component: pr3

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma
index 9619ee1d7124cde1c55ce5ad38ff333ed3ca8976..7430c894de358c930c8e86387392b1e9cd390cec 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr2/defs.ma".
+include "basic_1/pr2/defs.ma".
 
 inductive pr3 (c: C): T \to (T \to Prop) \def
 | pr3_refl: \forall (t: T).(pr3 c t t)

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma
index 726c420f002cf13e64e15d03e34705f56c46e75d..ca979dd820044c2a5db5480e87d6cef31852672f 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma
@@ -14,29 +14,39 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/props.ma".
+include "basic_1/pr3/defs.ma".
 
-include "Basic-1/pr2/fwd.ma".
+include "basic_1/pr2/fwd.ma".
+
+let rec pr3_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t 
+t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: 
+T).((pr3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr3 
+c t t0) on p: P t t0 \def match p with [(pr3_refl t1) \Rightarrow (f t1) | 
+(pr3_sing t2 t1 p0 t3 p1) \Rightarrow (let TMP_1 \def ((pr3_ind c P f f0) t2 
+t3 p1) in (f0 t2 t1 p0 t3 p1 TMP_1))].
 
 theorem pr3_gen_sort:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to 
 (eq T x (TSort n)))))
 \def
  \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort 
-n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: 
-T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda 
-(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: 
-T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda 
-(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 
-t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda 
-(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t: 
-T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n)) 
-\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T 
-(TSort n)))) t1 H4))))))))) y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 253
-END *)
+n) x)).(let TMP_1 \def (TSort n) in (let TMP_2 \def (\lambda (t: T).(pr3 c t 
+x)) in (let TMP_3 \def (\lambda (t: T).(eq T x t)) in (let TMP_17 \def 
+(\lambda (y: T).(\lambda (H0: (pr3 c y x)).(let TMP_4 \def (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) in (let TMP_5 \def 
+(\lambda (t: T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) in (let 
+TMP_16 \def (\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 
+t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 
+(TSort n)) \to (eq T t3 t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let TMP_6 
+\def (\lambda (t: T).(pr2 c t t2)) in (let TMP_7 \def (TSort n) in (let H5 
+\def (eq_ind T t1 TMP_6 H1 TMP_7 H4) in (let TMP_8 \def (TSort n) in (let 
+TMP_9 \def (\lambda (t: T).(eq T t3 t)) in (let TMP_10 \def (\lambda (t: 
+T).((eq T t (TSort n)) \to (eq T t3 t))) in (let TMP_11 \def (TSort n) in 
+(let TMP_12 \def (pr2_gen_sort c t2 n H5) in (let H6 \def (eq_ind T t2 TMP_10 
+H3 TMP_11 TMP_12) in (let TMP_13 \def (TSort n) in (let TMP_14 \def 
+(refl_equal T TMP_13) in (let TMP_15 \def (H6 TMP_14) in (eq_ind_r T TMP_8 
+TMP_9 TMP_15 t1 H4)))))))))))))))))))) in (pr3_ind c TMP_4 TMP_5 TMP_16 y x 
+H0)))))) in (insert_eq T TMP_1 TMP_2 TMP_3 TMP_17 H)))))))).
 
 theorem pr3_gen_abst:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
@@ -46,73 +56,112 @@ c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
 T).(pr3 (CHead c (Bind b) u) t1 t2))))))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: 
-T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead 
-(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: 
-T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c 
+(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(let TMP_1 \def (Bind Abst) in (let 
+TMP_2 \def (THead TMP_1 u1 t1) in (let TMP_3 \def (\lambda (t: T).(pr3 c t 
+x)) in (let TMP_11 \def (\lambda (_: T).(let TMP_6 \def (\lambda (u2: 
+T).(\lambda (t2: T).(let TMP_4 \def (Bind Abst) in (let TMP_5 \def (THead 
+TMP_4 u2 t2) in (eq T x TMP_5))))) in (let TMP_7 \def (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) in (let TMP_10 \def (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(let TMP_8 \def (Bind b) 
+in (let TMP_9 \def (CHead c TMP_8 u) in (pr3 TMP_9 t1 t2))))))) in (ex3_2 T T 
+TMP_6 TMP_7 TMP_10))))) in (let TMP_112 \def (\lambda (y: T).(\lambda (H0: 
+(pr3 c y x)).(let TMP_19 \def (\lambda (t: T).((eq T y (THead (Bind Abst) u1 
+t)) \to (let TMP_14 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_12 \def 
+(Bind Abst) in (let TMP_13 \def (THead TMP_12 u2 t2) in (eq T x TMP_13))))) 
+in (let TMP_15 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) in (let 
+TMP_18 \def (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: 
+T).(let TMP_16 \def (Bind b) in (let TMP_17 \def (CHead c TMP_16 u) in (pr3 
+TMP_17 t t2))))))) in (ex3_2 T T TMP_14 TMP_15 TMP_18)))))) in (let TMP_27 
+\def (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Abst) t x0)) \to 
+(let TMP_22 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_20 \def (Bind 
+Abst) in (let TMP_21 \def (THead TMP_20 u2 t2) in (eq T x TMP_21))))) in (let 
+TMP_23 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) in (let TMP_26 
+\def (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(let 
+TMP_24 \def (Bind b) in (let TMP_25 \def (CHead c TMP_24 u) in (pr3 TMP_25 x0 
+t2))))))) in (ex3_2 T T TMP_22 TMP_23 TMP_26))))))) in (let TMP_35 \def 
 (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
-(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 
-x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind 
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl 
-(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda 
-(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda 
-(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) 
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
-(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
-t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 
-t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
-(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 
-x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: 
-T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 
-t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3 
-(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
-(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst) 
-x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) 
-x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: 
-T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) 
-H11)))))))) H6)))))))))))) y x H0))))) H))))).
