(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr0/defs.ma".
+include "basic_1/pr0/defs.ma".
inductive pr1: T \to (T \to Prop) \def
| pr1_refl: \forall (t: T).(pr1 t t)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1/pr1/defs.ma".
+
+let rec pr1_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t t))) (f0:
+(\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pr1 t2
+t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr1 t t0) on p: P
+t t0 \def match p with [(pr1_refl t1) \Rightarrow (f t1) | (pr1_sing t2 t1 p0
+t3 p1) \Rightarrow (let TMP_1 \def ((pr1_ind P f f0) t2 t3 p1) in (f0 t2 t1
+p0 t3 p1 TMP_1))].
+
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr1/props.ma".
+include "basic_1/pr1/props.ma".
-include "Basic-1/pr0/pr0.ma".
+include "basic_1/pr0/pr0.ma".
theorem pr1_strip:
\forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0
t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda
-(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4:
-T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
-(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3))
-(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda
-(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda
-(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T
-(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5:
-T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda
-(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
-t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let
-H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t:
-T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)))
-(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x
-x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0
-H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3
-t2 H0)))))))))) t0 t1 H))).
-(* COMMENTS
-Initial nodes: 317
-END *)
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(let TMP_3 \def
+(\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (let TMP_1
+\def (\lambda (t4: T).(pr1 t2 t4)) in (let TMP_2 \def (\lambda (t4: T).(pr1
+t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let TMP_8 \def (\lambda (t:
+T).(\lambda (t2: T).(\lambda (H0: (pr0 t t2)).(let TMP_4 \def (\lambda (t3:
+T).(pr1 t t3)) in (let TMP_5 \def (\lambda (t3: T).(pr1 t2 t3)) in (let TMP_6
+\def (pr1_pr0 t t2 H0) in (let TMP_7 \def (pr1_refl t2) in (ex_intro2 T TMP_4
+TMP_5 t2 TMP_6 TMP_7)))))))) in (let TMP_26 \def (\lambda (t2: T).(\lambda
+(t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: (pr1 t2
+t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T (\lambda (t:
+T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda
+(H3: (pr0 t3 t5)).(let TMP_9 \def (\lambda (t: T).(pr0 t5 t)) in (let TMP_10
+\def (\lambda (t: T).(pr0 t2 t)) in (let TMP_11 \def (\lambda (t: T).(pr1 t4
+t)) in (let TMP_12 \def (\lambda (t: T).(pr1 t5 t)) in (let TMP_13 \def (ex2
+T TMP_11 TMP_12) in (let TMP_24 \def (\lambda (x: T).(\lambda (H4: (pr0 t5
+x)).(\lambda (H5: (pr0 t2 x)).(let H6 \def (H2 x H5) in (let TMP_14 \def
+(\lambda (t: T).(pr1 t4 t)) in (let TMP_15 \def (\lambda (t: T).(pr1 x t)) in
+(let TMP_16 \def (\lambda (t: T).(pr1 t4 t)) in (let TMP_17 \def (\lambda (t:
+T).(pr1 t5 t)) in (let TMP_18 \def (ex2 T TMP_16 TMP_17) in (let TMP_23 \def
+(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x x0)).(let
+TMP_19 \def (\lambda (t: T).(pr1 t4 t)) in (let TMP_20 \def (\lambda (t:
+T).(pr1 t5 t)) in (let TMP_21 \def (pr1_pr0 t5 x H4) in (let TMP_22 \def
+(pr1_t x t5 TMP_21 x0 H8) in (ex_intro2 T TMP_19 TMP_20 x0 H7 TMP_22))))))))
+in (ex2_ind T TMP_14 TMP_15 TMP_18 TMP_23 H6))))))))))) in (let TMP_25 \def
+(pr0_confluence t3 t5 H3 t2 H0) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_24
+TMP_25)))))))))))))))) in (pr1_ind TMP_3 TMP_8 TMP_26 t0 t1 H)))))).
theorem pr1_confluence:
\forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0
t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda
-(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4:
-T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
-(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3))
-(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2:
-T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_:
-(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda
-(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda
-(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def
-H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t))
-(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x:
-T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x
-H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda
-(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
-t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x
-x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0
-H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))).
-(* COMMENTS
-Initial nodes: 311
-END *)
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(let TMP_3 \def
+(\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (let TMP_1
+\def (\lambda (t4: T).(pr1 t2 t4)) in (let TMP_2 \def (\lambda (t4: T).(pr1
+t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let TMP_7 \def (\lambda (t:
+T).(\lambda (t2: T).(\lambda (H0: (pr1 t t2)).(let TMP_4 \def (\lambda (t3:
+T).(pr1 t t3)) in (let TMP_5 \def (\lambda (t3: T).(pr1 t2 t3)) in (let TMP_6
+\def (pr1_refl t2) in (ex_intro2 T TMP_4 TMP_5 t2 H0 TMP_6))))))) in (let
+TMP_23 \def (\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3
+t2)).(\lambda (t4: T).(\lambda (_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5:
+T).((pr1 t2 t5) \to (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1
+t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr1 t3 t5)).(let H_x \def
+(pr1_strip t3 t5 H3 t2 H0) in (let H4 \def H_x in (let TMP_8 \def (\lambda
+(t: T).(pr1 t5 t)) in (let TMP_9 \def (\lambda (t: T).(pr1 t2 t)) in (let
+TMP_10 \def (\lambda (t: T).(pr1 t4 t)) in (let TMP_11 \def (\lambda (t:
+T).(pr1 t5 t)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) in (let TMP_22 \def
+(\lambda (x: T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0
+\def (H2 x H6) in (let H7 \def H_x0 in (let TMP_13 \def (\lambda (t: T).(pr1
+t4 t)) in (let TMP_14 \def (\lambda (t: T).(pr1 x t)) in (let TMP_15 \def
+(\lambda (t: T).(pr1 t4 t)) in (let TMP_16 \def (\lambda (t: T).(pr1 t5 t))
+in (let TMP_17 \def (ex2 T TMP_15 TMP_16) in (let TMP_21 \def (\lambda (x0:
+T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x x0)).(let TMP_18 \def
+(\lambda (t: T).(pr1 t4 t)) in (let TMP_19 \def (\lambda (t: T).(pr1 t5 t))
+in (let TMP_20 \def (pr1_t x t5 H5 x0 H9) in (ex_intro2 T TMP_18 TMP_19 x0 H8
+TMP_20))))))) in (ex2_ind T TMP_13 TMP_14 TMP_17 TMP_21 H7)))))))))))) in
+(ex2_ind T TMP_8 TMP_9 TMP_12 TMP_22 H4))))))))))))))))) in (pr1_ind TMP_3
+TMP_7 TMP_23 t0 t1 H)))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr1/defs.ma".
+include "basic_1/pr1/fwd.ma".
-include "Basic-1/pr0/subst1.ma".
+include "basic_1/pr0/subst1.ma".
-include "Basic-1/subst1/props.ma".
+include "basic_1/subst1/props.ma".
-include "Basic-1/T/props.ma".
+include "basic_1/T/props.ma".
theorem pr1_pr0:
\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H
-t2 (pr1_refl t2)))).
-(* COMMENTS
-Initial nodes: 23
-END *)
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(let TMP_1 \def
+(pr1_refl t2) in (pr1_sing t2 t1 H t2 TMP_1)))).
theorem pr1_t:
\forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2
t3) \to (pr1 t1 t3)))))
\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
-(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3)))))
-(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda
-(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0
-t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2
-t5 H3)))))))))) t1 t2 H))).
-(* COMMENTS
-Initial nodes: 103
-END *)
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(let TMP_1 \def
+(\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t
+t3))))) in (let TMP_2 \def (\lambda (t: T).(\lambda (t3: T).(\lambda (H0:
+(pr1 t t3)).H0))) in (let TMP_4 \def (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0
+t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0
+t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(let TMP_3 \def (H2 t5
+H3) in (pr1_sing t0 t3 H0 t5 TMP_3)))))))))) in (pr1_ind TMP_1 TMP_2 TMP_4 t1
+t2 H)))))).
theorem pr1_head_1:
\forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall
(k: K).(pr1 (THead k u1 t) (THead k u2 t))))))
\def
\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t:
-T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k
-t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda
-(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda
-(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing
-(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k
-t3 t) H2))))))) u1 u2 H))))).
-(* COMMENTS
-Initial nodes: 137
-END *)
+T).(\lambda (k: K).(let TMP_3 \def (\lambda (t0: T).(\lambda (t1: T).(let
+TMP_1 \def (THead k t0 t) in (let TMP_2 \def (THead k t1 t) in (pr1 TMP_1
+TMP_2))))) in (let TMP_5 \def (\lambda (t0: T).(let TMP_4 \def (THead k t0 t)
+in (pr1_refl TMP_4))) in (let TMP_11 \def (\lambda (t2: T).(\lambda (t1:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda (_: (pr1 t2
+t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(let TMP_6 \def
+(THead k t2 t) in (let TMP_7 \def (THead k t1 t) in (let TMP_8 \def (pr0_refl
+t) in (let TMP_9 \def (pr0_comp t1 t2 H0 t t TMP_8 k) in (let TMP_10 \def
+(THead k t3 t) in (pr1_sing TMP_6 TMP_7 TMP_9 TMP_10 H2)))))))))))) in
+(pr1_ind TMP_3 TMP_5 TMP_11 u1 u2 H)))))))).
theorem pr1_head_2:
\forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall
(k: K).(pr1 (THead k u t1) (THead k u t2))))))
\def
\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u:
-T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u
-t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_:
-(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing
-(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k
-u t4) H2))))))) t1 t2 H))))).
-(* COMMENTS
-Initial nodes: 137
-END *)
+T).(\lambda (k: K).(let TMP_3 \def (\lambda (t: T).(\lambda (t0: T).(let
+TMP_1 \def (THead k u t) in (let TMP_2 \def (THead k u t0) in (pr1 TMP_1
+TMP_2))))) in (let TMP_5 \def (\lambda (t: T).(let TMP_4 \def (THead k u t)
+in (pr1_refl TMP_4))) in (let TMP_11 \def (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0
+t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(let TMP_6 \def
+(THead k u t0) in (let TMP_7 \def (THead k u t3) in (let TMP_8 \def (pr0_refl
+u) in (let TMP_9 \def (pr0_comp u u TMP_8 t3 t0 H0 k) in (let TMP_10 \def
+(THead k u t4) in (pr1_sing TMP_6 TMP_7 TMP_9 TMP_10 H2)))))))))))) in
+(pr1_ind TMP_3 TMP_5 TMP_11 t1 t2 H)))))))).
theorem pr1_comp:
\forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u:
T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u))))))))
\def
- \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t:
-T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall
-(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda
-(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2
-t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1
-t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t:
-T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead
-k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t
-u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k
-t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1
-H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4
-t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k
-t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp
-t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v
-w H))).
-(* COMMENTS
-Initial nodes: 273
-END *)
+ \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(let TMP_3 \def
+(\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u)
+\to (\forall (k: K).(let TMP_1 \def (THead k t t1) in (let TMP_2 \def (THead
+k t0 u) in (pr1 TMP_1 TMP_2))))))))) in (let TMP_4 \def (\lambda (t:
+T).(\lambda (t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k:
+K).(pr1_head_2 t0 u H0 t k)))))) in (let TMP_16 \def (\lambda (t2:
+T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda (H1:
+(pr1 t2 t3)).(\lambda (_: ((\forall (t: T).(\forall (u: T).((pr1 t u) \to
+(\forall (k: K).(pr1 (THead k t2 t) (THead k t3 u)))))))).(\lambda (t:
+T).(\lambda (u: T).(\lambda (H3: (pr1 t u)).(\lambda (k: K).(let TMP_7 \def
+(\lambda (t0: T).(\lambda (t4: T).(let TMP_5 \def (THead k t1 t0) in (let
+TMP_6 \def (THead k t3 t4) in (pr1 TMP_5 TMP_6))))) in (let TMP_9 \def
+(\lambda (t0: T).(let TMP_8 \def (pr1_sing t2 t1 H0 t3 H1) in (pr1_head_1 t1
+t3 TMP_8 t0 k))) in (let TMP_15 \def (\lambda (t0: T).(\lambda (t4:
+T).(\lambda (H4: (pr0 t4 t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0
+t5)).(\lambda (H6: (pr1 (THead k t1 t0) (THead k t3 t5))).(let TMP_10 \def
+(THead k t1 t0) in (let TMP_11 \def (THead k t1 t4) in (let TMP_12 \def
+(pr0_refl t1) in (let TMP_13 \def (pr0_comp t1 t1 TMP_12 t4 t0 H4 k) in (let
+TMP_14 \def (THead k t3 t5) in (pr1_sing TMP_10 TMP_11 TMP_13 TMP_14
+H6)))))))))))) in (pr1_ind TMP_7 TMP_9 TMP_15 t u H3)))))))))))))) in
+(pr1_ind TMP_3 TMP_4 TMP_16 v w H)))))).
theorem pr1_eta:
\forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in
(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl)
(TLRef O) (lift (S O) O t))) t)))))
\def
- \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in
-(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift
-(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v
-(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H
-(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O)
-(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u))
-(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O)
-(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef
-O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O)
-u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind
-Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O)
-(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl
-(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_n O))) u
-(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr
-not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O
-(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))).
-(* COMMENTS
-Initial nodes: 463
-END *)
+ \lambda (w: T).(\lambda (u: T).(let TMP_1 \def (Bind Abst) in (let t \def
+(THead TMP_1 w u) in (\lambda (v: T).(\lambda (H: (pr1 v w)).(let TMP_2 \def
+(Bind Abst) in (let TMP_3 \def (S O) in (let TMP_4 \def (lift TMP_3 O w) in
+(let TMP_5 \def (S O) in (let TMP_6 \def (S O) in (let TMP_7 \def (lift TMP_5
+TMP_6 u) in (let TMP_8 \def (THead TMP_2 TMP_4 TMP_7) in (let TMP_16 \def
+(\lambda (t0: T).(let TMP_9 \def (Bind Abst) in (let TMP_10 \def (Flat Appl)
+in (let TMP_11 \def (TLRef O) in (let TMP_12 \def (THead TMP_10 TMP_11 t0) in
+(let TMP_13 \def (THead TMP_9 v TMP_12) in (let TMP_14 \def (Bind Abst) in
+(let TMP_15 \def (THead TMP_14 w u) in (pr1 TMP_13 TMP_15))))))))) in (let
+TMP_17 \def (Flat Appl) in (let TMP_18 \def (TLRef O) in (let TMP_19 \def
+(Bind Abst) in (let TMP_20 \def (S O) in (let TMP_21 \def (lift TMP_20 O w)
+in (let TMP_22 \def (S O) in (let TMP_23 \def (S O) in (let TMP_24 \def (lift
+TMP_22 TMP_23 u) in (let TMP_25 \def (THead TMP_19 TMP_21 TMP_24) in (let
+TMP_26 \def (THead TMP_17 TMP_18 TMP_25) in (let TMP_27 \def (Bind Abbr) in
+(let TMP_28 \def (TLRef O) in (let TMP_29 \def (S O) in (let TMP_30 \def (S
+O) in (let TMP_31 \def (lift TMP_29 TMP_30 u) in (let TMP_32 \def (THead
+TMP_27 TMP_28 TMP_31) in (let TMP_33 \def (Flat Appl) in (let TMP_34 \def
+(TLRef O) in (let TMP_35 \def (Bind Abst) in (let TMP_36 \def (S O) in (let
+TMP_37 \def (lift TMP_36 O w) in (let TMP_38 \def (S O) in (let TMP_39 \def
+(S O) in (let TMP_40 \def (lift TMP_38 TMP_39 u) in (let TMP_41 \def (THead
+TMP_35 TMP_37 TMP_40) in (let TMP_42 \def (THead TMP_33 TMP_34 TMP_41) in
+(let TMP_43 \def (S O) in (let TMP_44 \def (lift TMP_43 O w) in (let TMP_45
+\def (TLRef O) in (let TMP_46 \def (TLRef O) in (let TMP_47 \def (TLRef O) in
+(let TMP_48 \def (pr0_refl TMP_47) in (let TMP_49 \def (S O) in (let TMP_50
+\def (S O) in (let TMP_51 \def (lift TMP_49 TMP_50 u) in (let TMP_52 \def (S
+O) in (let TMP_53 \def (S O) in (let TMP_54 \def (lift TMP_52 TMP_53 u) in
+(let TMP_55 \def (S O) in (let TMP_56 \def (S O) in (let TMP_57 \def (lift
+TMP_55 TMP_56 u) in (let TMP_58 \def (pr0_refl TMP_57) in (let TMP_59 \def
+(pr0_beta TMP_44 TMP_45 TMP_46 TMP_48 TMP_51 TMP_54 TMP_58) in (let TMP_60
+\def (Bind Abbr) in (let TMP_61 \def (TLRef O) in (let TMP_62 \def (S O) in
+(let TMP_63 \def (lift TMP_62 O u) in (let TMP_64 \def (THead TMP_60 TMP_61
+TMP_63) in (let TMP_65 \def (Bind Abbr) in (let TMP_66 \def (TLRef O) in (let
+TMP_67 \def (S O) in (let TMP_68 \def (S O) in (let TMP_69 \def (lift TMP_67
+TMP_68 u) in (let TMP_70 \def (THead TMP_65 TMP_66 TMP_69) in (let TMP_71
+\def (TLRef O) in (let TMP_72 \def (TLRef O) in (let TMP_73 \def (TLRef O) in
+(let TMP_74 \def (pr0_refl TMP_73) in (let TMP_75 \def (S O) in (let TMP_76
+\def (S O) in (let TMP_77 \def (lift TMP_75 TMP_76 u) in (let TMP_78 \def (S
+O) in (let TMP_79 \def (S O) in (let TMP_80 \def (lift TMP_78 TMP_79 u) in
+(let TMP_81 \def (S O) in (let TMP_82 \def (S O) in (let TMP_83 \def (lift
+TMP_81 TMP_82 u) in (let TMP_84 \def (pr0_refl TMP_83) in (let TMP_85 \def (S
+O) in (let TMP_86 \def (lift TMP_85 O u) in (let TMP_87 \def (le_O_n O) in
+(let TMP_88 \def (subst1_lift_S u O O TMP_87) in (let TMP_89 \def (pr0_delta1
+TMP_71 TMP_72 TMP_74 TMP_77 TMP_80 TMP_84 TMP_86 TMP_88) in (let TMP_90 \def
+(Bind Abbr) in (let TMP_91 \def (TLRef O) in (let TMP_92 \def (S O) in (let
+TMP_93 \def (lift TMP_92 O u) in (let TMP_94 \def (THead TMP_90 TMP_91
+TMP_93) in (let TMP_95 \def (pr0_refl u) in (let TMP_96 \def (TLRef O) in
+(let TMP_97 \def (pr0_zeta Abbr not_abbr_abst u u TMP_95 TMP_96) in (let
+TMP_98 \def (pr1_pr0 TMP_94 u TMP_97) in (let TMP_99 \def (pr1_sing TMP_64
+TMP_70 TMP_89 u TMP_98) in (let TMP_100 \def (pr1_sing TMP_32 TMP_42 TMP_59 u
+TMP_99) in (let TMP_101 \def (Bind Abst) in (let TMP_102 \def (pr1_comp v w H
+TMP_26 u TMP_100 TMP_101) in (let TMP_103 \def (S O) in (let TMP_104 \def
+(Bind Abst) in (let TMP_105 \def (THead TMP_104 w u) in (let TMP_106 \def
+(lift TMP_103 O TMP_105) in (let TMP_107 \def (S O) in (let TMP_108 \def
+(lift_bind Abst w u TMP_107 O) in (eq_ind_r T TMP_8 TMP_16 TMP_102 TMP_106
+TMP_108)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+))))))))))))))))))))))))))))))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr2/props.ma".
