambda (z: T).(let TMP_121 \def (s k d0) in (let
+TMP_122 \def (lift h0 TMP_121 z) in (eq T t2 TMP_122))))) in (ex3_2 T T
+TMP_118 TMP_120 TMP_123)))))))) in (let TMP_125 \def (s k d) in (let TMP_126
+\def (lift h TMP_125 t1) in (let H9 \def (eq_ind T t TMP_124 H0 TMP_126 H8)
+in (let TMP_134 \def (\lambda (t2: T).(\forall (h0: nat).(\forall (d0:
+nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (let TMP_128 \def (\lambda
+(y: T).(\lambda (z: T).(let TMP_127 \def (THead k y z) in (eq T t0
+TMP_127)))) in (let TMP_130 \def (\lambda (y: T).(\lambda (_: T).(let TMP_129
+\def (lift h0 d0 y) in (eq T u TMP_129)))) in (let TMP_133 \def (\lambda (_:
+T).(\lambda (z: T).(let TMP_131 \def (s k d0) in (let TMP_132 \def (lift h0
+TMP_131 z) in (eq T t2 TMP_132))))) in (ex3_2 T T TMP_128 TMP_130
+TMP_133)))))))) in (let TMP_135 \def (s k d) in (let TMP_136 \def (lift h
+TMP_135 t1) in (let H10 \def (eq_ind T t TMP_134 H TMP_136 H8) in (let
+TMP_137 \def (s k d) in (let TMP_138 \def (lift h TMP_137 t1) in (let TMP_147
+\def (\lambda (t2: T).(let TMP_141 \def (\lambda (y: T).(\lambda (z: T).(let
+TMP_139 \def (THead k t0 t1) in (let TMP_140 \def (THead k y z) in (eq T
+TMP_139 TMP_140))))) in (let TMP_143 \def (\lambda (y: T).(\lambda (_:
+T).(let TMP_142 \def (lift h d y) in (eq T u TMP_142)))) in (let TMP_146 \def
+(\lambda (_: T).(\lambda (z: T).(let TMP_144 \def (s k d) in (let TMP_145
+\def (lift h TMP_144 z) in (eq T t2 TMP_145))))) in (ex3_2 T T TMP_141
+TMP_143 TMP_146))))) in (let TMP_157 \def (\lambda (t2: T).(\forall (h0:
nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
-t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9
-(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z))))))
-(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y))))
-(\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d)
-z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0))
-(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4))
-H3))))))))))) x)))).
-(* COMMENTS
-Initial nodes: 2083
-END *)
+t0)) \to (let TMP_149 \def (\lambda (y: T).(\lambda (z: T).(let TMP_148 \def
+(THead k y z) in (eq T t0 TMP_148)))) in (let TMP_151 \def (\lambda (y:
+T).(\lambda (_: T).(let TMP_150 \def (lift h0 d0 y) in (eq T t2 TMP_150))))
+in (let TMP_156 \def (\lambda (_: T).(\lambda (z: T).(let TMP_152 \def (s k
+d) in (let TMP_153 \def (lift h TMP_152 t1) in (let TMP_154 \def (s k d0) in
+(let TMP_155 \def (lift h0 TMP_154 z) in (eq T TMP_153 TMP_155))))))) in
+(ex3_2 T T TMP_149 TMP_151 TMP_156)))))))) in (let TMP_158 \def (lift h d t0)
+in (let H11 \def (eq_ind T u TMP_157 H10 TMP_158 H6) in (let TMP_168 \def
+(\lambda (t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq T (THead k t2
+(lift h (s k d) t1)) (lift h0 d0 t1)) \to (let TMP_160 \def (\lambda (y:
+T).(\lambda (z: T).(let TMP_159 \def (THead k y z) in (eq T t1 TMP_159)))) in
+(let TMP_162 \def (\lambda (y: T).(\lambda (_: T).(let TMP_161 \def (lift h0
+d0 y) in (eq T t2 TMP_161)))) in (let TMP_167 \def (\lambda (_: T).(\lambda
+(z: T).(let TMP_163 \def (s k d) in (let TMP_164 \def (lift h TMP_163 t1) in
+(let TMP_165 \def (s k d0) in (let TMP_166 \def (lift h0 TMP_165 z) in (eq T
+TMP_164 TMP_166))))))) in (ex3_2 T T TMP_160 TMP_162 TMP_167)))))))) in (let
+TMP_169 \def (lift h d t0) in (let H12 \def (eq_ind T u TMP_168 H9 TMP_169
+H6) in (let TMP_170 \def (lift h d t0) in (let TMP_181 \def (\lambda (t2:
+T).(let TMP_173 \def (\lambda (y: T).(\lambda (z: T).(let TMP_171 \def (THead
+k t0 t1) in (let TMP_172 \def (THead k y z) in (eq T TMP_171 TMP_172))))) in
+(let TMP_175 \def (\lambda (y: T).(\lambda (_: T).(let TMP_174 \def (lift h d
+y) in (eq T t2 TMP_174)))) in (let TMP_180 \def (\lambda (_: T).(\lambda (z:
+T).(let TMP_176 \def (s k d) in (let TMP_177 \def (lift h TMP_176 t1) in (let
+TMP_178 \def (s k d) in (let TMP_179 \def (lift h TMP_178 z) in (eq T TMP_177
+TMP_179))))))) in (ex3_2 T T TMP_173 TMP_175 TMP_180))))) in (let TMP_184
+\def (\lambda (y: T).(\lambda (z: T).(let TMP_182 \def (THead k t0 t1) in
+(let TMP_183 \def (THead k y z) in (eq T TMP_182 TMP_183))))) in (let TMP_187
+\def (\lambda (y: T).(\lambda (_: T).(let TMP_185 \def (lift h d t0) in (let
+TMP_186 \def (lift h d y) in (eq T TMP_185 TMP_186))))) in (let TMP_192 \def
+(\lambda (_: T).(\lambda (z: T).(let TMP_188 \def (s k d) in (let TMP_189
+\def (lift h TMP_188 t1) in (let TMP_190 \def (s k d) in (let TMP_191 \def
+(lift h TMP_190 z) in (eq T TMP_189 TMP_191))))))) in (let TMP_193 \def
+(THead k t0 t1) in (let TMP_194 \def (refl_equal T TMP_193) in (let TMP_195
+\def (lift h d t0) in (let TMP_196 \def (refl_equal T TMP_195) in (let
+TMP_197 \def (s k d) in (let TMP_198 \def (lift h TMP_197 t1) in (let TMP_199
+\def (refl_equal T TMP_198) in (let TMP_200 \def (ex3_2_intro T T TMP_184
+TMP_187 TMP_192 t0 t1 TMP_194 TMP_196 TMP_199) in (let TMP_201 \def (eq_ind_r
+T TMP_170 TMP_181 TMP_200 u H6) in (let TMP_202 \def (eq_ind_r T TMP_138
+TMP_147 TMP_201 t H8) in (eq_ind K k TMP_116 TMP_202 k0
+H7)))))))))))))))))))))))))))))))))))))) in (let TMP_204 \def (TMP_203 H4) in
+(TMP_204 H3))))))))))))))))))))))))))))))))))))))))) in (T_ind TMP_8 TMP_27
+TMP_77 TMP_205 x)))))))).
theorem lift_gen_bind:
\forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
\def
\lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d
-x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
-(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
-h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
+x))).(let TMP_1 \def (Bind b) in (let H_x \def (lift_gen_head TMP_1 u t x h d
+H) in (let H0 \def H_x in (let TMP_4 \def (\lambda (y: T).(\lambda (z:
+T).(let TMP_2 \def (Bind b) in (let TMP_3 \def (THead TMP_2 y z) in (eq T x
+TMP_3))))) in (let TMP_6 \def (\lambda (y: T).(\lambda (_: T).(let TMP_5 \def
+(lift h d y) in (eq T u TMP_5)))) in (let TMP_9 \def (\lambda (_: T).(\lambda
+(z: T).(let TMP_7 \def (S d) in (let TMP_8 \def (lift h TMP_7 z) in (eq T t
+TMP_8))))) in (let TMP_12 \def (\lambda (y: T).(\lambda (z: T).(let TMP_10
+\def (Bind b) in (let TMP_11 \def (THead TMP_10 y z) in (eq T x TMP_11)))))
+in (let TMP_14 \def (\lambda (y: T).(\lambda (_: T).(let TMP_13 \def (lift h
+d y) in (eq T u TMP_13)))) in (let TMP_17 \def (\lambda (_: T).(\lambda (z:
+T).(let TMP_15 \def (S d) in (let TMP_16 \def (lift h TMP_15 z) in (eq T t
+TMP_16))))) in (let TMP_18 \def (ex3_2 T T TMP_12 TMP_14 TMP_17) in (let
+TMP_81 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
-(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d)
-x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
-(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
-u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d)
-z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro
-T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind
-b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d)
-z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d
-x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
-(* COMMENTS
-Initial nodes: 637
-END *)
+(lift h (S d) x1))).(let TMP_19 \def (Bind b) in (let TMP_20 \def (THead
+TMP_19 x0 x1) in (let TMP_29 \def (\lambda (t0: T).(let TMP_23 \def (\lambda
+(y: T).(\lambda (z: T).(let TMP_21 \def (Bind b) in (let TMP_22 \def (THead
+TMP_21 y z) in (eq T t0 TMP_22))))) in (let TMP_25 \def (\lambda (y:
+T).(\lambda (_: T).(let TMP_24 \def (lift h d y) in (eq T u TMP_24)))) in
+(let TMP_28 \def (\lambda (_: T).(\lambda (z: T).(let TMP_26 \def (S d) in
+(let TMP_27 \def (lift h TMP_26 z) in (eq T t TMP_27))))) in (ex3_2 T T
+TMP_23 TMP_25 TMP_28))))) in (let TMP_30 \def (S d) in (let TMP_31 \def (lift
+h TMP_30 x1) in (let TMP_42 \def (\lambda (t0: T).(let TMP_36 \def (\lambda
+(y: T).(\lambda (z: T).(let TMP_32 \def (Bind b) in (let TMP_33 \def (THead
+TMP_32 x0 x1) in (let TMP_34 \def (Bind b) in (let TMP_35 \def (THead TMP_34
+y z) in (eq T TMP_33 TMP_35))))))) in (let TMP_38 \def (\lambda (y:
+T).(\lambda (_: T).(let TMP_37 \def (lift h d y) in (eq T u TMP_37)))) in
+(let TMP_41 \def (\lambda (_: T).(\lambda (z: T).(let TMP_39 \def (S d) in
+(let TMP_40 \def (lift h TMP_39 z) in (eq T t0 TMP_40))))) in (ex3_2 T T
+TMP_36 TMP_38 TMP_41))))) in (let TMP_43 \def (lift h d x0) in (let TMP_56
+\def (\lambda (t0: T).(let TMP_48 \def (\lambda (y: T).(\lambda (z: T).(let
+TMP_44 \def (Bind b) in (let TMP_45 \def (THead TMP_44 x0 x1) in (let TMP_46
+\def (Bind b) in (let TMP_47 \def (THead TMP_46 y z) in (eq T TMP_45
+TMP_47))))))) in (let TMP_50 \def (\lambda (y: T).(\lambda (_: T).(let TMP_49
+\def (lift h d y) in (eq T t0 TMP_49)))) in (let TMP_55 \def (\lambda (_:
+T).(\lambda (z: T).(let TMP_51 \def (S d) in (let TMP_52 \def (lift h TMP_51
+x1) in (let TMP_53 \def (S d) in (let TMP_54 \def (lift h TMP_53 z) in (eq T
+TMP_52 TMP_54))))))) in (ex3_2 T T TMP_48 TMP_50 TMP_55))))) in (let TMP_61
+\def (\lambda (y: T).(\lambda (z: T).(let TMP_57 \def (Bind b) in (let TMP_58
+\def (THead TMP_57 x0 x1) in (let TMP_59 \def (Bind b) in (let TMP_60 \def
+(THead TMP_59 y z) in (eq T TMP_58 TMP_60))))))) in (let TMP_64 \def (\lambda
+(y: T).(\lambda (_: T).(let TMP_62 \def (lift h d x0) in (let TMP_63 \def
+(lift h d y) in (eq T TMP_62 TMP_63))))) in (let TMP_69 \def (\lambda (_:
+T).(\lambda (z: T).(let TMP_65 \def (S d) in (let TMP_66 \def (lift h TMP_65
+x1) in (let TMP_67 \def (S d) in (let TMP_68 \def (lift h TMP_67 z) in (eq T
+TMP_66 TMP_68))))))) in (let TMP_70 \def (Bind b) in (let TMP_71 \def (THead
+TMP_70 x0 x1) in (let TMP_72 \def (refl_equal T TMP_71) in (let TMP_73 \def
+(lift h d x0) in (let TMP_74 \def (refl_equal T TMP_73) in (let TMP_75 \def
+(S d) in (let TMP_76 \def (lift h TMP_75 x1) in (let TMP_77 \def (refl_equal
+T TMP_76) in (let TMP_78 \def (ex3_2_intro T T TMP_61 TMP_64 TMP_69 x0 x1
+TMP_72 TMP_74 TMP_77) in (let TMP_79 \def (eq_ind_r T TMP_43 TMP_56 TMP_78 u
+H2) in (let TMP_80 \def (eq_ind_r T TMP_31 TMP_42 TMP_79 t H3) in (eq_ind_r T
+TMP_20 TMP_29 TMP_80 x H1)))))))))))))))))))))))))))) in (ex3_2_ind T T TMP_4
+TMP_6 TMP_9 TMP_18 TMP_81 H0)))))))))))))))))).
theorem lift_gen_flat:
\forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
\def
\lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d
-x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
-(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
-h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
-(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
-(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T
-T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0:
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1)
-(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T
-(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq
-T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d
-x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T
-(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
-(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1))
-(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x
-H1)))))) H0))))))))).
-(* COMMENTS
-Initial nodes: 615
-END *)
+x))).(let TMP_1 \def (Flat f) in (let H_x \def (lift_gen_head TMP_1 u t x h d
+H) in (let H0 \def H_x in (let TMP_4 \def (\lambda (y: T).(\lambda (z:
+T).(let TMP_2 \def (Flat f) in (let TMP_3 \def (THead TMP_2 y z) in (eq T x
+TMP_3))))) in (let TMP_6 \def (\lambda (y: T).(\lambda (_: T).(let TMP_5 \def
+(lift h d y) in (eq T u TMP_5)))) in (let TMP_8 \def (\lambda (_: T).(\lambda
+(z: T).(let TMP_7 \def (lift h d z) in (eq T t TMP_7)))) in (let TMP_11 \def
+(\lambda (y: T).(\lambda (z: T).(let TMP_9 \def (Flat f) in (let TMP_10 \def
+(THead TMP_9 y z) in (eq T x TMP_10))))) in (let TMP_13 \def (\lambda (y:
+T).(\lambda (_: T).(let TMP_12 \def (lift h d y) in (eq T u TMP_12)))) in
+(let TMP_15 \def (\lambda (_: T).(\lambda (z: T).(let TMP_14 \def (lift h d
+z) in (eq T t TMP_14)))) in (let TMP_16 \def (ex3_2 T T TMP_11 TMP_13 TMP_15)
+in (let TMP_71 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x
+(THead (Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3:
+(eq T t (lift h d x1))).(let TMP_17 \def (Flat f) in (let TMP_18 \def (THead
+TMP_17 x0 x1) in (let TMP_26 \def (\lambda (t0: T).(let TMP_21 \def (\lambda
+(y: T).(\lambda (z: T).(let TMP_19 \def (Flat f) in (let TMP_20 \def (THead
+TMP_19 y z) in (eq T t0 TMP_20))))) in (let TMP_23 \def (\lambda (y:
+T).(\lambda (_: T).(let TMP_22 \def (lift h d y) in (eq T u TMP_22)))) in
+(let TMP_25 \def (\lambda (_: T).(\lambda (z: T).(let TMP_24 \def (lift h d
+z) in (eq T t TMP_24)))) in (ex3_2 T T TMP_21 TMP_23 TMP_25))))) in (let
+TMP_27 \def (lift h d x1) in (let TMP_37 \def (\lambda (t0: T).(let TMP_32
+\def (\lambda (y: T).(\lambda (z: T).(let TMP_28 \def (Flat f) in (let TMP_29
+\def (THead TMP_28 x0 x1) in (let TMP_30 \def (Flat f) in (let TMP_31 \def
+(THead TMP_30 y z) in (eq T TMP_29 TMP_31))))))) in (let TMP_34 \def (\lambda
+(y: T).(\lambda (_: T).(let TMP_33 \def (lift h d y) in (eq T u TMP_33)))) in
+(let TMP_36 \def (\lambda (_: T).(\lambda (z: T).(let TMP_35 \def (lift h d
+z) in (eq T t0 TMP_35)))) in (ex3_2 T T TMP_32 TMP_34 TMP_36))))) in (let
+TMP_38 \def (lift h d x0) in (let TMP_49 \def (\lambda (t0: T).(let TMP_43
+\def (\lambda (y: T).(\lambda (z: T).(let TMP_39 \def (Flat f) in (let TMP_40
+\def (THead TMP_39 x0 x1) in (let TMP_41 \def (Flat f) in (let TMP_42 \def
+(THead TMP_41 y z) in (eq T TMP_40 TMP_42))))))) in (let TMP_45 \def (\lambda
+(y: T).(\lambda (_: T).(let TMP_44 \def (lift h d y) in (eq T t0 TMP_44))))
+in (let TMP_48 \def (\lambda (_: T).(\lambda (z: T).(let TMP_46 \def (lift h
+d x1) in (let TMP_47 \def (lift h d z) in (eq T TMP_46 TMP_47))))) in (ex3_2
+T T TMP_43 TMP_45 TMP_48))))) in (let TMP_54 \def (\lambda (y: T).(\lambda
+(z: T).(let TMP_50 \def (Flat f) in (let TMP_51 \def (THead TMP_50 x0 x1) in
+(let TMP_52 \def (Flat f) in (let TMP_53 \def (THead TMP_52 y z) in (eq T
+TMP_51 TMP_53))))))) in (let TMP_57 \def (\lambda (y: T).(\lambda (_: T).(let
+TMP_55 \def (lift h d x0) in (let TMP_56 \def (lift h d y) in (eq T TMP_55
+TMP_56))))) in (let TMP_60 \def (\lambda (_: T).(\lambda (z: T).(let TMP_58
+\def (lift h d x1) in (let TMP_59 \def (lift h d z) in (eq T TMP_58
+TMP_59))))) in (let TMP_61 \def (Flat f) in (let TMP_62 \def (THead TMP_61 x0
+x1) in (let TMP_63 \def (refl_equal T TMP_62) in (let TMP_64 \def (lift h d
+x0) in (let TMP_65 \def (refl_equal T TMP_64) in (let TMP_66 \def (lift h d
+x1) in (let TMP_67 \def (refl_equal T TMP_66) in (let TMP_68 \def
+(ex3_2_intro T T TMP_54 TMP_57 TMP_60 x0 x1 TMP_63 TMP_65 TMP_67) in (let
+TMP_69 \def (eq_ind_r T TMP_38 TMP_49 TMP_68 u H2) in (let TMP_70 \def
+(eq_ind_r T TMP_27 TMP_37 TMP_69 t H3) in (eq_ind_r T TMP_18 TMP_26 TMP_70 x
+H1)))))))))))))))))))))))))) in (ex3_2_ind T T TMP_4 TMP_6 TMP_8 TMP_16
+TMP_71 H0)))))))))))))))))).
+
+theorem lift_inj:
+ \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
+(lift h d x) (lift h d t)) \to (eq T x t)))))
+\def
+ \lambda (x: T).(let TMP_1 \def (\lambda (t: T).(\forall (t0: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
+t0)))))) in (let TMP_10 \def (\lambda (n: nat).(\lambda (t: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d
+t))).(let TMP_2 \def (TSort n) in (let TMP_3 \def (lift h d TMP_2) in (let
+TMP_5 \def (\lambda (t0: T).(let TMP_4 \def (lift h d t) in (eq T t0 TMP_4)))
+in (let TMP_6 \def (TSort n) in (let TMP_7 \def (lift_sort n h d) in (let H0
+\def (eq_ind T TMP_3 TMP_5 H TMP_6 TMP_7) in (let TMP_8 \def (TSort n) in
+(let TMP_9 \def (lift_gen_sort h d n t H0) in (sym_eq T t TMP_8
+TMP_9)))))))))))))) in (let TMP_34 \def (\lambda (n: nat).(\lambda (t:
+T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (lift h d (TLRef
+n)) (lift h d t))).(let TMP_11 \def (TLRef n) in (let TMP_12 \def (eq T
+TMP_11 t) in (let TMP_23 \def (\lambda (H0: (lt n d)).(let TMP_13 \def (TLRef
+n) in (let TMP_14 \def (lift h d TMP_13) in (let TMP_16 \def (\lambda (t0:
+T).(let TMP_15 \def (lift h d t) in (eq T t0 TMP_15))) in (let TMP_17 \def
+(TLRef n) in (let TMP_18 \def (lift_lref_lt n h d H0) in (let H1 \def (eq_ind
+T TMP_14 TMP_16 H TMP_17 TMP_18) in (let TMP_19 \def (TLRef n) in (let TMP_20
+\def (le_n d) in (let TMP_21 \def (lt_le_trans n d d H0 TMP_20) in (let
+TMP_22 \def (lift_gen_lref_lt h d n TMP_21 t H1) in (sym_eq T t TMP_19
+TMP_22)))))))))))) in (let TMP_33 \def (\lambda (H0: (le d n)).(let TMP_24
+\def (TLRef n) in (let TMP_25 \def (lift h d TMP_24) in (let TMP_27 \def
+(\lambda (t0: T).(let TMP_26 \def (lift h d t) in (eq T t0 TMP_26))) in (let
+TMP_28 \def (plus n h) in (let TMP_29 \def (TLRef TMP_28) in (let TMP_30 \def
+(lift_lref_ge n h d H0) in (let H1 \def (eq_ind T TMP_25 TMP_27 H TMP_29
+TMP_30) in (let TMP_31 \def (TLRef n) in (let TMP_32 \def (lift_gen_lref_ge h
+d n H0 t H1) in (sym_eq T t TMP_31 TMP_32))))))))))) in (lt_le_e n d TMP_12
+TMP_23 TMP_33)))))))))) in (let TMP_140 \def (\lambda (k: K).(let TMP_36 \def
+(\lambda (k0: K).(\forall (t: T).(((\forall (t0: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
+t0)))))) \to (\forall (t0: T).(((\forall (t1: T).(\forall (h: nat).(\forall
+(d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 t1)))))) \to
+(\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d (THead
+k0 t t0)) (lift h d t1)) \to (let TMP_35 \def (THead k0 t t0) in (eq T TMP_35
+t1))))))))))) in (let TMP_90 \def (\lambda (b: B).(\lambda (t: T).(\lambda
+(H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
+(lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
+(t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d
+t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (eq T (lift h d (THead (Bind b) t t0)) (lift h d
+t1))).(let TMP_37 \def (Bind b) in (let TMP_38 \def (THead TMP_37 t t0) in
+(let TMP_39 \def (lift h d TMP_38) in (let TMP_41 \def (\lambda (t2: T).(let
+TMP_40 \def (lift h d t1) in (eq T t2 TMP_40))) in (let TMP_42 \def (Bind b)
+in (let TMP_43 \def (lift h d t) in (let TMP_44 \def (S d) in (let TMP_45
+\def (lift h TMP_44 t0) in (let TMP_46 \def (THead TMP_42 TMP_43 TMP_45) in
+(let TMP_47 \def (lift_bind b t t0 h d) in (let H2 \def (eq_ind T TMP_39
+TMP_41 H1 TMP_46 TMP_47) in (let TMP_50 \def (\lambda (y: T).(\lambda (z:
+T).(let TMP_48 \def (Bind b) in (let TMP_49 \def (THead TMP_48 y z) in (eq T
+t1 TMP_49))))) in (let TMP_53 \def (\lambda (y: T).(\lambda (_: T).(let
+TMP_51 \def (lift h d t) in (let TMP_52 \def (lift h d y) in (eq T TMP_51
+TMP_52))))) in (let TMP_58 \def (\lambda (_: T).(\lambda (z: T).(let TMP_54
+\def (S d) in (let TMP_55 \def (lift h TMP_54 t0) in (let TMP_56 \def (S d)
+in (let TMP_57 \def (lift h TMP_56 z) in (eq T TMP_55 TMP_57))))))) in (let
+TMP_59 \def (Bind b) in (let TMP_60 \def (THead TMP_59 t t0) in (let TMP_61
+\def (eq T TMP_60 t1) in (let TMP_85 \def (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift
+h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d)
+x1))).(let TMP_62 \def (Bind b) in (let TMP_63 \def (THead TMP_62 x0 x1) in
+(let TMP_66 \def (\lambda (t2: T).(let TMP_64 \def (Bind b) in (let TMP_65
+\def (THead TMP_64 t t0) in (eq T TMP_65 t2)))) in (let TMP_67 \def (Bind b)
+in (let TMP_68 \def (THead TMP_67 x0 x1) in (let TMP_69 \def (Bind b) in (let
+TMP_70 \def (THead TMP_69 t t0) in (let TMP_71 \def (Bind b) in (let TMP_72
+\def (THead TMP_71 t t0) in (let TMP_73 \def (Bind b) in (let TMP_74 \def
+(THead TMP_73 x0 x1) in (let TMP_75 \def (Bind b) in (let TMP_76 \def (Bind
+b) in (let TMP_77 \def (Bind b) in (let TMP_78 \def (refl_equal K TMP_77) in
+(let TMP_79 \def (H x0 h d H4) in (let TMP_80 \def (S d) in (let TMP_81 \def
+(H0 x1 h TMP_80 H5) in (let TMP_82 \def (f_equal3 K T T T THead TMP_75 TMP_76
+t x0 t0 x1 TMP_78 TMP_79 TMP_81) in (let TMP_83 \def (sym_eq T TMP_72 TMP_74
+TMP_82) in (let TMP_84 \def (sym_eq T TMP_68 TMP_70 TMP_83) in (eq_ind_r T
+TMP_63 TMP_66 TMP_84 t1 H3))))))))))))))))))))))))))) in (let TMP_86 \def
+(lift h d t) in (let TMP_87 \def (S d) in (let TMP_88 \def (lift h TMP_87 t0)
+in (let TMP_89 \def (lift_gen_bind b TMP_86 TMP_88 t1 h d H2) in (ex3_2_ind T
+T TMP_50 TMP_53 TMP_58 TMP_61 TMP_85 TMP_89))))))))))))))))))))))))))))))))
+in (let TMP_139 \def (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall
+(t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d
+t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1))
+\to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) (lift h d
+t1))).(let TMP_91 \def (Flat f) in (let TMP_92 \def (THead TMP_91 t t0) in
+(let TMP_93 \def (lift h d TMP_92) in (let TMP_95 \def (\lambda (t2: T).(let
+TMP_94 \def (lift h d t1) in (eq T t2 TMP_94))) in (let TMP_96 \def (Flat f)
+in (let TMP_97 \def (lift h d t) in (let TMP_98 \def (lift h d t0) in (let
+TMP_99 \def (THead TMP_96 TMP_97 TMP_98) in (let TMP_100 \def (lift_flat f t
+t0 h d) in (let H2 \def (eq_ind T TMP_93 TMP_95 H1 TMP_99 TMP_100) in (let
+TMP_103 \def (\lambda (y: T).(\lambda (z: T).(let TMP_101 \def (Flat f) in
+(let TMP_102 \def (THead TMP_101 y z) in (eq T t1 TMP_102))))) in (let
+TMP_106 \def (\lambda (y: T).(\lambda (_: T).(let TMP_104 \def (lift h d t)
+in (let TMP_105 \def (lift h d y) in (eq T TMP_104 TMP_105))))) in (let
+TMP_109 \def (\lambda (_: T).(\lambda (z: T).(let TMP_107 \def (lift h d t0)
+in (let TMP_108 \def (lift h d z) in (eq T TMP_107 TMP_108))))) in (let
+TMP_110 \def (Flat f) in (let TMP_111 \def (THead TMP_110 t t0) in (let
+TMP_112 \def (eq T TMP_111 t1) in (let TMP_135 \def (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T
+(lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d
+x1))).(let TMP_113 \def (Flat f) in (let TMP_114 \def (THead TMP_113 x0 x1)
+in (let TMP_117 \def (\lambda (t2: T).(let TMP_115 \def (Flat f) in (let
+TMP_116 \def (THead TMP_115 t t0) in (eq T TMP_116 t2)))) in (let TMP_118
+\def (Flat f) in (let TMP_119 \def (THead TMP_118 x0 x1) in (let TMP_120 \def
+(Flat f) in (let TMP_121 \def (THead TMP_120 t t0) in (let TMP_122 \def (Flat
+f) in (let TMP_123 \def (THead TMP_122 t t0) in (let TMP_124 \def (Flat f) in
+(let TMP_125 \def (THead TMP_124 x0 x1) in (let TMP_126 \def (Flat f) in (let
+TMP_127 \def (Flat f) in (let TMP_128 \def (Flat f) in (let TMP_129 \def
+(refl_equal K TMP_128) in (let TMP_130 \def (H x0 h d H4) in (let TMP_131
+\def (H0 x1 h d H5) in (let TMP_132 \def (f_equal3 K T T T THead TMP_126
+TMP_127 t x0 t0 x1 TMP_129 TMP_130 TMP_131) in (let TMP_133 \def (sym_eq T
+TMP_123 TMP_125 TMP_132) in (let TMP_134 \def (sym_eq T TMP_119 TMP_121
+TMP_133) in (eq_ind_r T TMP_114 TMP_117 TMP_134 t1
+H3)))))))))))))))))))))))))) in (let TMP_136 \def (lift h d t) in (let
+TMP_137 \def (lift h d t0) in (let TMP_138 \def (lift_gen_flat f TMP_136
+TMP_137 t1 h d H2) in (ex3_2_ind T T TMP_103 TMP_106 TMP_109 TMP_112 TMP_135
+TMP_138)))))))))))))))))))))))))))))) in (K_ind TMP_36 TMP_90 TMP_139 k)))))
+in (T_ind TMP_1 TMP_10 TMP_34 TMP_140 x))))).