-(* COMMENTS
-Initial nodes: 1261
-END *)
+(THead (Bind Abst) x0 x1)) \to (let TMP_30 \def (\lambda (u2: T).(\lambda 
+(t2: T).(let TMP_28 \def (Bind Abst) in (let TMP_29 \def (THead TMP_28 u2 t2) 
+in (eq T t0 TMP_29))))) in (let TMP_31 \def (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) in (let TMP_34 \def (\lambda (_: T).(\lambda (t2: 
+T).(\forall (b: B).(\forall (u: T).(let TMP_32 \def (Bind b) in (let TMP_33 
+\def (CHead c TMP_32 u) in (pr3 TMP_33 x1 t2))))))) in (ex3_2 T T TMP_30 
+TMP_31 TMP_34))))))))) in (let TMP_47 \def (\lambda (t: T).(\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 x1))).(let 
+TMP_38 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_36 \def (Bind Abst) in 
+(let TMP_37 \def (THead TMP_36 u2 t2) in (eq T t TMP_37))))) in (let TMP_39 
+\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_42 \def 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(let TMP_40 
+\def (Bind b) in (let TMP_41 \def (CHead c TMP_40 u) in (pr3 TMP_41 x1 
+t2))))))) in (let TMP_43 \def (pr3_refl c x0) in (let TMP_46 \def (\lambda 
+(b: B).(\lambda (u: T).(let TMP_44 \def (Bind b) in (let TMP_45 \def (CHead c 
+TMP_44 u) in (pr3_refl TMP_45 x1))))) in (ex3_2_intro T T TMP_38 TMP_39 
+TMP_42 x0 x1 H1 TMP_43 TMP_46)))))))))) in (let TMP_109 \def (\lambda (t2: 
+T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda 
+(_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 
+(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))))))).(\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (H4: (eq T t3 (THead (Bind Abst) x0 x1))).(let TMP_48 \def 
+(\lambda (t: T).(pr2 c t t2)) in (let TMP_49 \def (Bind Abst) in (let TMP_50 
+\def (THead TMP_49 x0 x1) in (let H5 \def (eq_ind T t3 TMP_48 H1 TMP_50 H4) 
+in (let H6 \def (pr2_gen_abst c x0 x1 t2 H5) in (let TMP_53 \def (\lambda 
+(u2: T).(\lambda (t5: T).(let TMP_51 \def (Bind Abst) in (let TMP_52 \def 
+(THead TMP_51 u2 t5) in (eq T t2 TMP_52))))) in (let TMP_54 \def (\lambda 
+(u2: T).(\lambda (_: T).(pr2 c x0 u2))) in (let TMP_57 \def (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(let TMP_55 \def (Bind b) 
+in (let TMP_56 \def (CHead c TMP_55 u) in (pr2 TMP_56 x1 t5))))))) in (let 
+TMP_60 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_58 \def (Bind Abst) in 
+(let TMP_59 \def (THead TMP_58 u2 t5) in (eq T t4 TMP_59))))) in (let TMP_61 
+\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_64 \def 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(let TMP_62 
+\def (Bind b) in (let TMP_63 \def (CHead c TMP_62 u) in (pr3 TMP_63 x1 
+t5))))))) in (let TMP_65 \def (ex3_2 T T TMP_60 TMP_61 TMP_64) in (let 
+TMP_108 \def (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t2 (THead 
+(Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 x2)).(\lambda (H9: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let TMP_73 \def 
+(\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind Abst) 
+x4 x5)) \to (let TMP_68 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_66 
+\def (Bind Abst) in (let TMP_67 \def (THead TMP_66 u2 t5) in (eq T t4 
+TMP_67))))) in (let TMP_69 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 
+u2))) in (let TMP_72 \def (\lambda (_: T).(\lambda (t5: T).(\forall (b: 
+B).(\forall (u: T).(let TMP_70 \def (Bind b) in (let TMP_71 \def (CHead c 
+TMP_70 u) in (pr3 TMP_71 x5 t5))))))) in (ex3_2 T T TMP_68 TMP_69 
+TMP_72)))))))) in (let TMP_74 \def (Bind Abst) in (let TMP_75 \def (THead 
+TMP_74 x2 x3) in (let H10 \def (eq_ind T t2 TMP_73 H3 TMP_75 H7) in (let 
+TMP_76 \def (Bind Abst) in (let TMP_77 \def (THead TMP_76 x2 x3) in (let 
+TMP_78 \def (refl_equal T TMP_77) in (let H11 \def (H10 x2 x3 TMP_78) in (let 
+TMP_81 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_79 \def (Bind Abst) in 
+(let TMP_80 \def (THead TMP_79 u2 t5) in (eq T t4 TMP_80))))) in (let TMP_82 
+\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) in (let TMP_85 \def 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(let TMP_83 
+\def (Bind b) in (let TMP_84 \def (CHead c TMP_83 u) in (pr3 TMP_84 x3 
+t5))))))) in (let TMP_88 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_86 
+\def (Bind Abst) in (let TMP_87 \def (THead TMP_86 u2 t5) in (eq T t4 
+TMP_87))))) in (let TMP_89 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) in (let TMP_92 \def (\lambda (_: T).(\lambda (t5: T).(\forall (b: 
+B).(\forall (u: T).(let TMP_90 \def (Bind b) in (let TMP_91 \def (CHead c 
+TMP_90 u) in (pr3 TMP_91 x1 t5))))))) in (let TMP_93 \def (ex3_2 T T TMP_88 
+TMP_89 TMP_92) in (let TMP_107 \def (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H12: (eq T t4 (THead (Bind Abst) x4 x5))).(\lambda (H13: (pr3 c 
+x2 x4)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
+b) u) x3 x5))))).(let TMP_96 \def (\lambda (u2: T).(\lambda (t5: T).(let 