+include "basic_1/pr2/props.ma".
-include "Basic-1/clen/getl.ma".
+include "basic_1/clen/getl.ma".
theorem pr2_gen_ctail:
\forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
(clen c) u t t2)))))))))
\def
\lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c)
-(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T
-(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda
-(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
-(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind
-Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3
-t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4)
-(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t))
-(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1)))))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
-(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4
-t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda
-(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let
-H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind
-(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c
-(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c)))
-(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
-(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
-T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
-T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d
-(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr)
-u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e:
-C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
-T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
-T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k
-u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c
-t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3
-t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4
-H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen
-c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
-(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq
-nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_:
-nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3
-T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
-(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8:
-(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T
-u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda
-(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0
-(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind
-Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind
-Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
-t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind
-Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
-t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda
-(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4
-(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1
-t2 H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1161
-END *)
+(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(let TMP_1 \def (CTail k u c)
+in (let TMP_2 \def (\lambda (c0: C).(pr2 c0 t1 t2)) in (let TMP_10 \def
+(\lambda (_: C).(let TMP_3 \def (pr2 c t1 t2) in (let TMP_5 \def (\lambda (_:
+T).(let TMP_4 \def (Bind Abbr) in (eq K k TMP_4))) in (let TMP_6 \def
+(\lambda (t: T).(pr0 t1 t)) in (let TMP_8 \def (\lambda (t: T).(let TMP_7
+\def (clen c) in (subst0 TMP_7 u t t2))) in (let TMP_9 \def (ex3 T TMP_5
+TMP_6 TMP_8) in (or TMP_3 TMP_9))))))) in (let TMP_126 \def (\lambda (y:
+C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_18 \def (\lambda (c0: C).(\lambda
+(t: T).(\lambda (t0: T).((eq C c0 (CTail k u c)) \to (let TMP_11 \def (pr2 c
+t t0) in (let TMP_13 \def (\lambda (_: T).(let TMP_12 \def (Bind Abbr) in (eq
+K k TMP_12))) in (let TMP_14 \def (\lambda (t3: T).(pr0 t t3)) in (let TMP_16
+\def (\lambda (t3: T).(let TMP_15 \def (clen c) in (subst0 TMP_15 u t3 t0)))
+in (let TMP_17 \def (ex3 T TMP_13 TMP_14 TMP_16) in (or TMP_11
+TMP_17)))))))))) in (let TMP_27 \def (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k
+u c))).(let TMP_19 \def (pr2 c t3 t4) in (let TMP_21 \def (\lambda (_:
+T).(let TMP_20 \def (Bind Abbr) in (eq K k TMP_20))) in (let TMP_22 \def
+(\lambda (t: T).(pr0 t3 t)) in (let TMP_24 \def (\lambda (t: T).(let TMP_23
+\def (clen c) in (subst0 TMP_23 u t t4))) in (let TMP_25 \def (ex3 T TMP_21
+TMP_22 TMP_24) in (let TMP_26 \def (pr2_free c t3 t4 H1) in (or_introl TMP_19
+TMP_25 TMP_26)))))))))))) in (let TMP_125 \def (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind
+Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 t)).(\lambda (H4: (eq C c0
+(CTail k u c))).(let TMP_30 \def (\lambda (c1: C).(let TMP_28 \def (Bind
+Abbr) in (let TMP_29 \def (CHead d TMP_28 u0) in (getl i c1 TMP_29)))) in
+(let TMP_31 \def (CTail k u c) in (let H5 \def (eq_ind C c0 TMP_30 H1 TMP_31
+H4) in (let H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x
+in (let TMP_33 \def (\lambda (e: C).(let TMP_32 \def (CTail k u e) in (eq C d
+TMP_32))) in (let TMP_36 \def (\lambda (e: C).(let TMP_34 \def (Bind Abbr) in
+(let TMP_35 \def (CHead e TMP_34 u0) in (getl i c TMP_35)))) in (let TMP_37
+\def (ex2 C TMP_33 TMP_36) in (let TMP_39 \def (\lambda (_: nat).(let TMP_38
+\def (clen c) in (eq nat i TMP_38))) in (let TMP_41 \def (\lambda (_:
+nat).(let TMP_40 \def (Bind Abbr) in (eq K k TMP_40))) in (let TMP_42 \def
+(\lambda (_: nat).(eq T u u0)) in (let TMP_44 \def (\lambda (n: nat).(let
+TMP_43 \def (CSort n) in (eq C d TMP_43))) in (let TMP_45 \def (ex4 nat
+TMP_39 TMP_41 TMP_42 TMP_44) in (let TMP_46 \def (pr2 c t3 t) in (let TMP_48
+\def (\lambda (_: T).(let TMP_47 \def (Bind Abbr) in (eq K k TMP_47))) in
+(let TMP_49 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_51 \def (\lambda
+(t0: T).(let TMP_50 \def (clen c) in (subst0 TMP_50 u t0 t))) in (let TMP_52
+\def (ex3 T TMP_48 TMP_49 TMP_51) in (let TMP_53 \def (or TMP_46 TMP_52) in
+(let TMP_76 \def (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d (CTail k u e)))
+(\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))))).(let TMP_55 \def
+(\lambda (e: C).(let TMP_54 \def (CTail k u e) in (eq C d TMP_54))) in (let
+TMP_58 \def (\lambda (e: C).(let TMP_56 \def (Bind Abbr) in (let TMP_57 \def
+(CHead e TMP_56 u0) in (getl i c TMP_57)))) in (let TMP_59 \def (pr2 c t3 t)
+in (let TMP_61 \def (\lambda (_: T).(let TMP_60 \def (Bind Abbr) in (eq K k
+TMP_60))) in (let TMP_62 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_64
+\def (\lambda (t0: T).(let TMP_63 \def (clen c) in (subst0 TMP_63 u t0 t)))
+in (let TMP_65 \def (ex3 T TMP_61 TMP_62 TMP_64) in (let TMP_66 \def (or
+TMP_59 TMP_65) in (let TMP_75 \def (\lambda (x: C).(\lambda (_: (eq C d
+(CTail k u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(let
+TMP_67 \def (pr2 c t3 t) in (let TMP_69 \def (\lambda (_: T).(let TMP_68 \def
+(Bind Abbr) in (eq K k TMP_68))) in (let TMP_70 \def (\lambda (t0: T).(pr0 t3
+t0)) in (let TMP_72 \def (\lambda (t0: T).(let TMP_71 \def (clen c) in
+(subst0 TMP_71 u t0 t))) in (let TMP_73 \def (ex3 T TMP_69 TMP_70 TMP_72) in
+(let TMP_74 \def (pr2_delta c x u0 i H9 t3 t4 H2 t H3) in (or_introl TMP_67
+TMP_73 TMP_74)))))))))) in (ex2_ind C TMP_55 TMP_58 TMP_66 TMP_75
+H7))))))))))) in (let TMP_124 \def (\lambda (H7: (ex4 nat (\lambda (_:
+nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda
+(_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))))).(let TMP_78
+\def (\lambda (_: nat).(let TMP_77 \def (clen c) in (eq nat i TMP_77))) in
+(let TMP_80 \def (\lambda (_: nat).(let TMP_79 \def (Bind Abbr) in (eq K k
+TMP_79))) in (let TMP_81 \def (\lambda (_: nat).(eq T u u0)) in (let TMP_83
+\def (\lambda (n: nat).(let TMP_82 \def (CSort n) in (eq C d TMP_82))) in
+(let TMP_84 \def (pr2 c t3 t) in (let TMP_86 \def (\lambda (_: T).(let TMP_85
+\def (Bind Abbr) in (eq K k TMP_85))) in (let TMP_87 \def (\lambda (t0:
+T).(pr0 t3 t0)) in (let TMP_89 \def (\lambda (t0: T).(let TMP_88 \def (clen
+c) in (subst0 TMP_88 u t0 t))) in (let TMP_90 \def (ex3 T TMP_86 TMP_87
+TMP_89) in (let TMP_91 \def (or TMP_84 TMP_90) in (let TMP_123 \def (\lambda
+(x0: nat).(\lambda (H8: (eq nat i (clen c))).(\lambda (H9: (eq K k (Bind
+Abbr))).(\lambda (H10: (eq T u u0)).(\lambda (_: (eq C d (CSort x0))).(let
+TMP_92 \def (\lambda (n: nat).(subst0 n u0 t4 t)) in (let TMP_93 \def (clen
+c) in (let H12 \def (eq_ind nat i TMP_92 H3 TMP_93 H8) in (let TMP_95 \def
+(\lambda (t0: T).(let TMP_94 \def (clen c) in (subst0 TMP_94 t0 t4 t))) in
+(let H13 \def (eq_ind_r T u0 TMP_95 H12 u H10) in (let TMP_96 \def (Bind
+Abbr) in (let TMP_104 \def (\lambda (k0: K).(let TMP_97 \def (pr2 c t3 t) in
+(let TMP_99 \def (\lambda (_: T).(let TMP_98 \def (Bind Abbr) in (eq K k0
+TMP_98))) in (let TMP_100 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_102
+\def (\lambda (t0: T).(let TMP_101 \def (clen c) in (subst0 TMP_101 u t0 t)))
+in (let TMP_103 \def (ex3 T TMP_99 TMP_100 TMP_102) in (or TMP_97
+TMP_103))))))) in (let TMP_105 \def (pr2 c t3 t) in (let TMP_108 \def
+(\lambda (_: T).(let TMP_106 \def (Bind Abbr) in (let TMP_107 \def (Bind
+Abbr) in (eq K TMP_106 TMP_107)))) in (let TMP_109 \def (\lambda (t0: T).(pr0
+t3 t0)) in (let TMP_111 \def (\lambda (t0: T).(let TMP_110 \def (clen c) in
+(subst0 TMP_110 u t0 t))) in (let TMP_112 \def (ex3 T TMP_108 TMP_109
+TMP_111) in (let TMP_115 \def (\lambda (_: T).(let TMP_113 \def (Bind Abbr)
+in (let TMP_114 \def (Bind Abbr) in (eq K TMP_113 TMP_114)))) in (let TMP_116
+\def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_118 \def (\lambda (t0: T).(let
+TMP_117 \def (clen c) in (subst0 TMP_117 u t0 t))) in (let TMP_119 \def (Bind
+Abbr) in (let TMP_120 \def (refl_equal K TMP_119) in (let TMP_121 \def
+(ex3_intro T TMP_115 TMP_116 TMP_118 t4 TMP_120 H2 H13) in (let TMP_122 \def
+(or_intror TMP_105 TMP_112 TMP_121) in (eq_ind_r K TMP_96 TMP_104 TMP_122 k
+H9))))))))))))))))))))))))) in (ex4_ind nat TMP_78 TMP_80 TMP_81 TMP_83
+TMP_91 TMP_123 H7))))))))))))) in (or_ind TMP_37 TMP_45 TMP_53 TMP_76 TMP_124
+H6))))))))))))))))))))))))))))))))) in (pr2_ind TMP_18 TMP_27 TMP_125 y t1 t2
+H0)))))) in (insert_eq C TMP_1 TMP_2 TMP_10 TMP_126 H)))))))))).
theorem pr2_gen_cbind:
\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
(THead (Bind b) v t2)))))))
\def
\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c
-(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead
-(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y
-t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
-(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v
-t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead
-(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1
-(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
-(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def
-(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
-c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u)
-v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d
-(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S
-j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead
-(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3)
-(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead
-d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d
-| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
-v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return
-(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
-(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v)
-H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def
-(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def
-(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr
-(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t)))
-(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v
-(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7:
-(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c
-(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j)))
-(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b)
-v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S
-x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal
-nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i
-(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t
-(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead
-c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t
-H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1085
-END *)
+(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let TMP_1 \def (Bind
+b) in (let TMP_2 \def (CHead c TMP_1 v) in (let TMP_3 \def (\lambda (c0:
+C).(pr2 c0 t1 t2)) in (let TMP_8 \def (\lambda (_: C).(let TMP_4 \def (Bind
+b) in (let TMP_5 \def (THead TMP_4 v t1) in (let TMP_6 \def (Bind b) in (let
+TMP_7 \def (THead TMP_6 v t2) in (pr2 c TMP_5 TMP_7)))))) in (let TMP_119
+\def (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_13 \def (\lambda
+(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v)) \to
+(let TMP_9 \def (Bind b) in (let TMP_10 \def (THead TMP_9 v t) in (let TMP_11
+\def (Bind b) in (let TMP_12 \def (THead TMP_11 v t0) in (pr2 c TMP_10
+TMP_12))))))))) in (let TMP_21 \def (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c
+(Bind b) v))).(let TMP_14 \def (Bind b) in (let TMP_15 \def (THead TMP_14 v
+t3) in (let TMP_16 \def (Bind b) in (let TMP_17 \def (THead TMP_16 v t4) in
+(let TMP_18 \def (pr0_refl v) in (let TMP_19 \def (Bind b) in (let TMP_20
+\def (pr0_comp v v TMP_18 t3 t4 H1 TMP_19) in (pr2_free c TMP_15 TMP_17
+TMP_20))))))))))))) in (let TMP_118 \def (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u t4 t)).(\lambda (H4: (eq C c0
+(CHead c (Bind b) v))).(let TMP_24 \def (\lambda (c1: C).(let TMP_22 \def
+(Bind Abbr) in (let TMP_23 \def (CHead d TMP_22 u) in (getl i c1 TMP_23))))
+in (let TMP_25 \def (Bind b) in (let TMP_26 \def (CHead c TMP_25 v) in (let
+H5 \def (eq_ind C c0 TMP_24 H1 TMP_26 H4) in (let TMP_27 \def (Bind Abbr) in
+(let TMP_28 \def (CHead d TMP_27 u) in (let H_x \def (getl_gen_bind b c
+TMP_28 v i H5) in (let H6 \def H_x in (let TMP_29 \def (eq nat i O) in (let
+TMP_30 \def (Bind Abbr) in (let TMP_31 \def (CHead d TMP_30 u) in (let TMP_32
+\def (Bind b) in (let TMP_33 \def (CHead c TMP_32 v) in (let TMP_34 \def (eq
+C TMP_31 TMP_33) in (let TMP_35 \def (land TMP_29 TMP_34) in (let TMP_37 \def
+(\lambda (j: nat).(let TMP_36 \def (S j) in (eq nat i TMP_36))) in (let
+TMP_40 \def (\lambda (j: nat).(let TMP_38 \def (Bind Abbr) in (let TMP_39
+\def (CHead d TMP_38 u) in (getl j c TMP_39)))) in (let TMP_41 \def (ex2 nat
+TMP_37 TMP_40) in (let TMP_42 \def (Bind b) in (let TMP_43 \def (THead TMP_42
+v t3) in (let TMP_44 \def (Bind b) in (let TMP_45 \def (THead TMP_44 v t) in
+(let TMP_46 \def (pr2 c TMP_43 TMP_45) in (let TMP_90 \def (\lambda (H7:
+(land (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(let
+TMP_47 \def (eq nat i O) in (let TMP_48 \def (Bind Abbr) in (let TMP_49 \def
+(CHead d TMP_48 u) in (let TMP_50 \def (Bind b) in (let TMP_51 \def (CHead c
+TMP_50 v) in (let TMP_52 \def (eq C TMP_49 TMP_51) in (let TMP_53 \def (Bind
+b) in (let TMP_54 \def (THead TMP_53 v t3) in (let TMP_55 \def (Bind b) in
+(let TMP_56 \def (THead TMP_55 v t) in (let TMP_57 \def (pr2 c TMP_54 TMP_56)
+in (let TMP_89 \def (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead d
+(Bind Abbr) u) (CHead c (Bind b) v))).(let TMP_58 \def (\lambda (e: C).(match
+e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) in (let
+TMP_59 \def (Bind Abbr) in (let TMP_60 \def (CHead d TMP_59 u) in (let TMP_61
+\def (Bind b) in (let TMP_62 \def (CHead c TMP_61 v) in (let H10 \def
+(f_equal C C TMP_58 TMP_60 TMP_62 H9) in (let TMP_63 \def (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) in
+(let TMP_64 \def (Bind Abbr) in (let TMP_65 \def (CHead d TMP_64 u) in (let
+TMP_66 \def (Bind b) in (let TMP_67 \def (CHead c TMP_66 v) in (let H11 \def
+(f_equal C B TMP_63 TMP_65 TMP_67 H9) in (let TMP_68 \def (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
+in (let TMP_69 \def (Bind Abbr) in (let TMP_70 \def (CHead d TMP_69 u) in
+(let TMP_71 \def (Bind b) in (let TMP_72 \def (CHead c TMP_71 v) in (let H12
+\def (f_equal C T TMP_68 TMP_70 TMP_72 H9) in (let TMP_87 \def (\lambda (H13:
+(eq B Abbr b)).(\lambda (_: (eq C d c)).(let TMP_73 \def (\lambda (n:
+nat).(subst0 n u t4 t)) in (let H15 \def (eq_ind nat i TMP_73 H3 O H8) in
+(let TMP_74 \def (\lambda (t0: T).(subst0 O t0 t4 t)) in (let H16 \def
+(eq_ind T u TMP_74 H15 v H12) in (let TMP_79 \def (\lambda (b0: B).(let
+TMP_75 \def (Bind b0) in (let TMP_76 \def (THead TMP_75 v t3) in (let TMP_77
+\def (Bind b0) in (let TMP_78 \def (THead TMP_77 v t) in (pr2 c TMP_76
+TMP_78)))))) in (let TMP_80 \def (Bind Abbr) in (let TMP_81 \def (THead
+TMP_80 v t3) in (let TMP_82 \def (Bind Abbr) in (let TMP_83 \def (THead
+TMP_82 v t) in (let TMP_84 \def (pr0_refl v) in (let TMP_85 \def (pr0_delta v
+v TMP_84 t3 t4 H2 t H16) in (let TMP_86 \def (pr2_free c TMP_81 TMP_83
+TMP_85) in (eq_ind B Abbr TMP_79 TMP_86 b H13))))))))))))))) in (let TMP_88
+\def (TMP_87 H11) in (TMP_88 H10))))))))))))))))))))))) in (land_ind TMP_47
+TMP_52 TMP_57 TMP_89 H7)))))))))))))) in (let TMP_117 \def (\lambda (H7: (ex2
+nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d
+(Bind Abbr) u))))).(let TMP_92 \def (\lambda (j: nat).(let TMP_91 \def (S j)
+in (eq nat i TMP_91))) in (let TMP_95 \def (\lambda (j: nat).(let TMP_93 \def
+(Bind Abbr) in (let TMP_94 \def (CHead d TMP_93 u) in (getl j c TMP_94)))) in
+(let TMP_96 \def (Bind b) in (let TMP_97 \def (THead TMP_96 v t3) in (let
+TMP_98 \def (Bind b) in (let TMP_99 \def (THead TMP_98 v t) in (let TMP_100
+\def (pr2 c TMP_97 TMP_99) in (let TMP_116 \def (\lambda (x: nat).(\lambda
+(H8: (eq nat i (S x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let
+TMP_101 \def (\lambda (e: nat).e) in (let TMP_102 \def (S x) in (let H10 \def
+(f_equal nat nat TMP_101 i TMP_102 H8) in (let TMP_103 \def (\lambda (n:
+nat).(subst0 n u t4 t)) in (let TMP_104 \def (S x) in (let H11 \def (eq_ind
+nat i TMP_103 H3 TMP_104 H10) in (let TMP_105 \def (Bind b) in (let TMP_106
+\def (Bind b) in (let TMP_107 \def (CHead c TMP_106 v) in (let TMP_108 \def
+(S x) in (let TMP_109 \def (Bind b) in (let TMP_110 \def (CHead c TMP_109 v)
+in (let TMP_111 \def (clear_bind b c v) in (let TMP_112 \def (Bind Abbr) in
+(let TMP_113 \def (CHead d TMP_112 u) in (let TMP_114 \def (getl_clear_bind b
+TMP_110 c v TMP_111 TMP_113 x H9) in (let TMP_115 \def (pr2_delta TMP_107 d u
+TMP_108 TMP_114 t3 t4 H2 t H11) in (pr2_head_2 c v t3 t TMP_105
+TMP_115))))))))))))))))))))) in (ex2_ind nat TMP_92 TMP_95 TMP_100 TMP_116
+H7)))))))))) in (or_ind TMP_35 TMP_41 TMP_46 TMP_90 TMP_117
+H6))))))))))))))))))))))))))))))))))))) in (pr2_ind TMP_13 TMP_21 TMP_118 y
+t1 t2 H0)))))) in (insert_eq C TMP_2 TMP_3 TMP_8 TMP_119 H))))))))))).
theorem pr2_gen_cflat:
\forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
\def
\lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c
-(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2))
-(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2
-c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4
-H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
-(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def
-(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
-c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u)
-v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0)))
-H)))))).