+
+theorem lift_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
+t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
+\def
+ \lambda (t1: T).(let TMP_5 \def (\lambda (t: T).(\forall (x: T).(\forall
+(h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1
+d2) \to ((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (let TMP_2 \def
+(\lambda (t2: T).(let TMP_1 \def (lift h1 d1 t2) in (eq T x TMP_1))) in (let
+TMP_4 \def (\lambda (t2: T).(let TMP_3 \def (lift h2 d2 t2) in (eq T t
+TMP_3))) in (ex2 T TMP_2 TMP_4))))))))))) in (let TMP_48 \def (\lambda (n:
+nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1:
+nat).(\lambda (d2: nat).(\lambda (_: (le d1 d2)).(\lambda (H0: (eq T (lift h1
+d1 (TSort n)) (lift h2 (plus d2 h1) x))).(let TMP_6 \def (TSort n) in (let
+TMP_7 \def (lift h1 d1 TMP_6) in (let TMP_10 \def (\lambda (t: T).(let TMP_8
+\def (plus d2 h1) in (let TMP_9 \def (lift h2 TMP_8 x) in (eq T t TMP_9))))
+in (let TMP_11 \def (TSort n) in (let TMP_12 \def (lift_sort n h1 d1) in (let
+H1 \def (eq_ind T TMP_7 TMP_10 H0 TMP_11 TMP_12) in (let TMP_13 \def (TSort
+n) in (let TMP_19 \def (\lambda (t: T).(let TMP_15 \def (\lambda (t2: T).(let
+TMP_14 \def (lift h1 d1 t2) in (eq T t TMP_14))) in (let TMP_18 \def (\lambda
+(t2: T).(let TMP_16 \def (TSort n) in (let TMP_17 \def (lift h2 d2 t2) in (eq
+T TMP_16 TMP_17)))) in (ex2 T TMP_15 TMP_18)))) in (let TMP_22 \def (\lambda
+(t2: T).(let TMP_20 \def (TSort n) in (let TMP_21 \def (lift h1 d1 t2) in (eq
+T TMP_20 TMP_21)))) in (let TMP_25 \def (\lambda (t2: T).(let TMP_23 \def
+(TSort n) in (let TMP_24 \def (lift h2 d2 t2) in (eq T TMP_23 TMP_24)))) in
+(let TMP_26 \def (TSort n) in (let TMP_27 \def (TSort n) in (let TMP_29 \def
+(\lambda (t: T).(let TMP_28 \def (TSort n) in (eq T TMP_28 t))) in (let
+TMP_30 \def (TSort n) in (let TMP_31 \def (refl_equal T TMP_30) in (let
+TMP_32 \def (TSort n) in (let TMP_33 \def (lift h1 d1 TMP_32) in (let TMP_34
+\def (lift_sort n h1 d1) in (let TMP_35 \def (eq_ind_r T TMP_27 TMP_29 TMP_31
+TMP_33 TMP_34) in (let TMP_36 \def (TSort n) in (let TMP_38 \def (\lambda (t:
+T).(let TMP_37 \def (TSort n) in (eq T TMP_37 t))) in (let TMP_39 \def (TSort
+n) in (let TMP_40 \def (refl_equal T TMP_39) in (let TMP_41 \def (TSort n) in
+(let TMP_42 \def (lift h2 d2 TMP_41) in (let TMP_43 \def (lift_sort n h2 d2)
+in (let TMP_44 \def (eq_ind_r T TMP_36 TMP_38 TMP_40 TMP_42 TMP_43) in (let
+TMP_45 \def (ex_intro2 T TMP_22 TMP_25 TMP_26 TMP_35 TMP_44) in (let TMP_46
+\def (plus d2 h1) in (let TMP_47 \def (lift_gen_sort h2 TMP_46 n x H1) in
+(eq_ind_r T TMP_13 TMP_19 TMP_45 x
+TMP_47))))))))))))))))))))))))))))))))))))))) in (let TMP_325 \def (\lambda
+(n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1:
+nat).(\lambda (d2: nat).(\lambda (H: (le d1 d2)).(\lambda (H0: (eq T (lift h1
+d1 (TLRef n)) (lift h2 (plus d2 h1) x))).(let TMP_50 \def (\lambda (t2:
+T).(let TMP_49 \def (lift h1 d1 t2) in (eq T x TMP_49))) in (let TMP_53 \def
+(\lambda (t2: T).(let TMP_51 \def (TLRef n) in (let TMP_52 \def (lift h2 d2
+t2) in (eq T TMP_51 TMP_52)))) in (let TMP_54 \def (ex2 T TMP_50 TMP_53) in
+(let TMP_101 \def (\lambda (H1: (lt n d1)).(let TMP_55 \def (TLRef n) in (let
+TMP_56 \def (lift h1 d1 TMP_55) in (let TMP_59 \def (\lambda (t: T).(let
+TMP_57 \def (plus d2 h1) in (let TMP_58 \def (lift h2 TMP_57 x) in (eq T t
+TMP_58)))) in (let TMP_60 \def (TLRef n) in (let TMP_61 \def (lift_lref_lt n
+h1 d1 H1) in (let H2 \def (eq_ind T TMP_56 TMP_59 H0 TMP_60 TMP_61) in (let
+TMP_62 \def (TLRef n) in (let TMP_68 \def (\lambda (t: T).(let TMP_64 \def
+(\lambda (t2: T).(let TMP_63 \def (lift h1 d1 t2) in (eq T t TMP_63))) in
+(let TMP_67 \def (\lambda (t2: T).(let TMP_65 \def (TLRef n) in (let TMP_66
+\def (lift h2 d2 t2) in (eq T TMP_65 TMP_66)))) in (ex2 T TMP_64 TMP_67))))
+in (let TMP_71 \def (\lambda (t2: T).(let TMP_69 \def (TLRef n) in (let
+TMP_70 \def (lift h1 d1 t2) in (eq T TMP_69 TMP_70)))) in (let TMP_74 \def
+(\lambda (t2: T).(let TMP_72 \def (TLRef n) in (let TMP_73 \def (lift h2 d2
+t2) in (eq T TMP_72 TMP_73)))) in (let TMP_75 \def (TLRef n) in (let TMP_76
+\def (TLRef n) in (let TMP_78 \def (\lambda (t: T).(let TMP_77 \def (TLRef n)
+in (eq T TMP_77 t))) in (let TMP_79 \def (TLRef n) in (let TMP_80 \def
+(refl_equal T TMP_79) in (let TMP_81 \def (TLRef n) in (let TMP_82 \def (lift
+h1 d1 TMP_81) in (let TMP_83 \def (lift_lref_lt n h1 d1 H1) in (let TMP_84
+\def (eq_ind_r T TMP_76 TMP_78 TMP_80 TMP_82 TMP_83) in (let TMP_85 \def
+(TLRef n) in (let TMP_87 \def (\lambda (t: T).(let TMP_86 \def (TLRef n) in
+(eq T TMP_86 t))) in (let TMP_88 \def (TLRef n) in (let TMP_89 \def
+(refl_equal T TMP_88) in (let TMP_90 \def (TLRef n) in (let TMP_91 \def (lift
+h2 d2 TMP_90) in (let TMP_92 \def (lt_le_trans n d1 d2 H1 H) in (let TMP_93
+\def (lift_lref_lt n h2 d2 TMP_92) in (let TMP_94 \def (eq_ind_r T TMP_85
+TMP_87 TMP_89 TMP_91 TMP_93) in (let TMP_95 \def (ex_intro2 T TMP_71 TMP_74
+TMP_75 TMP_84 TMP_94) in (let TMP_96 \def (plus d2 h1) in (let TMP_97 \def
+(plus d2 h1) in (let TMP_98 \def (le_plus_trans d1 d2 h1 H) in (let TMP_99
+\def (lt_le_trans n d1 TMP_97 H1 TMP_98) in (let TMP_100 \def
+(lift_gen_lref_lt h2 TMP_96 n TMP_99 x H2) in (eq_ind_r T TMP_62 TMP_68
+TMP_95 x TMP_100)))))))))))))))))))))))))))))))))))) in (let TMP_324 \def
+(\lambda (H1: (le d1 n)).(let TMP_102 \def (TLRef n) in (let TMP_103 \def
+(lift h1 d1 TMP_102) in (let TMP_106 \def (\lambda (t: T).(let TMP_104 \def
+(plus d2 h1) in (let TMP_105 \def (lift h2 TMP_104 x) in (eq T t TMP_105))))
+in (let TMP_107 \def (plus n h1) in (let TMP_108 \def (TLRef TMP_107) in (let
+TMP_109 \def (lift_lref_ge n h1 d1 H1) in (let H2 \def (eq_ind T TMP_103
+TMP_106 H0 TMP_108 TMP_109) in (let TMP_111 \def (\lambda (t2: T).(let
+TMP_110 \def (lift h1 d1 t2) in (eq T x TMP_110))) in (let TMP_114 \def
+(\lambda (t2: T).(let TMP_112 \def (TLRef n) in (let TMP_113 \def (lift h2 d2
+t2) in (eq T TMP_112 TMP_113)))) in (let TMP_115 \def (ex2 T TMP_111 TMP_114)
+in (let TMP_158 \def (\lambda (H3: (lt n d2)).(let TMP_116 \def (plus n h1)
+in (let TMP_117 \def (TLRef TMP_116) in (let TMP_123 \def (\lambda (t:
+T).(let TMP_119 \def (\lambda (t2: T).(let TMP_118 \def (lift h1 d1 t2) in
+(eq T t TMP_118))) in (let TMP_122 \def (\lambda (t2: T).(let TMP_120 \def
+(TLRef n) in (let TMP_121 \def (lift h2 d2 t2) in (eq T TMP_120 TMP_121))))
+in (ex2 T TMP_119 TMP_122)))) in (let TMP_127 \def (\lambda (t2: T).(let
+TMP_124 \def (plus n h1) in (let TMP_125 \def (TLRef TMP_124) in (let TMP_126
+\def (lift h1 d1 t2) in (eq T TMP_125 TMP_126))))) in (let TMP_130 \def
+(\lambda (t2: T).(let TMP_128 \def (TLRef n) in (let TMP_129 \def (lift h2 d2
+t2) in (eq T TMP_128 TMP_129)))) in (let TMP_131 \def (TLRef n) in (let
+TMP_132 \def (plus n h1) in (let TMP_133 \def (TLRef TMP_132) in (let TMP_136
+\def (\lambda (t: T).(let TMP_134 \def (plus n h1) in (let TMP_135 \def
+(TLRef TMP_134) in (eq T TMP_135 t)))) in (let TMP_137 \def (plus n h1) in
+(let TMP_138 \def (TLRef TMP_137) in (let TMP_139 \def (refl_equal T TMP_138)
+in (let TMP_140 \def (TLRef n) in (let TMP_141 \def (lift h1 d1 TMP_140) in
+(let TMP_142 \def (lift_lref_ge n h1 d1 H1) in (let TMP_143 \def (eq_ind_r T
+TMP_133 TMP_136 TMP_139 TMP_141 TMP_142) in (let TMP_144 \def (TLRef n) in
+(let TMP_146 \def (\lambda (t: T).(let TMP_145 \def (TLRef n) in (eq T
+TMP_145 t))) in (let TMP_147 \def (TLRef n) in (let TMP_148 \def (refl_equal
+T TMP_147) in (let TMP_149 \def (TLRef n) in (let TMP_150 \def (lift h2 d2
+TMP_149) in (let TMP_151 \def (lift_lref_lt n h2 d2 H3) in (let TMP_152 \def
+(eq_ind_r T TMP_144 TMP_146 TMP_148 TMP_150 TMP_151) in (let TMP_153 \def
+(ex_intro2 T TMP_127 TMP_130 TMP_131 TMP_143 TMP_152) in (let TMP_154 \def
+(plus d2 h1) in (let TMP_155 \def (plus n h1) in (let TMP_156 \def (lt_reg_r
+n d2 h1 H3) in (let TMP_157 \def (lift_gen_lref_lt h2 TMP_154 TMP_155 TMP_156
+x H2) in (eq_ind_r T TMP_117 TMP_123 TMP_153 x
+TMP_157))))))))))))))))))))))))))))))) in (let TMP_323 \def (\lambda (H3: (le
+d2 n)).(let TMP_159 \def (plus d2 h2) in (let TMP_161 \def (\lambda (t2:
+T).(let TMP_160 \def (lift h1 d1 t2) in (eq T x TMP_160))) in (let TMP_164
+\def (\lambda (t2: T).(let TMP_162 \def (TLRef n) in (let TMP_163 \def (lift
+h2 d2 t2) in (eq T TMP_162 TMP_163)))) in (let TMP_165 \def (ex2 T TMP_161
+TMP_164) in (let TMP_186 \def (\lambda (H4: (lt n (plus d2 h2))).(let TMP_166
+\def (plus d2 h1) in (let TMP_167 \def (plus n h1) in (let TMP_168 \def (le_n
+h1) in (let TMP_169 \def (le_plus_plus d2 n h1 h1 H3 TMP_168) in (let TMP_170
+\def (plus d2 h2) in (let TMP_171 \def (plus TMP_170 h1) in (let TMP_173 \def
+(\lambda (n0: nat).(let TMP_172 \def (plus n h1) in (lt TMP_172 n0))) in (let
+TMP_174 \def (plus d2 h2) in (let TMP_175 \def (lt_reg_r n TMP_174 h1 H4) in
+(let TMP_176 \def (plus d2 h1) in (let TMP_177 \def (plus TMP_176 h2) in (let
+TMP_178 \def (plus_permute_2_in_3 d2 h1 h2) in (let TMP_179 \def (eq_ind_r
+nat TMP_171 TMP_173 TMP_175 TMP_177 TMP_178) in (let TMP_181 \def (\lambda
+(t2: T).(let TMP_180 \def (lift h1 d1 t2) in (eq T x TMP_180))) in (let
+TMP_184 \def (\lambda (t2: T).(let TMP_182 \def (TLRef n) in (let TMP_183
+\def (lift h2 d2 t2) in (eq T TMP_182 TMP_183)))) in (let TMP_185 \def (ex2 T
+TMP_181 TMP_184) in (lift_gen_lref_false h2 TMP_166 TMP_167 TMP_169 TMP_179 x
+H2 TMP_185)))))))))))))))))) in (let TMP_322 \def (\lambda (H4: (le (plus d2
+h2) n)).(let TMP_187 \def (plus n h1) in (let TMP_191 \def (\lambda (n0:
+nat).(let TMP_188 \def (TLRef n0) in (let TMP_189 \def (plus d2 h1) in (let
+TMP_190 \def (lift h2 TMP_189 x) in (eq T TMP_188 TMP_190))))) in (let
+TMP_192 \def (plus n h1) in (let TMP_193 \def (minus TMP_192 h2) in (let
+TMP_194 \def (plus TMP_193 h2) in (let TMP_195 \def (plus n h1) in (let
+TMP_196 \def (plus d2 h2) in (let TMP_197 \def (le_plus_r d2 h2) in (let
+TMP_198 \def (le_trans h2 TMP_196 n TMP_197 H4) in (let TMP_199 \def
+(le_plus_trans h2 n h1 TMP_198) in (let TMP_200 \def (le_plus_minus_sym h2
+TMP_195 TMP_199) in (let H5 \def (eq_ind nat TMP_187 TMP_191 H2 TMP_194
+TMP_200) in (let TMP_201 \def (plus n h1) in (let TMP_202 \def (minus TMP_201
+h2) in (let TMP_203 \def (TLRef TMP_202) in (let TMP_209 \def (\lambda (t:
+T).(let TMP_205 \def (\lambda (t2: T).(let TMP_204 \def (lift h1 d1 t2) in
+(eq T t TMP_204))) in (let TMP_208 \def (\lambda (t2: T).(let TMP_206 \def
+(TLRef n) in (let TMP_207 \def (lift h2 d2 t2) in (eq T TMP_206 TMP_207))))
+in (ex2 T TMP_205 TMP_208)))) in (let TMP_214 \def (\lambda (t2: T).(let
+TMP_210 \def (plus n h1) in (let TMP_211 \def (minus TMP_210 h2) in (let
+TMP_212 \def (TLRef TMP_211) in (let TMP_213 \def (lift h1 d1 t2) in (eq T
+TMP_212 TMP_213)))))) in (let TMP_217 \def (\lambda (t2: T).(let TMP_215 \def
+(TLRef n) in (let TMP_216 \def (lift h2 d2 t2) in (eq T TMP_215 TMP_216))))
+in (let TMP_218 \def (minus n h2) in (let TMP_219 \def (TLRef TMP_218) in
+(let TMP_220 \def (minus n h2) in (let TMP_221 \def (plus TMP_220 h1) in (let
+TMP_226 \def (\lambda (n0: nat).(let TMP_222 \def (TLRef n0) in (let TMP_223
+\def (minus n h2) in (let TMP_224 \def (TLRef TMP_223) in (let TMP_225 \def
+(lift h1 d1 TMP_224) in (eq T TMP_222 TMP_225)))))) in (let TMP_227 \def
+(minus n h2) in (let TMP_228 \def (plus TMP_227 h1) in (let TMP_229 \def
+(TLRef TMP_228) in (let TMP_233 \def (\lambda (t: T).(let TMP_230 \def (minus
+n h2) in (let TMP_231 \def (plus TMP_230 h1) in (let TMP_232 \def (TLRef
+TMP_231) in (eq T TMP_232 t))))) in (let TMP_234 \def (minus n h2) in (let
+TMP_235 \def (plus TMP_234 h1) in (let TMP_236 \def (TLRef TMP_235) in (let
+TMP_237 \def (refl_equal T TMP_236) in (let TMP_238 \def (minus n h2) in (let
+TMP_239 \def (TLRef TMP_238) in (let TMP_240 \def (lift h1 d1 TMP_239) in
+(let TMP_241 \def (minus n h2) in (let TMP_242 \def (minus n h2) in (let
+TMP_243 \def (le_minus d2 n h2 H4) in (let TMP_244 \def (le_trans d1 d2
+TMP_242 H TMP_243) in (let TMP_245 \def (lift_lref_ge TMP_241 h1 d1 TMP_244)
+in (let TMP_246 \def (eq_ind_r T TMP_229 TMP_233 TMP_237 TMP_240 TMP_245) in
+(let TMP_247 \def (plus n h1) in (let TMP_248 \def (minus TMP_247 h2) in (let
+TMP_249 \def (plus d2 h2) in (let TMP_250 \def (le_plus_r d2 h2) in (let
+TMP_251 \def (le_trans h2 TMP_249 n TMP_250 H4) in (let TMP_252 \def
+(le_minus_plus h2 n TMP_251 h1) in (let TMP_253 \def (eq_ind_r nat TMP_221
+TMP_226 TMP_246 TMP_248 TMP_252) in (let TMP_254 \def (minus n h2) in (let
+TMP_255 \def (plus TMP_254 h2) in (let TMP_260 \def (\lambda (n0: nat).(let
+TMP_256 \def (TLRef n0) in (let TMP_257 \def (minus n0 h2) in (let TMP_258
+\def (TLRef TMP_257) in (let TMP_259 \def (lift h2 d2 TMP_258) in (eq T
+TMP_256 TMP_259)))))) in (let TMP_261 \def (minus n h2) in (let TMP_262 \def
+(plus TMP_261 h2) in (let TMP_263 \def (minus TMP_262 h2) in (let TMP_264
+\def (plus TMP_263 h2) in (let TMP_265 \def (TLRef TMP_264) in (let TMP_269
+\def (\lambda (t: T).(let TMP_266 \def (minus n h2) in (let TMP_267 \def
+(plus TMP_266 h2) in (let TMP_268 \def (TLRef TMP_267) in (eq T TMP_268
+t))))) in (let TMP_270 \def (minus n h2) in (let TMP_271 \def (plus TMP_270
+h2) in (let TMP_272 \def (minus TMP_271 h2) in (let TMP_273 \def (plus
+TMP_272 h2) in (let TMP_274 \def (TLRef TMP_273) in (let TMP_275 \def (minus
+n h2) in (let TMP_276 \def (plus TMP_275 h2) in (let TMP_277 \def (TLRef
+TMP_276) in (let TMP_278 \def (minus n h2) in (let TMP_279 \def (plus TMP_278
+h2) in (let TMP_280 \def (minus TMP_279 h2) in (let TMP_281 \def (plus
+TMP_280 h2) in (let TMP_282 \def (minus n h2) in (let TMP_283 \def (plus
+TMP_282 h2) in (let TMP_284 \def (minus n h2) in (let TMP_285 \def (plus
+TMP_284 h2) in (let TMP_286 \def (minus TMP_285 h2) in (let TMP_287 \def
+(minus n h2) in (let TMP_288 \def (minus n h2) in (let TMP_289 \def
+(minus_plus_r TMP_288 h2) in (let TMP_290 \def (refl_equal nat h2) in (let
+TMP_291 \def (f_equal2 nat nat nat plus TMP_286 TMP_287 h2 h2 TMP_289
+TMP_290) in (let TMP_292 \def (f_equal nat T TLRef TMP_281 TMP_283 TMP_291)
+in (let TMP_293 \def (sym_eq T TMP_274 TMP_277 TMP_292) in (let TMP_294 \def
+(minus n h2) in (let TMP_295 \def (plus TMP_294 h2) in (let TMP_296 \def
+(minus TMP_295 h2) in (let TMP_297 \def (TLRef TMP_296) in (let TMP_298 \def
+(lift h2 d2 TMP_297) in (let TMP_299 \def (minus n h2) in (let TMP_300 \def
+(plus TMP_299 h2) in (let TMP_301 \def (minus TMP_300 h2) in (let TMP_302
+\def (minus n h2) in (let TMP_303 \def (plus TMP_302 h2) in (let TMP_304 \def
+(minus n h2) in (let TMP_305 \def (le_minus d2 n h2 H4) in (let TMP_306 \def
+(le_n h2) in (let TMP_307 \def (le_plus_plus d2 TMP_304 h2 h2 TMP_305
+TMP_306) in (let TMP_308 \def (le_minus d2 TMP_303 h2 TMP_307) in (let
+TMP_309 \def (lift_lref_ge TMP_301 h2 d2 TMP_308) in (let TMP_310 \def
+(eq_ind_r T TMP_265 TMP_269 TMP_293 TMP_298 TMP_309) in (let TMP_311 \def
+(plus d2 h2) in (let TMP_312 \def (le_plus_r d2 h2) in (let TMP_313 \def
+(le_trans h2 TMP_311 n TMP_312 H4) in (let TMP_314 \def (le_plus_minus_sym h2
+n TMP_313) in (let TMP_315 \def (eq_ind_r nat TMP_255 TMP_260 TMP_310 n
+TMP_314) in (let TMP_316 \def (ex_intro2 T TMP_214 TMP_217 TMP_219 TMP_253
+TMP_315) in (let TMP_317 \def (plus d2 h1) in (let TMP_318 \def (plus n h1)
+in (let TMP_319 \def (minus TMP_318 h2) in (let TMP_320 \def (arith0 h2 d2 n
+H4 h1) in (let TMP_321 \def (lift_gen_lref_ge h2 TMP_317 TMP_319 TMP_320 x
+H5) in (eq_ind_r T TMP_203 TMP_209 TMP_316 x
+TMP_321)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))))))) in (lt_le_e n TMP_159 TMP_165 TMP_186
+TMP_322)))))))) in (lt_le_e n d2 TMP_115 TMP_158 TMP_323)))))))))))))) in
+(lt_le_e n d1 TMP_54 TMP_101 TMP_324)))))))))))))) in (let TMP_720 \def
+(\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1:
+nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
+((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
+t2))))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1:
+nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
+((eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2
+t2))))))))))))).(\lambda (x: T).(\lambda (h1: nat).(\lambda (h2:
+nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (H1: (le d1 d2)).(\lambda
+(H2: (eq T (lift h1 d1 (THead k t t0)) (lift h2 (plus d2 h1) x))).(let
+TMP_331 \def (\lambda (k0: K).((eq T (lift h1 d1 (THead k0 t t0)) (lift h2
+(plus d2 h1) x)) \to (let TMP_327 \def (\lambda (t2: T).(let TMP_326 \def
+(lift h1 d1 t2) in (eq T x TMP_326))) in (let TMP_330 \def (\lambda (t2:
+T).(let TMP_328 \def (THead k0 t t0) in (let TMP_329 \def (lift h2 d2 t2) in
+(eq T TMP_328 TMP_329)))) in (ex2 T TMP_327 TMP_330))))) in (let TMP_540 \def
+(\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t t0)) (lift
+h2 (plus d2 h1) x))).(let TMP_332 \def (Bind b) in (let TMP_333 \def (THead
+TMP_332 t t0) in (let TMP_334 \def (lift h1 d1 TMP_333) in (let TMP_337 \def
+(\lambda (t2: T).(let TMP_335 \def (plus d2 h1) in (let TMP_336 \def (lift h2
+TMP_335 x) in (eq T t2 TMP_336)))) in (let TMP_338 \def (Bind b) in (let
+TMP_339 \def (lift h1 d1 t) in (let TMP_340 \def (S d1) in (let TMP_341 \def
+(lift h1 TMP_340 t0) in (let TMP_342 \def (THead TMP_338 TMP_339 TMP_341) in
+(let TMP_343 \def (lift_bind b t t0 h1 d1) in (let H4 \def (eq_ind T TMP_334
+TMP_337 H3 TMP_342 TMP_343) in (let TMP_346 \def (\lambda (y: T).(\lambda (z:
+T).(let TMP_344 \def (Bind b) in (let TMP_345 \def (THead TMP_344 y z) in (eq
+T x TMP_345))))) in (let TMP_350 \def (\lambda (y: T).(\lambda (_: T).(let
+TMP_347 \def (lift h1 d1 t) in (let TMP_348 \def (plus d2 h1) in (let TMP_349
+\def (lift h2 TMP_348 y) in (eq T TMP_347 TMP_349)))))) in (let TMP_356 \def
+(\lambda (_: T).(\lambda (z: T).(let TMP_351 \def (S d1) in (let TMP_352 \def
+(lift h1 TMP_351 t0) in (let TMP_353 \def (plus d2 h1) in (let TMP_354 \def
+(S TMP_353) in (let TMP_355 \def (lift h2 TMP_354 z) in (eq T TMP_352
+TMP_355)))))))) in (let TMP_358 \def (\lambda (t2: T).(let TMP_357 \def (lift
+h1 d1 t2) in (eq T x TMP_357))) in (let TMP_362 \def (\lambda (t2: T).(let
+TMP_359 \def (Bind b) in (let TMP_360 \def (THead TMP_359 t t0) in (let
+TMP_361 \def (lift h2 d2 t2) in (eq T TMP_360 TMP_361))))) in (let TMP_363
+\def (ex2 T TMP_358 TMP_362) in (let TMP_534 \def (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda (H6: (eq T
+(lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T (lift h1 (S
+d1) t0) (lift h2 (S (plus d2 h1)) x1))).(let TMP_364 \def (Bind b) in (let
+TMP_365 \def (THead TMP_364 x0 x1) in (let TMP_372 \def (\lambda (t2: T).(let
+TMP_367 \def (\lambda (t3: T).(let TMP_366 \def (lift h1 d1 t3) in (eq T t2
+TMP_366))) in (let TMP_371 \def (\lambda (t3: T).(let TMP_368 \def (Bind b)
+in (let TMP_369 \def (THead TMP_368 t t0) in (let TMP_370 \def (lift h2 d2
+t3) in (eq T TMP_369 TMP_370))))) in (ex2 T TMP_367 TMP_371)))) in (let
+TMP_374 \def (\lambda (t2: T).(let TMP_373 \def (lift h1 d1 t2) in (eq T x0
+TMP_373))) in (let TMP_376 \def (\lambda (t2: T).(let TMP_375 \def (lift h2
+d2 t2) in (eq T t TMP_375))) in (let TMP_380 \def (\lambda (t2: T).(let
+TMP_377 \def (Bind b) in (let TMP_378 \def (THead TMP_377 x0 x1) in (let
+TMP_379 \def (lift h1 d1 t2) in (eq T TMP_378 TMP_379))))) in (let TMP_384
+\def (\lambda (t2: T).(let TMP_381 \def (Bind b) in (let TMP_382 \def (THead
+TMP_381 t t0) in (let TMP_383 \def (lift h2 d2 t2) in (eq T TMP_382
+TMP_383))))) in (let TMP_385 \def (ex2 T TMP_380 TMP_384) in (let TMP_531
+\def (\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9:
+(eq T t (lift h2 d2 x2))).(let TMP_386 \def (lift h1 d1 x2) in (let TMP_395
+\def (\lambda (t2: T).(let TMP_390 \def (\lambda (t3: T).(let TMP_387 \def
+(Bind b) in (let TMP_388 \def (THead TMP_387 t2 x1) in (let TMP_389 \def
+(lift h1 d1 t3) in (eq T TMP_388 TMP_389))))) in (let TMP_394 \def (\lambda