+TMP_94 \def (Bind Abst) in (let TMP_95 \def (THead TMP_94 u2 t5) in (eq T t4 
+TMP_95))))) in (let TMP_97 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) in (let TMP_100 \def (\lambda (_: T).(\lambda (t5: T).(\forall (b: 
+B).(\forall (u: T).(let TMP_98 \def (Bind b) in (let TMP_99 \def (CHead c 
+TMP_98 u) in (pr3 TMP_99 x1 t5))))))) in (let TMP_101 \def (pr3_sing c x2 x0 
+H8 x4 H13) in (let TMP_106 \def (\lambda (b: B).(\lambda (u: T).(let TMP_102 
+\def (Bind b) in (let TMP_103 \def (CHead c TMP_102 u) in (let TMP_104 \def 
+(H9 b u) in (let TMP_105 \def (H14 b u) in (pr3_sing TMP_103 x3 x1 TMP_104 x5 
+TMP_105))))))) in (ex3_2_intro T T TMP_96 TMP_97 TMP_100 x4 x5 H12 TMP_101 
+TMP_106))))))))))) in (ex3_2_ind T T TMP_81 TMP_82 TMP_85 TMP_93 TMP_107 
+H11)))))))))))))))))))))) in (ex3_2_ind T T TMP_53 TMP_54 TMP_57 TMP_65 
+TMP_108 H6))))))))))))))))))))))) in (let TMP_110 \def (pr3_ind c TMP_35 
+TMP_47 TMP_109 y x H0) in (let TMP_111 \def (unintro T u1 TMP_27 TMP_110) in 
+(unintro T t1 TMP_19 TMP_111)))))))))) in (insert_eq T TMP_2 TMP_3 TMP_11 
+TMP_112 H)))))))))).
 
 theorem pr3_gen_cast:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
@@ -122,102 +171,178 @@ T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c
 t1 x))))))
 \def
  \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 
-t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T 
-t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall 
-(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 
-t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall 
-(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 
-x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1)) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) 
-x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
-x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T 
-(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) 
-(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
-T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
-T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
-t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: 
-(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: 
-T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def 
-(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c 
-x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 
-c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda 
-(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) 
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) 
-x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 
-x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) 
+(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(let TMP_1 \def (Flat Cast) in (let 
+TMP_2 \def (THead TMP_1 u1 t1) in (let TMP_3 \def (\lambda (t: T).(pr3 c t 
+x)) in (let TMP_11 \def (\lambda (_: T).(let TMP_6 \def (\lambda (u2: 
+T).(\lambda (t2: T).(let TMP_4 \def (Flat Cast) in (let TMP_5 \def (THead 
+TMP_4 u2 t2) in (eq T x TMP_5))))) in (let TMP_7 \def (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) in (let TMP_8 \def (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c t1 t2))) in (let TMP_9 \def (ex3_2 T T TMP_6 TMP_7 
+TMP_8) in (let TMP_10 \def (pr3 c t1 x) in (or TMP_9 TMP_10))))))) in (let 
+TMP_204 \def (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(let TMP_19 \def 
+(\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (let TMP_14 \def 
+(\lambda (u2: T).(\lambda (t2: T).(let TMP_12 \def (Flat Cast) in (let TMP_13 
+\def (THead TMP_12 u2 t2) in (eq T x TMP_13))))) in (let TMP_15 \def (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))) in (let TMP_16 \def (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c t t2))) in (let TMP_17 \def (ex3_2 T T TMP_14 
+TMP_15 TMP_16) in (let TMP_18 \def (pr3 c t x) in (or TMP_17 TMP_18)))))))) 
+in (let TMP_27 \def (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Flat 
+Cast) t x0)) \to (let TMP_22 \def (\lambda (u2: T).(\lambda (t2: T).(let 
+TMP_20 \def (Flat Cast) in (let TMP_21 \def (THead TMP_20 u2 t2) in (eq T x 
+TMP_21))))) in (let TMP_23 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c t 
+u2))) in (let TMP_24 \def (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2))) in 
+(let TMP_25 \def (ex3_2 T T TMP_22 TMP_23 TMP_24) in (let TMP_26 \def (pr3 c 
+x0 x) in (or TMP_25 TMP_26))))))))) in (let TMP_35 \def (\lambda (t: 
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Flat 
+Cast) x0 x1)) \to (let TMP_30 \def (\lambda (u2: T).(\lambda (t2: T).(let 
+TMP_28 \def (Flat Cast) in (let TMP_29 \def (THead TMP_28 u2 t2) in (eq T t0 
+TMP_29))))) in (let TMP_31 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) in (let TMP_32 \def (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) in 