-(* COMMENTS
-Initial nodes: 293
-END *)
+(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let TMP_1 \def (Flat
+f) in (let TMP_2 \def (CHead c TMP_1 v) in (let TMP_3 \def (\lambda (c0:
+C).(pr2 c0 t1 t2)) in (let TMP_4 \def (\lambda (_: C).(pr2 c t1 t2)) in (let
+TMP_15 \def (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_5 \def
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f)
+v)) \to (pr2 c t t0))))) in (let TMP_6 \def (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c
+(Flat f) v))).(pr2_free c t3 t4 H1)))))) in (let TMP_14 \def (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u t4 t)).(\lambda (H4:
+(eq C c0 (CHead c (Flat f) v))).(let TMP_9 \def (\lambda (c1: C).(let TMP_7
+\def (Bind Abbr) in (let TMP_8 \def (CHead d TMP_7 u) in (getl i c1 TMP_8))))
+in (let TMP_10 \def (Flat f) in (let TMP_11 \def (CHead c TMP_10 v) in (let
+H5 \def (eq_ind C c0 TMP_9 H1 TMP_11 H4) in (let TMP_12 \def (Bind Abbr) in
+(let TMP_13 \def (CHead d TMP_12 u) in (let H_y \def (getl_gen_flat f c
+TMP_13 v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3))))))))))))))))))) in
+(pr2_ind TMP_5 TMP_6 TMP_14 y t1 t2 H0)))))) in (insert_eq C TMP_2 TMP_3
+TMP_4 TMP_15 H))))))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr0/defs.ma".
+include "basic_1/pr0/defs.ma".
-include "Basic-1/getl/defs.ma".
+include "basic_1/getl/defs.ma".
inductive pr2: C \to (T \to (T \to Prop)) \def
| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr2/defs.ma".
+include "basic_1/pr2/defs.ma".
-include "Basic-1/pr0/fwd.ma".
+include "basic_1/pr0/fwd.ma".
-include "Basic-1/getl/drop.ma".
+include "basic_1/getl/clear.ma".
-include "Basic-1/getl/clear.ma".
+include "basic_1/getl/drop.ma".
+
+theorem pr2_ind:
+ \forall (P: ((C \to (T \to (T \to Prop))))).(((\forall (c: C).(\forall (t1:
+T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 t2)))))) \to (((\forall (c:
+C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
+(Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to
+(\forall (t: T).((subst0 i u t2 t) \to (P c t1 t)))))))))))) \to (\forall (c:
+C).(\forall (t: T).(\forall (t0: T).((pr2 c t t0) \to (P c t t0)))))))
+\def
+ \lambda (P: ((C \to (T \to (T \to Prop))))).(\lambda (f: ((\forall (c:
+C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1
+t2))))))).(\lambda (f0: ((\forall (c: C).(\forall (d: C).(\forall (u:
+T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1:
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to
+(P c t1 t))))))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\lambda (p: (pr2 c t t0)).(match p with [(pr2_free x x0 x1 x2)
+\Rightarrow (f x x0 x1 x2) | (pr2_delta x x0 x1 x2 x3 x4 x5 x6 x7 x8)
+\Rightarrow (f0 x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))).
theorem pr2_gen_sort:
\forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to
(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n)
(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n))))
t1 H4))))))))))))) c y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 347
-END *)
theorem pr2_gen_lref:
\forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to
Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S
n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8)))
(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 1003
-END *)
theorem pr2_gen_abst:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3
H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9))))))))
(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 2383
-END *)
theorem pr2_gen_cast:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2
H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 2659
-END *)
theorem pr2_gen_csort:
\forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2)
(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d
(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr)
u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
-(* COMMENTS
-Initial nodes: 221
-END *)
theorem pr2_gen_appl:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
-x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
-x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O
-x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22
-\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
-x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3:
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+y2) z1 z2))))))))) (\lambda (H20: (lt (s (Bind x0) i) (S
+O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind
+x0) i)) H20 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O)
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i)
+(S O) O H19 Hle) in (let H20 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
-(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
-O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H21: (eq T x7 (lift (S O) O
+x8))).(\lambda (H22: (subst0 (minus i O) u x3 x8)).(eq_ind_r T (lift (S O) O
+x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5))
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H23 \def (eq_ind_r
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x8)) H22 i (minus_n_O i)) in
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift
+(S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))
+(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat
+Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
-(Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl)
+y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead (Bind x0) x1 x2))
+(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)))
+(pr2_delta c0 d u i H1 u1 x3 H10 x8 H23) (pr2_free c0 x1 x4 H11) (pr2_free
+(CHead c0 (Bind x0) x4) x2 x5 H12)))) x7 H21)))) H20))))) x6 H18)))) H17))
+(\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S
+O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O
+x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4
+x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
-(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4
-H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20)))))
-(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n
-i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead
-(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
-(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead
-(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
-(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T
+x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: (subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O
+x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
+T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda
-(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19:
-(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat
-Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
-O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2)
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
+Appl) (lift (S O) O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
-T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
-i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i)
+(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) x7 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S
+O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind
+x0) i)) H21 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3))))
(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda
-(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0)
-i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
-(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T
-(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8))
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
-O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O)
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i)
+(S O) O H19 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
-(Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
-(S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4
-H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead
-c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2
-x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S
-O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat
-Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14))
-(\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl)
-(lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
-i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+(Flat Appl) x7 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t
-(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17:
-(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
-x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x7 (lift (S O) O
+x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O
+x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat Appl) u2 t4))))
(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8))
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift
+(S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))
+(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat
+Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead (Bind x0) x1 x2))
+(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)))
+(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_free c0 x1 x4 H11) (pr2_delta
+(CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4)
+c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8
+H20)))) x7 H22)))) H21))))) x6 H18)))))) H17)) (subst0_gen_head (Flat Appl) u
+(lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) (\lambda (H14:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind x0) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O)
+O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat
+Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0)
+x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t
+(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17:
+(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
+x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
-x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
-x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O
-x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23
-\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
-x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3:
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S
+O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind
+x0) i)) H21 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O)
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i)
+(S O) O H20 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
-(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
-O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
-(Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
-(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u
-i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8
-H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i
-(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T
-x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s
-(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7
-(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
-Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda
-(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x8 (lift (S O) O
+x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O
+x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7)
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5))
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift
+(S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5))
(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift
-(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
-x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3:
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
-u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2)
-t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))
+(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat
+Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead (Bind x0) x1 x2))
+(refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)))
+(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_delta c0 d u i H1 x1 x4 H11 x6
+H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12)))) x8 H22)))) H21))))) x7
+H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat
+Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s
+(Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 (THead (Flat
+Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s
+(Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
-Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda
+(H19: (eq T x7 (THead (Flat Appl) (lift (S O) O x3) x8))).(\lambda (H20:
+(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat
+Appl) (lift (S O) O x3) x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Flat Appl) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
+x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
+O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2)
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
-T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
-i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i)
+(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (H22: (lt (s (Bind x0) i) (S
+O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind
+x0) i)) H22 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3))))
(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda
-(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind
-x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S
-O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in
-(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
-z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O)
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i)
+(S O) O H20 Hle) in (let H22 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x10: T).(\lambda (H23: (eq T x8 (lift (S O) O
+x10))).(\lambda (H24: (subst0 (minus i O) u x3 x10)).(eq_ind_r T (lift (S O)
+O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq
+T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H25 \def (eq_ind_r
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x10)) H24 i (minus_n_O i))
+in (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
+(Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T
(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl)
-(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta
+(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H25) (pr2_delta
c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i)
(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6)
-(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22)))))
-(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n
-i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5
-x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4
-(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6))
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21)))) x8 H23)))) H22))))) x7
+H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s
+(Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 (THead
+(Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6))
(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 38859
-END *)
-
-theorem pr2_gen_abbr:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
-(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u:
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T
-T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
-t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
-(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
-(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
-Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2))))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
-t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
-(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in
-(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
-b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
-c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
-t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
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-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13
-(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_:
-T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z:
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr)
-u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl
-Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S
-i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl
-x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda
-(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t
-(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15:
-(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1
-x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
-(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
-c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13
-(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
-u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T
-(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0:
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
-c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O
-(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr)
-u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1
-x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8)))))))
-H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x))))))
-H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
-(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
-(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2)
-H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i)))))))
-(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 11671
-END *)
-
-theorem pr2_gen_void:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
-(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
-(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
-(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift
-(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1
-t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind
-Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
-t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
-b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u)
-t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
-c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1
-x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1)
-(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
-c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1
-x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1
-H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
-u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
-(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0
-(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3))))))))
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
-(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
-T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
-t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T
-t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1
-x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
-T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T
-(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
-(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1)))
-(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t
-(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind
-Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8
-x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1
-x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead
-(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
-(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda
-(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0:
-T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0
-(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
-(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t
-(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14:
-(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
-u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b:
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
-(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11))
-(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
-t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
-T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
-T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b:
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
-(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S
-O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1
-t2 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 3467
-END *)
theorem pr2_gen_lift:
\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1
H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1
t2 h d H6)))))))))))))))) c y x H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1579
-END *)
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr2/defs.ma".
+include "basic_1/pr2/defs.ma".
-include "Basic-1/pr0/pr0.ma".
+include "basic_1/pr0/pr0.ma".
-include "Basic-1/getl/props.ma".
+include "basic_1/getl/fwd.ma".
theorem pr2_confluence__pr2_free_free:
\forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0
T).(pr2 c t2 t))))))))
\def
\lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0
-t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t))
-(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2
-x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
-(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1)))))
-(pr0_confluence t0 t2 H0 t1 H))))))).
-(* COMMENTS
-Initial nodes: 135
-END *)
+(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(let TMP_1 \def (\lambda (t:
+T).(pr0 t2 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t1 t)) in (let TMP_3
+\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t: T).(pr2 c
+t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_10 \def (\lambda
+(x: T).(\lambda (H1: (pr0 t2 x)).(\lambda (H2: (pr0 t1 x)).(let TMP_6 \def
+(\lambda (t: T).(pr2 c t1 t)) in (let TMP_7 \def (\lambda (t: T).(pr2 c t2
+t)) in (let TMP_8 \def (pr2_free c t1 x H2) in (let TMP_9 \def (pr2_free c t2
+x H1) in (ex_intro2 T TMP_6 TMP_7 x TMP_8 TMP_9)))))))) in (let TMP_11 \def
+(pr0_confluence t0 t2 H0 t1 H) in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_10
+TMP_11))))))))))))).
theorem pr2_confluence__pr2_free_delta:
\forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall
\lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda
(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0
t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0
-t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4
-t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda
-(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4:
-(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda
-(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2
-c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2
-x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2:
-T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda
-(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7:
-(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0
-H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u))))))
-(pr0_confluence t0 t4 H1 t1 H))))))))))))).
-(* COMMENTS
-Initial nodes: 403
-END *)
+t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(let TMP_1 \def (\lambda (t:
+T).(pr0 t4 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t1 t)) in (let TMP_3
+\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t: T).(pr2 c
+t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_31 \def (\lambda
+(x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: (pr0 t1 x)).(let TMP_6 \def
+(pr0 t2 x) in (let TMP_7 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_8
+\def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_9 \def (ex2 T TMP_7
+TMP_8) in (let TMP_10 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_11 \def
+(\lambda (t: T).(pr2 c t2 t)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) in
+(let TMP_17 \def (\lambda (H5: (pr0 t2 x)).(let TMP_13 \def (\lambda (t:
+T).(pr2 c t1 t)) in (let TMP_14 \def (\lambda (t: T).(pr2 c t2 t)) in (let
+TMP_15 \def (pr2_free c t1 x H4) in (let TMP_16 \def (pr2_free c t2 x H5) in
+(ex_intro2 T TMP_13 TMP_14 x TMP_15 TMP_16)))))) in (let TMP_28 \def (\lambda
+(H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i u x
+w2)))).(let TMP_18 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_19 \def
+(\lambda (w2: T).(subst0 i u x w2)) in (let TMP_20 \def (\lambda (t: T).(pr2
+c t1 t)) in (let TMP_21 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_22
+\def (ex2 T TMP_20 TMP_21) in (let TMP_27 \def (\lambda (x0: T).(\lambda (H6:
+(pr0 t2 x0)).(\lambda (H7: (subst0 i u x x0)).(let TMP_23 \def (\lambda (t:
+T).(pr2 c t1 t)) in (let TMP_24 \def (\lambda (t: T).(pr2 c t2 t)) in (let
+TMP_25 \def (pr2_delta c d u i H0 t1 x H4 x0 H7) in (let TMP_26 \def
+(pr2_free c t2 x0 H6) in (ex_intro2 T TMP_23 TMP_24 x0 TMP_25 TMP_26))))))))
+in (ex2_ind T TMP_18 TMP_19 TMP_22 TMP_27 H5)))))))) in (let TMP_29 \def
+(pr0_refl u) in (let TMP_30 \def (pr0_subst0 t4 x H3 u t2 i H2 u TMP_29) in
+(or_ind TMP_6 TMP_9 TMP_12 TMP_17 TMP_28 TMP_30))))))))))))))) in (let TMP_32
+\def (pr0_confluence t0 t4 H1 t1 H) in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_31
+TMP_32))))))))))))))))))).