+(t3: T).(let TMP_391 \def (Bind b) in (let TMP_392 \def (THead TMP_391 t t0)
+in (let TMP_393 \def (lift h2 d2 t3) in (eq T TMP_392 TMP_393))))) in (ex2 T
+TMP_390 TMP_394)))) in (let TMP_396 \def (lift h2 d2 x2) in (let TMP_406 \def
+(\lambda (t2: T).(let TMP_401 \def (\lambda (t3: T).(let TMP_397 \def (Bind
+b) in (let TMP_398 \def (lift h1 d1 x2) in (let TMP_399 \def (THead TMP_397
+TMP_398 x1) in (let TMP_400 \def (lift h1 d1 t3) in (eq T TMP_399
+TMP_400)))))) in (let TMP_405 \def (\lambda (t3: T).(let TMP_402 \def (Bind
+b) in (let TMP_403 \def (THead TMP_402 t2 t0) in (let TMP_404 \def (lift h2
+d2 t3) in (eq T TMP_403 TMP_404))))) in (ex2 T TMP_401 TMP_405)))) in (let
+TMP_407 \def (S d2) in (let TMP_408 \def (plus TMP_407 h1) in (let H10 \def
+(refl_equal nat TMP_408) in (let TMP_409 \def (plus d2 h1) in (let TMP_410
+\def (S TMP_409) in (let TMP_414 \def (\lambda (n: nat).(let TMP_411 \def (S
+d1) in (let TMP_412 \def (lift h1 TMP_411 t0) in (let TMP_413 \def (lift h2 n
+x1) in (eq T TMP_412 TMP_413))))) in (let TMP_415 \def (S d2) in (let TMP_416
+\def (plus TMP_415 h1) in (let H11 \def (eq_ind nat TMP_410 TMP_414 H7
+TMP_416 H10) in (let TMP_419 \def (\lambda (t2: T).(let TMP_417 \def (S d1)
+in (let TMP_418 \def (lift h1 TMP_417 t2) in (eq T x1 TMP_418)))) in (let
+TMP_422 \def (\lambda (t2: T).(let TMP_420 \def (S d2) in (let TMP_421 \def
+(lift h2 TMP_420 t2) in (eq T t0 TMP_421)))) in (let TMP_427 \def (\lambda
+(t2: T).(let TMP_423 \def (Bind b) in (let TMP_424 \def (lift h1 d1 x2) in
+(let TMP_425 \def (THead TMP_423 TMP_424 x1) in (let TMP_426 \def (lift h1 d1
+t2) in (eq T TMP_425 TMP_426)))))) in (let TMP_432 \def (\lambda (t2: T).(let
+TMP_428 \def (Bind b) in (let TMP_429 \def (lift h2 d2 x2) in (let TMP_430
+\def (THead TMP_428 TMP_429 t0) in (let TMP_431 \def (lift h2 d2 t2) in (eq T
+TMP_430 TMP_431)))))) in (let TMP_433 \def (ex2 T TMP_427 TMP_432) in (let
+TMP_524 \def (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1)
+x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(let TMP_434 \def (S d1)
+in (let TMP_435 \def (lift h1 TMP_434 x3) in (let TMP_446 \def (\lambda (t2:
+T).(let TMP_440 \def (\lambda (t3: T).(let TMP_436 \def (Bind b) in (let
+TMP_437 \def (lift h1 d1 x2) in (let TMP_438 \def (THead TMP_436 TMP_437 t2)
+in (let TMP_439 \def (lift h1 d1 t3) in (eq T TMP_438 TMP_439)))))) in (let
+TMP_445 \def (\lambda (t3: T).(let TMP_441 \def (Bind b) in (let TMP_442 \def
+(lift h2 d2 x2) in (let TMP_443 \def (THead TMP_441 TMP_442 t0) in (let
+TMP_444 \def (lift h2 d2 t3) in (eq T TMP_443 TMP_444)))))) in (ex2 T TMP_440
+TMP_445)))) in (let TMP_447 \def (S d2) in (let TMP_448 \def (lift h2 TMP_447
+x3) in (let TMP_461 \def (\lambda (t2: T).(let TMP_455 \def (\lambda (t3:
+T).(let TMP_449 \def (Bind b) in (let TMP_450 \def (lift h1 d1 x2) in (let
+TMP_451 \def (S d1) in (let TMP_452 \def (lift h1 TMP_451 x3) in (let TMP_453
+\def (THead TMP_449 TMP_450 TMP_452) in (let TMP_454 \def (lift h1 d1 t3) in
+(eq T TMP_453 TMP_454)))))))) in (let TMP_460 \def (\lambda (t3: T).(let
+TMP_456 \def (Bind b) in (let TMP_457 \def (lift h2 d2 x2) in (let TMP_458
+\def (THead TMP_456 TMP_457 t2) in (let TMP_459 \def (lift h2 d2 t3) in (eq T
+TMP_458 TMP_459)))))) in (ex2 T TMP_455 TMP_460)))) in (let TMP_468 \def
+(\lambda (t2: T).(let TMP_462 \def (Bind b) in (let TMP_463 \def (lift h1 d1
+x2) in (let TMP_464 \def (S d1) in (let TMP_465 \def (lift h1 TMP_464 x3) in
+(let TMP_466 \def (THead TMP_462 TMP_463 TMP_465) in (let TMP_467 \def (lift
+h1 d1 t2) in (eq T TMP_466 TMP_467)))))))) in (let TMP_475 \def (\lambda (t2:
+T).(let TMP_469 \def (Bind b) in (let TMP_470 \def (lift h2 d2 x2) in (let
+TMP_471 \def (S d2) in (let TMP_472 \def (lift h2 TMP_471 x3) in (let TMP_473
+\def (THead TMP_469 TMP_470 TMP_472) in (let TMP_474 \def (lift h2 d2 t2) in
+(eq T TMP_473 TMP_474)))))))) in (let TMP_476 \def (Bind b) in (let TMP_477
+\def (THead TMP_476 x2 x3) in (let TMP_478 \def (Bind b) in (let TMP_479 \def
+(lift h1 d1 x2) in (let TMP_480 \def (S d1) in (let TMP_481 \def (lift h1
+TMP_480 x3) in (let TMP_482 \def (THead TMP_478 TMP_479 TMP_481) in (let
+TMP_488 \def (\lambda (t2: T).(let TMP_483 \def (Bind b) in (let TMP_484 \def
+(lift h1 d1 x2) in (let TMP_485 \def (S d1) in (let TMP_486 \def (lift h1
+TMP_485 x3) in (let TMP_487 \def (THead TMP_483 TMP_484 TMP_486) in (eq T
+TMP_487 t2))))))) in (let TMP_489 \def (Bind b) in (let TMP_490 \def (lift h1
+d1 x2) in (let TMP_491 \def (S d1) in (let TMP_492 \def (lift h1 TMP_491 x3)
+in (let TMP_493 \def (THead TMP_489 TMP_490 TMP_492) in (let TMP_494 \def
+(refl_equal T TMP_493) in (let TMP_495 \def (Bind b) in (let TMP_496 \def
+(THead TMP_495 x2 x3) in (let TMP_497 \def (lift h1 d1 TMP_496) in (let
+TMP_498 \def (lift_bind b x2 x3 h1 d1) in (let TMP_499 \def (eq_ind_r T
+TMP_482 TMP_488 TMP_494 TMP_497 TMP_498) in (let TMP_500 \def (Bind b) in
+(let TMP_501 \def (lift h2 d2 x2) in (let TMP_502 \def (S d2) in (let TMP_503
+\def (lift h2 TMP_502 x3) in (let TMP_504 \def (THead TMP_500 TMP_501
+TMP_503) in (let TMP_510 \def (\lambda (t2: T).(let TMP_505 \def (Bind b) in
+(let TMP_506 \def (lift h2 d2 x2) in (let TMP_507 \def (S d2) in (let TMP_508
+\def (lift h2 TMP_507 x3) in (let TMP_509 \def (THead TMP_505 TMP_506
+TMP_508) in (eq T TMP_509 t2))))))) in (let TMP_511 \def (Bind b) in (let
+TMP_512 \def (lift h2 d2 x2) in (let TMP_513 \def (S d2) in (let TMP_514 \def
+(lift h2 TMP_513 x3) in (let TMP_515 \def (THead TMP_511 TMP_512 TMP_514) in
+(let TMP_516 \def (refl_equal T TMP_515) in (let TMP_517 \def (Bind b) in
+(let TMP_518 \def (THead TMP_517 x2 x3) in (let TMP_519 \def (lift h2 d2
+TMP_518) in (let TMP_520 \def (lift_bind b x2 x3 h2 d2) in (let TMP_521 \def
+(eq_ind_r T TMP_504 TMP_510 TMP_516 TMP_519 TMP_520) in (let TMP_522 \def
+(ex_intro2 T TMP_468 TMP_475 TMP_477 TMP_499 TMP_521) in (let TMP_523 \def
+(eq_ind_r T TMP_448 TMP_461 TMP_522 t0 H13) in (eq_ind_r T TMP_435 TMP_446
+TMP_523 x1 H12)))))))))))))))))))))))))))))))))))))))))))))))))) in (let
+TMP_525 \def (S d1) in (let TMP_526 \def (S d2) in (let TMP_527 \def (le_n_S
+d1 d2 H1) in (let TMP_528 \def (H0 x1 h1 h2 TMP_525 TMP_526 TMP_527 H11) in
+(let TMP_529 \def (ex2_ind T TMP_419 TMP_422 TMP_433 TMP_524 TMP_528) in (let
+TMP_530 \def (eq_ind_r T TMP_396 TMP_406 TMP_529 t H9) in (eq_ind_r T TMP_386
+TMP_395 TMP_530 x0 H8))))))))))))))))))))))))))))) in (let TMP_532 \def (H x0
+h1 h2 d1 d2 H1 H6) in (let TMP_533 \def (ex2_ind T TMP_374 TMP_376 TMP_385
+TMP_531 TMP_532) in (eq_ind_r T TMP_365 TMP_372 TMP_533 x H5)))))))))))))))))
+in (let TMP_535 \def (lift h1 d1 t) in (let TMP_536 \def (S d1) in (let
+TMP_537 \def (lift h1 TMP_536 t0) in (let TMP_538 \def (plus d2 h1) in (let
+TMP_539 \def (lift_gen_bind b TMP_535 TMP_537 x h2 TMP_538 H4) in (ex3_2_ind
+T T TMP_346 TMP_350 TMP_356 TMP_363 TMP_534 TMP_539))))))))))))))))))))))))))
+in (let TMP_719 \def (\lambda (f: F).(\lambda (H3: (eq T (lift h1 d1 (THead
+(Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let TMP_541 \def (Flat f) in (let
+TMP_542 \def (THead TMP_541 t t0) in (let TMP_543 \def (lift h1 d1 TMP_542)
+in (let TMP_546 \def (\lambda (t2: T).(let TMP_544 \def (plus d2 h1) in (let
+TMP_545 \def (lift h2 TMP_544 x) in (eq T t2 TMP_545)))) in (let TMP_547 \def
+(Flat f) in (let TMP_548 \def (lift h1 d1 t) in (let TMP_549 \def (lift h1 d1
+t0) in (let TMP_550 \def (THead TMP_547 TMP_548 TMP_549) in (let TMP_551 \def
+(lift_flat f t t0 h1 d1) in (let H4 \def (eq_ind T TMP_543 TMP_546 H3 TMP_550
+TMP_551) in (let TMP_554 \def (\lambda (y: T).(\lambda (z: T).(let TMP_552
+\def (Flat f) in (let TMP_553 \def (THead TMP_552 y z) in (eq T x
+TMP_553))))) in (let TMP_558 \def (\lambda (y: T).(\lambda (_: T).(let
+TMP_555 \def (lift h1 d1 t) in (let TMP_556 \def (plus d2 h1) in (let TMP_557
+\def (lift h2 TMP_556 y) in (eq T TMP_555 TMP_557)))))) in (let TMP_562 \def
+(\lambda (_: T).(\lambda (z: T).(let TMP_559 \def (lift h1 d1 t0) in (let
+TMP_560 \def (plus d2 h1) in (let TMP_561 \def (lift h2 TMP_560 z) in (eq T
+TMP_559 TMP_561)))))) in (let TMP_564 \def (\lambda (t2: T).(let TMP_563 \def
+(lift h1 d1 t2) in (eq T x TMP_563))) in (let TMP_568 \def (\lambda (t2:
+T).(let TMP_565 \def (Flat f) in (let TMP_566 \def (THead TMP_565 t t0) in
+(let TMP_567 \def (lift h2 d2 t2) in (eq T TMP_566 TMP_567))))) in (let
+TMP_569 \def (ex2 T TMP_564 TMP_568) in (let TMP_714 \def (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda
+(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
+(lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(let TMP_570 \def (Flat f) in
+(let TMP_571 \def (THead TMP_570 x0 x1) in (let TMP_578 \def (\lambda (t2:
+T).(let TMP_573 \def (\lambda (t3: T).(let TMP_572 \def (lift h1 d1 t3) in
+(eq T t2 TMP_572))) in (let TMP_577 \def (\lambda (t3: T).(let TMP_574 \def
+(Flat f) in (let TMP_575 \def (THead TMP_574 t t0) in (let TMP_576 \def (lift
+h2 d2 t3) in (eq T TMP_575 TMP_576))))) in (ex2 T TMP_573 TMP_577)))) in (let
+TMP_580 \def (\lambda (t2: T).(let TMP_579 \def (lift h1 d1 t2) in (eq T x0
+TMP_579))) in (let TMP_582 \def (\lambda (t2: T).(let TMP_581 \def (lift h2
+d2 t2) in (eq T t TMP_581))) in (let TMP_586 \def (\lambda (t2: T).(let
+TMP_583 \def (Flat f) in (let TMP_584 \def (THead TMP_583 x0 x1) in (let
+TMP_585 \def (lift h1 d1 t2) in (eq T TMP_584 TMP_585))))) in (let TMP_590
+\def (\lambda (t2: T).(let TMP_587 \def (Flat f) in (let TMP_588 \def (THead
+TMP_587 t t0) in (let TMP_589 \def (lift h2 d2 t2) in (eq T TMP_588
+TMP_589))))) in (let TMP_591 \def (ex2 T TMP_586 TMP_590) in (let TMP_711
+\def (\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9:
+(eq T t (lift h2 d2 x2))).(let TMP_592 \def (lift h1 d1 x2) in (let TMP_601
+\def (\lambda (t2: T).(let TMP_596 \def (\lambda (t3: T).(let TMP_593 \def
+(Flat f) in (let TMP_594 \def (THead TMP_593 t2 x1) in (let TMP_595 \def
+(lift h1 d1 t3) in (eq T TMP_594 TMP_595))))) in (let TMP_600 \def (\lambda
+(t3: T).(let TMP_597 \def (Flat f) in (let TMP_598 \def (THead TMP_597 t t0)
+in (let TMP_599 \def (lift h2 d2 t3) in (eq T TMP_598 TMP_599))))) in (ex2 T
+TMP_596 TMP_600)))) in (let TMP_602 \def (lift h2 d2 x2) in (let TMP_612 \def
+(\lambda (t2: T).(let TMP_607 \def (\lambda (t3: T).(let TMP_603 \def (Flat
+f) in (let TMP_604 \def (lift h1 d1 x2) in (let TMP_605 \def (THead TMP_603
+TMP_604 x1) in (let TMP_606 \def (lift h1 d1 t3) in (eq T TMP_605
+TMP_606)))))) in (let TMP_611 \def (\lambda (t3: T).(let TMP_608 \def (Flat
+f) in (let TMP_609 \def (THead TMP_608 t2 t0) in (let TMP_610 \def (lift h2
+d2 t3) in (eq T TMP_609 TMP_610))))) in (ex2 T TMP_607 TMP_611)))) in (let
+TMP_614 \def (\lambda (t2: T).(let TMP_613 \def (lift h1 d1 t2) in (eq T x1
+TMP_613))) in (let TMP_616 \def (\lambda (t2: T).(let TMP_615 \def (lift h2
+d2 t2) in (eq T t0 TMP_615))) in (let TMP_621 \def (\lambda (t2: T).(let
+TMP_617 \def (Flat f) in (let TMP_618 \def (lift h1 d1 x2) in (let TMP_619
+\def (THead TMP_617 TMP_618 x1) in (let TMP_620 \def (lift h1 d1 t2) in (eq T
+TMP_619 TMP_620)))))) in (let TMP_626 \def (\lambda (t2: T).(let TMP_622 \def
+(Flat f) in (let TMP_623 \def (lift h2 d2 x2) in (let TMP_624 \def (THead
+TMP_622 TMP_623 t0) in (let TMP_625 \def (lift h2 d2 t2) in (eq T TMP_624
+TMP_625)))))) in (let TMP_627 \def (ex2 T TMP_621 TMP_626) in (let TMP_707
+\def (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda
+(H11: (eq T t0 (lift h2 d2 x3))).(let TMP_628 \def (lift h1 d1 x3) in (let
+TMP_639 \def (\lambda (t2: T).(let TMP_633 \def (\lambda (t3: T).(let TMP_629
+\def (Flat f) in (let TMP_630 \def (lift h1 d1 x2) in (let TMP_631 \def
+(THead TMP_629 TMP_630 t2) in (let TMP_632 \def (lift h1 d1 t3) in (eq T
+TMP_631 TMP_632)))))) in (let TMP_638 \def (\lambda (t3: T).(let TMP_634 \def
+(Flat f) in (let TMP_635 \def (lift h2 d2 x2) in (let TMP_636 \def (THead
+TMP_634 TMP_635 t0) in (let TMP_637 \def (lift h2 d2 t3) in (eq T TMP_636
+TMP_637)))))) in (ex2 T TMP_633 TMP_638)))) in (let TMP_640 \def (lift h2 d2
+x3) in (let TMP_652 \def (\lambda (t2: T).(let TMP_646 \def (\lambda (t3:
+T).(let TMP_641 \def (Flat f) in (let TMP_642 \def (lift h1 d1 x2) in (let
+TMP_643 \def (lift h1 d1 x3) in (let TMP_644 \def (THead TMP_641 TMP_642
+TMP_643) in (let TMP_645 \def (lift h1 d1 t3) in (eq T TMP_644 TMP_645)))))))
+in (let TMP_651 \def (\lambda (t3: T).(let TMP_647 \def (Flat f) in (let
+TMP_648 \def (lift h2 d2 x2) in (let TMP_649 \def (THead TMP_647 TMP_648 t2)
+in (let TMP_650 \def (lift h2 d2 t3) in (eq T TMP_649 TMP_650)))))) in (ex2 T
+TMP_646 TMP_651)))) in (let TMP_658 \def (\lambda (t2: T).(let TMP_653 \def
+(Flat f) in (let TMP_654 \def (lift h1 d1 x2) in (let TMP_655 \def (lift h1
+d1 x3) in (let TMP_656 \def (THead TMP_653 TMP_654 TMP_655) in (let TMP_657
+\def (lift h1 d1 t2) in (eq T TMP_656 TMP_657))))))) in (let TMP_664 \def
+(\lambda (t2: T).(let TMP_659 \def (Flat f) in (let TMP_660 \def (lift h2 d2
+x2) in (let TMP_661 \def (lift h2 d2 x3) in (let TMP_662 \def (THead TMP_659
+TMP_660 TMP_661) in (let TMP_663 \def (lift h2 d2 t2) in (eq T TMP_662
+TMP_663))))))) in (let TMP_665 \def (Flat f) in (let TMP_666 \def (THead
+TMP_665 x2 x3) in (let TMP_667 \def (Flat f) in (let TMP_668 \def (lift h1 d1
+x2) in (let TMP_669 \def (lift h1 d1 x3) in (let TMP_670 \def (THead TMP_667
+TMP_668 TMP_669) in (let TMP_675 \def (\lambda (t2: T).(let TMP_671 \def
+(Flat f) in (let TMP_672 \def (lift h1 d1 x2) in (let TMP_673 \def (lift h1
+d1 x3) in (let TMP_674 \def (THead TMP_671 TMP_672 TMP_673) in (eq T TMP_674
+t2)))))) in (let TMP_676 \def (Flat f) in (let TMP_677 \def (lift h1 d1 x2)
+in (let TMP_678 \def (lift h1 d1 x3) in (let TMP_679 \def (THead TMP_676
+TMP_677 TMP_678) in (let TMP_680 \def (refl_equal T TMP_679) in (let TMP_681
+\def (Flat f) in (let TMP_682 \def (THead TMP_681 x2 x3) in (let TMP_683 \def
+(lift h1 d1 TMP_682) in (let TMP_684 \def (lift_flat f x2 x3 h1 d1) in (let
+TMP_685 \def (eq_ind_r T TMP_670 TMP_675 TMP_680 TMP_683 TMP_684) in (let
+TMP_686 \def (Flat f) in (let TMP_687 \def (lift h2 d2 x2) in (let TMP_688
+\def (lift h2 d2 x3) in (let TMP_689 \def (THead TMP_686 TMP_687 TMP_688) in
+(let TMP_694 \def (\lambda (t2: T).(let TMP_690 \def (Flat f) in (let TMP_691
+\def (lift h2 d2 x2) in (let TMP_692 \def (lift h2 d2 x3) in (let TMP_693
+\def (THead TMP_690 TMP_691 TMP_692) in (eq T TMP_693 t2)))))) in (let
+TMP_695 \def (Flat f) in (let TMP_696 \def (lift h2 d2 x2) in (let TMP_697
+\def (lift h2 d2 x3) in (let TMP_698 \def (THead TMP_695 TMP_696 TMP_697) in
+(let TMP_699 \def (refl_equal T TMP_698) in (let TMP_700 \def (Flat f) in
+(let TMP_701 \def (THead TMP_700 x2 x3) in (let TMP_702 \def (lift h2 d2
+TMP_701) in (let TMP_703 \def (lift_flat f x2 x3 h2 d2) in (let TMP_704 \def
+(eq_ind_r T TMP_689 TMP_694 TMP_699 TMP_702 TMP_703) in (let TMP_705 \def
+(ex_intro2 T TMP_658 TMP_664 TMP_666 TMP_685 TMP_704) in (let TMP_706 \def
+(eq_ind_r T TMP_640 TMP_652 TMP_705 t0 H11) in (eq_ind_r T TMP_628 TMP_639
+TMP_706 x1 H10)))))))))))))))))))))))))))))))))))))))))))) in (let TMP_708
+\def (H0 x1 h1 h2 d1 d2 H1 H7) in (let TMP_709 \def (ex2_ind T TMP_614
+TMP_616 TMP_627 TMP_707 TMP_708) in (let TMP_710 \def (eq_ind_r T TMP_602
+TMP_612 TMP_709 t H9) in (eq_ind_r T TMP_592 TMP_601 TMP_710 x0
+H8))))))))))))))))) in (let TMP_712 \def (H x0 h1 h2 d1 d2 H1 H6) in (let
+TMP_713 \def (ex2_ind T TMP_580 TMP_582 TMP_591 TMP_711 TMP_712) in (eq_ind_r
+T TMP_571 TMP_578 TMP_713 x H5))))))))))))))))) in (let TMP_715 \def (lift h1
+d1 t) in (let TMP_716 \def (lift h1 d1 t0) in (let TMP_717 \def (plus d2 h1)
+in (let TMP_718 \def (lift_gen_flat f TMP_715 TMP_716 x h2 TMP_717 H4) in
+(ex3_2_ind T T TMP_554 TMP_558 TMP_562 TMP_569 TMP_714
+TMP_718)))))))))))))))))))))))) in (K_ind TMP_331 TMP_540 TMP_719 k
+H2)))))))))))))))) in (T_ind TMP_5 TMP_48 TMP_325 TMP_720 t1))))).
+
+theorem lifts_inj:
+ \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d:
+nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
+\def
+ \lambda (xs: TList).(let TMP_1 \def (\lambda (t: TList).(\forall (ts:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h
+d ts)) \to (eq TList t ts)))))) in (let TMP_11 \def (\lambda (ts: TList).(let
+TMP_2 \def (\lambda (t: TList).(\forall (h: nat).(\forall (d: nat).((eq TList
+(lifts h d TNil) (lifts h d t)) \to (eq TList TNil t))))) in (let TMP_3 \def
+(\lambda (_: nat).(\lambda (_: nat).(\lambda (_: (eq TList TNil
+TNil)).(refl_equal TList TNil)))) in (let TMP_10 \def (\lambda (t:
+T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d:
+nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t)
+(lifts h d t0)))).(let TMP_4 \def (\lambda (ee: TList).(match ee with [TNil
+\Rightarrow True | (TCons _ _) \Rightarrow False])) in (let TMP_5 \def (lift
+h d t) in (let TMP_6 \def (lifts h d t0) in (let TMP_7 \def (TCons TMP_5
+TMP_6) in (let H1 \def (eq_ind TList TNil TMP_4 I TMP_7 H0) in (let TMP_8
+\def (TCons t t0) in (let TMP_9 \def (eq TList TNil TMP_8) in (False_ind
+TMP_9 H1)))))))))))))) in (TList_ind TMP_2 TMP_3 TMP_10 ts))))) in (let
+TMP_53 \def (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h
+d ts)) \to (eq TList t0 ts))))))).(\lambda (ts: TList).(let TMP_13 \def
+(\lambda (t1: TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h
+d (TCons t t0)) (lifts h d t1)) \to (let TMP_12 \def (TCons t t0) in (eq
+TList TMP_12 t1)))))) in (let TMP_20 \def (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d t0)) TNil)).(let
+TMP_14 \def (lift h d t) in (let TMP_15 \def (lifts h d t0) in (let TMP_16
+\def (TCons TMP_14 TMP_15) in (let TMP_17 \def (\lambda (ee: TList).(match ee
+with [TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) in (let H1
+\def (eq_ind TList TMP_16 TMP_17 I TNil H0) in (let TMP_18 \def (TCons t t0)
+in (let TMP_19 \def (eq TList TMP_18 TNil) in (False_ind TMP_19 H1)))))))))))
+in (let TMP_52 \def (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_:
+((\forall (h: nat).(\forall (d: nat).((eq TList (TCons (lift h d t) (lifts h
+d t0)) (lifts h d t2)) \to (eq TList (TCons t t0) t2)))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (eq TList (TCons (lift h d t) (lifts h d
+t0)) (TCons (lift h d t1) (lifts h d t2)))).(let TMP_27 \def (\lambda (e:
+TList).(match e with [TNil \Rightarrow (let TMP_25 \def lref_map in (let
+TMP_26 \def (\lambda (x: nat).(plus x h)) in (TMP_25 TMP_26 d t))) | (TCons
+t3 _) \Rightarrow t3])) in (let TMP_28 \def (lift h d t) in (let TMP_29 \def
+(lifts h d t0) in (let TMP_30 \def (TCons TMP_28 TMP_29) in (let TMP_31 \def
+(lift h d t1) in (let TMP_32 \def (lifts h d t2) in (let TMP_33 \def (TCons
+TMP_31 TMP_32) in (let H2 \def (f_equal TList T TMP_27 TMP_30 TMP_33 H1) in
+(let TMP_37 \def (\lambda (e: TList).(match e with [TNil \Rightarrow (let
+TMP_36 \def lifts in (TMP_36 h d t0)) | (TCons _ t3) \Rightarrow t3])) in
+(let TMP_38 \def (lift h d t) in (let TMP_39 \def (lifts h d t0) in (let
+TMP_40 \def (TCons TMP_38 TMP_39) in (let TMP_41 \def (lift h d t1) in (let
+TMP_42 \def (lifts h d t2) in (let TMP_43 \def (TCons TMP_41 TMP_42) in (let
+H3 \def (f_equal TList TList TMP_37 TMP_40 TMP_43 H1) in (let TMP_51 \def
+(\lambda (H4: (eq T (lift h d t) (lift h d t1))).(let TMP_46 \def (\lambda
+(t3: T).(let TMP_44 \def (TCons t t0) in (let TMP_45 \def (TCons t3 t2) in
+(eq TList TMP_44 TMP_45)))) in (let TMP_47 \def (refl_equal T t) in (let
+TMP_48 \def (H t2 h d H3) in (let TMP_49 \def (f_equal2 T TList TList TCons t
+t t0 t2 TMP_47 TMP_48) in (let TMP_50 \def (lift_inj t t1 h d H4) in (eq_ind
+T t TMP_46 TMP_49 t1 TMP_50))))))) in (TMP_51 H2)))))))))))))))))))))))) in
+(TList_ind TMP_13 TMP_20 TMP_52 ts)))))))) in (TList_ind TMP_1 TMP_11 TMP_53
+xs)))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/lift/fwd.ma".