+(let TMP_33 \def (ex3_2 T T TMP_30 TMP_31 TMP_32) in (let TMP_34 \def (pr3 c 
+x1 t0) in (or TMP_33 TMP_34))))))))))) in (let TMP_71 \def (\lambda (t: 
+T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) 
+x0 x1))).(let TMP_36 \def (Flat Cast) in (let TMP_37 \def (THead TMP_36 x0 
+x1) in (let TMP_45 \def (\lambda (t0: T).(let TMP_40 \def (\lambda (u2: 
+T).(\lambda (t2: T).(let TMP_38 \def (Flat Cast) in (let TMP_39 \def (THead 
+TMP_38 u2 t2) in (eq T t0 TMP_39))))) in (let TMP_41 \def (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_42 \def (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c x1 t2))) in (let TMP_43 \def (ex3_2 T T TMP_40 
+TMP_41 TMP_42) in (let TMP_44 \def (pr3 c x1 t0) in (or TMP_43 TMP_44))))))) 
+in (let TMP_50 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_46 \def (Flat 
+Cast) in (let TMP_47 \def (THead TMP_46 x0 x1) in (let TMP_48 \def (Flat 
+Cast) in (let TMP_49 \def (THead TMP_48 u2 t2) in (eq T TMP_47 TMP_49))))))) 
+in (let TMP_51 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let 
+TMP_52 \def (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) in (let TMP_53 
+\def (ex3_2 T T TMP_50 TMP_51 TMP_52) in (let TMP_54 \def (Flat Cast) in (let 
+TMP_55 \def (THead TMP_54 x0 x1) in (let TMP_56 \def (pr3 c x1 TMP_55) in 
+(let TMP_61 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_57 \def (Flat 
+Cast) in (let TMP_58 \def (THead TMP_57 x0 x1) in (let TMP_59 \def (Flat 
+Cast) in (let TMP_60 \def (THead TMP_59 u2 t2) in (eq T TMP_58 TMP_60))))))) 
+in (let TMP_62 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let 
+TMP_63 \def (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) in (let TMP_64 
+\def (Flat Cast) in (let TMP_65 \def (THead TMP_64 x0 x1) in (let TMP_66 \def 
+(refl_equal T TMP_65) in (let TMP_67 \def (pr3_refl c x0) in (let TMP_68 \def 
+(pr3_refl c x1) in (let TMP_69 \def (ex3_2_intro T T TMP_61 TMP_62 TMP_63 x0 
+x1 TMP_66 TMP_67 TMP_68) in (let TMP_70 \def (or_introl TMP_53 TMP_56 TMP_69) 
+in (eq_ind_r T TMP_37 TMP_45 TMP_70 t H1))))))))))))))))))))))))) in (let 
+TMP_201 \def (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 
+t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
-t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
-t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T 
-t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: 
-(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead 
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat 
-Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead 
-(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 
-(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) 
-(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 
-t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c 
-x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 
-H2))) H6)))))))))))) y x H0))))) H))))).
-(* COMMENTS
-Initial nodes: 2001
-END *)
+(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H4: (eq T t3 (THead (Flat Cast) x0 x1))).(let TMP_72 \def 
+(\lambda (t: T).(pr2 c t t2)) in (let TMP_73 \def (Flat Cast) in (let TMP_74 
+\def (THead TMP_73 x0 x1) in (let H5 \def (eq_ind T t3 TMP_72 H1 TMP_74 H4) 
+in (let H6 \def (pr2_gen_cast c x0 x1 t2 H5) in (let TMP_77 \def (\lambda 
+(u2: T).(\lambda (t5: T).(let TMP_75 \def (Flat Cast) in (let TMP_76 \def 
+(THead TMP_75 u2 t5) in (eq T t2 TMP_76))))) in (let TMP_78 \def (\lambda 
+(u2: T).(\lambda (_: T).(pr2 c x0 u2))) in (let TMP_79 \def (\lambda (_: 
+T).(\lambda (t5: T).(pr2 c x1 t5))) in (let TMP_80 \def (ex3_2 T T TMP_77 
+TMP_78 TMP_79) in (let TMP_81 \def (pr2 c x1 t2) in (let TMP_84 \def (\lambda 
+(u2: T).(\lambda (t5: T).(let TMP_82 \def (Flat Cast) in (let TMP_83 \def 
+(THead TMP_82 u2 t5) in (eq T t4 TMP_83))))) in (let TMP_85 \def (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_86 \def (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5))) in (let TMP_87 \def (ex3_2 T T TMP_84 
+TMP_85 TMP_86) in (let TMP_88 \def (pr3 c x1 t4) in (let TMP_89 \def (or 
+TMP_87 TMP_88) in (let TMP_191 \def (\lambda (H7: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 
+t5))))).(let TMP_92 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_90 \def 
+(Flat Cast) in (let TMP_91 \def (THead TMP_90 u2 t5) in (eq T t2 TMP_91))))) 
+in (let TMP_93 \def (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) in (let 
+TMP_94 \def (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) in (let TMP_97 
+\def (\lambda (u2: T).(\lambda (t5: T).(let TMP_95 \def (Flat Cast) in (let 
+TMP_96 \def (THead TMP_95 u2 t5) in (eq T t4 TMP_96))))) in (let TMP_98 \def 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_99 \def (\lambda 
+(_: T).(\lambda (t5: T).(pr3 c x1 t5))) in (let TMP_100 \def (ex3_2 T T 