theorem pr2_confluence__pr2_delta_delta:
\forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall
T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i
c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0
i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda
-(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t:
-T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4
-x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0
-t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2
-x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
-w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
-(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda
-(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda
-(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0
-u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
-(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x
-x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))
-x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8))
-(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T
-(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind
-T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T
-(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0:
-T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0
-t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
-w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
-(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
-(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H
-t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2))
-(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2
-w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2
-x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t:
-T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i
-i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0
-i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
-t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15:
-(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
-T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d
-u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9
-(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def
-(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15
-\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0)))
-H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0:
-C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind
-Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u)
-(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0
-(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e
-in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
-\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono
-c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda
-(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x
-x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c
-(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0
-(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind
-(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t:
-T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda
-(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1
-x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in
-(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0
-(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T
-(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0
-t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i
-u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(let TMP_1 \def
+(\lambda (t: T).(pr0 t4 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t3 t)) in
+(let TMP_3 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t:
+T).(pr2 c t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_165 \def
+(\lambda (x: T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (pr0 t3 x)).(let
+TMP_6 \def (pr0 t1 x) in (let TMP_7 \def (\lambda (w2: T).(pr0 t1 w2)) in
+(let TMP_8 \def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_9 \def (ex2 T
+TMP_7 TMP_8) in (let TMP_10 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_11
+\def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11)
+in (let TMP_38 \def (\lambda (H7: (pr0 t1 x)).(let TMP_13 \def (pr0 t2 x) in
+(let TMP_14 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_15 \def (\lambda
+(w2: T).(subst0 i0 u0 x w2)) in (let TMP_16 \def (ex2 T TMP_14 TMP_15) in
+(let TMP_17 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_18 \def (\lambda
+(t: T).(pr2 c t2 t)) in (let TMP_19 \def (ex2 T TMP_17 TMP_18) in (let TMP_24
+\def (\lambda (H8: (pr0 t2 x)).(let TMP_20 \def (\lambda (t: T).(pr2 c t1 t))
+in (let TMP_21 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_22 \def
+(pr2_free c t1 x H7) in (let TMP_23 \def (pr2_free c t2 x H8) in (ex_intro2 T
+TMP_20 TMP_21 x TMP_22 TMP_23)))))) in (let TMP_35 \def (\lambda (H8: (ex2 T
+(\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)))).(let
+TMP_25 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_26 \def (\lambda (w2:
+T).(subst0 i0 u0 x w2)) in (let TMP_27 \def (\lambda (t: T).(pr2 c t1 t)) in
+(let TMP_28 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_29 \def (ex2 T
+TMP_27 TMP_28) in (let TMP_34 \def (\lambda (x0: T).(\lambda (H9: (pr0 t2
+x0)).(\lambda (H10: (subst0 i0 u0 x x0)).(let TMP_30 \def (\lambda (t:
+T).(pr2 c t1 t)) in (let TMP_31 \def (\lambda (t: T).(pr2 c t2 t)) in (let
+TMP_32 \def (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) in (let TMP_33 \def
+(pr2_free c t2 x0 H9) in (ex_intro2 T TMP_30 TMP_31 x0 TMP_32 TMP_33))))))))
+in (ex2_ind T TMP_25 TMP_26 TMP_29 TMP_34 H8)))))))) in (let TMP_36 \def
+(pr0_refl u0) in (let TMP_37 \def (pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 TMP_36)
+in (or_ind TMP_13 TMP_16 TMP_19 TMP_24 TMP_35 TMP_37))))))))))))) in (let
+TMP_162 \def (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2:
+T).(subst0 i u x w2)))).(let TMP_39 \def (\lambda (w2: T).(pr0 t1 w2)) in
+(let TMP_40 \def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_41 \def
+(\lambda (t: T).(pr2 c t1 t)) in (let TMP_42 \def (\lambda (t: T).(pr2 c t2
+t)) in (let TMP_43 \def (ex2 T TMP_41 TMP_42) in (let TMP_161 \def (\lambda
+(x0: T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(let
+TMP_44 \def (pr0 t2 x) in (let TMP_45 \def (\lambda (w2: T).(pr0 t2 w2)) in
+(let TMP_46 \def (\lambda (w2: T).(subst0 i0 u0 x w2)) in (let TMP_47 \def
+(ex2 T TMP_45 TMP_46) in (let TMP_48 \def (\lambda (t: T).(pr2 c t1 t)) in
+(let TMP_49 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_50 \def (ex2 T
+TMP_48 TMP_49) in (let TMP_55 \def (\lambda (H10: (pr0 t2 x)).(let TMP_51
+\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_52 \def (\lambda (t: T).(pr2 c
+t2 t)) in (let TMP_53 \def (pr2_free c t1 x0 H8) in (let TMP_54 \def
+(pr2_delta c d u i H t2 x H10 x0 H9) in (ex_intro2 T TMP_51 TMP_52 x0 TMP_53
+TMP_54)))))) in (let TMP_158 \def (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0
+t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)))).(let TMP_56 \def (\lambda
+(w2: T).(pr0 t2 w2)) in (let TMP_57 \def (\lambda (w2: T).(subst0 i0 u0 x
+w2)) in (let TMP_58 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_59 \def
+(\lambda (t: T).(pr2 c t2 t)) in (let TMP_60 \def (ex2 T TMP_58 TMP_59) in
+(let TMP_157 \def (\lambda (x1: T).(\lambda (H11: (pr0 t2 x1)).(\lambda (H12:
+(subst0 i0 u0 x x1)).(let TMP_61 \def (\lambda (t: T).(pr2 c t1 t)) in (let
+TMP_62 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_63 \def (ex2 T TMP_61
+TMP_62) in (let TMP_76 \def (\lambda (H13: (not (eq nat i i0))).(let TMP_64
+\def (\lambda (t: T).(subst0 i u x1 t)) in (let TMP_65 \def (\lambda (t:
+T).(subst0 i0 u0 x0 t)) in (let TMP_66 \def (\lambda (t: T).(pr2 c t1 t)) in
+(let TMP_67 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_68 \def (ex2 T
+TMP_66 TMP_67) in (let TMP_73 \def (\lambda (x2: T).(\lambda (H14: (subst0 i
+u x1 x2)).(\lambda (H15: (subst0 i0 u0 x0 x2)).(let TMP_69 \def (\lambda (t:
+T).(pr2 c t1 t)) in (let TMP_70 \def (\lambda (t: T).(pr2 c t2 t)) in (let
+TMP_71 \def (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) in (let TMP_72 \def
+(pr2_delta c d u i H t2 x1 H11 x2 H14) in (ex_intro2 T TMP_69 TMP_70 x2
+TMP_71 TMP_72)))))))) in (let TMP_74 \def (sym_not_eq nat i i0 H13) in (let
+TMP_75 \def (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 TMP_74) in
+(ex2_ind T TMP_64 TMP_65 TMP_68 TMP_73 TMP_75)))))))))) in (let TMP_156 \def
+(\lambda (H13: (eq nat i i0)).(let TMP_77 \def (\lambda (n: nat).(subst0 n u0
+x x1)) in (let H14 \def (eq_ind_r nat i0 TMP_77 H12 i H13) in (let TMP_80
+\def (\lambda (n: nat).(let TMP_78 \def (Bind Abbr) in (let TMP_79 \def
+(CHead d0 TMP_78 u0) in (getl n c TMP_79)))) in (let H15 \def (eq_ind_r nat
+i0 TMP_80 H2 i H13) in (let TMP_81 \def (Bind Abbr) in (let TMP_82 \def
+(CHead d TMP_81 u) in (let TMP_83 \def (\lambda (c0: C).(getl i c c0)) in
+(let TMP_84 \def (Bind Abbr) in (let TMP_85 \def (CHead d0 TMP_84 u0) in (let
+TMP_86 \def (Bind Abbr) in (let TMP_87 \def (CHead d TMP_86 u) in (let TMP_88
+\def (Bind Abbr) in (let TMP_89 \def (CHead d0 TMP_88 u0) in (let TMP_90 \def
+(getl_mono c TMP_87 i H TMP_89 H15) in (let H16 \def (eq_ind C TMP_82 TMP_83
+H TMP_85 TMP_90) in (let TMP_91 \def (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) in (let TMP_92 \def (Bind
+Abbr) in (let TMP_93 \def (CHead d TMP_92 u) in (let TMP_94 \def (Bind Abbr)
+in (let TMP_95 \def (CHead d0 TMP_94 u0) in (let TMP_96 \def (Bind Abbr) in
+(let TMP_97 \def (CHead d TMP_96 u) in (let TMP_98 \def (Bind Abbr) in (let
+TMP_99 \def (CHead d0 TMP_98 u0) in (let TMP_100 \def (getl_mono c TMP_97 i H
+TMP_99 H15) in (let H17 \def (f_equal C C TMP_91 TMP_93 TMP_95 TMP_100) in
+(let TMP_101 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) in (let TMP_102 \def (Bind Abbr) in (let
+TMP_103 \def (CHead d TMP_102 u) in (let TMP_104 \def (Bind Abbr) in (let
+TMP_105 \def (CHead d0 TMP_104 u0) in (let TMP_106 \def (Bind Abbr) in (let
+TMP_107 \def (CHead d TMP_106 u) in (let TMP_108 \def (Bind Abbr) in (let
+TMP_109 \def (CHead d0 TMP_108 u0) in (let TMP_110 \def (getl_mono c TMP_107
+i H TMP_109 H15) in (let H18 \def (f_equal C T TMP_101 TMP_103 TMP_105
+TMP_110) in (let TMP_155 \def (\lambda (H19: (eq C d d0)).(let TMP_111 \def
+(\lambda (t: T).(subst0 i t x x1)) in (let H20 \def (eq_ind_r T u0 TMP_111
+H14 u H18) in (let TMP_114 \def (\lambda (t: T).(let TMP_112 \def (Bind Abbr)
+in (let TMP_113 \def (CHead d0 TMP_112 t) in (getl i c TMP_113)))) in (let
+H21 \def (eq_ind_r T u0 TMP_114 H16 u H18) in (let TMP_117 \def (\lambda (c0:
+C).(let TMP_115 \def (Bind Abbr) in (let TMP_116 \def (CHead c0 TMP_115 u) in
+(getl i c TMP_116)))) in (let H22 \def (eq_ind_r C d0 TMP_117 H21 d H19) in
+(let TMP_118 \def (eq T x1 x0) in (let TMP_119 \def (\lambda (t: T).(subst0 i
+u x1 t)) in (let TMP_120 \def (\lambda (t: T).(subst0 i u x0 t)) in (let
+TMP_121 \def (ex2 T TMP_119 TMP_120) in (let TMP_122 \def (subst0 i u x1 x0)
+in (let TMP_123 \def (subst0 i u x0 x1) in (let TMP_124 \def (\lambda (t:
+T).(pr2 c t1 t)) in (let TMP_125 \def (\lambda (t: T).(pr2 c t2 t)) in (let
+TMP_126 \def (ex2 T TMP_124 TMP_125) in (let TMP_132 \def (\lambda (H23: (eq
+T x1 x0)).(let TMP_127 \def (\lambda (t: T).(pr0 t2 t)) in (let H24 \def
+(eq_ind T x1 TMP_127 H11 x0 H23) in (let TMP_128 \def (\lambda (t: T).(pr2 c
+t1 t)) in (let TMP_129 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_130
+\def (pr2_free c t1 x0 H8) in (let TMP_131 \def (pr2_free c t2 x0 H24) in
+(ex_intro2 T TMP_128 TMP_129 x0 TMP_130 TMP_131)))))))) in (let TMP_143 \def
+(\lambda (H23: (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t:
+T).(subst0 i u x0 t)))).(let TMP_133 \def (\lambda (t: T).(subst0 i u x1 t))
+in (let TMP_134 \def (\lambda (t: T).(subst0 i u x0 t)) in (let TMP_135 \def
+(\lambda (t: T).(pr2 c t1 t)) in (let TMP_136 \def (\lambda (t: T).(pr2 c t2
+t)) in (let TMP_137 \def (ex2 T TMP_135 TMP_136) in (let TMP_142 \def
(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i
-u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c
-t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2
-x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T
-(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1
-x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u
-x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
-t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11)))
-(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10))
-(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x
-H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3
-H0))))))))))))))))))).
-(* COMMENTS
-Initial nodes: 1901
-END *)
+u x0 x2)).(let TMP_138 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_139
+\def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_140 \def (pr2_delta c d u i
+H22 t1 x0 H8 x2 H25) in (let TMP_141 \def (pr2_delta c d u i H22 t2 x1 H11 x2
+H24) in (ex_intro2 T TMP_138 TMP_139 x2 TMP_140 TMP_141)))))))) in (ex2_ind T
+TMP_133 TMP_134 TMP_137 TMP_142 H23)))))))) in (let TMP_148 \def (\lambda
+(H23: (subst0 i u x1 x0)).(let TMP_144 \def (\lambda (t: T).(pr2 c t1 t)) in
+(let TMP_145 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_146 \def
+(pr2_free c t1 x0 H8) in (let TMP_147 \def (pr2_delta c d u i H22 t2 x1 H11
+x0 H23) in (ex_intro2 T TMP_144 TMP_145 x0 TMP_146 TMP_147)))))) in (let
+TMP_153 \def (\lambda (H23: (subst0 i u x0 x1)).(let TMP_149 \def (\lambda
+(t: T).(pr2 c t1 t)) in (let TMP_150 \def (\lambda (t: T).(pr2 c t2 t)) in
+(let TMP_151 \def (pr2_delta c d u i H22 t1 x0 H8 x1 H23) in (let TMP_152
+\def (pr2_free c t2 x1 H11) in (ex_intro2 T TMP_149 TMP_150 x1 TMP_151
+TMP_152)))))) in (let TMP_154 \def (subst0_confluence_eq x x1 u i H20 x0 H9)
+in (or4_ind TMP_118 TMP_121 TMP_122 TMP_123 TMP_126 TMP_132 TMP_143 TMP_148
+TMP_153 TMP_154)))))))))))))))))))))) in (TMP_155
+H17)))))))))))))))))))))))))))))))))))))))) in (neq_eq_e i i0 TMP_63 TMP_76
+TMP_156))))))))) in (ex2_ind T TMP_56 TMP_57 TMP_60 TMP_157 H10)))))))) in
+(let TMP_159 \def (pr0_refl u0) in (let TMP_160 \def (pr0_subst0 t4 x H5 u0
+t2 i0 H4 u0 TMP_159) in (or_ind TMP_44 TMP_47 TMP_50 TMP_55 TMP_158
+TMP_160))))))))))))))) in (ex2_ind T TMP_39 TMP_40 TMP_43 TMP_161 H7))))))))
+in (let TMP_163 \def (pr0_refl u) in (let TMP_164 \def (pr0_subst0 t3 x H6 u
+t1 i H1 u TMP_163) in (or_ind TMP_6 TMP_9 TMP_12 TMP_38 TMP_162
+TMP_164))))))))))))))) in (let TMP_166 \def (pr0_confluence t0 t4 H3 t3 H0)
+in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_165 TMP_166))))))))))))))))))))))))).
theorem pr2_confluence:
\forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
T).(pr2 c t2 t))))))))
\def
\lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
-t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in
-pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_:
-(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T
-(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with
+t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H with
[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3:
-(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T
-t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c
-t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind
-T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5:
-T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4
-t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5:
-T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0
-t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t:
-T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0)
-\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6:
-T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda
-(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6
-t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5
-t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
-t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6
-t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7:
-T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t:
-T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7:
-T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0
-t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2
-H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) |
-(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1
-c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c
-(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d
-(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda
-(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14:
-(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c
-(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T
-(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))))
-(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d
-(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda
-(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16:
-(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda
-(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i
-H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1
-(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c)
-(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T
-t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4
-H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3
-t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0)
-\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4)
-\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda
-(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0
-(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to
-((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1
-t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind
-T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4)
-\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda
-(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr)
-u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12
-\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6:
-T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6
-t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2
-t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C
-c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c
-(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda
-(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7
-t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
-t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0
-t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
-t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1)
-(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6
-(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14
-H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow
-(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T
-t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to
-((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0
-t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2
-t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T
-t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to
-((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda
-(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2
-(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to
-((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda
-(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr)
-u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6
+(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(let TMP_54 \def (\lambda (_:
+C).((eq T t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (let TMP_52 \def
+(\lambda (t: T).(pr2 c t1 t)) in (let TMP_53 \def (\lambda (t: T).(pr2 c t2
+t)) in (ex2 T TMP_52 TMP_53))))))) in (let TMP_98 \def (\lambda (H5: (eq T t3
+t0)).(let TMP_57 \def (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (let
+TMP_55 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_56 \def (\lambda (t5:
+T).(pr2 c t2 t5)) in (ex2 T TMP_55 TMP_56)))))) in (let TMP_96 \def (\lambda
+(H6: (eq T t4 t1)).(let TMP_60 \def (\lambda (t: T).((pr0 t0 t) \to (let
+TMP_58 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_59 \def (\lambda (t5:
+T).(pr2 c t2 t5)) in (ex2 T TMP_58 TMP_59))))) in (let TMP_94 \def (\lambda
+(H7: (pr0 t0 t1)).(let H8 \def (match H0 with [(pr2_free c1 t5 t6 H8)
+\Rightarrow (\lambda (H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda
+(H11: (eq T t6 t2)).(let TMP_78 \def (\lambda (_: C).((eq T t5 t0) \to ((eq T
+t6 t2) \to ((pr0 t5 t6) \to (let TMP_76 \def (\lambda (t: T).(pr2 c t1 t)) in
+(let TMP_77 \def (\lambda (t: T).(pr2 c t2 t)) in (ex2 T TMP_76 TMP_77)))))))
+in (let TMP_89 \def (\lambda (H12: (eq T t5 t0)).(let TMP_81 \def (\lambda
+(t: T).((eq T t6 t2) \to ((pr0 t t6) \to (let TMP_79 \def (\lambda (t7:
+T).(pr2 c t1 t7)) in (let TMP_80 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2
+T TMP_79 TMP_80)))))) in (let TMP_87 \def (\lambda (H13: (eq T t6 t2)).(let
+TMP_84 \def (\lambda (t: T).((pr0 t0 t) \to (let TMP_82 \def (\lambda (t7:
+T).(pr2 c t1 t7)) in (let TMP_83 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2
+T TMP_82 TMP_83))))) in (let TMP_85 \def (\lambda (H14: (pr0 t0
+t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) in (let TMP_86 \def
+(sym_eq T t6 t2 H13) in (eq_ind T t2 TMP_84 TMP_85 t6 TMP_86))))) in (let
+TMP_88 \def (sym_eq T t5 t0 H12) in (eq_ind T t0 TMP_81 TMP_87 t5 TMP_88)))))
+in (let TMP_90 \def (sym_eq C c1 c H9) in (eq_ind C c TMP_78 TMP_89 c1 TMP_90
+H10 H11 H8))))))) | (pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow
+(\lambda (H11: (eq C c1 c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T
+t t2)).(let TMP_63 \def (\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to
+((getl i c2 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t)
+\to (let TMP_61 \def (\lambda (t7: T).(pr2 c t1 t7)) in (let TMP_62 \def
+(\lambda (t7: T).(pr2 c t2 t7)) in (ex2 T TMP_61 TMP_62))))))))) in (let
+TMP_74 \def (\lambda (H14: (eq T t5 t0)).(let TMP_66 \def (\lambda (t7:
+T).((eq T t t2) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to
+((subst0 i u t6 t) \to (let TMP_64 \def (\lambda (t8: T).(pr2 c t1 t8)) in
+(let TMP_65 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T TMP_64
+TMP_65)))))))) in (let TMP_72 \def (\lambda (H15: (eq T t t2)).(let TMP_69
+\def (\lambda (t7: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6)
+\to ((subst0 i u t6 t7) \to (let TMP_67 \def (\lambda (t8: T).(pr2 c t1 t8))
+in (let TMP_68 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T TMP_67
+TMP_68))))))) in (let TMP_70 \def (\lambda (H16: (getl i c (CHead d (Bind
+Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda (H18: (subst0 i u t6
+t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i H7 H16 H17 H18))))
+in (let TMP_71 \def (sym_eq T t t2 H15) in (eq_ind T t2 TMP_69 TMP_70 t
+TMP_71))))) in (let TMP_73 \def (sym_eq T t5 t0 H14) in (eq_ind T t0 TMP_66
+TMP_72 t5 TMP_73))))) in (let TMP_75 \def (sym_eq C c1 c H11) in (eq_ind C c
+TMP_63 TMP_74 c1 TMP_75 H12 H13 H8 H9 H10)))))))]) in (let TMP_91 \def
+(refl_equal C c) in (let TMP_92 \def (refl_equal T t0) in (let TMP_93 \def
+(refl_equal T t2) in (H8 TMP_91 TMP_92 TMP_93)))))) in (let TMP_95 \def
+(sym_eq T t4 t1 H6) in (eq_ind T t1 TMP_60 TMP_94 t4 TMP_95))))) in (let
+TMP_97 \def (sym_eq T t3 t0 H5) in (eq_ind T t0 TMP_57 TMP_96 t3 TMP_97)))))
+in (let TMP_99 \def (sym_eq C c0 c H2) in (eq_ind C c TMP_54 TMP_98 c0 TMP_99
+H3 H4 H1))))))) | (pr2_delta c0 d u i H1 t3 t4 H2 t H3) \Rightarrow (\lambda
+(H4: (eq C c0 c)).(\lambda (H5: (eq T t3 t0)).(\lambda (H6: (eq T t t1)).(let
+TMP_3 \def (\lambda (c1: C).((eq T t3 t0) \to ((eq T t t1) \to ((getl i c1
+(CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) \to ((subst0 i u t4 t) \to (let
+TMP_1 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_2 \def (\lambda (t5:
+T).(pr2 c t2 t5)) in (ex2 T TMP_1 TMP_2))))))))) in (let TMP_50 \def (\lambda
+(H7: (eq T t3 t0)).(let TMP_6 \def (\lambda (t5: T).((eq T t t1) \to ((getl i
+c (CHead d (Bind Abbr) u)) \to ((pr0 t5 t4) \to ((subst0 i u t4 t) \to (let
+TMP_4 \def (\lambda (t6: T).(pr2 c t1 t6)) in (let TMP_5 \def (\lambda (t6:
+T).(pr2 c t2 t6)) in (ex2 T TMP_4 TMP_5)))))))) in (let TMP_48 \def (\lambda
+(H8: (eq T t t1)).(let TMP_9 \def (\lambda (t5: T).((getl i c (CHead d (Bind
+Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t5) \to (let TMP_7 \def
+(\lambda (t6: T).(pr2 c t1 t6)) in (let TMP_8 \def (\lambda (t6: T).(pr2 c t2
+t6)) in (ex2 T TMP_7 TMP_8))))))) in (let TMP_46 \def (\lambda (H9: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0
+i u t4 t1)).(let H12 \def (match H0 with [(pr2_free c1 t5 t6 H12) \Rightarrow
+(\lambda (H13: (eq C c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T
+t6 t2)).(let TMP_27 \def (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to
+((pr0 t5 t6) \to (let TMP_25 \def (\lambda (t7: T).(pr2 c t1 t7)) in (let
+TMP_26 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2 T TMP_25 TMP_26))))))) in
+(let TMP_41 \def (\lambda (H16: (eq T t5 t0)).(let TMP_30 \def (\lambda (t7:
+T).((eq T t6 t2) \to ((pr0 t7 t6) \to (let TMP_28 \def (\lambda (t8: T).(pr2
+c t1 t8)) in (let TMP_29 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T
+TMP_28 TMP_29)))))) in (let TMP_39 \def (\lambda (H17: (eq T t6 t2)).(let
+TMP_33 \def (\lambda (t7: T).((pr0 t0 t7) \to (let TMP_31 \def (\lambda (t8:
+T).(pr2 c t1 t8)) in (let TMP_32 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2
+T TMP_31 TMP_32))))) in (let TMP_37 \def (\lambda (H18: (pr0 t0 t2)).(let
+TMP_34 \def (pr2 c t2) in (let TMP_35 \def (pr2 c t1) in (let TMP_36 \def
+(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11) in
+(ex2_sym T TMP_34 TMP_35 TMP_36))))) in (let TMP_38 \def (sym_eq T t6 t2 H17)
+in (eq_ind T t2 TMP_33 TMP_37 t6 TMP_38))))) in (let TMP_40 \def (sym_eq T t5
+t0 H16) in (eq_ind T t0 TMP_30 TMP_39 t5 TMP_40))))) in (let TMP_42 \def
+(sym_eq C c1 c H13) in (eq_ind C c TMP_27 TMP_41 c1 TMP_42 H14 H15 H12)))))))
+| (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow (\lambda (H15: (eq
+C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T t7 t2)).(let
+TMP_12 \def (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to ((getl i0 c2
+(CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 t6 t7) \to
+(let TMP_10 \def (\lambda (t8: T).(pr2 c t1 t8)) in (let TMP_11 \def (\lambda
+(t8: T).(pr2 c t2 t8)) in (ex2 T TMP_10 TMP_11))))))))) in (let TMP_23 \def
+(\lambda (H18: (eq T t5 t0)).(let TMP_15 \def (\lambda (t8: T).((eq T t7 t2)
+\to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to ((subst0 i0
+u0 t6 t7) \to (let TMP_13 \def (\lambda (t9: T).(pr2 c t1 t9)) in (let TMP_14
+\def (\lambda (t9: T).(pr2 c t2 t9)) in (ex2 T TMP_13 TMP_14)))))))) in (let
+TMP_21 \def (\lambda (H19: (eq T t7 t2)).(let TMP_18 \def (\lambda (t8:
+T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to ((subst0 i0 u0
+t6 t8) \to (let TMP_16 \def (\lambda (t9: T).(pr2 c t1 t9)) in (let TMP_17
+\def (\lambda (t9: T).(pr2 c t2 t9)) in (ex2 T TMP_16 TMP_17))))))) in (let
+TMP_19 \def (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda
+(H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6
t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10
-H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1
-(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c)
-(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T
-t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C
-c) (refl_equal T t0) (refl_equal T t1)))))))).