+include "basic_1/lift/defs.ma".
-include "Basic-1/s/props.ma".
+include "basic_1/s/props.ma".
+
+include "basic_1/T/fwd.ma".
+
+theorem lift_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
+n)) (TSort n))))
+\def
+ \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(let TMP_1 \def (TSort
+n) in (refl_equal T TMP_1)))).
+
+theorem lift_lref_lt:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T
+(lift h d (TLRef n)) (TLRef n)))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n
+d)).(let TMP_4 \def (\lambda (b: bool).(let TMP_1 \def (match b with [true
+\Rightarrow n | false \Rightarrow (plus n h)]) in (let TMP_2 \def (TLRef
+TMP_1) in (let TMP_3 \def (TLRef n) in (eq T TMP_2 TMP_3))))) in (let TMP_5
+\def (TLRef n) in (let TMP_6 \def (refl_equal T TMP_5) in (let TMP_7 \def
+(blt n d) in (let TMP_8 \def (blt n d) in (let TMP_9 \def (lt_blt d n H) in
+(let TMP_10 \def (sym_eq bool TMP_8 true TMP_9) in (eq_ind bool true TMP_4
+TMP_6 TMP_7 TMP_10))))))))))).
+
+theorem lift_lref_ge:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T
+(lift h d (TLRef n)) (TLRef (plus n h))))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d
+n)).(let TMP_5 \def (\lambda (b: bool).(let TMP_1 \def (match b with [true
+\Rightarrow n | false \Rightarrow (plus n h)]) in (let TMP_2 \def (TLRef
+TMP_1) in (let TMP_3 \def (plus n h) in (let TMP_4 \def (TLRef TMP_3) in (eq
+T TMP_2 TMP_4)))))) in (let TMP_6 \def (plus n h) in (let TMP_7 \def (TLRef
+TMP_6) in (let TMP_8 \def (refl_equal T TMP_7) in (let TMP_9 \def (blt n d)
+in (let TMP_10 \def (blt n d) in (let TMP_11 \def (le_bge d n H) in (let
+TMP_12 \def (sym_eq bool TMP_10 false TMP_11) in (eq_ind bool false TMP_5
+TMP_8 TMP_9 TMP_12)))))))))))).
+
+theorem lift_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d)
+t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(let TMP_1 \def (lift h d u) in (let TMP_2 \def (s k d) in (let
+TMP_3 \def (lift h TMP_2 t) in (let TMP_4 \def (THead k TMP_1 TMP_3) in
+(refl_equal T TMP_4))))))))).
+
+theorem lift_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u)
+(lift h (S d) t)))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(let TMP_1 \def (Bind b) in (let TMP_2 \def (lift h d u) in (let
+TMP_3 \def (S d) in (let TMP_4 \def (lift h TMP_3 t) in (let TMP_5 \def
+(THead TMP_1 TMP_2 TMP_4) in (refl_equal T TMP_5)))))))))).
+
+theorem lift_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u)
+(lift h d t)))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(let TMP_1 \def (Flat f) in (let TMP_2 \def (lift h d u) in (let
+TMP_3 \def (lift h d t) in (let TMP_4 \def (THead TMP_1 TMP_2 TMP_3) in
+(refl_equal T TMP_4))))))))).
theorem thead_x_lift_y_y:
\forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall
(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
\def
- \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v:
+ \lambda (k: K).(\lambda (t: T).(let TMP_1 \def (\lambda (t0: T).(\forall (v:
T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0)
-\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n)))
-(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d
-(TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n:
-nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
-(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def
-(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
-(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_:
-((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
-h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0:
-((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
-h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0
-t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1]))
-(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _)
-\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1)
-H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map
-(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort
-n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
-with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
-\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
-lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat
-\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
-\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
-lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow
-(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T
-\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f
-i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f
-(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec
-lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with
-[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
-d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3)
-\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in
-lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2)
-\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1)
-H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def
-(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall
-(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0:
-Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0
-t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0
-d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P))))))
-H3)) H2)))))))))))) t)).
-(* COMMENTS
-Initial nodes: 887
-END *)
+\to (\forall (P: Prop).P)))))) in (let TMP_7 \def (\lambda (n: nat).(\lambda
+(v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v
+(lift h d (TSort n))) (TSort n))).(\lambda (P: Prop).(let TMP_2 \def (TSort
+n) in (let TMP_3 \def (lift h d TMP_2) in (let TMP_4 \def (THead k v TMP_3)
+in (let TMP_5 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in
+(let TMP_6 \def (TSort n) in (let H0 \def (eq_ind T TMP_4 TMP_5 I TMP_6 H) in
+(False_ind P H0))))))))))))) in (let TMP_13 \def (\lambda (n: nat).(\lambda
+(v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v
+(lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let TMP_8 \def (TLRef
+n) in (let TMP_9 \def (lift h d TMP_8) in (let TMP_10 \def (THead k v TMP_9)
+in (let TMP_11 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in
+(let TMP_12 \def (TLRef n) in (let H0 \def (eq_ind T TMP_10 TMP_11 I TMP_12
+H) in (False_ind P H0))))))))))))) in (let TMP_72 \def (\lambda (k0:
+K).(\lambda (t0: T).(\lambda (_: ((\forall (v: T).(\forall (h: nat).(\forall
+(d: nat).((eq T (THead k v (lift h d t0)) t0) \to (\forall (P:
+Prop).P))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (v: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead k v (lift h d t1)) t1) \to (\forall (P:
+Prop).P))))))).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H1: (eq T (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0
+t1))).(\lambda (P: Prop).(let TMP_14 \def (\lambda (e: T).(match e with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _)
+\Rightarrow k1])) in (let TMP_15 \def (THead k0 t0 t1) in (let TMP_16 \def
+(lift h d TMP_15) in (let TMP_17 \def (THead k v TMP_16) in (let TMP_18 \def
+(THead k0 t0 t1) in (let H2 \def (f_equal T K TMP_14 TMP_17 TMP_18 H1) in
+(let TMP_19 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow v |
+(TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) in (let TMP_20
+\def (THead k0 t0 t1) in (let TMP_21 \def (lift h d TMP_20) in (let TMP_22
+\def (THead k v TMP_21) in (let TMP_23 \def (THead k0 t0 t1) in (let H3 \def
+(f_equal T T TMP_19 TMP_22 TMP_23 H1) in (let TMP_54 \def (\lambda (e:
+T).(match e with [(TSort _) \Rightarrow (let TMP_43 \def lref_map in (let
+TMP_44 \def (\lambda (x: nat).(plus x h)) in (let TMP_45 \def (TMP_43 TMP_44
+d t0) in (let TMP_50 \def lref_map in (let TMP_51 \def (\lambda (x:
+nat).(plus x h)) in (let TMP_52 \def (s k0 d) in (let TMP_53 \def (TMP_50
+TMP_51 TMP_52 t1) in (THead k0 TMP_45 TMP_53)))))))) | (TLRef _) \Rightarrow
+(let TMP_28 \def lref_map in (let TMP_29 \def (\lambda (x: nat).(plus x h))
+in (let TMP_30 \def (TMP_28 TMP_29 d t0) in (let TMP_35 \def lref_map in (let
+TMP_36 \def (\lambda (x: nat).(plus x h)) in (let TMP_37 \def (s k0 d) in
+(let TMP_38 \def (TMP_35 TMP_36 TMP_37 t1) in (THead k0 TMP_30 TMP_38))))))))
+| (THead _ _ t2) \Rightarrow t2])) in (let TMP_55 \def (THead k0 t0 t1) in
+(let TMP_56 \def (lift h d TMP_55) in (let TMP_57 \def (THead k v TMP_56) in
+(let TMP_58 \def (THead k0 t0 t1) in (let H4 \def (f_equal T T TMP_54 TMP_57
+TMP_58 H1) in (let TMP_70 \def (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K
+k k0)).(let TMP_59 \def (\lambda (k1: K).(\forall (v0: T).(\forall (h0:
+nat).(\forall (d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall
+(P0: Prop).P0)))))) in (let H7 \def (eq_ind K k TMP_59 H0 k0 H6) in (let
+TMP_60 \def (THead k0 t0 t1) in (let TMP_61 \def (lift h d TMP_60) in (let
+TMP_62 \def (\lambda (t2: T).(eq T t2 t1)) in (let TMP_63 \def (lift h d t0)
+in (let TMP_64 \def (s k0 d) in (let TMP_65 \def (lift h TMP_64 t1) in (let
+TMP_66 \def (THead k0 TMP_63 TMP_65) in (let TMP_67 \def (lift_head k0 t0 t1
+h d) in (let H8 \def (eq_ind T TMP_61 TMP_62 H4 TMP_66 TMP_67) in (let TMP_68
+\def (lift h d t0) in (let TMP_69 \def (s k0 d) in (H7 TMP_68 h TMP_69 H8
+P)))))))))))))))) in (let TMP_71 \def (TMP_70 H3) in (TMP_71
+H2))))))))))))))))))))))))))))))) in (T_ind TMP_1 TMP_7 TMP_13 TMP_72 t)))))).
theorem lift_r:
\forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0)
-t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda
-(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n))
-(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef
-n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H)))
-(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T
-t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
-(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k:
-K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0)
-t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1)
-t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d)
-t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k
-(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d)))
-(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t).
-(* COMMENTS
-Initial nodes: 367
-END *)
+ \lambda (t: T).(let TMP_2 \def (\lambda (t0: T).(\forall (d: nat).(let TMP_1
+\def (lift O d t0) in (eq T TMP_1 t0)))) in (let TMP_4 \def (\lambda (n:
+nat).(\lambda (_: nat).(let TMP_3 \def (TSort n) in (refl_equal T TMP_3))))
+in (let TMP_31 \def (\lambda (n: nat).(\lambda (d: nat).(let TMP_5 \def
+(TLRef n) in (let TMP_6 \def (lift O d TMP_5) in (let TMP_7 \def (TLRef n) in
+(let TMP_8 \def (eq T TMP_6 TMP_7) in (let TMP_17 \def (\lambda (H: (lt n
+d)).(let TMP_9 \def (TLRef n) in (let TMP_11 \def (\lambda (t0: T).(let
+TMP_10 \def (TLRef n) in (eq T t0 TMP_10))) in (let TMP_12 \def (TLRef n) in
+(let TMP_13 \def (refl_equal T TMP_12) in (let TMP_14 \def (TLRef n) in (let
+TMP_15 \def (lift O d TMP_14) in (let TMP_16 \def (lift_lref_lt n O d H) in
+(eq_ind_r T TMP_9 TMP_11 TMP_13 TMP_15 TMP_16))))))))) in (let TMP_30 \def
+(\lambda (H: (le d n)).(let TMP_18 \def (plus n O) in (let TMP_19 \def (TLRef
+TMP_18) in (let TMP_21 \def (\lambda (t0: T).(let TMP_20 \def (TLRef n) in
+(eq T t0 TMP_20))) in (let TMP_22 \def (plus n O) in (let TMP_23 \def (plus n
+O) in (let TMP_24 \def (plus_n_O n) in (let TMP_25 \def (sym_eq nat n TMP_23
+TMP_24) in (let TMP_26 \def (f_equal nat T TLRef TMP_22 n TMP_25) in (let
+TMP_27 \def (TLRef n) in (let TMP_28 \def (lift O d TMP_27) in (let TMP_29
+\def (lift_lref_ge n O d H) in (eq_ind_r T TMP_19 TMP_21 TMP_26 TMP_28
+TMP_29))))))))))))) in (lt_le_e n d TMP_8 TMP_17 TMP_30))))))))) in (let
+TMP_61 \def (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (d:
+nat).(eq T (lift O d t0) t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d:
+nat).(eq T (lift O d t1) t1)))).(\lambda (d: nat).(let TMP_32 \def (lift O d
+t0) in (let TMP_33 \def (s k d) in (let TMP_34 \def (lift O TMP_33 t1) in
+(let TMP_35 \def (THead k TMP_32 TMP_34) in (let TMP_37 \def (\lambda (t2:
+T).(let TMP_36 \def (THead k t0 t1) in (eq T t2 TMP_36))) in (let TMP_38 \def
+(THead k t0 t1) in (let TMP_39 \def (lift O d t0) in (let TMP_40 \def (s k d)
+in (let TMP_41 \def (lift O TMP_40 t1) in (let TMP_42 \def (THead k TMP_39
+TMP_41) in (let TMP_43 \def (lift O d t0) in (let TMP_44 \def (s k d) in (let
+TMP_45 \def (lift O TMP_44 t1) in (let TMP_46 \def (THead k TMP_43 TMP_45) in
+(let TMP_47 \def (THead k t0 t1) in (let TMP_48 \def (lift O d t0) in (let
+TMP_49 \def (s k d) in (let TMP_50 \def (lift O TMP_49 t1) in (let TMP_51
+\def (refl_equal K k) in (let TMP_52 \def (H d) in (let TMP_53 \def (s k d)
+in (let TMP_54 \def (H0 TMP_53) in (let TMP_55 \def (f_equal3 K T T T THead k
+k TMP_48 t0 TMP_50 t1 TMP_51 TMP_52 TMP_54) in (let TMP_56 \def (sym_eq T
+TMP_46 TMP_47 TMP_55) in (let TMP_57 \def (sym_eq T TMP_38 TMP_42 TMP_56) in
+(let TMP_58 \def (THead k t0 t1) in (let TMP_59 \def (lift O d TMP_58) in
+(let TMP_60 \def (lift_head k t0 t1 O d) in (eq_ind_r T TMP_35 TMP_37 TMP_57
+TMP_59 TMP_60))))))))))))))))))))))))))))))))))) in (T_ind TMP_2 TMP_4 TMP_31
+TMP_61 t))))).
theorem lift_lref_gt:
\forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef
(pred n))) (TLRef n))))
\def
- \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef
-(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus
-(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n
-(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S
-(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n)))
-(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d
-(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n))
-(S_pred n d H))))))).
-(* COMMENTS
-Initial nodes: 193
-END *)
+ \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(let TMP_1 \def
+(pred n) in (let TMP_2 \def (S O) in (let TMP_3 \def (plus TMP_1 TMP_2) in
+(let TMP_4 \def (TLRef TMP_3) in (let TMP_6 \def (\lambda (t: T).(let TMP_5
+\def (TLRef n) in (eq T t TMP_5))) in (let TMP_7 \def (S O) in (let TMP_8
+\def (pred n) in (let TMP_9 \def (plus TMP_7 TMP_8) in (let TMP_12 \def
+(\lambda (n0: nat).(let TMP_10 \def (TLRef n0) in (let TMP_11 \def (TLRef n)
+in (eq T TMP_10 TMP_11)))) in (let TMP_15 \def (\lambda (n0: nat).(let TMP_13
+\def (TLRef n0) in (let TMP_14 \def (TLRef n) in (eq T TMP_13 TMP_14)))) in
+(let TMP_16 \def (TLRef n) in (let TMP_17 \def (refl_equal T TMP_16) in (let
+TMP_18 \def (pred n) in (let TMP_19 \def (S TMP_18) in (let TMP_20 \def
+(S_pred n d H) in (let TMP_21 \def (eq_ind nat n TMP_15 TMP_17 TMP_19 TMP_20)
+in (let TMP_22 \def (pred n) in (let TMP_23 \def (S O) in (let TMP_24 \def
+(plus TMP_22 TMP_23) in (let TMP_25 \def (S O) in (let TMP_26 \def (pred n)
+in (let TMP_27 \def (plus_sym TMP_25 TMP_26) in (let TMP_28 \def (eq_ind nat
+TMP_9 TMP_12 TMP_21 TMP_24 TMP_27) in (let TMP_29 \def (S O) in (let TMP_30
+\def (pred n) in (let TMP_31 \def (TLRef TMP_30) in (let TMP_32 \def (lift
+TMP_29 d TMP_31) in (let TMP_33 \def (pred n) in (let TMP_34 \def (S O) in
+(let TMP_35 \def (pred n) in (let TMP_37 \def (\lambda (n0: nat).(let TMP_36
+\def (S d) in (le TMP_36 n0))) in (let TMP_38 \def (pred n) in (let TMP_39
+\def (S TMP_38) in (let TMP_40 \def (S_pred n d H) in (let TMP_41 \def
+(eq_ind nat n TMP_37 H TMP_39 TMP_40) in (let TMP_42 \def (le_S_n d TMP_35
+TMP_41) in (let TMP_43 \def (lift_lref_ge TMP_33 TMP_34 d TMP_42) in
+(eq_ind_r T TMP_4 TMP_6 TMP_28 TMP_32
+TMP_43)))))))))))))))))))))))))))))))))))))))).
+
+theorem lift_tle:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(tle t (lift h d t))))
+\def
+ \lambda (t: T).(let TMP_4 \def (\lambda (t0: T).(\forall (h: nat).(\forall
+(d: nat).(let TMP_1 \def (tweight t0) in (let TMP_2 \def (lift h d t0) in
+(let TMP_3 \def (tweight TMP_2) in (le TMP_1 TMP_3))))))) in (let TMP_6 \def
+(\lambda (_: nat).(\lambda (_: nat).(\lambda (_: nat).(let TMP_5 \def (S O)
+in (le_n TMP_5))))) in (let TMP_8 \def (\lambda (_: nat).(\lambda (_:
+nat).(\lambda (_: nat).(let TMP_7 \def (S O) in (le_n TMP_7))))) in (let
+TMP_31 \def (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h:
+nat).(\forall (d: nat).(le (tweight t0) (tweight (lift h d t0))))))).(\lambda
+(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).(le (tweight t1)
+(tweight (lift h d t1))))))).(\lambda (h: nat).(\lambda (d: nat).(let H_y
+\def (H h d) in (let TMP_9 \def (s k d) in (let H_y0 \def (H0 h TMP_9) in
+(let TMP_10 \def (tweight t0) in (let TMP_11 \def (tweight t1) in (let TMP_12
+\def (plus TMP_10 TMP_11) in (let TMP_13 \def (\lambda (x: nat).(plus x h))
+in (let TMP_14 \def (lref_map TMP_13 d t0) in (let TMP_15 \def (tweight
+TMP_14) in (let TMP_16 \def (\lambda (x: nat).(plus x h)) in (let TMP_17 \def
+(s k d) in (let TMP_18 \def (lref_map TMP_16 TMP_17 t1) in (let TMP_19 \def
+(tweight TMP_18) in (let TMP_20 \def (plus TMP_15 TMP_19) in (let TMP_21 \def
+(tweight t0) in (let TMP_22 \def (\lambda (x: nat).(plus x h)) in (let TMP_23
+\def (lref_map TMP_22 d t0) in (let TMP_24 \def (tweight TMP_23) in (let
+TMP_25 \def (tweight t1) in (let TMP_26 \def (\lambda (x: nat).(plus x h)) in
+(let TMP_27 \def (s k d) in (let TMP_28 \def (lref_map TMP_26 TMP_27 t1) in
+(let TMP_29 \def (tweight TMP_28) in (let TMP_30 \def (le_plus_plus TMP_21
+TMP_24 TMP_25 TMP_29 H_y H_y0) in (le_n_S TMP_12 TMP_20
+TMP_30)))))))))))))))))))))))))))))))) in (T_ind TMP_4 TMP_6 TMP_8 TMP_31
+t))))).
theorem lifts_tapp:
\forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq
TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs:
-TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp
-(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil))
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp
-t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d
-t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1)
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0
-v)) H)))) vs)))).
-(* COMMENTS
-Initial nodes: 215
-END *)
-
-theorem lift_inj:
- \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
-(lift h d x) (lift h d t)) \to (eq T x t)))))
-\def
- \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h:
-nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
-t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def
-(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H
-(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t
-H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq
-T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d
-(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt
-n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d
-d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift
-h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
-(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0
-t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
-T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
-(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1:
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1))
-\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d:
-nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t
-t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0:
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
-(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall
-(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0
-t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
-(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
-(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1
-(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
-(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z))))
-(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda
-(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift
-h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r
-T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2))
-(f_equal3 K T T T THead (Bind b) (Bind b) t x0 t0 x1 (refl_equal K (Bind b))
-(H x0 h d H4) (H0 x1 h (S d) H5)) t1 H3)))))) (lift_gen_bind b (lift h d t)
-(lift h (S d) t0) t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t:
-T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T
-(lift h d t) (lift h d t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda
-(H0: ((\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d
-t0) (lift h d t1)) \to (eq T t0 t1))))))).(\lambda (t1: T).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0))
-(lift h d t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0))
-(\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Flat f) (lift h d t)
-(lift h d t0)) (lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y:
-T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
-(_: T).(eq T (lift h d t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq
-T (lift h d t0) (lift h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda
-(H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0)
-(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T
-(THead (Flat f) t t0) t2)) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0
-x1 (refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)) t1 H3))))))
-(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x).
-(* COMMENTS
-Initial nodes: 1391
-END *)
-
-theorem lift_gen_lift:
- \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
-nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
-t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
-t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1:
-nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
-((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
-T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
-t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda
-(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1
-d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1)
-x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t
-(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T
-(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2)))
-(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda
-(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n)
-(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T
-(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1
-d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
-(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2
-(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda
-(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda
-(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2
-h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
-(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n
-d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t
-(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in
-(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift
-h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T
-(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
-(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t:
-T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n))
-(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef
-n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2
-(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n
-(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2))))
-(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n))
-(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1))
-(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x
-(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
-(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2
-T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
-(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1))
-(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef
-n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1))
-t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
-h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
-(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
-(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2)))
-(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq
-T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
-(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus
-n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2)
-h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4)
-(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda
-(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2
-d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus
-n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus
-(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans
-h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T
-(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T
-t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))
-(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1
-d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n
-h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0)
-(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2)
-h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T
-(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2)))
-(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2
-n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2
-h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2)
-(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2)))))
-(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t:
-T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus
-n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat
-plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus
-(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2))
-(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2)))
-(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus
-(minus n h2) h2) h2 (le_plus_plus d2 (minus n h2) h2 h2 (le_minus d2 n h2 H4)
-(le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n
-(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n
-h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k:
-K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall
-(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift
-h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift
-h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda
-(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2:
-nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
-t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
-t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x:
-T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2:
-nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t
-t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1
-(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T
-x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2
-t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t
-t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead
-(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3
-(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1))
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2
-h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2
-(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
-(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda
-(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
-(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind
-b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
-(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T
-(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
-d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1
-t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2))))
-(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
-t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
-(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
-T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
-x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1
-d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0)
-(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let
-H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1)
-t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2:
-T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2)
-t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift
-h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift
-h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1)
-x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S
-d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift
-h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
-h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda
-(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift
-h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
-h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead
-(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2:
-T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2
-t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2)
-(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2)
-(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift
-h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1
-d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
-(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
-t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)))
-(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1
-H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_n_S d1 d2 H1) H11)))) t H9) x0 H8))))
-(H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1
-(S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T
-(lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def
-(eq_ind T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift
-h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0))
-(lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z:
-T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
-(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2:
-T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0)
-(lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T x
-(THead (Flat f) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus d2
-h1) x0))).(\lambda (H7: (eq T (lift h1 d1 t0) (lift h2 (plus d2 h1)
-x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(ex2 T (\lambda
-(t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t
-t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h1 d1
-t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq
-T (THead (Flat f) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead
-(Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T x0
-(lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T (lift
-h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) t2
-x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2
-d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T (\lambda (t3:
-T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3:
-T).(eq T (THead (Flat f) t2 t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2:
-T).(eq T x1 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2)))
-(ex2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1
-t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2
-t2)))) (\lambda (x3: T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda
-(H11: (eq T t0 (lift h2 d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2:
-T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1
-d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2
-d2 t3))))) (eq_ind_r T (lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
-T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3)))
-(\lambda (t3: T).(eq T (THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2
-t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 x2)
-(lift h1 d1 x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Flat f)
-(lift h2 d2 x2) (lift h2 d2 x3)) (lift h2 d2 t2))) (THead (Flat f) x2 x3)
-(eq_ind_r T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (\lambda (t2:
-T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) t2)) (refl_equal T
-(THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3))) (lift h1 d1 (THead (Flat f)
-x2 x3)) (lift_flat f x2 x3 h1 d1)) (eq_ind_r T (THead (Flat f) (lift h2 d2
-x2) (lift h2 d2 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2)
-(lift h2 d2 x3)) t2)) (refl_equal T (THead (Flat f) (lift h2 d2 x2) (lift h2
-d2 x3))) (lift h2 d2 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h2 d2))) t0
-H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2
-H1 H6)) x H5)))))) (lift_gen_flat f (lift h1 d1 t) (lift h1 d1 t0) x h2 (plus
-d2 h1) H4))))) k H2))))))))))))) t1).