+TMP_97 TMP_98 TMP_99) in (let TMP_101 \def (pr3 c x1 t4) in (let TMP_102 \def 
+(or TMP_100 TMP_101) in (let TMP_190 \def (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 c x0 
+x2)).(\lambda (H10: (pr2 c x1 x3)).(let TMP_110 \def (\lambda (t: T).(\forall 
+(x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) \to (let TMP_105 
+\def (\lambda (u2: T).(\lambda (t5: T).(let TMP_103 \def (Flat Cast) in (let 
+TMP_104 \def (THead TMP_103 u2 t5) in (eq T t4 TMP_104))))) in (let TMP_106 
+\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) in (let TMP_107 \def 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5))) in (let TMP_108 \def (ex3_2 
+T T TMP_105 TMP_106 TMP_107) in (let TMP_109 \def (pr3 c x5 t4) in (or 
+TMP_108 TMP_109)))))))))) in (let TMP_111 \def (Flat Cast) in (let TMP_112 
+\def (THead TMP_111 x2 x3) in (let H11 \def (eq_ind T t2 TMP_110 H3 TMP_112 
+H8) in (let TMP_113 \def (Flat Cast) in (let TMP_114 \def (THead TMP_113 x2 
+x3) in (let TMP_115 \def (refl_equal T TMP_114) in (let H12 \def (H11 x2 x3 
+TMP_115) in (let TMP_118 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_116 
+\def (Flat Cast) in (let TMP_117 \def (THead TMP_116 u2 t5) in (eq T t4 
+TMP_117))))) in (let TMP_119 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
+u2))) in (let TMP_120 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) 
+in (let TMP_121 \def (ex3_2 T T TMP_118 TMP_119 TMP_120) in (let TMP_122 \def 
+(pr3 c x3 t4) in (let TMP_125 \def (\lambda (u2: T).(\lambda (t5: T).(let 
+TMP_123 \def (Flat Cast) in (let TMP_124 \def (THead TMP_123 u2 t5) in (eq T 
+t4 TMP_124))))) in (let TMP_126 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c 
+x0 u2))) in (let TMP_127 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5))) in (let TMP_128 \def (ex3_2 T T TMP_125 TMP_126 TMP_127) in (let 
+TMP_129 \def (pr3 c x1 t4) in (let TMP_130 \def (or TMP_128 TMP_129) in (let 
+TMP_180 \def (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))))).(let TMP_133 \def 
+(\lambda (u2: T).(\lambda (t5: T).(let TMP_131 \def (Flat Cast) in (let 
+TMP_132 \def (THead TMP_131 u2 t5) in (eq T t4 TMP_132))))) in (let TMP_134 
+\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) in (let TMP_135 \def 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) in (let TMP_138 \def 
+(\lambda (u2: T).(\lambda (t5: T).(let TMP_136 \def (Flat Cast) in (let 
+TMP_137 \def (THead TMP_136 u2 t5) in (eq T t4 TMP_137))))) in (let TMP_139 
+\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_140 \def 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) in (let TMP_141 \def (ex3_2 
+T T TMP_138 TMP_139 TMP_140) in (let TMP_142 \def (pr3 c x1 t4) in (let 
+TMP_143 \def (or TMP_141 TMP_142) in (let TMP_179 \def (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Flat Cast) x4 
+x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: (pr3 c x3 x5)).(let 
+TMP_144 \def (Flat Cast) in (let TMP_145 \def (THead TMP_144 x4 x5) in (let 
+TMP_153 \def (\lambda (t: T).(let TMP_148 \def (\lambda (u2: T).(\lambda (t5: 
+T).(let TMP_146 \def (Flat Cast) in (let TMP_147 \def (THead TMP_146 u2 t5) 
+in (eq T t TMP_147))))) in (let TMP_149 \def (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) in (let TMP_150 \def (\lambda (_: T).(\lambda (t5: T).(pr3 
+c x1 t5))) in (let TMP_151 \def (ex3_2 T T TMP_148 TMP_149 TMP_150) in (let 
+TMP_152 \def (pr3 c x1 t) in (or TMP_151 TMP_152))))))) in (let TMP_158 \def 
+(\lambda (u2: T).(\lambda (t5: T).(let TMP_154 \def (Flat Cast) in (let 
+TMP_155 \def (THead TMP_154 x4 x5) in (let TMP_156 \def (Flat Cast) in (let 
+TMP_157 \def (THead TMP_156 u2 t5) in (eq T TMP_155 TMP_157))))))) in (let 
+TMP_159 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_160 
+\def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) in (let TMP_161 \def 
+(ex3_2 T T TMP_158 TMP_159 TMP_160) in (let TMP_162 \def (Flat Cast) in (let 
+TMP_163 \def (THead TMP_162 x4 x5) in (let TMP_164 \def (pr3 c x1 TMP_163) in 
+(let TMP_169 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_165 \def (Flat 
+Cast) in (let TMP_166 \def (THead TMP_165 x4 x5) in (let TMP_167 \def (Flat 
+Cast) in (let TMP_168 \def (THead TMP_167 u2 t5) in (eq T TMP_166 
+TMP_168))))))) in (let TMP_170 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c 
+x0 u2))) in (let TMP_171 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5))) in (let TMP_172 \def (Flat Cast) in (let TMP_173 \def (THead TMP_172 x4 
+x5) in (let TMP_174 \def (refl_equal T TMP_173) in (let TMP_175 \def 
+(pr3_sing c x2 x0 H9 x4 H15) in (let TMP_176 \def (pr3_sing c x3 x1 H10 x5 
+H16) in (let TMP_177 \def (ex3_2_intro T T TMP_169 TMP_170 TMP_171 x4 x5 
+TMP_174 TMP_175 TMP_176) in (let TMP_178 \def (or_introl TMP_161 TMP_164 
+TMP_177) in (eq_ind_r T TMP_145 TMP_153 TMP_178 t4 