-(* COMMENTS
-Initial nodes: 2087
-END *)
+H11 H20 H21 H22)))) in (let TMP_20 \def (sym_eq T t7 t2 H19) in (eq_ind T t2
+TMP_18 TMP_19 t7 TMP_20))))) in (let TMP_22 \def (sym_eq T t5 t0 H18) in
+(eq_ind T t0 TMP_15 TMP_21 t5 TMP_22))))) in (let TMP_24 \def (sym_eq C c1 c
+H15) in (eq_ind C c TMP_12 TMP_23 c1 TMP_24 H16 H17 H12 H13 H14)))))))]) in
+(let TMP_43 \def (refl_equal C c) in (let TMP_44 \def (refl_equal T t0) in
+(let TMP_45 \def (refl_equal T t2) in (H12 TMP_43 TMP_44 TMP_45)))))))) in
+(let TMP_47 \def (sym_eq T t t1 H8) in (eq_ind T t1 TMP_9 TMP_46 t
+TMP_47))))) in (let TMP_49 \def (sym_eq T t3 t0 H7) in (eq_ind T t0 TMP_6
+TMP_48 t3 TMP_49))))) in (let TMP_51 \def (sym_eq C c0 c H4) in (eq_ind C c
+TMP_3 TMP_50 c0 TMP_51 H5 H6 H1 H2 H3)))))))]) in (let TMP_100 \def
+(refl_equal C c) in (let TMP_101 \def (refl_equal T t0) in (let TMP_102 \def
+(refl_equal T t1) in (H1 TMP_100 TMP_101 TMP_102)))))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr2/defs.ma".
+include "basic_1/pr2/fwd.ma".
-include "Basic-1/pr0/props.ma".
-
-include "Basic-1/getl/drop.ma".
-
-include "Basic-1/getl/clear.ma".
+include "basic_1/pr0/subst0.ma".
theorem pr2_thin_dx:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0
t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2
u)))))))))))) c t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 239
-END *)
theorem pr2_head_1:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1
t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c
u1 u2 H)))))).
-(* COMMENTS
-Initial nodes: 219
-END *)
theorem pr2_head_2:
\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4
t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0
(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b)
-u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
-c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c
-(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind
-Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
-\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
+u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u
-(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d
-(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr)
-u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in
-(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T
-u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda
-(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c
-(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u)
-t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1:
-(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
-T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
-c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u
-t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
-(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b)
-u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
-Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
-(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
-(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
-(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u
-t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n)
-(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3)
-(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead
-(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u)))))))))))))
-i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
-T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind
-Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall
-(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c
-(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0
-(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2:
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4:
-(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1:
-C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in
-(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c)
-(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f)
-u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4)
-(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t)
-(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1:
-(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
-T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
-c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u
-t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
-(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
-(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f)
-u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
-Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
-(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
-(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
-(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u
-t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n)
-(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3)
-(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead
-(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u)))))))))))))
-i)))))) k) y t1 t2 H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1947
-END *)
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H7
+\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr |
+(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat
+_) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u)
+(clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b)
+u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 \def (f_equal C T (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0]))
+(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d
+(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)
+H5))) in (\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def
+(eq_ind T u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr
+(\lambda (b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t)))
+(pr2_free c (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u
+(pr0_refl u) t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n:
+nat).(\lambda (H1: (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3:
+T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to
+((eq C c0 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind
+b) u t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr)
+u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c
+(Bind b) u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1
+(CHead d (Bind Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def
+(eq_ind C c0 (\lambda (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to
+(\forall (t5: T).(\forall (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0
+n u0 t6 t0) \to ((eq C c1 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u
+t5) (THead (Bind b) u t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta
+c d u0 (r (Bind b) n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6)
+(THead (Bind b) u t3) (THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4
+H3 (Bind b)) (THead (Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n)
+H4 u))))))))))))) i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0
+(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4)
+\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u))
+\to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1:
+(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4
+t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0
+(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f)
+u) H4) in (pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c
+(drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O
+(CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3)
+(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead
+(Flat f) u t) (subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n:
+nat).(\lambda (H1: (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3:
+T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to
+((eq C c0 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat
+f) u t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr)
+u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c
+(Flat f) u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1
+(CHead d (Bind Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def
+(eq_ind C c0 (\lambda (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to
+(\forall (t5: T).(\forall (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0
+n u0 t6 t0) \to ((eq C c1 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u
+t5) (THead (Flat f) u t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta
+c d u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6)
+(THead (Flat f) u t3) (THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4
+H3 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n)
+H4 u))))))))))))) i)))))) k) y t1 t2 H0))) H)))))).
theorem clear_pr2_trans:
\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to
t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1:
C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c
(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 171
-END *)
theorem pr2_cflat:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u
i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c
t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 175
-END *)
theorem pr2_ctail:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2:
(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail
Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 171
-END *)
theorem pr2_change:
\forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))
(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr)
u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
-v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind
-b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H10 \def (f_equal C B (\lambda
-(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind
-Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
+C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d
+(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)
+H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d
-(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr)
-u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in
-(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind
-T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def
-(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B
-Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match
-(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c
-(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda
-(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr)
-u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
+v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind
+b) v1) (CHead d (Bind Abbr) u) H7))) in (\lambda (H12: (eq B Abbr
+b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind T u (\lambda (t0:
+T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def (eq_ind_r B b (\lambda
+(b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B Abbr (\lambda (b0:
+B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match (H15 (refl_equal B
+Abbr)) in False with []) in H16) b H12)))))) H10)) H9)))) (\lambda (i0:
+nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))
+\to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda
(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda
(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0)
(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c
(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4)))))))))))))
y t1 t2 H1))) H0)))))))).
-(* COMMENTS
-Initial nodes: 913
-END *)
theorem pr2_lift:
\forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1:
C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c
(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def
-(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0
-(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
-O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
-(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c
-(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9:
-(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11:
-(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i)
-(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i
-H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u
-H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h
-(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c
-(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x
-(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S
-i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead
-x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d
+(drop_getl_trans_le i d (le_S_n i d (le_S_n (S i) (S d) (le_S (S (S i)) (S d)
+(le_n_S (S i) d H7)))) c e h H (CHead d0 (Bind Abbr) u) H6) in (ex3_2_ind C C
+(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda
+(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear
+e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t3) (lift h d t)) (\lambda
+(x0: C).(\lambda (x1: C).(\lambda (H9: (drop i O c x0)).(\lambda (H10: (drop
+h (minus d i) x0 x1)).(\lambda (H11: (clear x1 (CHead d0 (Bind Abbr)
+u))).(let H12 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0
+x1)) H10 (S (minus d (S i))) (minus_x_Sy d i H7)) in (let H13 \def
+(drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u H11) in (ex2_ind C
+(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i))
+u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t3)
+(lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x (Bind Abbr)
+(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x
+d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x
+(Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d
t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7
h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h)
(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3)
(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h
H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
-(* COMMENTS
-Initial nodes: 849
-END *)
+
+theorem pr2_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u:
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T
+T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
+(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
+t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
+(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
+(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind
+Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0
+(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2))))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O
+t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
+(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in
+(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
+b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
+c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
+t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
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+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13
+(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_:
+T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z:
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr)
+u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl
+Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S
+i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl
+x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda
+(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
+(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
+t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t
+(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15:
+(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1
+x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13
+(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
+u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O
+(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr)
+u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1
+x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8)))))))
+H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x))))))
+H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
+(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2)
+H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i)))))))
+(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
+
+theorem pr2_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))
+(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda
+(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift
+(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1
+t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind
+Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind
+b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u)
+t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
+c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1
+x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1)
+(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead
+c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1
+x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1
+H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0
+(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
+t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T
+t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1
+x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
+T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T
+(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t
+(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind
+Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8
+x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1
+x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead
+(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda
+(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0:
+T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0
+(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t
+(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14:
+(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11))
+(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
+t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S
+O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1
+t2 H5)))))))))))))) c y x H0))) H))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr2/defs.ma".
+include "basic_1/pr2/fwd.ma".
-include "Basic-1/pr0/subst1.ma".
+include "basic_1/pr0/subst1.ma".
-include "Basic-1/pr0/fwd.ma".
+include "basic_1/csubst1/getl.ma".
-include "Basic-1/csubst1/getl.ma".
-
-include "Basic-1/csubst1/fwd.ma".
-
-include "Basic-1/subst1/subst1.ma".
-
-include "Basic-1/getl/drop.ma".
+include "basic_1/subst1/subst1.ma".
theorem pr2_delta1:
\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0)
(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2
H0 t0 H2))) t H1)))))))))).
-(* COMMENTS
-Initial nodes: 111
-END *)
theorem pr2_subst1:
\forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v)
(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead
e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal
-C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v)
-(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d
-(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _
-t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u)
-(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in
+C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow e | (CHead c1 _ _)
+\Rightarrow c1])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono
+c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in ((let H15 \def
+(f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow v |
+(CHead _ _ t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr)
+u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in
(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl
i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u
(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C
(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9)))))))
H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0
H5))))))))))))))) y t1 t2 H1))) H0)))))))).
-(* COMMENTS
-Initial nodes: 1311
-END *)
theorem pr2_gen_cabbr:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1))
H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead
e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match
-e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _
-_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0)
-(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in
-((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind
-Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d
+e0 with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d
+(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u)
+i H0 (CHead e (Bind Abbr) u0) H16)) in ((let H19 \def (f_equal C T (\lambda
+(e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow
+t0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d
+(Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d
e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind
Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0:
T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r
(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t
(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u
(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0
-(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d
-(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n:
-nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3
-H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S
-O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S
-O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift
-(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12))))))))))
-(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0
-x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 3757
-END *)
+(csubst1_getl_ge d0 i (le_S_n d0 i (le_S_n (S d0) (S i) (le_S (S (S d0)) (S
+i) (le_n_S (S d0) i H12)))) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a (S O)
+d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0
+(S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 H16)))))) (subst1_gen_lift_ge u x0
+x2 i (S O) d0 H14 (eq_ind_r nat (plus (S O) d0) (\lambda (n: nat).(le n i))
+H12 (plus d0 (S O)) (plus_sym d0 (S O)))))))) (subst1_confluence_neq t4 t u i
+(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i
+(lt_neq d0 i H12)))))))))) (pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3
+t4 H1 u0 (lift (S O) d0 x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1
+t2 H)))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/pr0/defs.ma".
+include "basic_1/pr0/defs.ma".
-include "Basic-1/C/defs.ma".
+include "basic_1/C/defs.ma".
inductive wcpr0: C \to (C \to Prop) \def
| wcpr0_refl: \forall (c: C).(wcpr0 c c)
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/wcpr0/defs.ma".
+include "basic_1/wcpr0/defs.ma".
+
+let rec wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P c c)))
+(f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) \to
+(\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P (CHead
+c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on w: P c
+c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp c1 c2
+w0 u1 u2 p k) \Rightarrow (let TMP_1 \def ((wcpr0_ind P f f0) c1 c2 w0) in
+(f0 c1 c2 w0 TMP_1 u1 u2 p k))].
theorem wcpr0_gen_sort:
\forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort
n))))
\def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n)
-x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C
-x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c:
-C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c:
-C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
-C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0))
-(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda
-(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2
-c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
-(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C
-(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
-with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
-(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1))
-H5))))))))))) y x H0))) H))).
-(* COMMENTS
-Initial nodes: 249
-END *)
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let
+TMP_1 \def (CSort n) in (let TMP_2 \def (\lambda (c: C).(wcpr0 c x)) in (let
+TMP_3 \def (\lambda (c: C).(eq C x c)) in (let TMP_19 \def (\lambda (y:
+C).(\lambda (H0: (wcpr0 y x)).(let TMP_4 \def (\lambda (c: C).(\lambda (c0:
+C).((eq C c (CSort n)) \to (eq C c0 c)))) in (let TMP_11 \def (\lambda (c:
+C).(\lambda (H1: (eq C c (CSort n))).(let TMP_5 \def (\lambda (e: C).e) in
+(let TMP_6 \def (CSort n) in (let H2 \def (f_equal C C TMP_5 c TMP_6 H1) in
+(let TMP_7 \def (CSort n) in (let TMP_8 \def (\lambda (c0: C).(eq C c0 c0))
+in (let TMP_9 \def (CSort n) in (let TMP_10 \def (refl_equal C TMP_9) in
+(eq_ind_r C TMP_7 TMP_8 TMP_10 c H2)))))))))) in (let TMP_18 \def (\lambda
+(c1: C).(\lambda (c2: C).(\lambda (_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1
+(CSort n)) \to (eq C c2 c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_:
+(pr0 u1 u2)).(\lambda (k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort
+n))).(let TMP_12 \def (CHead c1 k u1) in (let TMP_13 \def (\lambda (ee:
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
+True])) in (let TMP_14 \def (CSort n) in (let H5 \def (eq_ind C TMP_12 TMP_13
+I TMP_14 H4) in (let TMP_15 \def (CHead c2 k u2) in (let TMP_16 \def (CHead
+c1 k u1) in (let TMP_17 \def (eq C TMP_15 TMP_16) in (False_ind TMP_17
+H5))))))))))))))))) in (wcpr0_ind TMP_4 TMP_11 TMP_18 y x H0)))))) in
+(insert_eq C TMP_1 TMP_2 TMP_3 TMP_19 H))))))).
theorem wcpr0_gen_head:
\forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0
T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
\def
\lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
-(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c:
-C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2:
-C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda
-(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0:
-C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2:
-C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))
-(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal
-C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1)
-(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
-T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1
-c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C
-(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
-T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))
-(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2:
+(H: (wcpr0 (CHead c1 k u1) x)).(let TMP_1 \def (CHead c1 k u1) in (let TMP_2
+\def (\lambda (c: C).(wcpr0 c x)) in (let TMP_9 \def (\lambda (c: C).(let
+TMP_3 \def (eq C x c) in (let TMP_5 \def (\lambda (c2: C).(\lambda (u2:
+T).(let TMP_4 \def (CHead c2 k u2) in (eq C x TMP_4)))) in (let TMP_6 \def
+(\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) in (let TMP_7 \def (\lambda
+(_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_8 \def (ex3_2 C T TMP_5
+TMP_6 TMP_7) in (or TMP_3 TMP_8))))))) in (let TMP_111 \def (\lambda (y:
+C).(\lambda (H0: (wcpr0 y x)).(let TMP_16 \def (\lambda (c: C).(\lambda (c0:
+C).((eq C c (CHead c1 k u1)) \to (let TMP_10 \def (eq C c0 c) in (let TMP_12
+\def (\lambda (c2: C).(\lambda (u2: T).(let TMP_11 \def (CHead c2 k u2) in
+(eq C c0 TMP_11)))) in (let TMP_13 \def (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) in (let TMP_14 \def (\lambda (_: C).(\lambda (u2: T).(pr0
+u1 u2))) in (let TMP_15 \def (ex3_2 C T TMP_12 TMP_13 TMP_14) in (or TMP_10
+TMP_15))))))))) in (let TMP_39 \def (\lambda (c: C).(\lambda (H1: (eq C c
+(CHead c1 k u1))).(let TMP_17 \def (\lambda (e: C).e) in (let TMP_18 \def
+(CHead c1 k u1) in (let H2 \def (f_equal C C TMP_17 c TMP_18 H1) in (let
+TMP_19 \def (CHead c1 k u1) in (let TMP_26 \def (\lambda (c0: C).(let TMP_20
+\def (eq C c0 c0) in (let TMP_22 \def (\lambda (c2: C).(\lambda (u2: T).(let
+TMP_21 \def (CHead c2 k u2) in (eq C c0 TMP_21)))) in (let TMP_23 \def
+(\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) in (let TMP_24 \def (\lambda
+(_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_25 \def (ex3_2 C T TMP_22
+TMP_23 TMP_24) in (or TMP_20 TMP_25))))))) in (let TMP_27 \def (CHead c1 k
+u1) in (let TMP_28 \def (CHead c1 k u1) in (let TMP_29 \def (eq C TMP_27
+TMP_28) in (let TMP_32 \def (\lambda (c2: C).(\lambda (u2: T).(let TMP_30
+\def (CHead c1 k u1) in (let TMP_31 \def (CHead c2 k u2) in (eq C TMP_30
+TMP_31))))) in (let TMP_33 \def (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1
+c2))) in (let TMP_34 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in
+(let TMP_35 \def (ex3_2 C T TMP_32 TMP_33 TMP_34) in (let TMP_36 \def (CHead
+c1 k u1) in (let TMP_37 \def (refl_equal C TMP_36) in (let TMP_38 \def
+(or_introl TMP_29 TMP_35 TMP_37) in (eq_ind_r C TMP_19 TMP_26 TMP_38 c
+H2)))))))))))))))))) in (let TMP_110 \def (\lambda (c0: C).(\lambda (c2:
C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to
(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead
c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_:
C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2:
T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0
-k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
-\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
-(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
-\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K
-k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C
-(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3:
-T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_:
-T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10
-\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1
-(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda
-(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
-(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k
-u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C
-c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def
-(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1
-(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda
-(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
-(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T
-(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3))))
-(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq
-C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0
-c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C
-(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x
-H0))) H))))).