-(* COMMENTS
-Initial nodes: 5037
-END *)
-
-theorem lifts_inj:
- \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d:
-nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
-\def
- \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts:
-TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h
-d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t:
-TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts
-h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_:
-nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda
-(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d:
-nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t)
-(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match
-ee in TList return (\lambda (_: TList).Prop) with [TNil \Rightarrow True |
-(TCons _ _) \Rightarrow False])) I (TCons (lift h d t) (lifts h d t0)) H0) in
-(False_ind (eq TList TNil (TCons t t0)) H1)))))))) ts)) (\lambda (t:
-T).(\lambda (t0: TList).(\lambda (H: ((\forall (ts: TList).(\forall (h:
-nat).(\forall (d: nat).((eq TList (lifts h d t0) (lifts h d ts)) \to (eq
-TList t0 ts))))))).(\lambda (ts: TList).(TList_ind (\lambda (t1:
-TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d (TCons t
-t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts h d
-t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d t0))
-(\lambda (ee: TList).(match ee in TList return (\lambda (_: TList).Prop) with
-[TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H0) in
-(False_ind (eq TList (TCons t t0) TNil) H1))))) (\lambda (t1: T).(\lambda
-(t2: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq TList
-(TCons (lift h d t) (lifts h d t0)) (lifts h d t2)) \to (eq TList (TCons t
-t0) t2)))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq TList
-(TCons (lift h d t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d
-t2)))).(let H2 \def (f_equal TList T (\lambda (e: TList).(match e in TList
-return (\lambda (_: TList).T) with [TNil \Rightarrow ((let rec lref_map (f:
-((nat \to nat))) (d0: nat) (t3: T) on t3: T \def (match t3 with [(TSort n)
-\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t4) \Rightarrow
-(THead k (lref_map f d0 u) (lref_map f (s k d0) t4))]) in lref_map) (\lambda
-(x: nat).(plus x h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d
-t) (lifts h d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def
-(f_equal TList TList (\lambda (e: TList).(match e in TList return (\lambda
-(_: TList).TList) with [TNil \Rightarrow ((let rec lifts (h0: nat) (d0: nat)
-(ts0: TList) on ts0: TList \def (match ts0 with [TNil \Rightarrow TNil |
-(TCons t3 ts1) \Rightarrow (TCons (lift h0 d0 t3) (lifts h0 d0 ts1))]) in
-lifts) h d t0) | (TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h
-d t0)) (TCons (lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift
-h d t) (lift h d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0)
-(TCons t3 t2))) (f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H
-t2 h d H3)) t1 (lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
-(* COMMENTS
-Initial nodes: 772
-END *)
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: TList).(let
+TMP_6 \def (\lambda (t: TList).(let TMP_1 \def (TApp t v) in (let TMP_2 \def
+(lifts h d TMP_1) in (let TMP_3 \def (lifts h d t) in (let TMP_4 \def (lift h
+d v) in (let TMP_5 \def (TApp TMP_3 TMP_4) in (eq TList TMP_2 TMP_5))))))) in
+(let TMP_7 \def (lift h d v) in (let TMP_8 \def (TCons TMP_7 TNil) in (let
+TMP_9 \def (refl_equal TList TMP_8) in (let TMP_29 \def (\lambda (t:
+T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp t0 v)) (TApp
+(lifts h d t0) (lift h d v)))).(let TMP_10 \def (lifts h d t0) in (let TMP_11
+\def (lift h d v) in (let TMP_12 \def (TApp TMP_10 TMP_11) in (let TMP_20
+\def (\lambda (t1: TList).(let TMP_13 \def (lift h d t) in (let TMP_14 \def
+(TCons TMP_13 t1) in (let TMP_15 \def (lift h d t) in (let TMP_16 \def (lifts
+h d t0) in (let TMP_17 \def (lift h d v) in (let TMP_18 \def (TApp TMP_16
+TMP_17) in (let TMP_19 \def (TCons TMP_15 TMP_18) in (eq TList TMP_14
+TMP_19))))))))) in (let TMP_21 \def (lift h d t) in (let TMP_22 \def (lifts h
+d t0) in (let TMP_23 \def (lift h d v) in (let TMP_24 \def (TApp TMP_22
+TMP_23) in (let TMP_25 \def (TCons TMP_21 TMP_24) in (let TMP_26 \def
+(refl_equal TList TMP_25) in (let TMP_27 \def (TApp t0 v) in (let TMP_28 \def
+(lifts h d TMP_27) in (eq_ind_r TList TMP_12 TMP_20 TMP_26 TMP_28
+H)))))))))))))))) in (TList_ind TMP_6 TMP_9 TMP_29 vs))))))))).
theorem lift_free:
\forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
(lift h d t)) (lift (plus k h) d t))))))))
\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
-nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
-(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n:
+ \lambda (t: T).(let TMP_5 \def (\lambda (t0: T).(\forall (h: nat).(\forall
+(k: nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e)
+\to (let TMP_1 \def (lift h d t0) in (let TMP_2 \def (lift k e TMP_1) in (let
+TMP_3 \def (plus k h) in (let TMP_4 \def (lift TMP_3 d t0) in (eq T TMP_2
+TMP_4)))))))))))) in (let TMP_35 \def (\lambda (n: nat).(\lambda (h:
+nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: (le e
+(plus d h))).(\lambda (_: (le d e)).(let TMP_6 \def (TSort n) in (let TMP_11
+\def (\lambda (t0: T).(let TMP_7 \def (lift k e t0) in (let TMP_8 \def (plus
+k h) in (let TMP_9 \def (TSort n) in (let TMP_10 \def (lift TMP_8 d TMP_9) in
+(eq T TMP_7 TMP_10)))))) in (let TMP_12 \def (TSort n) in (let TMP_16 \def
+(\lambda (t0: T).(let TMP_13 \def (plus k h) in (let TMP_14 \def (TSort n) in
+(let TMP_15 \def (lift TMP_13 d TMP_14) in (eq T t0 TMP_15))))) in (let
+TMP_17 \def (TSort n) in (let TMP_19 \def (\lambda (t0: T).(let TMP_18 \def
+(TSort n) in (eq T TMP_18 t0))) in (let TMP_20 \def (TSort n) in (let TMP_21
+\def (refl_equal T TMP_20) in (let TMP_22 \def (plus k h) in (let TMP_23 \def
+(TSort n) in (let TMP_24 \def (lift TMP_22 d TMP_23) in (let TMP_25 \def
+(plus k h) in (let TMP_26 \def (lift_sort n TMP_25 d) in (let TMP_27 \def
+(eq_ind_r T TMP_17 TMP_19 TMP_21 TMP_24 TMP_26) in (let TMP_28 \def (TSort n)
+in (let TMP_29 \def (lift k e TMP_28) in (let TMP_30 \def (lift_sort n k e)
+in (let TMP_31 \def (eq_ind_r T TMP_12 TMP_16 TMP_27 TMP_29 TMP_30) in (let
+TMP_32 \def (TSort n) in (let TMP_33 \def (lift h d TMP_32) in (let TMP_34
+\def (lift_sort n h d) in (eq_ind_r T TMP_6 TMP_11 TMP_31 TMP_33
+TMP_34))))))))))))))))))))))))))))) in (let TMP_122 \def (\lambda (n:
nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
-nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T
-(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort
-n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
-(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0))
-(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k
-h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n))
-(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k:
-nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d
-h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef
-n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T
-(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef
-n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
-(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
-(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus
-k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1
-H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d
-n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift
-(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda
-(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n
-(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal
-nat T TLRef (plus (plus n h) k) (plus n (plus k h))
-(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n))
-(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge
-(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1
-(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda
-(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0:
+nat).(\lambda (H: (le e (plus d h))).(\lambda (H0: (le d e)).(let TMP_36 \def
+(TLRef n) in (let TMP_37 \def (lift h d TMP_36) in (let TMP_38 \def (lift k e
+TMP_37) in (let TMP_39 \def (plus k h) in (let TMP_40 \def (TLRef n) in (let
+TMP_41 \def (lift TMP_39 d TMP_40) in (let TMP_42 \def (eq T TMP_38 TMP_41)
+in (let TMP_73 \def (\lambda (H1: (lt n d)).(let TMP_43 \def (TLRef n) in
+(let TMP_48 \def (\lambda (t0: T).(let TMP_44 \def (lift k e t0) in (let
+TMP_45 \def (plus k h) in (let TMP_46 \def (TLRef n) in (let TMP_47 \def
+(lift TMP_45 d TMP_46) in (eq T TMP_44 TMP_47)))))) in (let TMP_49 \def
+(TLRef n) in (let TMP_53 \def (\lambda (t0: T).(let TMP_50 \def (plus k h) in
+(let TMP_51 \def (TLRef n) in (let TMP_52 \def (lift TMP_50 d TMP_51) in (eq
+T t0 TMP_52))))) in (let TMP_54 \def (TLRef n) in (let TMP_56 \def (\lambda
+(t0: T).(let TMP_55 \def (TLRef n) in (eq T TMP_55 t0))) in (let TMP_57 \def
+(TLRef n) in (let TMP_58 \def (refl_equal T TMP_57) in (let TMP_59 \def (plus
+k h) in (let TMP_60 \def (TLRef n) in (let TMP_61 \def (lift TMP_59 d TMP_60)
+in (let TMP_62 \def (plus k h) in (let TMP_63 \def (lift_lref_lt n TMP_62 d
+H1) in (let TMP_64 \def (eq_ind_r T TMP_54 TMP_56 TMP_58 TMP_61 TMP_63) in
+(let TMP_65 \def (TLRef n) in (let TMP_66 \def (lift k e TMP_65) in (let
+TMP_67 \def (lt_le_trans n d e H1 H0) in (let TMP_68 \def (lift_lref_lt n k e
+TMP_67) in (let TMP_69 \def (eq_ind_r T TMP_49 TMP_53 TMP_64 TMP_66 TMP_68)
+in (let TMP_70 \def (TLRef n) in (let TMP_71 \def (lift h d TMP_70) in (let
+TMP_72 \def (lift_lref_lt n h d H1) in (eq_ind_r T TMP_43 TMP_48 TMP_69
+TMP_71 TMP_72)))))))))))))))))))))))) in (let TMP_121 \def (\lambda (H1: (le
+d n)).(let TMP_74 \def (plus n h) in (let TMP_75 \def (TLRef TMP_74) in (let
+TMP_80 \def (\lambda (t0: T).(let TMP_76 \def (lift k e t0) in (let TMP_77
+\def (plus k h) in (let TMP_78 \def (TLRef n) in (let TMP_79 \def (lift
+TMP_77 d TMP_78) in (eq T TMP_76 TMP_79)))))) in (let TMP_81 \def (plus n h)
+in (let TMP_82 \def (plus TMP_81 k) in (let TMP_83 \def (TLRef TMP_82) in
+(let TMP_87 \def (\lambda (t0: T).(let TMP_84 \def (plus k h) in (let TMP_85
+\def (TLRef n) in (let TMP_86 \def (lift TMP_84 d TMP_85) in (eq T t0
+TMP_86))))) in (let TMP_88 \def (plus k h) in (let TMP_89 \def (plus n
+TMP_88) in (let TMP_90 \def (TLRef TMP_89) in (let TMP_94 \def (\lambda (t0:
+T).(let TMP_91 \def (plus n h) in (let TMP_92 \def (plus TMP_91 k) in (let
+TMP_93 \def (TLRef TMP_92) in (eq T TMP_93 t0))))) in (let TMP_95 \def (plus
+n h) in (let TMP_96 \def (plus TMP_95 k) in (let TMP_97 \def (plus k h) in
+(let TMP_98 \def (plus n TMP_97) in (let TMP_99 \def
+(plus_permute_2_in_3_assoc n h k) in (let TMP_100 \def (f_equal nat T TLRef
+TMP_96 TMP_98 TMP_99) in (let TMP_101 \def (plus k h) in (let TMP_102 \def
+(TLRef n) in (let TMP_103 \def (lift TMP_101 d TMP_102) in (let TMP_104 \def
+(plus k h) in (let TMP_105 \def (lift_lref_ge n TMP_104 d H1) in (let TMP_106
+\def (eq_ind_r T TMP_90 TMP_94 TMP_100 TMP_103 TMP_105) in (let TMP_107 \def
+(plus n h) in (let TMP_108 \def (TLRef TMP_107) in (let TMP_109 \def (lift k
+e TMP_108) in (let TMP_110 \def (plus n h) in (let TMP_111 \def (plus d h) in
+(let TMP_112 \def (plus n h) in (let TMP_113 \def (le_n h) in (let TMP_114
+\def (le_plus_plus d n h h H1 TMP_113) in (let TMP_115 \def (le_trans e
+TMP_111 TMP_112 H TMP_114) in (let TMP_116 \def (lift_lref_ge TMP_110 k e
+TMP_115) in (let TMP_117 \def (eq_ind_r T TMP_83 TMP_87 TMP_106 TMP_109
+TMP_116) in (let TMP_118 \def (TLRef n) in (let TMP_119 \def (lift h d
+TMP_118) in (let TMP_120 \def (lift_lref_ge n h d H1) in (eq_ind_r T TMP_75
+TMP_80 TMP_117 TMP_119 TMP_120))))))))))))))))))))))))))))))))))))))) in
+(lt_le_e n d TMP_42 TMP_73 TMP_121))))))))))))))))) in (let TMP_204 \def
+(\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall
+(k0: nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d
+e) \to (eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d
+t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0:
nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
-(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda
-(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d:
-nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e
-(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda
-(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d
-h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k
-d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0
-t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift
-h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0
-t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d)
-t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k
-e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h
-d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift
-(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d)
-(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le
-k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift
-(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e
-(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift
-h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h
-d))))))))))))) t).
-(* COMMENTS
-Initial nodes: 1407
-END *)
+(eq T (lift k0 e (lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h:
+nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le
+e (plus d h))).(\lambda (H2: (le d e)).(let TMP_123 \def (lift h d t0) in
+(let TMP_124 \def (s k d) in (let TMP_125 \def (lift h TMP_124 t1) in (let
+TMP_126 \def (THead k TMP_123 TMP_125) in (let TMP_131 \def (\lambda (t2:
+T).(let TMP_127 \def (lift k0 e t2) in (let TMP_128 \def (plus k0 h) in (let
+TMP_129 \def (THead k t0 t1) in (let TMP_130 \def (lift TMP_128 d TMP_129) in
+(eq T TMP_127 TMP_130)))))) in (let TMP_132 \def (lift h d t0) in (let
+TMP_133 \def (lift k0 e TMP_132) in (let TMP_134 \def (s k e) in (let TMP_135
+\def (s k d) in (let TMP_136 \def (lift h TMP_135 t1) in (let TMP_137 \def
+(lift k0 TMP_134 TMP_136) in (let TMP_138 \def (THead k TMP_133 TMP_137) in
+(let TMP_142 \def (\lambda (t2: T).(let TMP_139 \def (plus k0 h) in (let
+TMP_140 \def (THead k t0 t1) in (let TMP_141 \def (lift TMP_139 d TMP_140) in
+(eq T t2 TMP_141))))) in (let TMP_143 \def (plus k0 h) in (let TMP_144 \def
+(lift TMP_143 d t0) in (let TMP_145 \def (plus k0 h) in (let TMP_146 \def (s
+k d) in (let TMP_147 \def (lift TMP_145 TMP_146 t1) in (let TMP_148 \def
+(THead k TMP_144 TMP_147) in (let TMP_156 \def (\lambda (t2: T).(let TMP_149
+\def (lift h d t0) in (let TMP_150 \def (lift k0 e TMP_149) in (let TMP_151
+\def (s k e) in (let TMP_152 \def (s k d) in (let TMP_153 \def (lift h
+TMP_152 t1) in (let TMP_154 \def (lift k0 TMP_151 TMP_153) in (let TMP_155
+\def (THead k TMP_150 TMP_154) in (eq T TMP_155 t2))))))))) in (let TMP_157
+\def (lift h d t0) in (let TMP_158 \def (lift k0 e TMP_157) in (let TMP_159
+\def (plus k0 h) in (let TMP_160 \def (lift TMP_159 d t0) in (let TMP_161
+\def (s k e) in (let TMP_162 \def (s k d) in (let TMP_163 \def (lift h
+TMP_162 t1) in (let TMP_164 \def (lift k0 TMP_161 TMP_163) in (let TMP_165
+\def (plus k0 h) in (let TMP_166 \def (s k d) in (let TMP_167 \def (lift
+TMP_165 TMP_166 t1) in (let TMP_168 \def (refl_equal K k) in (let TMP_169
+\def (H h k0 d e H1 H2) in (let TMP_170 \def (s k d) in (let TMP_171 \def (s
+k e) in (let TMP_172 \def (plus d h) in (let TMP_173 \def (s k TMP_172) in
+(let TMP_175 \def (\lambda (n: nat).(let TMP_174 \def (s k e) in (le TMP_174
+n))) in (let TMP_176 \def (plus d h) in (let TMP_177 \def (s_le k e TMP_176
+H1) in (let TMP_178 \def (s k d) in (let TMP_179 \def (plus TMP_178 h) in
+(let TMP_180 \def (s_plus k d h) in (let TMP_181 \def (eq_ind nat TMP_173
+TMP_175 TMP_177 TMP_179 TMP_180) in (let TMP_182 \def (s_le k d e H2) in (let
+TMP_183 \def (H0 h k0 TMP_170 TMP_171 TMP_181 TMP_182) in (let TMP_184 \def
+(f_equal3 K T T T THead k k TMP_158 TMP_160 TMP_164 TMP_167 TMP_168 TMP_169
+TMP_183) in (let TMP_185 \def (plus k0 h) in (let TMP_186 \def (THead k t0
+t1) in (let TMP_187 \def (lift TMP_185 d TMP_186) in (let TMP_188 \def (plus
+k0 h) in (let TMP_189 \def (lift_head k t0 t1 TMP_188 d) in (let TMP_190 \def
+(eq_ind_r T TMP_148 TMP_156 TMP_184 TMP_187 TMP_189) in (let TMP_191 \def
+(lift h d t0) in (let TMP_192 \def (s k d) in (let TMP_193 \def (lift h
+TMP_192 t1) in (let TMP_194 \def (THead k TMP_191 TMP_193) in (let TMP_195
+\def (lift k0 e TMP_194) in (let TMP_196 \def (lift h d t0) in (let TMP_197
+\def (s k d) in (let TMP_198 \def (lift h TMP_197 t1) in (let TMP_199 \def
+(lift_head k TMP_196 TMP_198 k0 e) in (let TMP_200 \def (eq_ind_r T TMP_138
+TMP_142 TMP_190 TMP_195 TMP_199) in (let TMP_201 \def (THead k t0 t1) in (let
+TMP_202 \def (lift h d TMP_201) in (let TMP_203 \def (lift_head k t0 t1 h d)
+in (eq_ind_r T TMP_126 TMP_131 TMP_200 TMP_202
+TMP_203)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))) in (T_ind TMP_5 TMP_35 TMP_122 TMP_204 t))))).
+
+theorem lift_free_sym:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
+(lift h d t)) (lift (plus h k) d t))))))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d:
+nat).(\lambda (e: nat).(\lambda (H: (le e (plus d h))).(\lambda (H0: (le d
+e)).(let TMP_1 \def (plus k h) in (let TMP_5 \def (\lambda (n: nat).(let
+TMP_2 \def (lift h d t) in (let TMP_3 \def (lift k e TMP_2) in (let TMP_4
+\def (lift n d t) in (eq T TMP_3 TMP_4))))) in (let TMP_6 \def (lift_free t h
+k d e H H0) in (let TMP_7 \def (plus h k) in (let TMP_8 \def (plus_sym h k)
+in (eq_ind_r nat TMP_1 TMP_5 TMP_6 TMP_7 TMP_8)))))))))))).
theorem lift_d:
\forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t))
(lift k e (lift h d t))))))))
\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
-nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
-d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda
-(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_:
-(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0)
-(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
-T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0:
-T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
-T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k
-e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n))
-(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e))))))))
-(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d:
-nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h
-(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda
-(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n)
-(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef
-n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d
-(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k
-e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
-(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift
-h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n))
-(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k
-d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e
-n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d)
-t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0:
-nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n)))))
-(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d
-(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda
-(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n)
-(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef
-(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
-(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
-(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
-(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1:
-(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0
-(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda
-(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef
-(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0))
-(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat
-(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k
-e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0)))
-(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus
-n k))) (lift_lref_ge (plus n k) h (plus d k) (le_plus_plus d n k k H1 (le_n
-k)))))) (plus k d) (plus_sym k d)) (lift k e (TLRef n)) (lift_lref_ge n k e
-H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h:
-nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq
-T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d
-t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0:
+ \lambda (t: T).(let TMP_6 \def (\lambda (t0: T).(\forall (h: nat).(\forall
+(k: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (let TMP_1 \def
+(plus k d) in (let TMP_2 \def (lift k e t0) in (let TMP_3 \def (lift h TMP_1
+TMP_2) in (let TMP_4 \def (lift h d t0) in (let TMP_5 \def (lift k e TMP_4)
+in (eq T TMP_3 TMP_5)))))))))))) in (let TMP_45 \def (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
+nat).(\lambda (_: (le e d)).(let TMP_7 \def (TSort n) in (let TMP_13 \def
+(\lambda (t0: T).(let TMP_8 \def (plus k d) in (let TMP_9 \def (lift h TMP_8
+t0) in (let TMP_10 \def (TSort n) in (let TMP_11 \def (lift h d TMP_10) in
+(let TMP_12 \def (lift k e TMP_11) in (eq T TMP_9 TMP_12))))))) in (let
+TMP_14 \def (TSort n) in (let TMP_18 \def (\lambda (t0: T).(let TMP_15 \def
+(TSort n) in (let TMP_16 \def (lift h d TMP_15) in (let TMP_17 \def (lift k e
+TMP_16) in (eq T t0 TMP_17))))) in (let TMP_19 \def (TSort n) in (let TMP_22
+\def (\lambda (t0: T).(let TMP_20 \def (TSort n) in (let TMP_21 \def (lift k
+e t0) in (eq T TMP_20 TMP_21)))) in (let TMP_23 \def (TSort n) in (let TMP_25
+\def (\lambda (t0: T).(let TMP_24 \def (TSort n) in (eq T TMP_24 t0))) in
+(let TMP_26 \def (TSort n) in (let TMP_27 \def (refl_equal T TMP_26) in (let
+TMP_28 \def (TSort n) in (let TMP_29 \def (lift k e TMP_28) in (let TMP_30
+\def (lift_sort n k e) in (let TMP_31 \def (eq_ind_r T TMP_23 TMP_25 TMP_27
+TMP_29 TMP_30) in (let TMP_32 \def (TSort n) in (let TMP_33 \def (lift h d
+TMP_32) in (let TMP_34 \def (lift_sort n h d) in (let TMP_35 \def (eq_ind_r T
+TMP_19 TMP_22 TMP_31 TMP_33 TMP_34) in (let TMP_36 \def (plus k d) in (let
+TMP_37 \def (TSort n) in (let TMP_38 \def (lift h TMP_36 TMP_37) in (let
+TMP_39 \def (plus k d) in (let TMP_40 \def (lift_sort n h TMP_39) in (let
+TMP_41 \def (eq_ind_r T TMP_14 TMP_18 TMP_35 TMP_38 TMP_40) in (let TMP_42
+\def (TSort n) in (let TMP_43 \def (lift k e TMP_42) in (let TMP_44 \def
+(lift_sort n k e) in (eq_ind_r T TMP_7 TMP_13 TMP_41 TMP_43
+TMP_44)))))))))))))))))))))))))))))))))) in (let TMP_212 \def (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
+nat).(\lambda (H: (le e d)).(let TMP_46 \def (plus k d) in (let TMP_47 \def
+(TLRef n) in (let TMP_48 \def (lift k e TMP_47) in (let TMP_49 \def (lift h
+TMP_46 TMP_48) in (let TMP_50 \def (TLRef n) in (let TMP_51 \def (lift h d
+TMP_50) in (let TMP_52 \def (lift k e TMP_51) in (let TMP_53 \def (eq T
+TMP_49 TMP_52) in (let TMP_95 \def (\lambda (H0: (lt n e)).(let H1 \def
+(lt_le_trans n e d H0 H) in (let TMP_54 \def (TLRef n) in (let TMP_60 \def
+(\lambda (t0: T).(let TMP_55 \def (plus k d) in (let TMP_56 \def (lift h
+TMP_55 t0) in (let TMP_57 \def (TLRef n) in (let TMP_58 \def (lift h d
+TMP_57) in (let TMP_59 \def (lift k e TMP_58) in (eq T TMP_56 TMP_59)))))))
+in (let TMP_61 \def (TLRef n) in (let TMP_65 \def (\lambda (t0: T).(let
+TMP_62 \def (TLRef n) in (let TMP_63 \def (lift h d TMP_62) in (let TMP_64
+\def (lift k e TMP_63) in (eq T t0 TMP_64))))) in (let TMP_66 \def (TLRef n)
+in (let TMP_69 \def (\lambda (t0: T).(let TMP_67 \def (TLRef n) in (let
+TMP_68 \def (lift k e t0) in (eq T TMP_67 TMP_68)))) in (let TMP_70 \def
+(TLRef n) in (let TMP_72 \def (\lambda (t0: T).(let TMP_71 \def (TLRef n) in
+(eq T TMP_71 t0))) in (let TMP_73 \def (TLRef n) in (let TMP_74 \def
+(refl_equal T TMP_73) in (let TMP_75 \def (TLRef n) in (let TMP_76 \def (lift
+k e TMP_75) in (let TMP_77 \def (lift_lref_lt n k e H0) in (let TMP_78 \def
+(eq_ind_r T TMP_70 TMP_72 TMP_74 TMP_76 TMP_77) in (let TMP_79 \def (TLRef n)
+in (let TMP_80 \def (lift h d TMP_79) in (let TMP_81 \def (lift_lref_lt n h d
+H1) in (let TMP_82 \def (eq_ind_r T TMP_66 TMP_69 TMP_78 TMP_80 TMP_81) in
+(let TMP_83 \def (plus k d) in (let TMP_84 \def (TLRef n) in (let TMP_85 \def
+(lift h TMP_83 TMP_84) in (let TMP_86 \def (plus k d) in (let TMP_87 \def
+(plus k d) in (let TMP_88 \def (le_plus_r k d) in (let TMP_89 \def
+(lt_le_trans n d TMP_87 H1 TMP_88) in (let TMP_90 \def (lift_lref_lt n h
+TMP_86 TMP_89) in (let TMP_91 \def (eq_ind_r T TMP_61 TMP_65 TMP_82 TMP_85
+TMP_90) in (let TMP_92 \def (TLRef n) in (let TMP_93 \def (lift k e TMP_92)
+in (let TMP_94 \def (lift_lref_lt n k e H0) in (eq_ind_r T TMP_54 TMP_60
+TMP_91 TMP_93 TMP_94))))))))))))))))))))))))))))))))) in (let TMP_211 \def
+(\lambda (H0: (le e n)).(let TMP_96 \def (plus n k) in (let TMP_97 \def
+(TLRef TMP_96) in (let TMP_103 \def (\lambda (t0: T).(let TMP_98 \def (plus k
+d) in (let TMP_99 \def (lift h TMP_98 t0) in (let TMP_100 \def (TLRef n) in
+(let TMP_101 \def (lift h d TMP_100) in (let TMP_102 \def (lift k e TMP_101)
+in (eq T TMP_99 TMP_102))))))) in (let TMP_104 \def (plus d k) in (let
+TMP_111 \def (\lambda (n0: nat).(let TMP_105 \def (plus n k) in (let TMP_106
+\def (TLRef TMP_105) in (let TMP_107 \def (lift h n0 TMP_106) in (let TMP_108
+\def (TLRef n) in (let TMP_109 \def (lift h d TMP_108) in (let TMP_110 \def
+(lift k e TMP_109) in (eq T TMP_107 TMP_110)))))))) in (let TMP_112 \def
+(plus d k) in (let TMP_113 \def (plus n k) in (let TMP_114 \def (TLRef
+TMP_113) in (let TMP_115 \def (lift h TMP_112 TMP_114) in (let TMP_116 \def
+(TLRef n) in (let TMP_117 \def (lift h d TMP_116) in (let TMP_118 \def (lift
+k e TMP_117) in (let TMP_119 \def (eq T TMP_115 TMP_118) in (let TMP_155 \def
+(\lambda (H1: (lt n d)).(let TMP_120 \def (plus n k) in (let TMP_121 \def
+(TLRef TMP_120) in (let TMP_125 \def (\lambda (t0: T).(let TMP_122 \def
+(TLRef n) in (let TMP_123 \def (lift h d TMP_122) in (let TMP_124 \def (lift
+k e TMP_123) in (eq T t0 TMP_124))))) in (let TMP_126 \def (TLRef n) in (let
+TMP_130 \def (\lambda (t0: T).(let TMP_127 \def (plus n k) in (let TMP_128
+\def (TLRef TMP_127) in (let TMP_129 \def (lift k e t0) in (eq T TMP_128
+TMP_129))))) in (let TMP_131 \def (plus n k) in (let TMP_132 \def (TLRef
+TMP_131) in (let TMP_135 \def (\lambda (t0: T).(let TMP_133 \def (plus n k)
+in (let TMP_134 \def (TLRef TMP_133) in (eq T TMP_134 t0)))) in (let TMP_136
+\def (plus n k) in (let TMP_137 \def (TLRef TMP_136) in (let TMP_138 \def
+(refl_equal T TMP_137) in (let TMP_139 \def (TLRef n) in (let TMP_140 \def
+(lift k e TMP_139) in (let TMP_141 \def (lift_lref_ge n k e H0) in (let
+TMP_142 \def (eq_ind_r T TMP_132 TMP_135 TMP_138 TMP_140 TMP_141) in (let
+TMP_143 \def (TLRef n) in (let TMP_144 \def (lift h d TMP_143) in (let
+TMP_145 \def (lift_lref_lt n h d H1) in (let TMP_146 \def (eq_ind_r T TMP_126
+TMP_130 TMP_142 TMP_144 TMP_145) in (let TMP_147 \def (plus d k) in (let
+TMP_148 \def (plus n k) in (let TMP_149 \def (TLRef TMP_148) in (let TMP_150
+\def (lift h TMP_147 TMP_149) in (let TMP_151 \def (plus n k) in (let TMP_152
+\def (plus d k) in (let TMP_153 \def (lt_reg_r n d k H1) in (let TMP_154 \def
+(lift_lref_lt TMP_151 h TMP_152 TMP_153) in (eq_ind_r T TMP_121 TMP_125
+TMP_146 TMP_150 TMP_154))))))))))))))))))))))))))))) in (let TMP_203 \def
+(\lambda (H1: (le d n)).(let TMP_156 \def (plus n k) in (let TMP_157 \def
+(plus TMP_156 h) in (let TMP_158 \def (TLRef TMP_157) in (let TMP_162 \def
+(\lambda (t0: T).(let TMP_159 \def (TLRef n) in (let TMP_160 \def (lift h d
+TMP_159) in (let TMP_161 \def (lift k e TMP_160) in (eq T t0 TMP_161))))) in
+(let TMP_163 \def (plus n h) in (let TMP_164 \def (TLRef TMP_163) in (let
+TMP_169 \def (\lambda (t0: T).(let TMP_165 \def (plus n k) in (let TMP_166
+\def (plus TMP_165 h) in (let TMP_167 \def (TLRef TMP_166) in (let TMP_168
+\def (lift k e t0) in (eq T TMP_167 TMP_168)))))) in (let TMP_170 \def (plus
+n h) in (let TMP_171 \def (plus TMP_170 k) in (let TMP_172 \def (TLRef
+TMP_171) in (let TMP_176 \def (\lambda (t0: T).(let TMP_173 \def (plus n k)
+in (let TMP_174 \def (plus TMP_173 h) in (let TMP_175 \def (TLRef TMP_174) in
+(eq T TMP_175 t0))))) in (let TMP_177 \def (plus n k) in (let TMP_178 \def
+(plus TMP_177 h) in (let TMP_179 \def (plus n h) in (let TMP_180 \def (plus
+TMP_179 k) in (let TMP_181 \def (plus_permute_2_in_3 n k h) in (let TMP_182
+\def (f_equal nat T TLRef TMP_178 TMP_180 TMP_181) in (let TMP_183 \def (plus
+n h) in (let TMP_184 \def (TLRef TMP_183) in (let TMP_185 \def (lift k e
+TMP_184) in (let TMP_186 \def (plus n h) in (let TMP_187 \def (le_plus_trans
+e n h H0) in (let TMP_188 \def (lift_lref_ge TMP_186 k e TMP_187) in (let
+TMP_189 \def (eq_ind_r T TMP_172 TMP_176 TMP_182 TMP_185 TMP_188) in (let
+TMP_190 \def (TLRef n) in (let TMP_191 \def (lift h d TMP_190) in (let
+TMP_192 \def (lift_lref_ge n h d H1) in (let TMP_193 \def (eq_ind_r T TMP_164
+TMP_169 TMP_189 TMP_191 TMP_192) in (let TMP_194 \def (plus d k) in (let
+TMP_195 \def (plus n k) in (let TMP_196 \def (TLRef TMP_195) in (let TMP_197
+\def (lift h TMP_194 TMP_196) in (let TMP_198 \def (plus n k) in (let TMP_199
+\def (plus d k) in (let TMP_200 \def (le_n k) in (let TMP_201 \def
+(le_plus_plus d n k k H1 TMP_200) in (let TMP_202 \def (lift_lref_ge TMP_198
+h TMP_199 TMP_201) in (eq_ind_r T TMP_158 TMP_162 TMP_193 TMP_197
+TMP_202))))))))))))))))))))))))))))))))))))))) in (let TMP_204 \def (lt_le_e
+n d TMP_119 TMP_155 TMP_203) in (let TMP_205 \def (plus k d) in (let TMP_206
+\def (plus_sym k d) in (let TMP_207 \def (eq_ind_r nat TMP_104 TMP_111
+TMP_204 TMP_205 TMP_206) in (let TMP_208 \def (TLRef n) in (let TMP_209 \def
+(lift k e TMP_208) in (let TMP_210 \def (lift_lref_ge n k e H0) in (eq_ind_r
+T TMP_97 TMP_103 TMP_207 TMP_209 TMP_210)))))))))))))))))))))))) in (lt_le_e
+n e TMP_53 TMP_95 TMP_211))))))))))))))))) in (let TMP_339 \def (\lambda (k:
+K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0:
nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0
-d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h:
-nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le
-e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2:
-T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1)))))
-(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
-k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d
-(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1))
-(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h
-(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead
-k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda
-(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
-k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda
-(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift
-k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h
-(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e
-t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e)
-t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1)
-(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0
-d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k
-(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1))
-(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0
-(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0
-d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t).