+H14)))))))))))))))))))))))))) in (ex3_2_ind T T TMP_133 TMP_134 TMP_135 
+TMP_143 TMP_179 H13)))))))))))) in (let TMP_189 \def (\lambda (H13: (pr3 c x3 
+t4)).(let TMP_183 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_181 \def 
+(Flat Cast) in (let TMP_182 \def (THead TMP_181 u2 t5) in (eq T t4 
+TMP_182))))) in (let TMP_184 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) in (let TMP_185 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) 
+in (let TMP_186 \def (ex3_2 T T TMP_183 TMP_184 TMP_185) in (let TMP_187 \def 
+(pr3 c x1 t4) in (let TMP_188 \def (pr3_sing c x3 x1 H10 t4 H13) in 
+(or_intror TMP_186 TMP_187 TMP_188)))))))) in (or_ind TMP_121 TMP_122 TMP_130 
+TMP_180 TMP_189 H12))))))))))))))))))))))))))) in (ex3_2_ind T T TMP_92 
+TMP_93 TMP_94 TMP_102 TMP_190 H7)))))))))))) in (let TMP_200 \def (\lambda 
+(H7: (pr2 c x1 t2)).(let TMP_194 \def (\lambda (u2: T).(\lambda (t5: T).(let 
+TMP_192 \def (Flat Cast) in (let TMP_193 \def (THead TMP_192 u2 t5) in (eq T 
+t4 TMP_193))))) in (let TMP_195 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c 
+x0 u2))) in (let TMP_196 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5))) in (let TMP_197 \def (ex3_2 T T TMP_194 TMP_195 TMP_196) in (let 
+TMP_198 \def (pr3 c x1 t4) in (let TMP_199 \def (pr3_sing c t2 x1 H7 t4 H2) 
+in (or_intror TMP_197 TMP_198 TMP_199)))))))) in (or_ind TMP_80 TMP_81 TMP_89 
+TMP_191 TMP_200 H6)))))))))))))))))))))))))))) in (let TMP_202 \def (pr3_ind 
+c TMP_35 TMP_71 TMP_201 y x H0) in (let TMP_203 \def (unintro T u1 TMP_27 
+TMP_202) in (unintro T t1 TMP_19 TMP_203)))))))))) in (insert_eq T TMP_2 
+TMP_3 TMP_11 TMP_204 H)))))))))).
 
 theorem pr3_gen_lift:
  \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
@@ -226,38 +351,51 @@ theorem pr3_gen_lift:
 t2))))))))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) 
-(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) 
-\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e 
-t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda 
-(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T 
-(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) 
-(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h 
-d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T 
-t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: 
-T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: 
-C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h 
-d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
+(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(let TMP_1 \def (lift h d t1) 
+in (let TMP_2 \def (\lambda (t: T).(pr3 c t x)) in (let TMP_6 \def (\lambda 
+(_: T).(\forall (e: C).((drop h d c e) \to (let TMP_4 \def (\lambda (t2: 
+T).(let TMP_3 \def (lift h d t2) in (eq T x TMP_3))) in (let TMP_5 \def 
+(\lambda (t2: T).(pr3 e t1 t2)) in (ex2 T TMP_4 TMP_5)))))) in (let TMP_45 
+\def (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(let TMP_10 \def (\lambda (t: 
+T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (let TMP_8 
+\def (\lambda (t2: T).(let TMP_7 \def (lift h d t2) in (eq T x TMP_7))) in 
+(let TMP_9 \def (\lambda (t2: T).(pr3 e t t2)) in (ex2 T TMP_8 TMP_9))))))) 
+in (let TMP_14 \def (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T 
+t (lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (let TMP_12 \def 
+(\lambda (t2: T).(let TMP_11 \def (lift h d t2) in (eq T t0 TMP_11))) in (let 
+TMP_13 \def (\lambda (t2: T).(pr3 e x0 t2)) in (ex2 T TMP_12 TMP_13))))))))) 
+in (let TMP_19 \def (\lambda (t: T).(\lambda (x0: T).(\lambda (H1: (eq T t 
+(lift h d x0))).(\lambda (e: C).(\lambda (_: (drop h d c e)).(let TMP_16 \def 
+(\lambda (t2: T).(let TMP_15 \def (lift h d t2) in (eq T t TMP_15))) in (let 
+TMP_17 \def (\lambda (t2: T).(pr3 e x0 t2)) in (let TMP_18 \def (pr3_refl e 
+x0) in (ex_intro2 T TMP_16 TMP_17 x0 H1 TMP_18))))))))) in (let TMP_43 \def 
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
 T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 
 (lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: 
 T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda 
 (x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: 
-(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 
-(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in 
-(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e 
-x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: 
-T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d 
-x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift 
-h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 
-(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda 
-(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T 
-(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 
-H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x 
-H0)))) H)))))).