-(* COMMENTS
-Initial nodes: 1133
-END *)
+k0 u0) (CHead c1 k u1))).(let TMP_40 \def (\lambda (e: C).(match e with
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) in (let TMP_41
+\def (CHead c0 k0 u0) in (let TMP_42 \def (CHead c1 k u1) in (let H5 \def
+(f_equal C C TMP_40 TMP_41 TMP_42 H4) in (let TMP_43 \def (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
+in (let TMP_44 \def (CHead c0 k0 u0) in (let TMP_45 \def (CHead c1 k u1) in
+(let H6 \def (f_equal C K TMP_43 TMP_44 TMP_45 H4) in (let TMP_46 \def
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) in (let TMP_47 \def (CHead c0 k0 u0) in (let TMP_48 \def
+(CHead c1 k u1) in (let H7 \def (f_equal C T TMP_46 TMP_47 TMP_48 H4) in (let
+TMP_108 \def (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 c1)).(let
+TMP_58 \def (\lambda (k1: K).(let TMP_49 \def (CHead c2 k1 u2) in (let TMP_50
+\def (CHead c0 k1 u0) in (let TMP_51 \def (eq C TMP_49 TMP_50) in (let TMP_54
+\def (\lambda (c3: C).(\lambda (u3: T).(let TMP_52 \def (CHead c2 k1 u2) in
+(let TMP_53 \def (CHead c3 k u3) in (eq C TMP_52 TMP_53))))) in (let TMP_55
+\def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_56 \def
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_57 \def (ex3_2 C T
+TMP_54 TMP_55 TMP_56) in (or TMP_51 TMP_57))))))))) in (let TMP_59 \def
+(\lambda (t: T).(pr0 t u2)) in (let H10 \def (eq_ind T u0 TMP_59 H3 u1 H7) in
+(let TMP_69 \def (\lambda (t: T).(let TMP_60 \def (CHead c2 k u2) in (let
+TMP_61 \def (CHead c0 k t) in (let TMP_62 \def (eq C TMP_60 TMP_61) in (let
+TMP_65 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_63 \def (CHead c2 k
+u2) in (let TMP_64 \def (CHead c3 k u3) in (eq C TMP_63 TMP_64))))) in (let
+TMP_66 \def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_67
+\def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_68 \def
+(ex3_2 C T TMP_65 TMP_66 TMP_67) in (or TMP_62 TMP_68))))))))) in (let TMP_76
+\def (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (let TMP_70 \def (eq C c2
+c) in (let TMP_72 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_71 \def
+(CHead c3 k u3) in (eq C c2 TMP_71)))) in (let TMP_73 \def (\lambda (c3:
+C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_74 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_75 \def (ex3_2 C T TMP_72
+TMP_73 TMP_74) in (or TMP_70 TMP_75)))))))) in (let H11 \def (eq_ind C c0
+TMP_76 H2 c1 H9) in (let TMP_77 \def (\lambda (c: C).(wcpr0 c c2)) in (let
+H12 \def (eq_ind C c0 TMP_77 H1 c1 H9) in (let TMP_87 \def (\lambda (c:
+C).(let TMP_78 \def (CHead c2 k u2) in (let TMP_79 \def (CHead c k u1) in
+(let TMP_80 \def (eq C TMP_78 TMP_79) in (let TMP_83 \def (\lambda (c3:
+C).(\lambda (u3: T).(let TMP_81 \def (CHead c2 k u2) in (let TMP_82 \def
+(CHead c3 k u3) in (eq C TMP_81 TMP_82))))) in (let TMP_84 \def (\lambda (c3:
+C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_85 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_86 \def (ex3_2 C T TMP_83
+TMP_84 TMP_85) in (or TMP_80 TMP_86))))))))) in (let TMP_88 \def (CHead c2 k
+u2) in (let TMP_89 \def (CHead c1 k u1) in (let TMP_90 \def (eq C TMP_88
+TMP_89) in (let TMP_93 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_91
+\def (CHead c2 k u2) in (let TMP_92 \def (CHead c3 k u3) in (eq C TMP_91
+TMP_92))))) in (let TMP_94 \def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1
+c3))) in (let TMP_95 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in
+(let TMP_96 \def (ex3_2 C T TMP_93 TMP_94 TMP_95) in (let TMP_99 \def
+(\lambda (c3: C).(\lambda (u3: T).(let TMP_97 \def (CHead c2 k u2) in (let
+TMP_98 \def (CHead c3 k u3) in (eq C TMP_97 TMP_98))))) in (let TMP_100 \def
+(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_101 \def
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_102 \def (CHead c2
+k u2) in (let TMP_103 \def (refl_equal C TMP_102) in (let TMP_104 \def
+(ex3_2_intro C T TMP_99 TMP_100 TMP_101 c2 u2 TMP_103 H12 H10) in (let
+TMP_105 \def (or_intror TMP_90 TMP_96 TMP_104) in (let TMP_106 \def (eq_ind_r
+C c1 TMP_87 TMP_105 c0 H9) in (let TMP_107 \def (eq_ind_r T u1 TMP_69 TMP_106
+u0 H7) in (eq_ind_r K k TMP_58 TMP_107 k0 H8)))))))))))))))))))))))))))) in
+(let TMP_109 \def (TMP_108 H6) in (TMP_109 H5)))))))))))))))))))))))) in
+(wcpr0_ind TMP_16 TMP_39 TMP_110 y x H0)))))) in (insert_eq C TMP_1 TMP_2
+TMP_9 TMP_111 H))))))))).
(* This file was automatically generated: do not edit *********************)
-include "Basic-1/wcpr0/defs.ma".
+include "basic_1/wcpr0/fwd.ma".
-include "Basic-1/getl/props.ma".
+include "basic_1/getl/props.ma".
theorem wcpr0_drop:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
(_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(let TMP_5 \def
(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
-(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
-u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
-(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
-e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
-u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead
-e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
-c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1)
-(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _)
-\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k
-u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0)
-(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1)
-(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in
-(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda
-(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k
-u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t:
-T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2)
-(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C
-T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k
-u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_:
-C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7))))
-H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0
-u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead
-e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
-(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b:
+(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (let TMP_2 \def
+(\lambda (e2: C).(\lambda (u2: T).(let TMP_1 \def (CHead e2 k u2) in (drop h
+O c0 TMP_1)))) in (let TMP_3 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) in (let TMP_4 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in
+(ex3_2 C T TMP_2 TMP_3 TMP_4))))))))))) in (let TMP_12 \def (\lambda (c:
+C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k:
+K).(\lambda (H0: (drop h O c (CHead e1 k u1))).(let TMP_7 \def (\lambda (e2:
+C).(\lambda (u2: T).(let TMP_6 \def (CHead e2 k u2) in (drop h O c TMP_6))))
+in (let TMP_8 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let
+TMP_9 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_10 \def
+(wcpr0_refl e1) in (let TMP_11 \def (pr0_refl u1) in (ex3_2_intro C T TMP_7
+TMP_8 TMP_9 e1 u1 H0 TMP_10 TMP_11)))))))))))) in (let TMP_132 \def (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 c4)).(\lambda (H1: ((\forall
+(h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c3
+(CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O
+c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h:
+nat).(let TMP_18 \def (\lambda (n: nat).(\forall (e1: C).(\forall (u3:
+T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead e1 k0 u3)) \to (let
+TMP_15 \def (\lambda (e2: C).(\lambda (u4: T).(let TMP_13 \def (CHead c4 k
+u2) in (let TMP_14 \def (CHead e2 k0 u4) in (drop n O TMP_13 TMP_14))))) in
+(let TMP_16 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let
+TMP_17 \def (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))) in (ex3_2 C T
+TMP_15 TMP_16 TMP_17))))))))) in (let TMP_67 \def (\lambda (e1: C).(\lambda
+(u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) (CHead e1 k0
+u0))).(let TMP_19 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow
+c3 | (CHead c _ _) \Rightarrow c])) in (let TMP_20 \def (CHead c3 k u1) in
+(let TMP_21 \def (CHead e1 k0 u0) in (let TMP_22 \def (CHead c3 k u1) in (let
+TMP_23 \def (CHead e1 k0 u0) in (let TMP_24 \def (drop_gen_refl TMP_22 TMP_23
+H3) in (let H4 \def (f_equal C C TMP_19 TMP_20 TMP_21 TMP_24) in (let TMP_25
+\def (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | (CHead _ k1 _)
+\Rightarrow k1])) in (let TMP_26 \def (CHead c3 k u1) in (let TMP_27 \def
+(CHead e1 k0 u0) in (let TMP_28 \def (CHead c3 k u1) in (let TMP_29 \def
+(CHead e1 k0 u0) in (let TMP_30 \def (drop_gen_refl TMP_28 TMP_29 H3) in (let
+H5 \def (f_equal C K TMP_25 TMP_26 TMP_27 TMP_30) in (let TMP_31 \def
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead _ _ t)
+\Rightarrow t])) in (let TMP_32 \def (CHead c3 k u1) in (let TMP_33 \def
+(CHead e1 k0 u0) in (let TMP_34 \def (CHead c3 k u1) in (let TMP_35 \def
+(CHead e1 k0 u0) in (let TMP_36 \def (drop_gen_refl TMP_34 TMP_35 H3) in (let
+H6 \def (f_equal C T TMP_31 TMP_32 TMP_33 TMP_36) in (let TMP_65 \def
+(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(let TMP_42 \def
+(\lambda (k1: K).(let TMP_39 \def (\lambda (e2: C).(\lambda (u3: T).(let
+TMP_37 \def (CHead c4 k u2) in (let TMP_38 \def (CHead e2 k1 u3) in (drop O O
+TMP_37 TMP_38))))) in (let TMP_40 \def (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) in (let TMP_41 \def (\lambda (_: C).(\lambda (u3: T).(pr0
+u0 u3))) in (ex3_2 C T TMP_39 TMP_40 TMP_41))))) in (let TMP_48 \def (\lambda
+(t: T).(let TMP_45 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_43 \def
+(CHead c4 k u2) in (let TMP_44 \def (CHead e2 k u3) in (drop O O TMP_43
+TMP_44))))) in (let TMP_46 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) in (let TMP_47 \def (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))) in
+(ex3_2 C T TMP_45 TMP_46 TMP_47))))) in (let TMP_54 \def (\lambda (c: C).(let
+TMP_51 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_49 \def (CHead c4 k
+u2) in (let TMP_50 \def (CHead e2 k u3) in (drop O O TMP_49 TMP_50))))) in
+(let TMP_52 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) in (let
+TMP_53 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (ex3_2 C T
+TMP_51 TMP_52 TMP_53))))) in (let TMP_57 \def (\lambda (e2: C).(\lambda (u3:
+T).(let TMP_55 \def (CHead c4 k u2) in (let TMP_56 \def (CHead e2 k u3) in
+(drop O O TMP_55 TMP_56))))) in (let TMP_58 \def (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 c3 e2))) in (let TMP_59 \def (\lambda (_: C).(\lambda (u3:
+T).(pr0 u1 u3))) in (let TMP_60 \def (CHead c4 k u2) in (let TMP_61 \def
+(drop_refl TMP_60) in (let TMP_62 \def (ex3_2_intro C T TMP_57 TMP_58 TMP_59
+c4 u2 TMP_61 H0 H2) in (let TMP_63 \def (eq_ind C c3 TMP_54 TMP_62 e1 H8) in
+(let TMP_64 \def (eq_ind T u1 TMP_48 TMP_63 u0 H6) in (eq_ind K k TMP_42
+TMP_64 k0 H7)))))))))))))) in (let TMP_66 \def (TMP_65 H5) in (TMP_66
+H4)))))))))))))))))))))))))))) in (let TMP_74 \def (\lambda (k0: K).(\forall
+(n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop n O
+(CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda
+(u4: T).(drop n O (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
+u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S
+n) O (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (let TMP_71 \def (\lambda (e2:
+C).(\lambda (u4: T).(let TMP_68 \def (S n) in (let TMP_69 \def (CHead c4 k0
+u2) in (let TMP_70 \def (CHead e2 k1 u4) in (drop TMP_68 O TMP_69
+TMP_70)))))) in (let TMP_72 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) in (let TMP_73 \def (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))) in
+(ex3_2 C T TMP_71 TMP_72 TMP_73))))))))))) in (let TMP_101 \def (\lambda (b:
B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2
k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead
-e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1
-k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
-O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
-n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0
-x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
-c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop
-(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f:
-F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2
-k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead
-e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead
-e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
-(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
-(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0
-u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
-(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1
-(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k)
-h)))))))))) c1 c2 H))).
-(* COMMENTS
-Initial nodes: 1755
-END *)
+e1 k0 u0))).(let TMP_75 \def (Bind b) in (let TMP_76 \def (CHead e1 k0 u0) in
+(let TMP_77 \def (drop_gen_drop TMP_75 c3 TMP_76 u1 n H4) in (let H5 \def (H1
+n e1 u0 k0 TMP_77) in (let TMP_79 \def (\lambda (e2: C).(\lambda (u3: T).(let
+TMP_78 \def (CHead e2 k0 u3) in (drop n O c4 TMP_78)))) in (let TMP_80 \def
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_81 \def (\lambda
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_86 \def (\lambda (e2:
+C).(\lambda (u3: T).(let TMP_82 \def (S n) in (let TMP_83 \def (Bind b) in
+(let TMP_84 \def (CHead c4 TMP_83 u2) in (let TMP_85 \def (CHead e2 k0 u3) in
+(drop TMP_82 O TMP_84 TMP_85))))))) in (let TMP_87 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_88 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_89 \def (ex3_2 C T TMP_86
+TMP_87 TMP_88) in (let TMP_100 \def (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (drop n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
+x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_94 \def (\lambda (e2: C).(\lambda
+(u3: T).(let TMP_90 \def (S n) in (let TMP_91 \def (Bind b) in (let TMP_92
+\def (CHead c4 TMP_91 u2) in (let TMP_93 \def (CHead e2 k0 u3) in (drop
+TMP_90 O TMP_92 TMP_93))))))) in (let TMP_95 \def (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e1 e2))) in (let TMP_96 \def (\lambda (_: C).(\lambda (u3:
+T).(pr0 u0 u3))) in (let TMP_97 \def (Bind b) in (let TMP_98 \def (CHead x0
+k0 x1) in (let TMP_99 \def (drop_drop TMP_97 n c4 TMP_98 H6 u2) in
+(ex3_2_intro C T TMP_94 TMP_95 TMP_96 x0 x1 TMP_99 H7 H8)))))))))))) in
+(ex3_2_ind C T TMP_79 TMP_80 TMP_81 TMP_89 TMP_100 H5)))))))))))))))))))) in
+(let TMP_130 \def (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 (Flat f) u1)
+(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O
+(CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
+u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
+(drop (S n) O (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let TMP_102 \def (S
+n) in (let TMP_103 \def (Flat f) in (let TMP_104 \def (CHead e1 k0 u0) in
+(let TMP_105 \def (drop_gen_drop TMP_103 c3 TMP_104 u1 n H4) in (let H5 \def
+(H1 TMP_102 e1 u0 k0 TMP_105) in (let TMP_108 \def (\lambda (e2: C).(\lambda
+(u3: T).(let TMP_106 \def (S n) in (let TMP_107 \def (CHead e2 k0 u3) in
+(drop TMP_106 O c4 TMP_107))))) in (let TMP_109 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_110 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_115 \def (\lambda (e2:
+C).(\lambda (u3: T).(let TMP_111 \def (S n) in (let TMP_112 \def (Flat f) in
+(let TMP_113 \def (CHead c4 TMP_112 u2) in (let TMP_114 \def (CHead e2 k0 u3)
+in (drop TMP_111 O TMP_113 TMP_114))))))) in (let TMP_116 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_117 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_118 \def (ex3_2 C T TMP_115
+TMP_116 TMP_117) in (let TMP_129 \def (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (drop (S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
+x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_123 \def (\lambda (e2: C).(\lambda
+(u3: T).(let TMP_119 \def (S n) in (let TMP_120 \def (Flat f) in (let TMP_121
+\def (CHead c4 TMP_120 u2) in (let TMP_122 \def (CHead e2 k0 u3) in (drop
+TMP_119 O TMP_121 TMP_122))))))) in (let TMP_124 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_125 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_126 \def (Flat f) in (let
+TMP_127 \def (CHead x0 k0 x1) in (let TMP_128 \def (drop_drop TMP_126 n c4
+TMP_127 H6 u2) in (ex3_2_intro C T TMP_123 TMP_124 TMP_125 x0 x1 TMP_128 H7
+H8)))))))))))) in (ex3_2_ind C T TMP_108 TMP_109 TMP_110 TMP_118 TMP_129
+H5))))))))))))))))))))) in (let TMP_131 \def (K_ind TMP_74 TMP_101 TMP_130 k)
+in (nat_ind TMP_18 TMP_67 TMP_131 h)))))))))))))))) in (wcpr0_ind TMP_5
+TMP_12 TMP_132 c1 c2 H)))))).