-(* COMMENTS
-Initial nodes: 2143
-END *)
+d) (lift k0 e t0)) (lift k0 e (lift h d t0)))))))))).(\lambda (t1:
+T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d:
+nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e
+t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda (k0:
+nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e d)).(let TMP_213
+\def (lift k0 e t0) in (let TMP_214 \def (s k e) in (let TMP_215 \def (lift
+k0 TMP_214 t1) in (let TMP_216 \def (THead k TMP_213 TMP_215) in (let TMP_222
+\def (\lambda (t2: T).(let TMP_217 \def (plus k0 d) in (let TMP_218 \def
+(lift h TMP_217 t2) in (let TMP_219 \def (THead k t0 t1) in (let TMP_220 \def
+(lift h d TMP_219) in (let TMP_221 \def (lift k0 e TMP_220) in (eq T TMP_218
+TMP_221))))))) in (let TMP_223 \def (plus k0 d) in (let TMP_224 \def (lift k0
+e t0) in (let TMP_225 \def (lift h TMP_223 TMP_224) in (let TMP_226 \def
+(plus k0 d) in (let TMP_227 \def (s k TMP_226) in (let TMP_228 \def (s k e)
+in (let TMP_229 \def (lift k0 TMP_228 t1) in (let TMP_230 \def (lift h
+TMP_227 TMP_229) in (let TMP_231 \def (THead k TMP_225 TMP_230) in (let
+TMP_235 \def (\lambda (t2: T).(let TMP_232 \def (THead k t0 t1) in (let
+TMP_233 \def (lift h d TMP_232) in (let TMP_234 \def (lift k0 e TMP_233) in
+(eq T t2 TMP_234))))) in (let TMP_236 \def (lift h d t0) in (let TMP_237 \def
+(s k d) in (let TMP_238 \def (lift h TMP_237 t1) in (let TMP_239 \def (THead
+k TMP_236 TMP_238) in (let TMP_250 \def (\lambda (t2: T).(let TMP_240 \def
+(plus k0 d) in (let TMP_241 \def (lift k0 e t0) in (let TMP_242 \def (lift h
+TMP_240 TMP_241) in (let TMP_243 \def (plus k0 d) in (let TMP_244 \def (s k
+TMP_243) in (let TMP_245 \def (s k e) in (let TMP_246 \def (lift k0 TMP_245
+t1) in (let TMP_247 \def (lift h TMP_244 TMP_246) in (let TMP_248 \def (THead
+k TMP_242 TMP_247) in (let TMP_249 \def (lift k0 e t2) in (eq T TMP_248
+TMP_249)))))))))))) in (let TMP_251 \def (lift h d t0) in (let TMP_252 \def
+(lift k0 e TMP_251) in (let TMP_253 \def (s k e) in (let TMP_254 \def (s k d)
+in (let TMP_255 \def (lift h TMP_254 t1) in (let TMP_256 \def (lift k0
+TMP_253 TMP_255) in (let TMP_257 \def (THead k TMP_252 TMP_256) in (let
+TMP_267 \def (\lambda (t2: T).(let TMP_258 \def (plus k0 d) in (let TMP_259
+\def (lift k0 e t0) in (let TMP_260 \def (lift h TMP_258 TMP_259) in (let
+TMP_261 \def (plus k0 d) in (let TMP_262 \def (s k TMP_261) in (let TMP_263
+\def (s k e) in (let TMP_264 \def (lift k0 TMP_263 t1) in (let TMP_265 \def
+(lift h TMP_262 TMP_264) in (let TMP_266 \def (THead k TMP_260 TMP_265) in
+(eq T TMP_266 t2))))))))))) in (let TMP_268 \def (s k d) in (let TMP_269 \def
+(plus k0 TMP_268) in (let TMP_284 \def (\lambda (n: nat).(let TMP_270 \def
+(plus k0 d) in (let TMP_271 \def (lift k0 e t0) in (let TMP_272 \def (lift h
+TMP_270 TMP_271) in (let TMP_273 \def (s k e) in (let TMP_274 \def (lift k0
+TMP_273 t1) in (let TMP_275 \def (lift h n TMP_274) in (let TMP_276 \def
+(THead k TMP_272 TMP_275) in (let TMP_277 \def (lift h d t0) in (let TMP_278
+\def (lift k0 e TMP_277) in (let TMP_279 \def (s k e) in (let TMP_280 \def (s
+k d) in (let TMP_281 \def (lift h TMP_280 t1) in (let TMP_282 \def (lift k0
+TMP_279 TMP_281) in (let TMP_283 \def (THead k TMP_278 TMP_282) in (eq T
+TMP_276 TMP_283)))))))))))))))) in (let TMP_285 \def (plus k0 d) in (let
+TMP_286 \def (lift k0 e t0) in (let TMP_287 \def (lift h TMP_285 TMP_286) in
+(let TMP_288 \def (lift h d t0) in (let TMP_289 \def (lift k0 e TMP_288) in
+(let TMP_290 \def (s k d) in (let TMP_291 \def (plus k0 TMP_290) in (let
+TMP_292 \def (s k e) in (let TMP_293 \def (lift k0 TMP_292 t1) in (let
+TMP_294 \def (lift h TMP_291 TMP_293) in (let TMP_295 \def (s k e) in (let
+TMP_296 \def (s k d) in (let TMP_297 \def (lift h TMP_296 t1) in (let TMP_298
+\def (lift k0 TMP_295 TMP_297) in (let TMP_299 \def (refl_equal K k) in (let
+TMP_300 \def (H h k0 d e H1) in (let TMP_301 \def (s k d) in (let TMP_302
+\def (s k e) in (let TMP_303 \def (s_le k e d H1) in (let TMP_304 \def (H0 h
+k0 TMP_301 TMP_302 TMP_303) in (let TMP_305 \def (f_equal3 K T T T THead k k
+TMP_287 TMP_289 TMP_294 TMP_298 TMP_299 TMP_300 TMP_304) in (let TMP_306 \def
+(plus k0 d) in (let TMP_307 \def (s k TMP_306) in (let TMP_308 \def
+(s_plus_sym k k0 d) in (let TMP_309 \def (eq_ind_r nat TMP_269 TMP_284
+TMP_305 TMP_307 TMP_308) in (let TMP_310 \def (lift h d t0) in (let TMP_311
+\def (s k d) in (let TMP_312 \def (lift h TMP_311 t1) in (let TMP_313 \def
+(THead k TMP_310 TMP_312) in (let TMP_314 \def (lift k0 e TMP_313) in (let
+TMP_315 \def (lift h d t0) in (let TMP_316 \def (s k d) in (let TMP_317 \def
+(lift h TMP_316 t1) in (let TMP_318 \def (lift_head k TMP_315 TMP_317 k0 e)
+in (let TMP_319 \def (eq_ind_r T TMP_257 TMP_267 TMP_309 TMP_314 TMP_318) in
+(let TMP_320 \def (THead k t0 t1) in (let TMP_321 \def (lift h d TMP_320) in
+(let TMP_322 \def (lift_head k t0 t1 h d) in (let TMP_323 \def (eq_ind_r T
+TMP_239 TMP_250 TMP_319 TMP_321 TMP_322) in (let TMP_324 \def (plus k0 d) in
+(let TMP_325 \def (lift k0 e t0) in (let TMP_326 \def (s k e) in (let TMP_327
+\def (lift k0 TMP_326 t1) in (let TMP_328 \def (THead k TMP_325 TMP_327) in
+(let TMP_329 \def (lift h TMP_324 TMP_328) in (let TMP_330 \def (lift k0 e
+t0) in (let TMP_331 \def (s k e) in (let TMP_332 \def (lift k0 TMP_331 t1) in
+(let TMP_333 \def (plus k0 d) in (let TMP_334 \def (lift_head k TMP_330
+TMP_332 h TMP_333) in (let TMP_335 \def (eq_ind_r T TMP_231 TMP_235 TMP_323
+TMP_329 TMP_334) in (let TMP_336 \def (THead k t0 t1) in (let TMP_337 \def
+(lift k0 e TMP_336) in (let TMP_338 \def (lift_head k t0 t1 k0 e) in
+(eq_ind_r T TMP_216 TMP_222 TMP_335 TMP_337
+TMP_338)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))) in (T_ind TMP_6 TMP_45 TMP_212 TMP_339 t))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/lift/fwd.ma".
+include "basic_1/lift/fwd.ma".
-include "Basic-1/tlt/props.ma".
+include "basic_1/tlt/props.ma".
theorem lift_weight_map:
\forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
(weight_map f (lift h d t)) (weight_map f t))))))
\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d:
-nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
-(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0)))))))
-(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
-nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m)
-O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n:
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
-nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m)
-O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f
-(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat
-(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0)))
-(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq
-nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda
-(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0))
-(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda
+ \lambda (t: T).(let TMP_4 \def (\lambda (t0: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq
+nat (f m) O)))) \to (let TMP_1 \def (lift h d t0) in (let TMP_2 \def
+(weight_map f TMP_1) in (let TMP_3 \def (weight_map f t0) in (eq nat TMP_2
+TMP_3))))))))) in (let TMP_7 \def (\lambda (n: nat).(\lambda (_:
+nat).(\lambda (d: nat).(\lambda (f: ((nat \to nat))).(\lambda (_: ((\forall
+(m: nat).((le d m) \to (eq nat (f m) O))))).(let TMP_5 \def (TSort n) in (let
+TMP_6 \def (weight_map f TMP_5) in (refl_equal nat TMP_6)))))))) in (let
+TMP_45 \def (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(f: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f
+m) O))))).(let TMP_8 \def (TLRef n) in (let TMP_9 \def (lift h d TMP_8) in
+(let TMP_10 \def (weight_map f TMP_9) in (let TMP_11 \def (TLRef n) in (let
+TMP_12 \def (weight_map f TMP_11) in (let TMP_13 \def (eq nat TMP_10 TMP_12)
+in (let TMP_25 \def (\lambda (H0: (lt n d)).(let TMP_14 \def (TLRef n) in
+(let TMP_18 \def (\lambda (t0: T).(let TMP_15 \def (weight_map f t0) in (let
+TMP_16 \def (TLRef n) in (let TMP_17 \def (weight_map f TMP_16) in (eq nat
+TMP_15 TMP_17))))) in (let TMP_19 \def (TLRef n) in (let TMP_20 \def
+(weight_map f TMP_19) in (let TMP_21 \def (refl_equal nat TMP_20) in (let
+TMP_22 \def (TLRef n) in (let TMP_23 \def (lift h d TMP_22) in (let TMP_24
+\def (lift_lref_lt n h d H0) in (eq_ind_r T TMP_14 TMP_18 TMP_21 TMP_23
+TMP_24)))))))))) in (let TMP_44 \def (\lambda (H0: (le d n)).(let TMP_26 \def
+(plus n h) in (let TMP_27 \def (TLRef TMP_26) in (let TMP_31 \def (\lambda
+(t0: T).(let TMP_28 \def (weight_map f t0) in (let TMP_29 \def (TLRef n) in
+(let TMP_30 \def (weight_map f TMP_29) in (eq nat TMP_28 TMP_30))))) in (let
+TMP_34 \def (\lambda (n0: nat).(let TMP_32 \def (plus n h) in (let TMP_33
+\def (f TMP_32) in (eq nat TMP_33 n0)))) in (let TMP_35 \def (plus n h) in
+(let TMP_36 \def (le_plus_trans d n h H0) in (let TMP_37 \def (H TMP_35
+TMP_36) in (let TMP_38 \def (f n) in (let TMP_39 \def (H n H0) in (let TMP_40
+\def (eq_ind_r nat O TMP_34 TMP_37 TMP_38 TMP_39) in (let TMP_41 \def (TLRef
+n) in (let TMP_42 \def (lift h d TMP_41) in (let TMP_43 \def (lift_lref_ge n
+h d H0) in (eq_ind_r T TMP_27 TMP_31 TMP_40 TMP_42 TMP_43))))))))))))))) in
+(lt_le_e n d TMP_13 TMP_25 TMP_44)))))))))))))) in (let TMP_325 \def (\lambda
(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d:
nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f
(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f
t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m)
-O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
-t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead
-(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat
-(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda
-(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift
-h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d)
-t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map
-(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f
-(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with
-[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S
-(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0)
-(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0)
-(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n:
-nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S
-(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))))
-(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n:
-nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0)
-(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus
-(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))
-(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d)
-(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d)
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
-n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda
-(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda
-(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3))))
-(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r
-nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map
-f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O)
-t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
-(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2
-nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map
-(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat
-(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h
-(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat
-(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd
-f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le
-d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x
-H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1)
-(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus
-(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus
-(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f
-t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f
-(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd
-f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1))))
-(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m:
-nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
-n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x:
-nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S
-x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d
-m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h
-d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s
-(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f
-(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d
-t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1))
-(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0)
-(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1)))
-(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d)))
-k)))))))))) t).
-(* COMMENTS
-Initial nodes: 1969
-END *)
+O))))).(let TMP_51 \def (\lambda (k0: K).(let TMP_46 \def (THead k0 t0 t1) in
+(let TMP_47 \def (lift h d TMP_46) in (let TMP_48 \def (weight_map f TMP_47)
+in (let TMP_49 \def (THead k0 t0 t1) in (let TMP_50 \def (weight_map f
+TMP_49) in (eq nat TMP_48 TMP_50))))))) in (let TMP_289 \def (\lambda (b:
+B).(let TMP_52 \def (Bind b) in (let TMP_53 \def (lift h d t0) in (let TMP_54
+\def (Bind b) in (let TMP_55 \def (s TMP_54 d) in (let TMP_56 \def (lift h
+TMP_55 t1) in (let TMP_57 \def (THead TMP_52 TMP_53 TMP_56) in (let TMP_62
+\def (\lambda (t2: T).(let TMP_58 \def (weight_map f t2) in (let TMP_59 \def
+(Bind b) in (let TMP_60 \def (THead TMP_59 t0 t1) in (let TMP_61 \def
+(weight_map f TMP_60) in (eq nat TMP_58 TMP_61)))))) in (let TMP_103 \def
+(\lambda (b0: B).(let TMP_87 \def (match b0 with [Abbr \Rightarrow (let
+TMP_77 \def (lift h d t0) in (let TMP_78 \def (weight_map f TMP_77) in (let
+TMP_79 \def (lift h d t0) in (let TMP_80 \def (weight_map f TMP_79) in (let
+TMP_81 \def (S TMP_80) in (let TMP_82 \def (wadd f TMP_81) in (let TMP_83
+\def (S d) in (let TMP_84 \def (lift h TMP_83 t1) in (let TMP_85 \def
+(weight_map TMP_82 TMP_84) in (let TMP_86 \def (plus TMP_78 TMP_85) in (S
+TMP_86))))))))))) | Abst \Rightarrow (let TMP_70 \def (lift h d t0) in (let
+TMP_71 \def (weight_map f TMP_70) in (let TMP_72 \def (wadd f O) in (let
+TMP_73 \def (S d) in (let TMP_74 \def (lift h TMP_73 t1) in (let TMP_75 \def
+(weight_map TMP_72 TMP_74) in (let TMP_76 \def (plus TMP_71 TMP_75) in (S
+TMP_76)))))))) | Void \Rightarrow (let TMP_63 \def (lift h d t0) in (let
+TMP_64 \def (weight_map f TMP_63) in (let TMP_65 \def (wadd f O) in (let
+TMP_66 \def (S d) in (let TMP_67 \def (lift h TMP_66 t1) in (let TMP_68 \def
+(weight_map TMP_65 TMP_67) in (let TMP_69 \def (plus TMP_64 TMP_68) in (S
+TMP_69))))))))]) in (let TMP_102 \def (match b0 with [Abbr \Rightarrow (let
+TMP_96 \def (weight_map f t0) in (let TMP_97 \def (weight_map f t0) in (let
+TMP_98 \def (S TMP_97) in (let TMP_99 \def (wadd f TMP_98) in (let TMP_100
+\def (weight_map TMP_99 t1) in (let TMP_101 \def (plus TMP_96 TMP_100) in (S
+TMP_101))))))) | Abst \Rightarrow (let TMP_92 \def (weight_map f t0) in (let
+TMP_93 \def (wadd f O) in (let TMP_94 \def (weight_map TMP_93 t1) in (let
+TMP_95 \def (plus TMP_92 TMP_94) in (S TMP_95))))) | Void \Rightarrow (let
+TMP_88 \def (weight_map f t0) in (let TMP_89 \def (wadd f O) in (let TMP_90
+\def (weight_map TMP_89 t1) in (let TMP_91 \def (plus TMP_88 TMP_90) in (S
+TMP_91)))))]) in (eq nat TMP_87 TMP_102)))) in (let TMP_104 \def (weight_map
+f t0) in (let TMP_119 \def (\lambda (n: nat).(let TMP_105 \def (S n) in (let
+TMP_106 \def (wadd f TMP_105) in (let TMP_107 \def (S d) in (let TMP_108 \def
+(lift h TMP_107 t1) in (let TMP_109 \def (weight_map TMP_106 TMP_108) in (let
+TMP_110 \def (plus n TMP_109) in (let TMP_111 \def (S TMP_110) in (let
+TMP_112 \def (weight_map f t0) in (let TMP_113 \def (weight_map f t0) in (let
+TMP_114 \def (S TMP_113) in (let TMP_115 \def (wadd f TMP_114) in (let
+TMP_116 \def (weight_map TMP_115 t1) in (let TMP_117 \def (plus TMP_112
+TMP_116) in (let TMP_118 \def (S TMP_117) in (eq nat TMP_111
+TMP_118)))))))))))))))) in (let TMP_120 \def (weight_map f t0) in (let
+TMP_121 \def (S TMP_120) in (let TMP_122 \def (wadd f TMP_121) in (let
+TMP_123 \def (weight_map TMP_122 t1) in (let TMP_134 \def (\lambda (n:
+nat).(let TMP_124 \def (weight_map f t0) in (let TMP_125 \def (plus TMP_124
+n) in (let TMP_126 \def (S TMP_125) in (let TMP_127 \def (weight_map f t0) in
+(let TMP_128 \def (weight_map f t0) in (let TMP_129 \def (S TMP_128) in (let
+TMP_130 \def (wadd f TMP_129) in (let TMP_131 \def (weight_map TMP_130 t1) in
+(let TMP_132 \def (plus TMP_127 TMP_131) in (let TMP_133 \def (S TMP_132) in
+(eq nat TMP_126 TMP_133)))))))))))) in (let TMP_135 \def (weight_map f t0) in
+(let TMP_136 \def (weight_map f t0) in (let TMP_137 \def (S TMP_136) in (let
+TMP_138 \def (wadd f TMP_137) in (let TMP_139 \def (weight_map TMP_138 t1) in
+(let TMP_140 \def (plus TMP_135 TMP_139) in (let TMP_141 \def (S TMP_140) in
+(let TMP_142 \def (refl_equal nat TMP_141) in (let TMP_143 \def (weight_map f
+t0) in (let TMP_144 \def (S TMP_143) in (let TMP_145 \def (wadd f TMP_144) in
+(let TMP_146 \def (S d) in (let TMP_147 \def (lift h TMP_146 t1) in (let
+TMP_148 \def (weight_map TMP_145 TMP_147) in (let TMP_149 \def (S d) in (let
+TMP_150 \def (weight_map f t0) in (let TMP_151 \def (S TMP_150) in (let
+TMP_152 \def (wadd f TMP_151) in (let TMP_168 \def (\lambda (m: nat).(\lambda
+(H2: (le (S d) m)).(let TMP_154 \def (\lambda (n: nat).(let TMP_153 \def (S
+n) in (eq nat m TMP_153))) in (let TMP_155 \def (\lambda (n: nat).(le d n))
+in (let TMP_156 \def (weight_map f t0) in (let TMP_157 \def (S TMP_156) in
+(let TMP_158 \def (wadd f TMP_157 m) in (let TMP_159 \def (eq nat TMP_158 O)
+in (let TMP_166 \def (\lambda (x: nat).(\lambda (H3: (eq nat m (S
+x))).(\lambda (H4: (le d x)).(let TMP_160 \def (S x) in (let TMP_164 \def
+(\lambda (n: nat).(let TMP_161 \def (weight_map f t0) in (let TMP_162 \def (S
+TMP_161) in (let TMP_163 \def (wadd f TMP_162 n) in (eq nat TMP_163 O))))) in
+(let TMP_165 \def (H1 x H4) in (eq_ind_r nat TMP_160 TMP_164 TMP_165 m
+H3))))))) in (let TMP_167 \def (le_gen_S d m H2) in (ex2_ind nat TMP_154
+TMP_155 TMP_159 TMP_166 TMP_167))))))))))) in (let TMP_169 \def (H0 h TMP_149
+TMP_152 TMP_168) in (let TMP_170 \def (eq_ind_r nat TMP_123 TMP_134 TMP_142
+TMP_148 TMP_169) in (let TMP_171 \def (lift h d t0) in (let TMP_172 \def
+(weight_map f TMP_171) in (let TMP_173 \def (H h d f H1) in (let TMP_174 \def
+(eq_ind_r nat TMP_104 TMP_119 TMP_170 TMP_172 TMP_173) in (let TMP_175 \def
+(wadd f O) in (let TMP_176 \def (weight_map TMP_175 t1) in (let TMP_186 \def
+(\lambda (n: nat).(let TMP_177 \def (lift h d t0) in (let TMP_178 \def
+(weight_map f TMP_177) in (let TMP_179 \def (plus TMP_178 n) in (let TMP_180
+\def (S TMP_179) in (let TMP_181 \def (weight_map f t0) in (let TMP_182 \def
+(wadd f O) in (let TMP_183 \def (weight_map TMP_182 t1) in (let TMP_184 \def
+(plus TMP_181 TMP_183) in (let TMP_185 \def (S TMP_184) in (eq nat TMP_180
+TMP_185))))))))))) in (let TMP_187 \def (lift h d t0) in (let TMP_188 \def
+(weight_map f TMP_187) in (let TMP_189 \def (wadd f O) in (let TMP_190 \def
+(weight_map TMP_189 t1) in (let TMP_191 \def (plus TMP_188 TMP_190) in (let
+TMP_192 \def (weight_map f t0) in (let TMP_193 \def (wadd f O) in (let
+TMP_194 \def (weight_map TMP_193 t1) in (let TMP_195 \def (plus TMP_192
+TMP_194) in (let TMP_196 \def (lift h d t0) in (let TMP_197 \def (weight_map
+f TMP_196) in (let TMP_198 \def (weight_map f t0) in (let TMP_199 \def (wadd
+f O) in (let TMP_200 \def (weight_map TMP_199 t1) in (let TMP_201 \def (wadd
+f O) in (let TMP_202 \def (weight_map TMP_201 t1) in (let TMP_203 \def (H h d
+f H1) in (let TMP_204 \def (wadd f O) in (let TMP_205 \def (weight_map
+TMP_204 t1) in (let TMP_206 \def (refl_equal nat TMP_205) in (let TMP_207
+\def (f_equal2 nat nat nat plus TMP_197 TMP_198 TMP_200 TMP_202 TMP_203
+TMP_206) in (let TMP_208 \def (f_equal nat nat S TMP_191 TMP_195 TMP_207) in
+(let TMP_209 \def (wadd f O) in (let TMP_210 \def (S d) in (let TMP_211 \def
+(lift h TMP_210 t1) in (let TMP_212 \def (weight_map TMP_209 TMP_211) in (let
+TMP_213 \def (S d) in (let TMP_214 \def (wadd f O) in (let TMP_226 \def
+(\lambda (m: nat).(\lambda (H2: (le (S d) m)).(let TMP_216 \def (\lambda (n:
+nat).(let TMP_215 \def (S n) in (eq nat m TMP_215))) in (let TMP_217 \def
+(\lambda (n: nat).(le d n)) in (let TMP_218 \def (wadd f O m) in (let TMP_219
+\def (eq nat TMP_218 O) in (let TMP_224 \def (\lambda (x: nat).(\lambda (H3:
+(eq nat m (S x))).(\lambda (H4: (le d x)).(let TMP_220 \def (S x) in (let
+TMP_222 \def (\lambda (n: nat).(let TMP_221 \def (wadd f O n) in (eq nat
+TMP_221 O))) in (let TMP_223 \def (H1 x H4) in (eq_ind_r nat TMP_220 TMP_222
+TMP_223 m H3))))))) in (let TMP_225 \def (le_gen_S d m H2) in (ex2_ind nat
+TMP_216 TMP_217 TMP_219 TMP_224 TMP_225))))))))) in (let TMP_227 \def (H0 h
+TMP_213 TMP_214 TMP_226) in (let TMP_228 \def (eq_ind_r nat TMP_176 TMP_186
+TMP_208 TMP_212 TMP_227) in (let TMP_229 \def (wadd f O) in (let TMP_230 \def
+(weight_map TMP_229 t1) in (let TMP_240 \def (\lambda (n: nat).(let TMP_231
+\def (lift h d t0) in (let TMP_232 \def (weight_map f TMP_231) in (let
+TMP_233 \def (plus TMP_232 n) in (let TMP_234 \def (S TMP_233) in (let
+TMP_235 \def (weight_map f t0) in (let TMP_236 \def (wadd f O) in (let
+TMP_237 \def (weight_map TMP_236 t1) in (let TMP_238 \def (plus TMP_235
+TMP_237) in (let TMP_239 \def (S TMP_238) in (eq nat TMP_234
+TMP_239))))))))))) in (let TMP_241 \def (lift h d t0) in (let TMP_242 \def
+(weight_map f TMP_241) in (let TMP_243 \def (wadd f O) in (let TMP_244 \def
+(weight_map TMP_243 t1) in (let TMP_245 \def (plus TMP_242 TMP_244) in (let
+TMP_246 \def (weight_map f t0) in (let TMP_247 \def (wadd f O) in (let
+TMP_248 \def (weight_map TMP_247 t1) in (let TMP_249 \def (plus TMP_246
+TMP_248) in (let TMP_250 \def (lift h d t0) in (let TMP_251 \def (weight_map
+f TMP_250) in (let TMP_252 \def (weight_map f t0) in (let TMP_253 \def (wadd
+f O) in (let TMP_254 \def (weight_map TMP_253 t1) in (let TMP_255 \def (wadd
+f O) in (let TMP_256 \def (weight_map TMP_255 t1) in (let TMP_257 \def (H h d
+f H1) in (let TMP_258 \def (wadd f O) in (let TMP_259 \def (weight_map
+TMP_258 t1) in (let TMP_260 \def (refl_equal nat TMP_259) in (let TMP_261
+\def (f_equal2 nat nat nat plus TMP_251 TMP_252 TMP_254 TMP_256 TMP_257
+TMP_260) in (let TMP_262 \def (f_equal nat nat S TMP_245 TMP_249 TMP_261) in
+(let TMP_263 \def (wadd f O) in (let TMP_264 \def (S d) in (let TMP_265 \def
+(lift h TMP_264 t1) in (let TMP_266 \def (weight_map TMP_263 TMP_265) in (let
+TMP_267 \def (S d) in (let TMP_268 \def (wadd f O) in (let TMP_280 \def
+(\lambda (m: nat).(\lambda (H2: (le (S d) m)).(let TMP_270 \def (\lambda (n:
+nat).(let TMP_269 \def (S n) in (eq nat m TMP_269))) in (let TMP_271 \def
+(\lambda (n: nat).(le d n)) in (let TMP_272 \def (wadd f O m) in (let TMP_273
+\def (eq nat TMP_272 O) in (let TMP_278 \def (\lambda (x: nat).(\lambda (H3:
+(eq nat m (S x))).(\lambda (H4: (le d x)).(let TMP_274 \def (S x) in (let
+TMP_276 \def (\lambda (n: nat).(let TMP_275 \def (wadd f O n) in (eq nat
+TMP_275 O))) in (let TMP_277 \def (H1 x H4) in (eq_ind_r nat TMP_274 TMP_276
+TMP_277 m H3))))))) in (let TMP_279 \def (le_gen_S d m H2) in (ex2_ind nat
+TMP_270 TMP_271 TMP_273 TMP_278 TMP_279))))))))) in (let TMP_281 \def (H0 h
+TMP_267 TMP_268 TMP_280) in (let TMP_282 \def (eq_ind_r nat TMP_230 TMP_240
+TMP_262 TMP_266 TMP_281) in (let TMP_283 \def (B_ind TMP_103 TMP_174 TMP_228
+TMP_282 b) in (let TMP_284 \def (Bind b) in (let TMP_285 \def (THead TMP_284
+t0 t1) in (let TMP_286 \def (lift h d TMP_285) in (let TMP_287 \def (Bind b)
+in (let TMP_288 \def (lift_head TMP_287 t0 t1 h d) in (eq_ind_r T TMP_57
+TMP_62 TMP_283 TMP_286
+TMP_288)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))))))))))))) in (let TMP_324 \def (\lambda
+(f0: F).(let TMP_290 \def (Flat f0) in (let TMP_291 \def (lift h d t0) in
+(let TMP_292 \def (Flat f0) in (let TMP_293 \def (s TMP_292 d) in (let
+TMP_294 \def (lift h TMP_293 t1) in (let TMP_295 \def (THead TMP_290 TMP_291
+TMP_294) in (let TMP_300 \def (\lambda (t2: T).(let TMP_296 \def (weight_map
+f t2) in (let TMP_297 \def (Flat f0) in (let TMP_298 \def (THead TMP_297 t0
+t1) in (let TMP_299 \def (weight_map f TMP_298) in (eq nat TMP_296
+TMP_299)))))) in (let TMP_301 \def (lift h d t0) in (let TMP_302 \def
+(weight_map f TMP_301) in (let TMP_303 \def (lift h d t1) in (let TMP_304
+\def (weight_map f TMP_303) in (let TMP_305 \def (plus TMP_302 TMP_304) in
+(let TMP_306 \def (weight_map f t0) in (let TMP_307 \def (weight_map f t1) in
+(let TMP_308 \def (plus TMP_306 TMP_307) in (let TMP_309 \def (lift h d t0)
+in (let TMP_310 \def (weight_map f TMP_309) in (let TMP_311 \def (weight_map
+f t0) in (let TMP_312 \def (lift h d t1) in (let TMP_313 \def (weight_map f
+TMP_312) in (let TMP_314 \def (weight_map f t1) in (let TMP_315 \def (H h d f
+H1) in (let TMP_316 \def (H0 h d f H1) in (let TMP_317 \def (f_equal2 nat nat
+nat plus TMP_310 TMP_311 TMP_313 TMP_314 TMP_315 TMP_316) in (let TMP_318
+\def (f_equal nat nat S TMP_305 TMP_308 TMP_317) in (let TMP_319 \def (Flat
+f0) in (let TMP_320 \def (THead TMP_319 t0 t1) in (let TMP_321 \def (lift h d
+TMP_320) in (let TMP_322 \def (Flat f0) in (let TMP_323 \def (lift_head
+TMP_322 t0 t1 h d) in (eq_ind_r T TMP_295 TMP_300 TMP_318 TMP_321
+TMP_323)))))))))))))))))))))))))))))))) in (K_ind TMP_51 TMP_289 TMP_324
+k))))))))))))) in (T_ind TMP_4 TMP_7 TMP_45 TMP_325 t))))).
theorem lift_weight:
\forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d
t)) (weight t))))
\def
- \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d
-(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat
-O)))))).
-(* COMMENTS
-Initial nodes: 31
-END *)
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(let TMP_1 \def (\lambda
+(_: nat).O) in (let TMP_2 \def (\lambda (m: nat).(\lambda (_: (le d
+m)).(refl_equal nat O))) in (lift_weight_map t h d TMP_1 TMP_2))))).
theorem lift_weight_add:
\forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d:
(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat
(weight_map f (lift h d t)) (weight_map g (lift (S h) d t)))))))))))
\def
- \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h:
-nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat
-(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))
-\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d
-t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+ \lambda (w: nat).(\lambda (t: T).(let TMP_6 \def (\lambda (t0: T).(\forall
+(h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat
+\to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq
+nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
+m))))) \to (let TMP_1 \def (lift h d t0) in (let TMP_2 \def (weight_map f
+TMP_1) in (let TMP_3 \def (S h) in (let TMP_4 \def (lift TMP_3 d t0) in (let
+TMP_5 \def (weight_map g TMP_4) in (eq nat TMP_2 TMP_5)))))))))))))) in (let
+TMP_11 \def (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda
(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m:
nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d)
w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
-m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n))))))))))))
-(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m
-d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1:
-((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d
-(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d
-(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n))
-(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef
-n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d
-H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0:
-T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
-(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f
-(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda
-(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f
-(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h))
-(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift
-h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda
-(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat
-\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to
-(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d
-m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0))
-(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0:
-((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall
-(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f
-m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g
-(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift
-(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m:
-nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d)
-w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
-m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
-t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b:
-B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1))
-(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead
-(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h)
-(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b)
-(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind
-(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus
-(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d
-t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h
-d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S
-(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
-t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h)
-d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h)
-(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0))
-(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
-t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
-(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift
-(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map
-f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S
-(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S
-(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2
-H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S
-(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S
-d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
-(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d
-t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m
-O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift
-(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat
-nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq
-nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g
-H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S
-m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat
-m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g
-(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda
-(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r
-nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d
-t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6))))
-H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
-n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x:
-nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
-x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0)))
-n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus
-(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
-t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g
-(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map
-(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O)
-(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O)
-(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))
-(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat
-O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m
-H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda
-(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0)))
-(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x:
-nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S
-x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6))))
-H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
-n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S
-x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g
-n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat
-S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
-t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S
-h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0))
-(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1))
-(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S
-d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S
-d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
-(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda
-(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n)
-(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0:
-nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda
-(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O
-m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda
-(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n)
-(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m:
-nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
-n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x:
-nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
-x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5))))
-(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head
-(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind
-b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0)
-(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2)
-(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead
-(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2:
-T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0)
-d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d
-t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0))
-(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f
-(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1))
-(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3)))
-(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d))
-(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d)))
-k))))))))))))) t)).
-(* COMMENTS
-Initial nodes: 3697
-END *)
+m)))))).(let TMP_7 \def (S h) in (let TMP_8 \def (TSort n) in (let TMP_9 \def
+(lift TMP_7 d TMP_8) in (let TMP_10 \def (weight_map g TMP_9) in (refl_equal
+nat TMP_10))))))))))))) in (let TMP_91 \def (\lambda (n: nat).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H: ((\forall (m: nat).((lt m d) \to (eq nat (g m) (f
+m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: ((\forall (m: nat).((le
+d m) \to (eq nat (g (S m)) (f m)))))).(let TMP_12 \def (TLRef n) in (let
+TMP_13 \def (lift h d TMP_12) in (let TMP_14 \def (weight_map f TMP_13) in
+(let TMP_15 \def (S h) in (let TMP_16 \def (TLRef n) in (let TMP_17 \def
+(lift TMP_15 d TMP_16) in (let TMP_18 \def (weight_map g TMP_17) in (let
+TMP_19 \def (eq nat TMP_14 TMP_18) in (let TMP_45 \def (\lambda (H2: (lt n
+d)).(let TMP_20 \def (TLRef n) in (let TMP_26 \def (\lambda (t0: T).(let
+TMP_21 \def (weight_map f t0) in (let TMP_22 \def (S h) in (let TMP_23 \def
+(TLRef n) in (let TMP_24 \def (lift TMP_22 d TMP_23) in (let TMP_25 \def
+(weight_map g TMP_24) in (eq nat TMP_21 TMP_25))))))) in (let TMP_27 \def
+(TLRef n) in (let TMP_31 \def (\lambda (t0: T).(let TMP_28 \def (TLRef n) in
+(let TMP_29 \def (weight_map f TMP_28) in (let TMP_30 \def (weight_map g t0)
+in (eq nat TMP_29 TMP_30))))) in (let TMP_32 \def (g n) in (let TMP_33 \def
+(f n) in (let TMP_34 \def (H n H2) in (let TMP_35 \def (sym_eq nat TMP_32
+TMP_33 TMP_34) in (let TMP_36 \def (S h) in (let TMP_37 \def (TLRef n) in
+(let TMP_38 \def (lift TMP_36 d TMP_37) in (let TMP_39 \def (S h) in (let
+TMP_40 \def (lift_lref_lt n TMP_39 d H2) in (let TMP_41 \def (eq_ind_r T
+TMP_27 TMP_31 TMP_35 TMP_38 TMP_40) in (let TMP_42 \def (TLRef n) in (let
+TMP_43 \def (lift h d TMP_42) in (let TMP_44 \def (lift_lref_lt n h d H2) in
+(eq_ind_r T TMP_20 TMP_26 TMP_41 TMP_43 TMP_44))))))))))))))))))) in (let
+TMP_90 \def (\lambda (H2: (le d n)).(let TMP_46 \def (plus n h) in (let
+TMP_47 \def (TLRef TMP_46) in (let TMP_53 \def (\lambda (t0: T).(let TMP_48
+\def (weight_map f t0) in (let TMP_49 \def (S h) in (let TMP_50 \def (TLRef
+n) in (let TMP_51 \def (lift TMP_49 d TMP_50) in (let TMP_52 \def (weight_map
+g TMP_51) in (eq nat TMP_48 TMP_52))))))) in (let TMP_54 \def (S h) in (let
+TMP_55 \def (plus n TMP_54) in (let TMP_56 \def (TLRef TMP_55) in (let TMP_61
+\def (\lambda (t0: T).(let TMP_57 \def (plus n h) in (let TMP_58 \def (TLRef
+TMP_57) in (let TMP_59 \def (weight_map f TMP_58) in (let TMP_60 \def
+(weight_map g t0) in (eq nat TMP_59 TMP_60)))))) in (let TMP_62 \def (plus n
+h) in (let TMP_63 \def (S TMP_62) in (let TMP_67 \def (\lambda (n0: nat).(let
+TMP_64 \def (plus n h) in (let TMP_65 \def (f TMP_64) in (let TMP_66 \def (g
+n0) in (eq nat TMP_65 TMP_66))))) in (let TMP_68 \def (plus n h) in (let
+TMP_69 \def (S TMP_68) in (let TMP_70 \def (g TMP_69) in (let TMP_71 \def
+(plus n h) in (let TMP_72 \def (f TMP_71) in (let TMP_73 \def (plus n h) in
+(let TMP_74 \def (le_plus_trans d n h H2) in (let TMP_75 \def (H1 TMP_73
+TMP_74) in (let TMP_76 \def (sym_eq nat TMP_70 TMP_72 TMP_75) in (let TMP_77
+\def (S h) in (let TMP_78 \def (plus n TMP_77) in (let TMP_79 \def (plus_n_Sm
+n h) in (let TMP_80 \def (eq_ind nat TMP_63 TMP_67 TMP_76 TMP_78 TMP_79) in
+(let TMP_81 \def (S h) in (let TMP_82 \def (TLRef n) in (let TMP_83 \def
+(lift TMP_81 d TMP_82) in (let TMP_84 \def (S h) in (let TMP_85 \def
+(lift_lref_ge n TMP_84 d H2) in (let TMP_86 \def (eq_ind_r T TMP_56 TMP_61
+TMP_80 TMP_83 TMP_85) in (let TMP_87 \def (TLRef n) in (let TMP_88 \def (lift
+h d TMP_87) in (let TMP_89 \def (lift_lref_ge n h d H2) in (eq_ind_r T TMP_47
+TMP_53 TMP_86 TMP_88 TMP_89)))))))))))))))))))))))))))))))))) in (lt_le_e n d
+TMP_19 TMP_45 TMP_90))))))))))))))))))) in (let TMP_577 \def (\lambda (k:
+K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to
+(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat
+(weight_map f (lift h d t0)) (weight_map g (lift (S h) d
+t0)))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to
+(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat
+(weight_map f (lift h d t1)) (weight_map g (lift (S h) d
+t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m: nat).((lt m
+d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d) w)).(\lambda (H3:
+((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(let TMP_99
+\def (\lambda (k0: K).(let TMP_92 \def (THead k0 t0 t1) in (let TMP_93 \def
+(lift h d TMP_92) in (let TMP_94 \def (weight_map f TMP_93) in (let TMP_95
+\def (S h) in (let TMP_96 \def (THead k0 t0 t1) in (let TMP_97 \def (lift
+TMP_95 d TMP_96) in (let TMP_98 \def (weight_map g TMP_97) in (eq nat TMP_94
+TMP_98))))))))) in (let TMP_506 \def (\lambda (b: B).(let TMP_100 \def (Bind
+b) in (let TMP_101 \def (lift h d t0) in (let TMP_102 \def (Bind b) in (let
+TMP_103 \def (s TMP_102 d) in (let TMP_104 \def (lift h TMP_103 t1) in (let
+TMP_105 \def (THead TMP_100 TMP_101 TMP_104) in (let TMP_112 \def (\lambda
+(t2: T).(let TMP_106 \def (weight_map f t2) in (let TMP_107 \def (S h) in
+(let TMP_108 \def (Bind b) in (let TMP_109 \def (THead TMP_108 t0 t1) in (let
+TMP_110 \def (lift TMP_107 d TMP_109) in (let TMP_111 \def (weight_map g
+TMP_110) in (eq nat TMP_106 TMP_111)))))))) in (let TMP_113 \def (Bind b) in
+(let TMP_114 \def (S h) in (let TMP_115 \def (lift TMP_114 d t0) in (let
+TMP_116 \def (S h) in (let TMP_117 \def (Bind b) in (let TMP_118 \def (s
+TMP_117 d) in (let TMP_119 \def (lift TMP_116 TMP_118 t1) in (let TMP_120
+\def (THead TMP_113 TMP_115 TMP_119) in (let TMP_129 \def (\lambda (t2:
+T).(let TMP_121 \def (Bind b) in (let TMP_122 \def (lift h d t0) in (let
+TMP_123 \def (Bind b) in (let TMP_124 \def (s TMP_123 d) in (let TMP_125 \def
+(lift h TMP_124 t1) in (let TMP_126 \def (THead TMP_121 TMP_122 TMP_125) in
+(let TMP_127 \def (weight_map f TMP_126) in (let TMP_128 \def (weight_map g
+t2) in (eq nat TMP_127 TMP_128)))))))))) in (let TMP_187 \def (\lambda (b0:
+B).(let TMP_154 \def (match b0 with [Abbr \Rightarrow (let TMP_144 \def (lift
+h d t0) in (let TMP_145 \def (weight_map f TMP_144) in (let TMP_146 \def
+(lift h d t0) in (let TMP_147 \def (weight_map f TMP_146) in (let TMP_148
+\def (S TMP_147) in (let TMP_149 \def (wadd f TMP_148) in (let TMP_150 \def
+(S d) in (let TMP_151 \def (lift h TMP_150 t1) in (let TMP_152 \def
+(weight_map TMP_149 TMP_151) in (let TMP_153 \def (plus TMP_145 TMP_152) in
+(S TMP_153))))))))))) | Abst \Rightarrow (let TMP_137 \def (lift h d t0) in
+(let TMP_138 \def (weight_map f TMP_137) in (let TMP_139 \def (wadd f O) in
+(let TMP_140 \def (S d) in (let TMP_141 \def (lift h TMP_140 t1) in (let
+TMP_142 \def (weight_map TMP_139 TMP_141) in (let TMP_143 \def (plus TMP_138
+TMP_142) in (S TMP_143)))))))) | Void \Rightarrow (let TMP_130 \def (lift h d
+t0) in (let TMP_131 \def (weight_map f TMP_130) in (let TMP_132 \def (wadd f
+O) in (let TMP_133 \def (S d) in (let TMP_134 \def (lift h TMP_133 t1) in
+(let TMP_135 \def (weight_map TMP_132 TMP_134) in (let TMP_136 \def (plus
+TMP_131 TMP_135) in (S TMP_136))))))))]) in (let TMP_186 \def (match b0 with
+[Abbr \Rightarrow (let TMP_173 \def (S h) in (let TMP_174 \def (lift TMP_173
+d t0) in (let TMP_175 \def (weight_map g TMP_174) in (let TMP_176 \def (S h)
+in (let TMP_177 \def (lift TMP_176 d t0) in (let TMP_178 \def (weight_map g
+TMP_177) in (let TMP_179 \def (S TMP_178) in (let TMP_180 \def (wadd g
+TMP_179) in (let TMP_181 \def (S h) in (let TMP_182 \def (S d) in (let
+TMP_183 \def (lift TMP_181 TMP_182 t1) in (let TMP_184 \def (weight_map
+TMP_180 TMP_183) in (let TMP_185 \def (plus TMP_175 TMP_184) in (S
+TMP_185)))))))))))))) | Abst \Rightarrow (let TMP_164 \def (S h) in (let
+TMP_165 \def (lift TMP_164 d t0) in (let TMP_166 \def (weight_map g TMP_165)
+in (let TMP_167 \def (wadd g O) in (let TMP_168 \def (S h) in (let TMP_169
+\def (S d) in (let TMP_170 \def (lift TMP_168 TMP_169 t1) in (let TMP_171
+\def (weight_map TMP_167 TMP_170) in (let TMP_172 \def (plus TMP_166 TMP_171)
+in (S TMP_172)))))))))) | Void \Rightarrow (let TMP_155 \def (S h) in (let
+TMP_156 \def (lift TMP_155 d t0) in (let TMP_157 \def (weight_map g TMP_156)
+in (let TMP_158 \def (wadd g O) in (let TMP_159 \def (S h) in (let TMP_160
+\def (S d) in (let TMP_161 \def (lift TMP_159 TMP_160 t1) in (let TMP_162
+\def (weight_map TMP_158 TMP_161) in (let TMP_163 \def (plus TMP_157 TMP_162)
+in (S TMP_163))))))))))]) in (eq nat TMP_154 TMP_186)))) in (let TMP_188 \def
+(lift h d t0) in (let TMP_189 \def (weight_map f TMP_188) in (let TMP_190
+\def (lift h d t0) in (let TMP_191 \def (weight_map f TMP_190) in (let
+TMP_192 \def (S TMP_191) in (let TMP_193 \def (wadd f TMP_192) in (let
+TMP_194 \def (S d) in (let TMP_195 \def (lift h TMP_194 t1) in (let TMP_196
+\def (weight_map TMP_193 TMP_195) in (let TMP_197 \def (plus TMP_189 TMP_196)
+in (let TMP_198 \def (S h) in (let TMP_199 \def (lift TMP_198 d t0) in (let
+TMP_200 \def (weight_map g TMP_199) in (let TMP_201 \def (S h) in (let
+TMP_202 \def (lift TMP_201 d t0) in (let TMP_203 \def (weight_map g TMP_202)
+in (let TMP_204 \def (S TMP_203) in (let TMP_205 \def (wadd g TMP_204) in
+(let TMP_206 \def (S h) in (let TMP_207 \def (S d) in (let TMP_208 \def (lift
+TMP_206 TMP_207 t1) in (let TMP_209 \def (weight_map TMP_205 TMP_208) in (let
+TMP_210 \def (plus TMP_200 TMP_209) in (let TMP_211 \def (lift h d t0) in
+(let TMP_212 \def (weight_map f TMP_211) in (let TMP_213 \def (S h) in (let
+TMP_214 \def (lift TMP_213 d t0) in (let TMP_215 \def (weight_map g TMP_214)
+in (let TMP_216 \def (lift h d t0) in (let TMP_217 \def (weight_map f
+TMP_216) in (let TMP_218 \def (S TMP_217) in (let TMP_219 \def (wadd f
+TMP_218) in (let TMP_220 \def (S d) in (let TMP_221 \def (lift h TMP_220 t1)
+in (let TMP_222 \def (weight_map TMP_219 TMP_221) in (let TMP_223 \def (S h)
+in (let TMP_224 \def (lift TMP_223 d t0) in (let TMP_225 \def (weight_map g
+TMP_224) in (let TMP_226 \def (S TMP_225) in (let TMP_227 \def (wadd g
+TMP_226) in (let TMP_228 \def (S h) in (let TMP_229 \def (S d) in (let
+TMP_230 \def (lift TMP_228 TMP_229 t1) in (let TMP_231 \def (weight_map
+TMP_227 TMP_230) in (let TMP_232 \def (H h d f g H1 H2 H3) in (let TMP_233
+\def (S d) in (let TMP_234 \def (lift h d t0) in (let TMP_235 \def
+(weight_map f TMP_234) in (let TMP_236 \def (S TMP_235) in (let TMP_237 \def
+(wadd f TMP_236) in (let TMP_238 \def (S h) in (let TMP_239 \def (lift
+TMP_238 d t0) in (let TMP_240 \def (weight_map g TMP_239) in (let TMP_241
+\def (S TMP_240) in (let TMP_242 \def (wadd g TMP_241) in (let TMP_310 \def
+(\lambda (m: nat).(\lambda (H4: (lt m (S d))).(let TMP_243 \def (eq nat m O)
+in (let TMP_245 \def (\lambda (m0: nat).(let TMP_244 \def (S m0) in (eq nat m
+TMP_244))) in (let TMP_246 \def (\lambda (m0: nat).(lt m0 d)) in (let TMP_247
+\def (ex2 nat TMP_245 TMP_246) in (let TMP_248 \def (S h) in (let TMP_249
+\def (lift TMP_248 d t0) in (let TMP_250 \def (weight_map g TMP_249) in (let
+TMP_251 \def (S TMP_250) in (let TMP_252 \def (wadd g TMP_251 m) in (let
+TMP_253 \def (lift h d t0) in (let TMP_254 \def (weight_map f TMP_253) in
+(let TMP_255 \def (S TMP_254) in (let TMP_256 \def (wadd f TMP_255 m) in (let
+TMP_257 \def (eq nat TMP_252 TMP_256) in (let TMP_281 \def (\lambda (H5: (eq
+nat m O)).(let TMP_267 \def (\lambda (n: nat).(let TMP_258 \def (S h) in (let
+TMP_259 \def (lift TMP_258 d t0) in (let TMP_260 \def (weight_map g TMP_259)
+in (let TMP_261 \def (S TMP_260) in (let TMP_262 \def (wadd g TMP_261 n) in
+(let TMP_263 \def (lift h d t0) in (let TMP_264 \def (weight_map f TMP_263)
+in (let TMP_265 \def (S TMP_264) in (let TMP_266 \def (wadd f TMP_265 n) in
+(eq nat TMP_262 TMP_266))))))))))) in (let TMP_268 \def (S h) in (let TMP_269
+\def (lift TMP_268 d t0) in (let TMP_270 \def (weight_map g TMP_269) in (let
+TMP_271 \def (lift h d t0) in (let TMP_272 \def (weight_map f TMP_271) in
+(let TMP_273 \def (lift h d t0) in (let TMP_274 \def (weight_map f TMP_273)
+in (let TMP_275 \def (S h) in (let TMP_276 \def (lift TMP_275 d t0) in (let
+TMP_277 \def (weight_map g TMP_276) in (let TMP_278 \def (H h d f g H1 H2 H3)
+in (let TMP_279 \def (sym_eq nat TMP_274 TMP_277 TMP_278) in (let TMP_280
+\def (f_equal nat nat S TMP_270 TMP_272 TMP_279) in (eq_ind_r nat O TMP_267
+TMP_280 m H5)))))))))))))))) in (let TMP_308 \def (\lambda (H5: (ex2 nat
+(\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(let
+TMP_283 \def (\lambda (m0: nat).(let TMP_282 \def (S m0) in (eq nat m
+TMP_282))) in (let TMP_284 \def (\lambda (m0: nat).(lt m0 d)) in (let TMP_285
+\def (S h) in (let TMP_286 \def (lift TMP_285 d t0) in (let TMP_287 \def
+(weight_map g TMP_286) in (let TMP_288 \def (S TMP_287) in (let TMP_289 \def
+(wadd g TMP_288 m) in (let TMP_290 \def (lift h d t0) in (let TMP_291 \def
+(weight_map f TMP_290) in (let TMP_292 \def (S TMP_291) in (let TMP_293 \def
+(wadd f TMP_292 m) in (let TMP_294 \def (eq nat TMP_289 TMP_293) in (let
+TMP_307 \def (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7:
+(lt x d)).(let TMP_295 \def (S x) in (let TMP_305 \def (\lambda (n: nat).(let
+TMP_296 \def (S h) in (let TMP_297 \def (lift TMP_296 d t0) in (let TMP_298
+\def (weight_map g TMP_297) in (let TMP_299 \def (S TMP_298) in (let TMP_300
+\def (wadd g TMP_299 n) in (let TMP_301 \def (lift h d t0) in (let TMP_302
+\def (weight_map f TMP_301) in (let TMP_303 \def (S TMP_302) in (let TMP_304
+\def (wadd f TMP_303 n) in (eq nat TMP_300 TMP_304))))))))))) in (let TMP_306
+\def (H1 x H7) in (eq_ind_r nat TMP_295 TMP_305 TMP_306 m H6))))))) in
+(ex2_ind nat TMP_283 TMP_284 TMP_294 TMP_307 H5))))))))))))))) in (let
+TMP_309 \def (lt_gen_xS m d H4) in (or_ind TMP_243 TMP_247 TMP_257 TMP_281
+TMP_308 TMP_309)))))))))))))))))))) in (let TMP_330 \def (\lambda (m:
+nat).(\lambda (H4: (le (S d) m)).(let TMP_312 \def (\lambda (n: nat).(let
+TMP_311 \def (S n) in (eq nat m TMP_311))) in (let TMP_313 \def (\lambda (n:
+nat).(le d n)) in (let TMP_314 \def (g m) in (let TMP_315 \def (lift h d t0)
+in (let TMP_316 \def (weight_map f TMP_315) in (let TMP_317 \def (S TMP_316)
+in (let TMP_318 \def (wadd f TMP_317 m) in (let TMP_319 \def (eq nat TMP_314
+TMP_318) in (let TMP_328 \def (\lambda (x: nat).(\lambda (H5: (eq nat m (S
+x))).(\lambda (H6: (le d x)).(let TMP_320 \def (S x) in (let TMP_326 \def
+(\lambda (n: nat).(let TMP_321 \def (g n) in (let TMP_322 \def (lift h d t0)
+in (let TMP_323 \def (weight_map f TMP_322) in (let TMP_324 \def (S TMP_323)
+in (let TMP_325 \def (wadd f TMP_324 n) in (eq nat TMP_321 TMP_325))))))) in
+(let TMP_327 \def (H3 x H6) in (eq_ind_r nat TMP_320 TMP_326 TMP_327 m
+H5))))))) in (let TMP_329 \def (le_gen_S d m H4) in (ex2_ind nat TMP_312
+TMP_313 TMP_319 TMP_328 TMP_329))))))))))))) in (let TMP_331 \def (H0 h
+TMP_233 TMP_237 TMP_242 TMP_310 H2 TMP_330) in (let TMP_332 \def (f_equal2
+nat nat nat plus TMP_212 TMP_215 TMP_222 TMP_231 TMP_232 TMP_331) in (let
+TMP_333 \def (f_equal nat nat S TMP_197 TMP_210 TMP_332) in (let TMP_334 \def
+(lift h d t0) in (let TMP_335 \def (weight_map f TMP_334) in (let TMP_336
+\def (wadd f O) in (let TMP_337 \def (S d) in (let TMP_338 \def (lift h
+TMP_337 t1) in (let TMP_339 \def (weight_map TMP_336 TMP_338) in (let TMP_340
+\def (plus TMP_335 TMP_339) in (let TMP_341 \def (S h) in (let TMP_342 \def
+(lift TMP_341 d t0) in (let TMP_343 \def (weight_map g TMP_342) in (let
+TMP_344 \def (wadd g O) in (let TMP_345 \def (S h) in (let TMP_346 \def (S d)
+in (let TMP_347 \def (lift TMP_345 TMP_346 t1) in (let TMP_348 \def
+(weight_map TMP_344 TMP_347) in (let TMP_349 \def (plus TMP_343 TMP_348) in
+(let TMP_350 \def (lift h d t0) in (let TMP_351 \def (weight_map f TMP_350)
+in (let TMP_352 \def (S h) in (let TMP_353 \def (lift TMP_352 d t0) in (let
+TMP_354 \def (weight_map g TMP_353) in (let TMP_355 \def (wadd f O) in (let
+TMP_356 \def (S d) in (let TMP_357 \def (lift h TMP_356 t1) in (let TMP_358