theorem wcpr0_drop_back:
\forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
(_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(let TMP_5 \def
(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
-(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0
-u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
-(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
-e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
-u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead
-e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
-c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2)
-(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _)
-\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k
-u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0)
-(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2)
-(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in
-(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda
-(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k
-u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t:
-T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1)
-(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C
-T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k
-u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_:
-C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7))))
-H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0
-u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead
-e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
-(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b:
+(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (let TMP_2 \def
+(\lambda (e2: C).(\lambda (u2: T).(let TMP_1 \def (CHead e2 k u2) in (drop h
+O c TMP_1)))) in (let TMP_3 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) in (let TMP_4 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) in
+(ex3_2 C T TMP_2 TMP_3 TMP_4))))))))))) in (let TMP_12 \def (\lambda (c:
+C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k:
+K).(\lambda (H0: (drop h O c (CHead e1 k u1))).(let TMP_7 \def (\lambda (e2:
+C).(\lambda (u2: T).(let TMP_6 \def (CHead e2 k u2) in (drop h O c TMP_6))))
+in (let TMP_8 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let
+TMP_9 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) in (let TMP_10 \def
+(wcpr0_refl e1) in (let TMP_11 \def (pr0_refl u1) in (ex3_2_intro C T TMP_7
+TMP_8 TMP_9 e1 u1 H0 TMP_10 TMP_11)))))))))))) in (let TMP_132 \def (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 c4)).(\lambda (H1: ((\forall
+(h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c4
+(CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O
+c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h:
+nat).(let TMP_18 \def (\lambda (n: nat).(\forall (e1: C).(\forall (u3:
+T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead e1 k0 u3)) \to (let
+TMP_15 \def (\lambda (e2: C).(\lambda (u4: T).(let TMP_13 \def (CHead c3 k
+u1) in (let TMP_14 \def (CHead e2 k0 u4) in (drop n O TMP_13 TMP_14))))) in
+(let TMP_16 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let
+TMP_17 \def (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))) in (ex3_2 C T
+TMP_15 TMP_16 TMP_17))))))))) in (let TMP_67 \def (\lambda (e1: C).(\lambda
+(u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) (CHead e1 k0
+u0))).(let TMP_19 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow
+c4 | (CHead c _ _) \Rightarrow c])) in (let TMP_20 \def (CHead c4 k u2) in
+(let TMP_21 \def (CHead e1 k0 u0) in (let TMP_22 \def (CHead c4 k u2) in (let
+TMP_23 \def (CHead e1 k0 u0) in (let TMP_24 \def (drop_gen_refl TMP_22 TMP_23
+H3) in (let H4 \def (f_equal C C TMP_19 TMP_20 TMP_21 TMP_24) in (let TMP_25
+\def (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | (CHead _ k1 _)
+\Rightarrow k1])) in (let TMP_26 \def (CHead c4 k u2) in (let TMP_27 \def
+(CHead e1 k0 u0) in (let TMP_28 \def (CHead c4 k u2) in (let TMP_29 \def
+(CHead e1 k0 u0) in (let TMP_30 \def (drop_gen_refl TMP_28 TMP_29 H3) in (let
+H5 \def (f_equal C K TMP_25 TMP_26 TMP_27 TMP_30) in (let TMP_31 \def
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead _ _ t)
+\Rightarrow t])) in (let TMP_32 \def (CHead c4 k u2) in (let TMP_33 \def
+(CHead e1 k0 u0) in (let TMP_34 \def (CHead c4 k u2) in (let TMP_35 \def
+(CHead e1 k0 u0) in (let TMP_36 \def (drop_gen_refl TMP_34 TMP_35 H3) in (let
+H6 \def (f_equal C T TMP_31 TMP_32 TMP_33 TMP_36) in (let TMP_65 \def
+(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(let TMP_42 \def
+(\lambda (k1: K).(let TMP_39 \def (\lambda (e2: C).(\lambda (u3: T).(let
+TMP_37 \def (CHead c3 k u1) in (let TMP_38 \def (CHead e2 k1 u3) in (drop O O
+TMP_37 TMP_38))))) in (let TMP_40 \def (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) in (let TMP_41 \def (\lambda (_: C).(\lambda (u3: T).(pr0
+u3 u0))) in (ex3_2 C T TMP_39 TMP_40 TMP_41))))) in (let TMP_48 \def (\lambda
+(t: T).(let TMP_45 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_43 \def
+(CHead c3 k u1) in (let TMP_44 \def (CHead e2 k u3) in (drop O O TMP_43
+TMP_44))))) in (let TMP_46 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) in (let TMP_47 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))) in
+(ex3_2 C T TMP_45 TMP_46 TMP_47))))) in (let TMP_54 \def (\lambda (c: C).(let
+TMP_51 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_49 \def (CHead c3 k
+u1) in (let TMP_50 \def (CHead e2 k u3) in (drop O O TMP_49 TMP_50))))) in
+(let TMP_52 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) in (let
+TMP_53 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) in (ex3_2 C T
+TMP_51 TMP_52 TMP_53))))) in (let TMP_57 \def (\lambda (e2: C).(\lambda (u3:
+T).(let TMP_55 \def (CHead c3 k u1) in (let TMP_56 \def (CHead e2 k u3) in
+(drop O O TMP_55 TMP_56))))) in (let TMP_58 \def (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e2 c4))) in (let TMP_59 \def (\lambda (_: C).(\lambda (u3:
+T).(pr0 u3 u2))) in (let TMP_60 \def (CHead c3 k u1) in (let TMP_61 \def
+(drop_refl TMP_60) in (let TMP_62 \def (ex3_2_intro C T TMP_57 TMP_58 TMP_59
+c3 u1 TMP_61 H0 H2) in (let TMP_63 \def (eq_ind C c4 TMP_54 TMP_62 e1 H8) in
+(let TMP_64 \def (eq_ind T u2 TMP_48 TMP_63 u0 H6) in (eq_ind K k TMP_42
+TMP_64 k0 H7)))))))))))))) in (let TMP_66 \def (TMP_65 H5) in (TMP_66
+H4)))))))))))))))))))))))))))) in (let TMP_74 \def (\lambda (k0: K).(\forall
+(n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop n O
+(CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda
+(u4: T).(drop n O (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
+u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S
+n) O (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (let TMP_71 \def (\lambda (e2:
+C).(\lambda (u4: T).(let TMP_68 \def (S n) in (let TMP_69 \def (CHead c3 k0
+u1) in (let TMP_70 \def (CHead e2 k1 u4) in (drop TMP_68 O TMP_69
+TMP_70)))))) in (let TMP_72 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) in (let TMP_73 \def (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))) in
+(ex3_2 C T TMP_71 TMP_72 TMP_73))))))))))) in (let TMP_101 \def (\lambda (b:
B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2
k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead
-e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1
-k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
-O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
-n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1
-u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
-c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop
-(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f:
-F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2
-k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead
-e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead
-e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
-(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
-(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0
-x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
-(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1
-(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k)
-h)))))))))) c2 c1 H))).
-(* COMMENTS
-Initial nodes: 1755
-END *)
+e1 k0 u0))).(let TMP_75 \def (Bind b) in (let TMP_76 \def (CHead e1 k0 u0) in
+(let TMP_77 \def (drop_gen_drop TMP_75 c4 TMP_76 u2 n H4) in (let H5 \def (H1
+n e1 u0 k0 TMP_77) in (let TMP_79 \def (\lambda (e2: C).(\lambda (u3: T).(let
+TMP_78 \def (CHead e2 k0 u3) in (drop n O c3 TMP_78)))) in (let TMP_80 \def
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_81 \def (\lambda
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_86 \def (\lambda (e2:
+C).(\lambda (u3: T).(let TMP_82 \def (S n) in (let TMP_83 \def (Bind b) in
+(let TMP_84 \def (CHead c3 TMP_83 u1) in (let TMP_85 \def (CHead e2 k0 u3) in
+(drop TMP_82 O TMP_84 TMP_85))))))) in (let TMP_87 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_88 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_89 \def (ex3_2 C T TMP_86
+TMP_87 TMP_88) in (let TMP_100 \def (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (drop n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
+e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_94 \def (\lambda (e2: C).(\lambda
+(u3: T).(let TMP_90 \def (S n) in (let TMP_91 \def (Bind b) in (let TMP_92
+\def (CHead c3 TMP_91 u1) in (let TMP_93 \def (CHead e2 k0 u3) in (drop
+TMP_90 O TMP_92 TMP_93))))))) in (let TMP_95 \def (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e2 e1))) in (let TMP_96 \def (\lambda (_: C).(\lambda (u3:
+T).(pr0 u3 u0))) in (let TMP_97 \def (Bind b) in (let TMP_98 \def (CHead x0
+k0 x1) in (let TMP_99 \def (drop_drop TMP_97 n c3 TMP_98 H6 u1) in
+(ex3_2_intro C T TMP_94 TMP_95 TMP_96 x0 x1 TMP_99 H7 H8)))))))))))) in
+(ex3_2_ind C T TMP_79 TMP_80 TMP_81 TMP_89 TMP_100 H5)))))))))))))))))))) in
+(let TMP_130 \def (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 (Flat f) u2)
+(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O
+(CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
+u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
+(drop (S n) O (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let TMP_102 \def (S
+n) in (let TMP_103 \def (Flat f) in (let TMP_104 \def (CHead e1 k0 u0) in
+(let TMP_105 \def (drop_gen_drop TMP_103 c4 TMP_104 u2 n H4) in (let H5 \def
+(H1 TMP_102 e1 u0 k0 TMP_105) in (let TMP_108 \def (\lambda (e2: C).(\lambda
+(u3: T).(let TMP_106 \def (S n) in (let TMP_107 \def (CHead e2 k0 u3) in
+(drop TMP_106 O c3 TMP_107))))) in (let TMP_109 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_110 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_115 \def (\lambda (e2:
+C).(\lambda (u3: T).(let TMP_111 \def (S n) in (let TMP_112 \def (Flat f) in
+(let TMP_113 \def (CHead c3 TMP_112 u1) in (let TMP_114 \def (CHead e2 k0 u3)
+in (drop TMP_111 O TMP_113 TMP_114))))))) in (let TMP_116 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_117 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_118 \def (ex3_2 C T TMP_115
+TMP_116 TMP_117) in (let TMP_129 \def (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (drop (S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
+e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_123 \def (\lambda (e2: C).(\lambda
+(u3: T).(let TMP_119 \def (S n) in (let TMP_120 \def (Flat f) in (let TMP_121
+\def (CHead c3 TMP_120 u1) in (let TMP_122 \def (CHead e2 k0 u3) in (drop
+TMP_119 O TMP_121 TMP_122))))))) in (let TMP_124 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_125 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_126 \def (Flat f) in (let
+TMP_127 \def (CHead x0 k0 x1) in (let TMP_128 \def (drop_drop TMP_126 n c3
+TMP_127 H6 u1) in (ex3_2_intro C T TMP_123 TMP_124 TMP_125 x0 x1 TMP_128 H7
+H8)))))))))))) in (ex3_2_ind C T TMP_108 TMP_109 TMP_110 TMP_118 TMP_129
+H5))))))))))))))))))))) in (let TMP_131 \def (K_ind TMP_74 TMP_101 TMP_130 k)
+in (nat_ind TMP_18 TMP_67 TMP_131 h)))))))))))))))) in (wcpr0_ind TMP_5
+TMP_12 TMP_132 c2 c1 H)))))).
theorem wcpr0_getl:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(let TMP_5 \def
(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
-(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
-u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
-(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
-(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
-u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k
-u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1)
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1
-u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda
-(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C
-C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3
-(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
-(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let
-H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
-u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in
-(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K
-(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
-O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r
-T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
-(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
-(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3)))))
-(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
-T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1
-u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4
-(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2
-(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5
-\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3)
-(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda
-(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
-u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f)
-u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
-x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
-(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
-u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k
-(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
-K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
-K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
-u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
-(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
-u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1)
+(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (let TMP_2 \def
+(\lambda (e2: C).(\lambda (u2: T).(let TMP_1 \def (CHead e2 k u2) in (getl h
+c0 TMP_1)))) in (let TMP_3 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) in (let TMP_4 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in
+(ex3_2 C T TMP_2 TMP_3 TMP_4))))))))))) in (let TMP_12 \def (\lambda (c:
+C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k:
+K).(\lambda (H0: (getl h c (CHead e1 k u1))).(let TMP_7 \def (\lambda (e2:
+C).(\lambda (u2: T).(let TMP_6 \def (CHead e2 k u2) in (getl h c TMP_6)))) in
+(let TMP_8 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let
+TMP_9 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_10 \def
+(wcpr0_refl e1) in (let TMP_11 \def (pr0_refl u1) in (ex3_2_intro C T TMP_7
+TMP_8 TMP_9 e1 u1 H0 TMP_10 TMP_11)))))))))))) in (let TMP_175 \def (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 c4)).(\lambda (H1: ((\forall
+(h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c3 (CHead
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c4 (CHead
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2))))))))))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(let TMP_18
+\def (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0:
+K).((getl n (CHead c3 k u1) (CHead e1 k0 u3)) \to (let TMP_15 \def (\lambda
+(e2: C).(\lambda (u4: T).(let TMP_13 \def (CHead c4 k u2) in (let TMP_14 \def
+(CHead e2 k0 u4) in (getl n TMP_13 TMP_14))))) in (let TMP_16 \def (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_17 \def (\lambda (_:
+C).(\lambda (u4: T).(pr0 u3 u4))) in (ex3_2 C T TMP_15 TMP_16 TMP_17)))))))))
+in (let TMP_110 \def (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H3: (getl O (CHead c3 k u1) (CHead e1 k0 u0))).(let TMP_24 \def
+(\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 k0 u0)) \to (let TMP_21
+\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_19 \def (CHead c4 k1 u2) in
+(let TMP_20 \def (CHead e2 k0 u3) in (getl O TMP_19 TMP_20))))) in (let
+TMP_22 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_23
+\def (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) in (ex3_2 C T TMP_21
+TMP_22 TMP_23)))))) in (let TMP_80 \def (\lambda (b: B).(\lambda (H4: (clear
+(CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let TMP_25 \def (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c]))
+in (let TMP_26 \def (CHead e1 k0 u0) in (let TMP_27 \def (Bind b) in (let
+TMP_28 \def (CHead c3 TMP_27 u1) in (let TMP_29 \def (CHead e1 k0 u0) in (let
+TMP_30 \def (clear_gen_bind b c3 TMP_29 u1 H4) in (let H5 \def (f_equal C C
+TMP_25 TMP_26 TMP_28 TMP_30) in (let TMP_31 \def (\lambda (e: C).(match e
+with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) in (let
+TMP_32 \def (CHead e1 k0 u0) in (let TMP_33 \def (Bind b) in (let TMP_34 \def
+(CHead c3 TMP_33 u1) in (let TMP_35 \def (CHead e1 k0 u0) in (let TMP_36 \def
+(clear_gen_bind b c3 TMP_35 u1 H4) in (let H6 \def (f_equal C K TMP_31 TMP_32
+TMP_34 TMP_36) in (let TMP_37 \def (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) in (let TMP_38 \def (CHead e1
+k0 u0) in (let TMP_39 \def (Bind b) in (let TMP_40 \def (CHead c3 TMP_39 u1)
+in (let TMP_41 \def (CHead e1 k0 u0) in (let TMP_42 \def (clear_gen_bind b c3
+TMP_41 u1 H4) in (let H7 \def (f_equal C T TMP_37 TMP_38 TMP_40 TMP_42) in
+(let TMP_78 \def (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1
+c3)).(let TMP_43 \def (Bind b) in (let TMP_50 \def (\lambda (k1: K).(let
+TMP_47 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_44 \def (Bind b) in
+(let TMP_45 \def (CHead c4 TMP_44 u2) in (let TMP_46 \def (CHead e2 k1 u3) in
+(getl O TMP_45 TMP_46)))))) in (let TMP_48 \def (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) in (let TMP_49 \def (\lambda (_: C).(\lambda (u3: T).(pr0
+u0 u3))) in (ex3_2 C T TMP_47 TMP_48 TMP_49))))) in (let TMP_58 \def (\lambda
+(t: T).(let TMP_55 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_51 \def
+(Bind b) in (let TMP_52 \def (CHead c4 TMP_51 u2) in (let TMP_53 \def (Bind
+b) in (let TMP_54 \def (CHead e2 TMP_53 u3) in (getl O TMP_52 TMP_54)))))))
+in (let TMP_56 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let
+TMP_57 \def (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))) in (ex3_2 C T
+TMP_55 TMP_56 TMP_57))))) in (let TMP_66 \def (\lambda (c: C).(let TMP_63
+\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_59 \def (Bind b) in (let
+TMP_60 \def (CHead c4 TMP_59 u2) in (let TMP_61 \def (Bind b) in (let TMP_62
+\def (CHead e2 TMP_61 u3) in (getl O TMP_60 TMP_62))))))) in (let TMP_64 \def
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) in (let TMP_65 \def (\lambda
+(_: C).(\lambda (u3: T).(pr0 u1 u3))) in (ex3_2 C T TMP_63 TMP_64 TMP_65)))))
+in (let TMP_71 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_67 \def (Bind
+b) in (let TMP_68 \def (CHead c4 TMP_67 u2) in (let TMP_69 \def (Bind b) in
+(let TMP_70 \def (CHead e2 TMP_69 u3) in (getl O TMP_68 TMP_70))))))) in (let
+TMP_72 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 c3 e2))) in (let TMP_73
+\def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_74 \def
+(getl_refl b c4 u2) in (let TMP_75 \def (ex3_2_intro C T TMP_71 TMP_72 TMP_73
+c4 u2 TMP_74 H0 H2) in (let TMP_76 \def (eq_ind_r C c3 TMP_66 TMP_75 e1 H9)
+in (let TMP_77 \def (eq_ind_r T u1 TMP_58 TMP_76 u0 H7) in (eq_ind_r K TMP_43
+TMP_50 TMP_77 k0 H8)))))))))))))) in (let TMP_79 \def (TMP_78 H6) in (TMP_79
+H5)))))))))))))))))))))))))) in (let TMP_106 \def (\lambda (f: F).(\lambda
+(H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let TMP_81 \def (CHead
+e1 k0 u0) in (let TMP_82 \def (drop_refl c3) in (let TMP_83 \def (CHead e1 k0
+u0) in (let TMP_84 \def (clear_gen_flat f c3 TMP_83 u1 H4) in (let TMP_85
+\def (getl_intro O c3 TMP_81 c3 TMP_82 TMP_84) in (let H5 \def (H1 O e1 u0 k0
+TMP_85) in (let TMP_87 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_86
+\def (CHead e2 k0 u3) in (getl O c4 TMP_86)))) in (let TMP_88 \def (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_89 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_93 \def (\lambda (e2:
+C).(\lambda (u3: T).(let TMP_90 \def (Flat f) in (let TMP_91 \def (CHead c4
+TMP_90 u2) in (let TMP_92 \def (CHead e2 k0 u3) in (getl O TMP_91
+TMP_92)))))) in (let TMP_94 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) in (let TMP_95 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) in
+(let TMP_96 \def (ex3_2 C T TMP_93 TMP_94 TMP_95) in (let TMP_105 \def
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0
+x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_100
+\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_97 \def (Flat f) in (let
+TMP_98 \def (CHead c4 TMP_97 u2) in (let TMP_99 \def (CHead e2 k0 u3) in
+(getl O TMP_98 TMP_99)))))) in (let TMP_101 \def (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e1 e2))) in (let TMP_102 \def (\lambda (_: C).(\lambda (u3:
+T).(pr0 u0 u3))) in (let TMP_103 \def (CHead x0 k0 x1) in (let TMP_104 \def
+(getl_flat c4 TMP_103 O H6 f u2) in (ex3_2_intro C T TMP_100 TMP_101 TMP_102
+x0 x1 TMP_104 H7 H8))))))))))) in (ex3_2_ind C T TMP_87 TMP_88 TMP_89 TMP_96
+TMP_105 H5))))))))))))))))) in (let TMP_107 \def (CHead c3 k u1) in (let
+TMP_108 \def (CHead e1 k0 u0) in (let TMP_109 \def (getl_gen_O TMP_107
+TMP_108 H3) in (K_ind TMP_24 TMP_80 TMP_106 k TMP_109))))))))))) in (let
+TMP_117 \def (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall
+(u3: T).(\forall (k1: K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to
+(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead
+e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: C).(\forall (u3:
+T).(\forall (k1: K).((getl (S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (let
+TMP_114 \def (\lambda (e2: C).(\lambda (u4: T).(let TMP_111 \def (S n) in
+(let TMP_112 \def (CHead c4 k0 u2) in (let TMP_113 \def (CHead e2 k1 u4) in
+(getl TMP_111 TMP_112 TMP_113)))))) in (let TMP_115 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_116 \def (\lambda (_:
+C).(\lambda (u4: T).(pr0 u3 u4))) in (ex3_2 C T TMP_114 TMP_115
+TMP_116))))))))))) in (let TMP_144 \def (\lambda (b: B).(\lambda (n:
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl
+n (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl n (CHead c4 (Bind b) u2) (CHead e2 k0 u4))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let
+TMP_118 \def (Bind b) in (let TMP_119 \def (CHead e1 k0 u0) in (let TMP_120
+\def (getl_gen_S TMP_118 c3 TMP_119 u1 n H4) in (let H5 \def (H1 n e1 u0 k0
+TMP_120) in (let TMP_122 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_121
+\def (CHead e2 k0 u3) in (getl n c4 TMP_121)))) in (let TMP_123 \def (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_124 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_129 \def (\lambda (e2:
+C).(\lambda (u3: T).(let TMP_125 \def (S n) in (let TMP_126 \def (Bind b) in
+(let TMP_127 \def (CHead c4 TMP_126 u2) in (let TMP_128 \def (CHead e2 k0 u3)
+in (getl TMP_125 TMP_127 TMP_128))))))) in (let TMP_130 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_131 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_132 \def (ex3_2 C T TMP_129
+TMP_130 TMP_131) in (let TMP_143 \def (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
+x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_137 \def (\lambda (e2: C).(\lambda
+(u3: T).(let TMP_133 \def (S n) in (let TMP_134 \def (Bind b) in (let TMP_135
+\def (CHead c4 TMP_134 u2) in (let TMP_136 \def (CHead e2 k0 u3) in (getl
+TMP_133 TMP_135 TMP_136))))))) in (let TMP_138 \def (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e1 e2))) in (let TMP_139 \def (\lambda (_: C).(\lambda (u3:
+T).(pr0 u0 u3))) in (let TMP_140 \def (Bind b) in (let TMP_141 \def (CHead x0
+k0 x1) in (let TMP_142 \def (getl_head TMP_140 n c4 TMP_141 H6 u2) in
+(ex3_2_intro C T TMP_137 TMP_138 TMP_139 x0 x1 TMP_142 H7 H8)))))))))))) in
+(ex3_2_ind C T TMP_122 TMP_123 TMP_124 TMP_132 TMP_143 H5))))))))))))))))))))
+in (let TMP_173 \def (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat f) u1)
(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
-(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
+(CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
-u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
-(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7:
-(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2)
-H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat
-f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
-T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
-u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
-e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
-(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda
-(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2)
-H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))).