+\def (weight_map TMP_355 TMP_357) in (let TMP_359 \def (wadd g O) in (let
+TMP_360 \def (S h) in (let TMP_361 \def (S d) in (let TMP_362 \def (lift
+TMP_360 TMP_361 t1) in (let TMP_363 \def (weight_map TMP_359 TMP_362) in (let
+TMP_364 \def (H h d f g H1 H2 H3) in (let TMP_365 \def (S d) in (let TMP_366
+\def (wadd f O) in (let TMP_367 \def (wadd g O) in (let TMP_395 \def (\lambda
+(m: nat).(\lambda (H4: (lt m (S d))).(let TMP_368 \def (eq nat m O) in (let
+TMP_370 \def (\lambda (m0: nat).(let TMP_369 \def (S m0) in (eq nat m
+TMP_369))) in (let TMP_371 \def (\lambda (m0: nat).(lt m0 d)) in (let TMP_372
+\def (ex2 nat TMP_370 TMP_371) in (let TMP_373 \def (wadd g O m) in (let
+TMP_374 \def (wadd f O m) in (let TMP_375 \def (eq nat TMP_373 TMP_374) in
+(let TMP_380 \def (\lambda (H5: (eq nat m O)).(let TMP_378 \def (\lambda (n:
+nat).(let TMP_376 \def (wadd g O n) in (let TMP_377 \def (wadd f O n) in (eq
+nat TMP_376 TMP_377)))) in (let TMP_379 \def (refl_equal nat O) in (eq_ind_r
+nat O TMP_378 TMP_379 m H5)))) in (let TMP_393 \def (\lambda (H5: (ex2 nat
+(\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(let
+TMP_382 \def (\lambda (m0: nat).(let TMP_381 \def (S m0) in (eq nat m
+TMP_381))) in (let TMP_383 \def (\lambda (m0: nat).(lt m0 d)) in (let TMP_384
+\def (wadd g O m) in (let TMP_385 \def (wadd f O m) in (let TMP_386 \def (eq
+nat TMP_384 TMP_385) in (let TMP_392 \def (\lambda (x: nat).(\lambda (H6: (eq
+nat m (S x))).(\lambda (H7: (lt x d)).(let TMP_387 \def (S x) in (let TMP_390
+\def (\lambda (n: nat).(let TMP_388 \def (wadd g O n) in (let TMP_389 \def
+(wadd f O n) in (eq nat TMP_388 TMP_389)))) in (let TMP_391 \def (H1 x H7) in
+(eq_ind_r nat TMP_387 TMP_390 TMP_391 m H6))))))) in (ex2_ind nat TMP_382
+TMP_383 TMP_386 TMP_392 H5)))))))) in (let TMP_394 \def (lt_gen_xS m d H4) in
+(or_ind TMP_368 TMP_372 TMP_375 TMP_380 TMP_393 TMP_394))))))))))))) in (let
+TMP_409 \def (\lambda (m: nat).(\lambda (H4: (le (S d) m)).(let TMP_397 \def
+(\lambda (n: nat).(let TMP_396 \def (S n) in (eq nat m TMP_396))) in (let
+TMP_398 \def (\lambda (n: nat).(le d n)) in (let TMP_399 \def (g m) in (let
+TMP_400 \def (wadd f O m) in (let TMP_401 \def (eq nat TMP_399 TMP_400) in
+(let TMP_407 \def (\lambda (x: nat).(\lambda (H5: (eq nat m (S x))).(\lambda
+(H6: (le d x)).(let TMP_402 \def (S x) in (let TMP_405 \def (\lambda (n:
+nat).(let TMP_403 \def (g n) in (let TMP_404 \def (wadd f O n) in (eq nat
+TMP_403 TMP_404)))) in (let TMP_406 \def (H3 x H6) in (eq_ind_r nat TMP_402
+TMP_405 TMP_406 m H5))))))) in (let TMP_408 \def (le_gen_S d m H4) in
+(ex2_ind nat TMP_397 TMP_398 TMP_401 TMP_407 TMP_408)))))))))) in (let
+TMP_410 \def (H0 h TMP_365 TMP_366 TMP_367 TMP_395 H2 TMP_409) in (let
+TMP_411 \def (f_equal2 nat nat nat plus TMP_351 TMP_354 TMP_358 TMP_363
+TMP_364 TMP_410) in (let TMP_412 \def (f_equal nat nat S TMP_340 TMP_349
+TMP_411) in (let TMP_413 \def (lift h d t0) in (let TMP_414 \def (weight_map
+f TMP_413) in (let TMP_415 \def (wadd f O) in (let TMP_416 \def (S d) in (let
+TMP_417 \def (lift h TMP_416 t1) in (let TMP_418 \def (weight_map TMP_415
+TMP_417) in (let TMP_419 \def (plus TMP_414 TMP_418) in (let TMP_420 \def (S
+h) in (let TMP_421 \def (lift TMP_420 d t0) in (let TMP_422 \def (weight_map
+g TMP_421) in (let TMP_423 \def (wadd g O) in (let TMP_424 \def (S h) in (let
+TMP_425 \def (S d) in (let TMP_426 \def (lift TMP_424 TMP_425 t1) in (let
+TMP_427 \def (weight_map TMP_423 TMP_426) in (let TMP_428 \def (plus TMP_422
+TMP_427) in (let TMP_429 \def (lift h d t0) in (let TMP_430 \def (weight_map
+f TMP_429) in (let TMP_431 \def (S h) in (let TMP_432 \def (lift TMP_431 d
+t0) in (let TMP_433 \def (weight_map g TMP_432) in (let TMP_434 \def (wadd f
+O) in (let TMP_435 \def (S d) in (let TMP_436 \def (lift h TMP_435 t1) in
+(let TMP_437 \def (weight_map TMP_434 TMP_436) in (let TMP_438 \def (wadd g
+O) in (let TMP_439 \def (S h) in (let TMP_440 \def (S d) in (let TMP_441 \def
+(lift TMP_439 TMP_440 t1) in (let TMP_442 \def (weight_map TMP_438 TMP_441)
+in (let TMP_443 \def (H h d f g H1 H2 H3) in (let TMP_444 \def (S d) in (let
+TMP_445 \def (wadd f O) in (let TMP_446 \def (wadd g O) in (let TMP_474 \def
+(\lambda (m: nat).(\lambda (H4: (lt m (S d))).(let TMP_447 \def (eq nat m O)
+in (let TMP_449 \def (\lambda (m0: nat).(let TMP_448 \def (S m0) in (eq nat m
+TMP_448))) in (let TMP_450 \def (\lambda (m0: nat).(lt m0 d)) in (let TMP_451
+\def (ex2 nat TMP_449 TMP_450) in (let TMP_452 \def (wadd g O m) in (let
+TMP_453 \def (wadd f O m) in (let TMP_454 \def (eq nat TMP_452 TMP_453) in
+(let TMP_459 \def (\lambda (H5: (eq nat m O)).(let TMP_457 \def (\lambda (n:
+nat).(let TMP_455 \def (wadd g O n) in (let TMP_456 \def (wadd f O n) in (eq
+nat TMP_455 TMP_456)))) in (let TMP_458 \def (refl_equal nat O) in (eq_ind_r
+nat O TMP_457 TMP_458 m H5)))) in (let TMP_472 \def (\lambda (H5: (ex2 nat
+(\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(let
+TMP_461 \def (\lambda (m0: nat).(let TMP_460 \def (S m0) in (eq nat m
+TMP_460))) in (let TMP_462 \def (\lambda (m0: nat).(lt m0 d)) in (let TMP_463
+\def (wadd g O m) in (let TMP_464 \def (wadd f O m) in (let TMP_465 \def (eq
+nat TMP_463 TMP_464) in (let TMP_471 \def (\lambda (x: nat).(\lambda (H6: (eq
+nat m (S x))).(\lambda (H7: (lt x d)).(let TMP_466 \def (S x) in (let TMP_469
+\def (\lambda (n: nat).(let TMP_467 \def (wadd g O n) in (let TMP_468 \def
+(wadd f O n) in (eq nat TMP_467 TMP_468)))) in (let TMP_470 \def (H1 x H7) in
+(eq_ind_r nat TMP_466 TMP_469 TMP_470 m H6))))))) in (ex2_ind nat TMP_461
+TMP_462 TMP_465 TMP_471 H5)))))))) in (let TMP_473 \def (lt_gen_xS m d H4) in
+(or_ind TMP_447 TMP_451 TMP_454 TMP_459 TMP_472 TMP_473))))))))))))) in (let
+TMP_488 \def (\lambda (m: nat).(\lambda (H4: (le (S d) m)).(let TMP_476 \def
+(\lambda (n: nat).(let TMP_475 \def (S n) in (eq nat m TMP_475))) in (let
+TMP_477 \def (\lambda (n: nat).(le d n)) in (let TMP_478 \def (g m) in (let
+TMP_479 \def (wadd f O m) in (let TMP_480 \def (eq nat TMP_478 TMP_479) in
+(let TMP_486 \def (\lambda (x: nat).(\lambda (H5: (eq nat m (S x))).(\lambda
+(H6: (le d x)).(let TMP_481 \def (S x) in (let TMP_484 \def (\lambda (n:
+nat).(let TMP_482 \def (g n) in (let TMP_483 \def (wadd f O n) in (eq nat
+TMP_482 TMP_483)))) in (let TMP_485 \def (H3 x H6) in (eq_ind_r nat TMP_481
+TMP_484 TMP_485 m H5))))))) in (let TMP_487 \def (le_gen_S d m H4) in
+(ex2_ind nat TMP_476 TMP_477 TMP_480 TMP_486 TMP_487)))))))))) in (let
+TMP_489 \def (H0 h TMP_444 TMP_445 TMP_446 TMP_474 H2 TMP_488) in (let
+TMP_490 \def (f_equal2 nat nat nat plus TMP_430 TMP_433 TMP_437 TMP_442
+TMP_443 TMP_489) in (let TMP_491 \def (f_equal nat nat S TMP_419 TMP_428
+TMP_490) in (let TMP_492 \def (B_ind TMP_187 TMP_333 TMP_412 TMP_491 b) in
+(let TMP_493 \def (S h) in (let TMP_494 \def (Bind b) in (let TMP_495 \def
+(THead TMP_494 t0 t1) in (let TMP_496 \def (lift TMP_493 d TMP_495) in (let
+TMP_497 \def (Bind b) in (let TMP_498 \def (S h) in (let TMP_499 \def
+(lift_head TMP_497 t0 t1 TMP_498 d) in (let TMP_500 \def (eq_ind_r T TMP_120
+TMP_129 TMP_492 TMP_496 TMP_499) in (let TMP_501 \def (Bind b) in (let
+TMP_502 \def (THead TMP_501 t0 t1) in (let TMP_503 \def (lift h d TMP_502) in
+(let TMP_504 \def (Bind b) in (let TMP_505 \def (lift_head TMP_504 t0 t1 h d)
+in (eq_ind_r T TMP_105 TMP_112 TMP_500 TMP_503
+TMP_505)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+)))))))))))))))))))))) in (let TMP_576 \def (\lambda (f0: F).(let TMP_507
+\def (Flat f0) in (let TMP_508 \def (lift h d t0) in (let TMP_509 \def (Flat
+f0) in (let TMP_510 \def (s TMP_509 d) in (let TMP_511 \def (lift h TMP_510
+t1) in (let TMP_512 \def (THead TMP_507 TMP_508 TMP_511) in (let TMP_519 \def
+(\lambda (t2: T).(let TMP_513 \def (weight_map f t2) in (let TMP_514 \def (S
+h) in (let TMP_515 \def (Flat f0) in (let TMP_516 \def (THead TMP_515 t0 t1)
+in (let TMP_517 \def (lift TMP_514 d TMP_516) in (let TMP_518 \def
+(weight_map g TMP_517) in (eq nat TMP_513 TMP_518)))))))) in (let TMP_520
+\def (Flat f0) in (let TMP_521 \def (S h) in (let TMP_522 \def (lift TMP_521
+d t0) in (let TMP_523 \def (S h) in (let TMP_524 \def (Flat f0) in (let
+TMP_525 \def (s TMP_524 d) in (let TMP_526 \def (lift TMP_523 TMP_525 t1) in
+(let TMP_527 \def (THead TMP_520 TMP_522 TMP_526) in (let TMP_536 \def
+(\lambda (t2: T).(let TMP_528 \def (Flat f0) in (let TMP_529 \def (lift h d
+t0) in (let TMP_530 \def (Flat f0) in (let TMP_531 \def (s TMP_530 d) in (let
+TMP_532 \def (lift h TMP_531 t1) in (let TMP_533 \def (THead TMP_528 TMP_529
+TMP_532) in (let TMP_534 \def (weight_map f TMP_533) in (let TMP_535 \def
+(weight_map g t2) in (eq nat TMP_534 TMP_535)))))))))) in (let TMP_537 \def
+(lift h d t0) in (let TMP_538 \def (weight_map f TMP_537) in (let TMP_539
+\def (lift h d t1) in (let TMP_540 \def (weight_map f TMP_539) in (let
+TMP_541 \def (plus TMP_538 TMP_540) in (let TMP_542 \def (S h) in (let
+TMP_543 \def (lift TMP_542 d t0) in (let TMP_544 \def (weight_map g TMP_543)
+in (let TMP_545 \def (S h) in (let TMP_546 \def (lift TMP_545 d t1) in (let
+TMP_547 \def (weight_map g TMP_546) in (let TMP_548 \def (plus TMP_544
+TMP_547) in (let TMP_549 \def (lift h d t0) in (let TMP_550 \def (weight_map
+f TMP_549) in (let TMP_551 \def (S h) in (let TMP_552 \def (lift TMP_551 d
+t0) in (let TMP_553 \def (weight_map g TMP_552) in (let TMP_554 \def (lift h
+d t1) in (let TMP_555 \def (weight_map f TMP_554) in (let TMP_556 \def (S h)
+in (let TMP_557 \def (lift TMP_556 d t1) in (let TMP_558 \def (weight_map g
+TMP_557) in (let TMP_559 \def (H h d f g H1 H2 H3) in (let TMP_560 \def (H0 h
+d f g H1 H2 H3) in (let TMP_561 \def (f_equal2 nat nat nat plus TMP_550
+TMP_553 TMP_555 TMP_558 TMP_559 TMP_560) in (let TMP_562 \def (f_equal nat
+nat S TMP_541 TMP_548 TMP_561) in (let TMP_563 \def (S h) in (let TMP_564
+\def (Flat f0) in (let TMP_565 \def (THead TMP_564 t0 t1) in (let TMP_566
+\def (lift TMP_563 d TMP_565) in (let TMP_567 \def (Flat f0) in (let TMP_568
+\def (S h) in (let TMP_569 \def (lift_head TMP_567 t0 t1 TMP_568 d) in (let
+TMP_570 \def (eq_ind_r T TMP_527 TMP_536 TMP_562 TMP_566 TMP_569) in (let
+TMP_571 \def (Flat f0) in (let TMP_572 \def (THead TMP_571 t0 t1) in (let
+TMP_573 \def (lift h d TMP_572) in (let TMP_574 \def (Flat f0) in (let
+TMP_575 \def (lift_head TMP_574 t0 t1 h d) in (eq_ind_r T TMP_512 TMP_519
+TMP_570 TMP_573
+TMP_575))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (K_ind
+TMP_99 TMP_506 TMP_576 k)))))))))))))))) in (T_ind TMP_6 TMP_11 TMP_91
+TMP_577 t)))))).
theorem lift_weight_add_O:
\forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to
O t))))))
\def
\lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to
-nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m:
-nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m)))))
-(plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal
-nat (f m)))))))).
-(* COMMENTS
-Initial nodes: 93
-END *)
+nat))).(let TMP_1 \def (wadd f w O) in (let TMP_2 \def (minus TMP_1 O) in
+(let TMP_3 \def (wadd f w) in (let TMP_7 \def (\lambda (m: nat).(\lambda (H:
+(lt m O)).(let TMP_4 \def (wadd f w m) in (let TMP_5 \def (f m) in (let TMP_6
+\def (eq nat TMP_4 TMP_5) in (lt_x_O m H TMP_6)))))) in (let TMP_8 \def (wadd
+f w O) in (let TMP_9 \def (minus_n_O TMP_8) in (let TMP_11 \def (\lambda (m:
+nat).(\lambda (_: (le O m)).(let TMP_10 \def (f m) in (refl_equal nat
+TMP_10)))) in (lift_weight_add TMP_2 t h O f TMP_3 TMP_7 TMP_9
+TMP_11))))))))))).
theorem lift_tlt_dx:
\forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
(d: nat).(tlt t (THead k u (lift h d t)))))))
\def
\lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
-(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight
-(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t)
-(lift_weight t h d)))))).
-(* COMMENTS
-Initial nodes: 71
-END *)
+(d: nat).(let TMP_1 \def (lift h d t) in (let TMP_2 \def (weight TMP_1) in
+(let TMP_6 \def (\lambda (n: nat).(let TMP_3 \def (lift h d t) in (let TMP_4
+\def (THead k u TMP_3) in (let TMP_5 \def (weight TMP_4) in (lt n TMP_5)))))
+in (let TMP_7 \def (lift h d t) in (let TMP_8 \def (tlt_head_dx k u TMP_7) in
+(let TMP_9 \def (weight t) in (let TMP_10 \def (lift_weight t h d) in (eq_ind
+nat TMP_2 TMP_6 TMP_8 TMP_9 TMP_10)))))))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/T/defs.ma".
+include "basic_1/T/defs.ma".
definition r:
K \to (nat \to nat)
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/r/defs.ma".
+include "basic_1/r/defs.ma".
-include "Basic-1/s/defs.ma".
+include "basic_1/s/defs.ma".
theorem r_S:
\forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S
-i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
-(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat
-f) i))))) k).
-(* COMMENTS
-Initial nodes: 65
-END *)
+ \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1
+\def (S i) in (let TMP_2 \def (r k0 TMP_1) in (let TMP_3 \def (r k0 i) in
+(let TMP_4 \def (S TMP_3) in (eq nat TMP_2 TMP_4))))))) in (let TMP_9 \def
+(\lambda (b: B).(\lambda (i: nat).(let TMP_6 \def (Bind b) in (let TMP_7 \def
+(r TMP_6 i) in (let TMP_8 \def (S TMP_7) in (refl_equal nat TMP_8)))))) in
+(let TMP_13 \def (\lambda (f: F).(\lambda (i: nat).(let TMP_10 \def (Flat f)
+in (let TMP_11 \def (r TMP_10 i) in (let TMP_12 \def (S TMP_11) in
+(refl_equal nat TMP_12)))))) in (K_ind TMP_5 TMP_9 TMP_13 k)))).
theorem r_plus:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
(plus (r k i) j))))
\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j)))))
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r
-(Flat f) i) j))))) k).
-(* COMMENTS
-Initial nodes: 79
-END *)
+ \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).(let TMP_1 \def (plus i j) in (let TMP_2 \def (r k0 TMP_1) in (let
+TMP_3 \def (r k0 i) in (let TMP_4 \def (plus TMP_3 j) in (eq nat TMP_2
+TMP_4)))))))) in (let TMP_9 \def (\lambda (b: B).(\lambda (i: nat).(\lambda
+(j: nat).(let TMP_6 \def (Bind b) in (let TMP_7 \def (r TMP_6 i) in (let
+TMP_8 \def (plus TMP_7 j) in (refl_equal nat TMP_8))))))) in (let TMP_13 \def
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(let TMP_10 \def (Flat f)
+in (let TMP_11 \def (r TMP_10 i) in (let TMP_12 \def (plus TMP_11 j) in
+(refl_equal nat TMP_12))))))) in (K_ind TMP_5 TMP_9 TMP_13 k)))).
theorem r_plus_sym:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
(plus i (r k j)))))
\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
-nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_:
-F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
-(* COMMENTS
-Initial nodes: 63
-END *)
+ \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).(let TMP_1 \def (plus i j) in (let TMP_2 \def (r k0 TMP_1) in (let
+TMP_3 \def (r k0 j) in (let TMP_4 \def (plus i TMP_3) in (eq nat TMP_2
+TMP_4)))))))) in (let TMP_7 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
+(j: nat).(let TMP_6 \def (plus i j) in (refl_equal nat TMP_6))))) in (let
+TMP_8 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i
+j)))) in (K_ind TMP_5 TMP_7 TMP_8 k)))).
theorem r_minus:
\forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
(minus (r k i) (S n)) (r k (minus i (S n)))))))
\def
\lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k:
-K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
-n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_:
-F).(minus_x_Sy i n H)) k)))).
-(* COMMENTS
-Initial nodes: 69
-END *)
+K).(let TMP_7 \def (\lambda (k0: K).(let TMP_1 \def (r k0 i) in (let TMP_2
+\def (S n) in (let TMP_3 \def (minus TMP_1 TMP_2) in (let TMP_4 \def (S n) in
+(let TMP_5 \def (minus i TMP_4) in (let TMP_6 \def (r k0 TMP_5) in (eq nat
+TMP_3 TMP_6)))))))) in (let TMP_10 \def (\lambda (_: B).(let TMP_8 \def (S n)
+in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat TMP_9)))) in (let
+TMP_11 \def (\lambda (_: F).(minus_x_Sy i n H)) in (K_ind TMP_7 TMP_10 TMP_11
+k))))))).
theorem r_dis:
\forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i:
-nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i)
-(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H:
-((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_:
-((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i:
-nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
-((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0:
-((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda
-(i: nat).(refl_equal nat (S i)))))))) k).
-(* COMMENTS
-Initial nodes: 151
-END *)
+ \lambda (k: K).(let TMP_1 \def (\lambda (k0: K).(\forall (P:
+Prop).(((((\forall (i: nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall
+(i: nat).(eq nat (r k0 i) (S i)))) \to P)) \to P)))) in (let TMP_3 \def
+(\lambda (b: B).(\lambda (P: Prop).(\lambda (H: ((((\forall (i: nat).(eq nat
+(r (Bind b) i) i))) \to P))).(\lambda (_: ((((\forall (i: nat).(eq nat (r
+(Bind b) i) (S i)))) \to P))).(let TMP_2 \def (\lambda (i: nat).(refl_equal
+nat i)) in (H TMP_2)))))) in (let TMP_6 \def (\lambda (f: F).(\lambda (P:
+Prop).(\lambda (_: ((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to
+P))).(\lambda (H0: ((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to
+P))).(let TMP_5 \def (\lambda (i: nat).(let TMP_4 \def (S i) in (refl_equal
+nat TMP_4))) in (H0 TMP_5)))))) in (K_ind TMP_1 TMP_3 TMP_6 k)))).
theorem s_r:
\forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0
-i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i))))
-(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
-(* COMMENTS
-Initial nodes: 51
-END *)
+ \lambda (k: K).(let TMP_4 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1
+\def (r k0 i) in (let TMP_2 \def (s k0 TMP_1) in (let TMP_3 \def (S i) in (eq
+nat TMP_2 TMP_3)))))) in (let TMP_6 \def (\lambda (_: B).(\lambda (i:
+nat).(let TMP_5 \def (S i) in (refl_equal nat TMP_5)))) in (let TMP_8 \def
+(\lambda (_: F).(\lambda (i: nat).(let TMP_7 \def (S i) in (refl_equal nat
+TMP_7)))) in (K_ind TMP_4 TMP_6 TMP_8 k)))).
theorem r_arith0:
\forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
\def
- \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n:
-nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
-nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O))
-(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
-(* COMMENTS
-Initial nodes: 105
-END *)
+ \lambda (k: K).(\lambda (i: nat).(let TMP_1 \def (r k i) in (let TMP_2 \def
+(S TMP_1) in (let TMP_6 \def (\lambda (n: nat).(let TMP_3 \def (S O) in (let
+TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in (eq nat TMP_4
+TMP_5))))) in (let TMP_7 \def (r k i) in (let TMP_9 \def (\lambda (n:
+nat).(let TMP_8 \def (r k i) in (eq nat n TMP_8))) in (let TMP_10 \def (r k
+i) in (let TMP_11 \def (refl_equal nat TMP_10) in (let TMP_12 \def (r k i) in
+(let TMP_13 \def (S TMP_12) in (let TMP_14 \def (S O) in (let TMP_15 \def
+(minus TMP_13 TMP_14) in (let TMP_16 \def (r k i) in (let TMP_17 \def
+(minus_Sx_SO TMP_16) in (let TMP_18 \def (eq_ind_r nat TMP_7 TMP_9 TMP_11
+TMP_15 TMP_17) in (let TMP_19 \def (S i) in (let TMP_20 \def (r k TMP_19) in
+(let TMP_21 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_6 TMP_18 TMP_20
+TMP_21))))))))))))))))))).
theorem r_arith1:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
i)) (S j)) (minus (r k i) j))))
\def
- \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i))
-(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat
-(minus (r k i) j)) (r k (S i)) (r_S k i)))).
-(* COMMENTS
-Initial nodes: 69
-END *)
+ \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(let TMP_1 \def (r k i)
+in (let TMP_2 \def (S TMP_1) in (let TMP_7 \def (\lambda (n: nat).(let TMP_3
+\def (S j) in (let TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in
+(let TMP_6 \def (minus TMP_5 j) in (eq nat TMP_4 TMP_6)))))) in (let TMP_8
+\def (r k i) in (let TMP_9 \def (minus TMP_8 j) in (let TMP_10 \def
+(refl_equal nat TMP_9) in (let TMP_11 \def (S i) in (let TMP_12 \def (r k
+TMP_11) in (let TMP_13 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_7 TMP_10
+TMP_12 TMP_13)))))))))))).
+
+theorem r_arith2:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (S i) (s k j)) \to
+(le (r k i) j))))
+\def
+ \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).((le (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (le TMP_1
+j)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le (S i) (S j))).(let H_y \def (le_S_n i j H) in H_y)))))
+in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem r_arith3:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k j) (S i)) \to
+(le j (r k i)))))
+\def
+ \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).((le (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (le j
+TMP_1)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le (S j) (S i))).(let H_y \def (le_S_n j i H) in H_y)))))
+in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le j (S i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem r_arith4:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (S i) (s k
+j)) (minus (r k i) j))))
+\def
+ \lambda (k: K).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).(let TMP_1 \def (S i) in (let TMP_2 \def (s k0 j) in (let TMP_3 \def
+(minus TMP_1 TMP_2) in (let TMP_4 \def (r k0 i) in (let TMP_5 \def (minus
+TMP_4 j) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8
+\def (r TMP_7 i) in (let TMP_9 \def (minus TMP_8 j) in (refl_equal nat
+TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda
+(j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (r TMP_11 i) in (let
+TMP_13 \def (minus TMP_12 j) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6
+TMP_10 TMP_14 k)))).
+
+theorem r_arith5:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k j) (S i)) \to
+(lt j (r k i)))))
+\def
+ \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).((lt (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (lt j
+TMP_1)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H))))) in (let TMP_4 \def
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S
+i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+
+theorem r_arith6:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k i) (S
+j)) (minus i (s k j)))))
+\def
+ \lambda (k: K).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).(let TMP_1 \def (r k0 i) in (let TMP_2 \def (S j) in (let TMP_3 \def
+(minus TMP_1 TMP_2) in (let TMP_4 \def (s k0 j) in (let TMP_5 \def (minus i
+TMP_4) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8
+\def (s TMP_7 j) in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat
+TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda
+(j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (s TMP_11 j) in (let
+TMP_13 \def (minus i TMP_12) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6
+TMP_10 TMP_14 k)))).
+
+theorem r_arith7:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (S i) (s k j))
+\to (eq nat (r k i) j))))
+\def
+ \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
+(j: nat).((eq nat (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (eq nat
+TMP_1 j)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
+(j: nat).(\lambda (H: (eq nat (S i) (S j))).(eq_add_S i j H))))) in (let
+TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H:
+(eq nat (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
\lambda (t1: T).(\lambda (t2: T).(let TMP_1 \def (weight t1) in (let TMP_2
\def (weight t2) in (lt TMP_1 TMP_2)))).
-definition tle:
- T \to (T \to Prop)
-\def
- \lambda (t1: T).(\lambda (t2: T).(let TMP_1 \def (tweight t1) in (let TMP_2
-\def (tweight t2) in (le TMP_1 TMP_2)))).
-
TMP_172) in (le_n_S TMP_163 TMP_168 TMP_173)))))))))))))))) in (K_ind TMP_6
TMP_161 TMP_174 k)))).
-theorem tle_r:
- \forall (t: T).(tle t t)
-\def
- \lambda (t: T).(let TMP_3 \def (\lambda (t0: T).(let TMP_1 \def (tweight t0)
-in (let TMP_2 \def (tweight t0) in (le TMP_1 TMP_2)))) in (let TMP_5 \def
-(\lambda (_: nat).(let TMP_4 \def (S O) in (le_n TMP_4))) in (let TMP_7 \def
-(\lambda (_: nat).(let TMP_6 \def (S O) in (le_n TMP_6))) in (let TMP_12 \def
-(\lambda (_: K).(\lambda (t0: T).(\lambda (_: (le (tweight t0) (tweight
-t0))).(\lambda (t1: T).(\lambda (_: (le (tweight t1) (tweight t1))).(let
-TMP_8 \def (tweight t0) in (let TMP_9 \def (tweight t1) in (let TMP_10 \def
-(plus TMP_8 TMP_9) in (let TMP_11 \def (S TMP_10) in (le_n TMP_11))))))))))
-in (T_ind TMP_3 TMP_5 TMP_7 TMP_12 t))))).
-
projects / helm.git / commitdiff