-(* COMMENTS
-Initial nodes: 2103
-END *)
+(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let TMP_145 \def (S n)
+in (let TMP_146 \def (Flat f) in (let TMP_147 \def (CHead e1 k0 u0) in (let
+TMP_148 \def (getl_gen_S TMP_146 c3 TMP_147 u1 n H4) in (let H5 \def (H1
+TMP_145 e1 u0 k0 TMP_148) in (let TMP_151 \def (\lambda (e2: C).(\lambda (u3:
+T).(let TMP_149 \def (S n) in (let TMP_150 \def (CHead e2 k0 u3) in (getl
+TMP_149 c4 TMP_150))))) in (let TMP_152 \def (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) in (let TMP_153 \def (\lambda (_: C).(\lambda (u3: T).(pr0
+u0 u3))) in (let TMP_158 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_154
+\def (S n) in (let TMP_155 \def (Flat f) in (let TMP_156 \def (CHead c4
+TMP_155 u2) in (let TMP_157 \def (CHead e2 k0 u3) in (getl TMP_154 TMP_156
+TMP_157))))))) in (let TMP_159 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0
+e1 e2))) in (let TMP_160 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))
+in (let TMP_161 \def (ex3_2 C T TMP_158 TMP_159 TMP_160) in (let TMP_172 \def
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0
+x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(let TMP_166
+\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_162 \def (S n) in (let
+TMP_163 \def (Flat f) in (let TMP_164 \def (CHead c4 TMP_163 u2) in (let
+TMP_165 \def (CHead e2 k0 u3) in (getl TMP_162 TMP_164 TMP_165))))))) in (let
+TMP_167 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) in (let TMP_168
+\def (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) in (let TMP_169 \def
+(Flat f) in (let TMP_170 \def (CHead x0 k0 x1) in (let TMP_171 \def
+(getl_head TMP_169 n c4 TMP_170 H6 u2) in (ex3_2_intro C T TMP_166 TMP_167
+TMP_168 x0 x1 TMP_171 H7 H8)))))))))))) in (ex3_2_ind C T TMP_151 TMP_152
+TMP_153 TMP_161 TMP_172 H5))))))))))))))))))))) in (let TMP_174 \def (K_ind
+TMP_117 TMP_144 TMP_173 k) in (nat_ind TMP_18 TMP_110 TMP_174
+h)))))))))))))))) in (wcpr0_ind TMP_5 TMP_12 TMP_175 c1 c2 H)))))).
theorem wcpr0_getl_back:
\forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(let TMP_5 \def
(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
-(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
-u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
-(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
-(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
-u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k
-u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2)
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1
-u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda
-(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C
-C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4
-(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
-(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let
-H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
-u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in
-(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K
-(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
-O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r
-T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
-(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
-(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t)))))
-(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
-T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3
-(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1
-(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5
-\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4)
-(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda
-(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f)
-u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
-e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
-(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k
-(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
-K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
-K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
-u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
-(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
-u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2)
+(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (let TMP_2 \def
+(\lambda (e2: C).(\lambda (u2: T).(let TMP_1 \def (CHead e2 k u2) in (getl h
+c TMP_1)))) in (let TMP_3 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) in (let TMP_4 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) in
+(ex3_2 C T TMP_2 TMP_3 TMP_4))))))))))) in (let TMP_12 \def (\lambda (c:
+C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k:
+K).(\lambda (H0: (getl h c (CHead e1 k u1))).(let TMP_7 \def (\lambda (e2:
+C).(\lambda (u2: T).(let TMP_6 \def (CHead e2 k u2) in (getl h c TMP_6)))) in
+(let TMP_8 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let
+TMP_9 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) in (let TMP_10 \def
+(wcpr0_refl e1) in (let TMP_11 \def (pr0_refl u1) in (ex3_2_intro C T TMP_7
+TMP_8 TMP_9 e1 u1 H0 TMP_10 TMP_11)))))))))))) in (let TMP_175 \def (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 c4)).(\lambda (H1: ((\forall
+(h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c4 (CHead
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c3 (CHead
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u1))))))))))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(let TMP_18
+\def (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0:
+K).((getl n (CHead c4 k u2) (CHead e1 k0 u3)) \to (let TMP_15 \def (\lambda
+(e2: C).(\lambda (u4: T).(let TMP_13 \def (CHead c3 k u1) in (let TMP_14 \def
+(CHead e2 k0 u4) in (getl n TMP_13 TMP_14))))) in (let TMP_16 \def (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_17 \def (\lambda (_:
+C).(\lambda (u4: T).(pr0 u4 u3))) in (ex3_2 C T TMP_15 TMP_16 TMP_17)))))))))
+in (let TMP_110 \def (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H3: (getl O (CHead c4 k u2) (CHead e1 k0 u0))).(let TMP_24 \def
+(\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 k0 u0)) \to (let TMP_21
+\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_19 \def (CHead c3 k1 u1) in
+(let TMP_20 \def (CHead e2 k0 u3) in (getl O TMP_19 TMP_20))))) in (let
+TMP_22 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_23
+\def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) in (ex3_2 C T TMP_21
+TMP_22 TMP_23)))))) in (let TMP_80 \def (\lambda (b: B).(\lambda (H4: (clear
+(CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let TMP_25 \def (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c]))
+in (let TMP_26 \def (CHead e1 k0 u0) in (let TMP_27 \def (Bind b) in (let
+TMP_28 \def (CHead c4 TMP_27 u2) in (let TMP_29 \def (CHead e1 k0 u0) in (let
+TMP_30 \def (clear_gen_bind b c4 TMP_29 u2 H4) in (let H5 \def (f_equal C C
+TMP_25 TMP_26 TMP_28 TMP_30) in (let TMP_31 \def (\lambda (e: C).(match e
+with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) in (let
+TMP_32 \def (CHead e1 k0 u0) in (let TMP_33 \def (Bind b) in (let TMP_34 \def
+(CHead c4 TMP_33 u2) in (let TMP_35 \def (CHead e1 k0 u0) in (let TMP_36 \def
+(clear_gen_bind b c4 TMP_35 u2 H4) in (let H6 \def (f_equal C K TMP_31 TMP_32
+TMP_34 TMP_36) in (let TMP_37 \def (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) in (let TMP_38 \def (CHead e1
+k0 u0) in (let TMP_39 \def (Bind b) in (let TMP_40 \def (CHead c4 TMP_39 u2)
+in (let TMP_41 \def (CHead e1 k0 u0) in (let TMP_42 \def (clear_gen_bind b c4
+TMP_41 u2 H4) in (let H7 \def (f_equal C T TMP_37 TMP_38 TMP_40 TMP_42) in
+(let TMP_78 \def (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1
+c4)).(let TMP_43 \def (Bind b) in (let TMP_50 \def (\lambda (k1: K).(let
+TMP_47 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_44 \def (Bind b) in
+(let TMP_45 \def (CHead c3 TMP_44 u1) in (let TMP_46 \def (CHead e2 k1 u3) in
+(getl O TMP_45 TMP_46)))))) in (let TMP_48 \def (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) in (let TMP_49 \def (\lambda (_: C).(\lambda (u3: T).(pr0
+u3 u0))) in (ex3_2 C T TMP_47 TMP_48 TMP_49))))) in (let TMP_58 \def (\lambda
+(t: T).(let TMP_55 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_51 \def
+(Bind b) in (let TMP_52 \def (CHead c3 TMP_51 u1) in (let TMP_53 \def (Bind
+b) in (let TMP_54 \def (CHead e2 TMP_53 u3) in (getl O TMP_52 TMP_54)))))))
+in (let TMP_56 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let
+TMP_57 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))) in (ex3_2 C T
+TMP_55 TMP_56 TMP_57))))) in (let TMP_66 \def (\lambda (c: C).(let TMP_63
+\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_59 \def (Bind b) in (let
+TMP_60 \def (CHead c3 TMP_59 u1) in (let TMP_61 \def (Bind b) in (let TMP_62
+\def (CHead e2 TMP_61 u3) in (getl O TMP_60 TMP_62))))))) in (let TMP_64 \def
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) in (let TMP_65 \def (\lambda
+(_: C).(\lambda (u3: T).(pr0 u3 u2))) in (ex3_2 C T TMP_63 TMP_64 TMP_65)))))
+in (let TMP_71 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_67 \def (Bind
+b) in (let TMP_68 \def (CHead c3 TMP_67 u1) in (let TMP_69 \def (Bind b) in
+(let TMP_70 \def (CHead e2 TMP_69 u3) in (getl O TMP_68 TMP_70))))))) in (let
+TMP_72 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c4))) in (let TMP_73
+\def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) in (let TMP_74 \def
+(getl_refl b c3 u1) in (let TMP_75 \def (ex3_2_intro C T TMP_71 TMP_72 TMP_73
+c3 u1 TMP_74 H0 H2) in (let TMP_76 \def (eq_ind_r C c4 TMP_66 TMP_75 e1 H9)
+in (let TMP_77 \def (eq_ind_r T u2 TMP_58 TMP_76 u0 H7) in (eq_ind_r K TMP_43
+TMP_50 TMP_77 k0 H8)))))))))))))) in (let TMP_79 \def (TMP_78 H6) in (TMP_79
+H5)))))))))))))))))))))))))) in (let TMP_106 \def (\lambda (f: F).(\lambda
+(H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let TMP_81 \def (CHead
+e1 k0 u0) in (let TMP_82 \def (drop_refl c4) in (let TMP_83 \def (CHead e1 k0
+u0) in (let TMP_84 \def (clear_gen_flat f c4 TMP_83 u2 H4) in (let TMP_85
+\def (getl_intro O c4 TMP_81 c4 TMP_82 TMP_84) in (let H5 \def (H1 O e1 u0 k0
+TMP_85) in (let TMP_87 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_86
+\def (CHead e2 k0 u3) in (getl O c3 TMP_86)))) in (let TMP_88 \def (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_89 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_93 \def (\lambda (e2:
+C).(\lambda (u3: T).(let TMP_90 \def (Flat f) in (let TMP_91 \def (CHead c3
+TMP_90 u1) in (let TMP_92 \def (CHead e2 k0 u3) in (getl O TMP_91
+TMP_92)))))) in (let TMP_94 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) in (let TMP_95 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) in
+(let TMP_96 \def (ex3_2 C T TMP_93 TMP_94 TMP_95) in (let TMP_105 \def
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0
+x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_100
+\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_97 \def (Flat f) in (let
+TMP_98 \def (CHead c3 TMP_97 u1) in (let TMP_99 \def (CHead e2 k0 u3) in
+(getl O TMP_98 TMP_99)))))) in (let TMP_101 \def (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e2 e1))) in (let TMP_102 \def (\lambda (_: C).(\lambda (u3:
+T).(pr0 u3 u0))) in (let TMP_103 \def (CHead x0 k0 x1) in (let TMP_104 \def
+(getl_flat c3 TMP_103 O H6 f u1) in (ex3_2_intro C T TMP_100 TMP_101 TMP_102
+x0 x1 TMP_104 H7 H8))))))))))) in (ex3_2_ind C T TMP_87 TMP_88 TMP_89 TMP_96
+TMP_105 H5))))))))))))))))) in (let TMP_107 \def (CHead c4 k u2) in (let
+TMP_108 \def (CHead e1 k0 u0) in (let TMP_109 \def (getl_gen_O TMP_107
+TMP_108 H3) in (K_ind TMP_24 TMP_80 TMP_106 k TMP_109))))))))))) in (let
+TMP_117 \def (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall
+(u3: T).(\forall (k1: K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to
+(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead
+e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: C).(\forall (u3:
+T).(\forall (k1: K).((getl (S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (let
+TMP_114 \def (\lambda (e2: C).(\lambda (u4: T).(let TMP_111 \def (S n) in
+(let TMP_112 \def (CHead c3 k0 u1) in (let TMP_113 \def (CHead e2 k1 u4) in
+(getl TMP_111 TMP_112 TMP_113)))))) in (let TMP_115 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_116 \def (\lambda (_:
+C).(\lambda (u4: T).(pr0 u4 u3))) in (ex3_2 C T TMP_114 TMP_115
+TMP_116))))))))))) in (let TMP_144 \def (\lambda (b: B).(\lambda (n:
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl
+n (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u1) (CHead e2 k0 u4))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H4: (getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let
+TMP_118 \def (Bind b) in (let TMP_119 \def (CHead e1 k0 u0) in (let TMP_120
+\def (getl_gen_S TMP_118 c4 TMP_119 u2 n H4) in (let H5 \def (H1 n e1 u0 k0
+TMP_120) in (let TMP_122 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_121
+\def (CHead e2 k0 u3) in (getl n c3 TMP_121)))) in (let TMP_123 \def (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_124 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_129 \def (\lambda (e2:
+C).(\lambda (u3: T).(let TMP_125 \def (S n) in (let TMP_126 \def (Bind b) in
+(let TMP_127 \def (CHead c3 TMP_126 u1) in (let TMP_128 \def (CHead e2 k0 u3)
+in (getl TMP_125 TMP_127 TMP_128))))))) in (let TMP_130 \def (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_131 \def (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_132 \def (ex3_2 C T TMP_129
+TMP_130 TMP_131) in (let TMP_143 \def (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
+e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_137 \def (\lambda (e2: C).(\lambda
+(u3: T).(let TMP_133 \def (S n) in (let TMP_134 \def (Bind b) in (let TMP_135
+\def (CHead c3 TMP_134 u1) in (let TMP_136 \def (CHead e2 k0 u3) in (getl
+TMP_133 TMP_135 TMP_136))))))) in (let TMP_138 \def (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e2 e1))) in (let TMP_139 \def (\lambda (_: C).(\lambda (u3:
+T).(pr0 u3 u0))) in (let TMP_140 \def (Bind b) in (let TMP_141 \def (CHead x0
+k0 x1) in (let TMP_142 \def (getl_head TMP_140 n c3 TMP_141 H6 u1) in
+(ex3_2_intro C T TMP_137 TMP_138 TMP_139 x0 x1 TMP_142 H7 H8)))))))))))) in
+(ex3_2_ind C T TMP_122 TMP_123 TMP_124 TMP_132 TMP_143 H5))))))))))))))))))))
+in (let TMP_173 \def (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat f) u2)
(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
-(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
+(CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
-u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
-(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7:
-(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1)
-H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat
-f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
-T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2:
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
-u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
-e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
-(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda
-(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1)
-H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))).
-(* COMMENTS
-Initial nodes: 2103
-END *)
+(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let TMP_145 \def (S n)
+in (let TMP_146 \def (Flat f) in (let TMP_147 \def (CHead e1 k0 u0) in (let
+TMP_148 \def (getl_gen_S TMP_146 c4 TMP_147 u2 n H4) in (let H5 \def (H1
+TMP_145 e1 u0 k0 TMP_148) in (let TMP_151 \def (\lambda (e2: C).(\lambda (u3:
+T).(let TMP_149 \def (S n) in (let TMP_150 \def (CHead e2 k0 u3) in (getl
+TMP_149 c3 TMP_150))))) in (let TMP_152 \def (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) in (let TMP_153 \def (\lambda (_: C).(\lambda (u3: T).(pr0
+u3 u0))) in (let TMP_158 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_154
+\def (S n) in (let TMP_155 \def (Flat f) in (let TMP_156 \def (CHead c3
+TMP_155 u1) in (let TMP_157 \def (CHead e2 k0 u3) in (getl TMP_154 TMP_156
+TMP_157))))))) in (let TMP_159 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0
+e2 e1))) in (let TMP_160 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))
+in (let TMP_161 \def (ex3_2 C T TMP_158 TMP_159 TMP_160) in (let TMP_172 \def
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0
+x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(let TMP_166
+\def (\lambda (e2: C).(\lambda (u3: T).(let TMP_162 \def (S n) in (let
+TMP_163 \def (Flat f) in (let TMP_164 \def (CHead c3 TMP_163 u1) in (let
+TMP_165 \def (CHead e2 k0 u3) in (getl TMP_162 TMP_164 TMP_165))))))) in (let
+TMP_167 \def (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) in (let TMP_168
+\def (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) in (let TMP_169 \def
+(Flat f) in (let TMP_170 \def (CHead x0 k0 x1) in (let TMP_171 \def
+(getl_head TMP_169 n c3 TMP_170 H6 u1) in (ex3_2_intro C T TMP_166 TMP_167
+TMP_168 x0 x1 TMP_171 H7 H8)))))))))))) in (ex3_2_ind C T TMP_151 TMP_152
+TMP_153 TMP_161 TMP_172 H5))))))))))))))))))))) in (let TMP_174 \def (K_ind
+TMP_117 TMP_144 TMP_173 k) in (nat_ind TMP_18 TMP_110 TMP_174
+h)))))))))))))))) in (wcpr0_ind TMP_5 TMP_12 TMP_175 c2 c1 H)))))).
projects / helm.git / commitdiff