-(* COMMENTS
-Initial nodes: 689
-END *)
+(drop h d c e)).(let TMP_20 \def (\lambda (t: T).(pr2 c t t2)) in (let TMP_21 
+\def (lift h d x0) in (let H6 \def (eq_ind T t3 TMP_20 H1 TMP_21 H4) in (let 
+H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in (let TMP_23 \def (\lambda (t5: 
+T).(let TMP_22 \def (lift h d t5) in (eq T t2 TMP_22))) in (let TMP_24 \def 
+(\lambda (t5: T).(pr2 e x0 t5)) in (let TMP_26 \def (\lambda (t5: T).(let 
+TMP_25 \def (lift h d t5) in (eq T t4 TMP_25))) in (let TMP_27 \def (\lambda 
+(t5: T).(pr3 e x0 t5)) in (let TMP_28 \def (ex2 T TMP_26 TMP_27) in (let 
+TMP_42 \def (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d x1))).(\lambda 
+(H9: (pr2 e x0 x1)).(let TMP_30 \def (\lambda (t5: T).(let TMP_29 \def (lift 
+h d t5) in (eq T t4 TMP_29))) in (let TMP_31 \def (\lambda (t5: T).(pr3 e x1 
+t5)) in (let TMP_33 \def (\lambda (t5: T).(let TMP_32 \def (lift h d t5) in 
+(eq T t4 TMP_32))) in (let TMP_34 \def (\lambda (t5: T).(pr3 e x0 t5)) in 
+(let TMP_35 \def (ex2 T TMP_33 TMP_34) in (let TMP_40 \def (\lambda (x2: 
+T).(\lambda (H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(let 
+TMP_37 \def (\lambda (t5: T).(let TMP_36 \def (lift h d t5) in (eq T t4 
+TMP_36))) in (let TMP_38 \def (\lambda (t5: T).(pr3 e x0 t5)) in (let TMP_39 
+\def (pr3_sing e x1 x0 H9 x2 H11) in (ex_intro2 T TMP_37 TMP_38 x2 H10 
+TMP_39))))))) in (let TMP_41 \def (H3 x1 H8 e H5) in (ex2_ind T TMP_30 TMP_31 
+TMP_35 TMP_40 TMP_41))))))))))) in (ex2_ind T TMP_23 TMP_24 TMP_28 TMP_42 
+H7))))))))))))))))))))) in (let TMP_44 \def (pr3_ind c TMP_14 TMP_19 TMP_43 y 
+x H0) in (unintro T t1 TMP_10 TMP_44)))))))) in (insert_eq T TMP_1 TMP_2 
+TMP_6 TMP_45 H)))))))))).
 
 theorem pr3_gen_lref:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to 
@@ -267,1338 +405,132 @@ T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
 (v: T).(eq T x (lift (S n) O v))))))))))
 \def
  \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef 
-n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: 
-T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y 
-x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or 
-(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T 
-t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t)))) 
-(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: 
-T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or 
-(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef 
-n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) 
-in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T 
+n) x)).(let TMP_1 \def (TLRef n) in (let TMP_2 \def (\lambda (t: T).(pr3 c t 
+x)) in (let TMP_12 \def (\lambda (t: T).(let TMP_3 \def (eq T x t) in (let 
+TMP_6 \def (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_4 \def 
+(Bind Abbr) in (let TMP_5 \def (CHead d TMP_4 u) in (getl n c TMP_5)))))) in 
+(let TMP_7 \def (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u 
+v)))) in (let TMP_10 \def (\lambda (_: C).(\lambda (_: T).(\lambda (v: 
+T).(let TMP_8 \def (S n) in (let TMP_9 \def (lift TMP_8 O v) in (eq T x 
+TMP_9)))))) in (let TMP_11 \def (ex3_3 C T T TMP_6 TMP_7 TMP_10) in (or TMP_3 
+TMP_11))))))) in (let TMP_155 \def (\lambda (y: T).(\lambda (H0: (pr3 c y 
+x)).(let TMP_22 \def (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to 
+(let TMP_13 \def (eq T t0 t) in (let TMP_16 \def (\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: T).(let TMP_14 \def (Bind Abbr) in (let TMP_15 \def (CHead d 
+TMP_14 u) in (getl n c TMP_15)))))) in (let TMP_17 \def (\lambda (d: 
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_20 \def 
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_18 \def (S n) in 
+(let TMP_19 \def (lift TMP_18 O v) in (eq T t0 TMP_19)))))) in (let TMP_21 
+\def (ex3_3 C T T TMP_16 TMP_17 TMP_20) in (or TMP_13 TMP_21))))))))) in (let 
+TMP_33 \def (\lambda (t: T).(\lambda (_: (eq T t (TLRef n))).(let TMP_23 \def 
+(eq T t t) in (let TMP_26 \def (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
+T).(let TMP_24 \def (Bind Abbr) in (let TMP_25 \def (CHead d TMP_24 u) in 
+(getl n c TMP_25)))))) in (let TMP_27 \def (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_30 \def (\lambda (_: 
+C).(\lambda (_: T).(\lambda (v: T).(let TMP_28 \def (S n) in (let TMP_29 \def 
+(lift TMP_28 O v) in (eq T t TMP_29)))))) in (let TMP_31 \def (ex3_3 C T T 
+TMP_26 TMP_27 TMP_30) in (let TMP_32 \def (refl_equal T t) in (or_introl 
+TMP_23 TMP_31 TMP_32))))))))) in (let TMP_154 \def (\lambda (t2: T).(\lambda 
+(t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H2: (pr3 c t2 
+t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or (eq T t3 t2) (ex3_3 C T T 
 (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind 
 Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) 
 (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
-v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef 
-n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T 
-t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef 
-n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or 
-(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def 
-(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8 
-(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d: 
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d: 
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) 
-(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead 
-d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u 
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
-v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 
-(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def 
-(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C 
-T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind 
-Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
-v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda 
-(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift 
-c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda 
-(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq 
-T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T 
-t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3 
-(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: 
-T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 
-(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13))))) 
-H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 1515
-END *)
-
-theorem pr3_gen_void:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 
-(lift (S O) O x)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
-t1) (\lambda (t: T).(pr3 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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c 
-(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y 
-x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \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).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 
-(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \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).(pr3 c t u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c 
-(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
-(THead (Bind Void) x0 x1)) \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).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
-(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind 
-Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
-(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c 
-(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead 
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) 
-(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) 
-u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c 
-t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall 
-(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
-(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let 
-H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 
-x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
-t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
-Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
-(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
-(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
-T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) 
-O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind 
-Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind 
-T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
-Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O 
-t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 
-(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c 
-(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
-(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 
-x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
-(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 
-(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead 
-c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: 
-(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
-(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind 
-Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift 
-(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: 
-((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
-t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) 
-(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S 
-O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) 
-O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
-(* COMMENTS
-Initial nodes: 2645
-END *)
-
-theorem pr3_gen_abbr:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
-t1) (\lambda (t: T).(pr3 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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
-O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda 
-(t: T).((eq T y (THead (Bind Abbr) u1 t)) \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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) 
-O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind 
-Abbr) t x0)) \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).(pr3 c t u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 
-(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: 
-T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind 
-Abbr) x0 x1)) \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).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 
-x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) 
-x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 
-x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) 
-(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
-T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
-T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
-(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
-t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 
-t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
-(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
-T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
-(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
-z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 
-(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead 
-c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z 
-t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
-T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
-(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
-z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 
-c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 
-(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: 
-T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
-z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
-(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
-(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind 
-T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
-Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 
-(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 
-x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) 
-in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
-Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c 
-(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 
-t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
-(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c 
-x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T 
-(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 
-x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 
-(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 
-H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
-t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
-x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) 
-O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: 
-T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 
-x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
-(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
-T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) 
-x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall 
-(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S 
-O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 
-(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
-O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 
-x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) 
-x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
-x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 
-x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) 
-(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 
-x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c 
-(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 
-x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O 
-t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T 
-(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) 
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
-(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
-T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 
-x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) 
-x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall 
-(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S 
-O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 
-(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
-O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
-(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 
-x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) 
-x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
-x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 
-x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 
-H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind 
-Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 
-(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) 
-(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
-t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
-x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 
-(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing 
-(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 
-(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) 
-H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing 
-(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) 
-(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c 
-(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
-(* COMMENTS
-Initial nodes: 5983
-END *)
-
-theorem pr3_gen_appl:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
-(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
-c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 
-t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (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).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 
-(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 c t (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda 
-(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (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).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall 
-(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) 
-(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t 
-(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda 
-(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) 
-(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat 
-Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 
-x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda 
-(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 
-t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat 
-Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
-z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
-(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
-t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 
-t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (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 x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 
-t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 
-x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 
-\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t 
-(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
-(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: 
-T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 
-(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5)))))))) (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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
-t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4 
-x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T 
-(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5)))))))) (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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4 
-x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5)))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) 
-x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
-x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T 
-(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 
-x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: 
-(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda 
-(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro 
-T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: 
-T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
-t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 
-(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind 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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: 
-T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 
-c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 
-(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 
-x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) 
-x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) 
-x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 
-(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) 
-(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
-Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
-Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq 
-T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 
-x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 
-x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
-(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 
-\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t 
-(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
-(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: 
-T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5)))))))) (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).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
-(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind 
-Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) 
-x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (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 x1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-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 x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t2 (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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: 
-T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T 
-x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 
-(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 
-x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) 
-x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c t (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 
-(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) 
-x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
-c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
-(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let 
-H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 
-(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5)))))))) (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).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (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).(pr3 c 
-(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) 
-H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) 
-x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
-(* COMMENTS
-Initial nodes: 12691
-END *)
-
-theorem pr3_gen_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
-T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind 
-b) u1) t1 (lift (S O) O x)))))))))
-\def
- \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
-(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind 
-b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 
-(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B 
-Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: 
-T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def 
-(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
-Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x 
-(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 
-(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 
-H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
-O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
-(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: 
-(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 
-\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c 
-(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_: 
-(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 
-\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c 
-(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda 
-(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
-u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 
-x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S 
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 
-H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 
-(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
-(* COMMENTS
-Initial nodes: 1721
-END *)
+v)))))))))).(\lambda (H4: (eq T t1 (TLRef n))).(let TMP_34 \def (\lambda (t: 
+T).(pr2 c t t2)) in (let TMP_35 \def (TLRef n) in (let H5 \def (eq_ind T t1 
+TMP_34 H1 TMP_35 H4) in (let TMP_36 \def (TLRef n) in (let TMP_46 \def 
+(\lambda (t: T).(let TMP_37 \def (eq T t3 t) in (let TMP_40 \def (\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: T).(let TMP_38 \def (Bind Abbr) in (let 
+TMP_39 \def (CHead d TMP_38 u) in (getl n c TMP_39)))))) in (let TMP_41 \def 
+(\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_44 
+\def (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_42 \def (S n) 
+in (let TMP_43 \def (lift TMP_42 O v) in (eq T t3 TMP_43)))))) in (let TMP_45 
+\def (ex3_3 C T T TMP_40 TMP_41 TMP_44) in (or TMP_37 TMP_45))))))) in (let 
+H6 \def (pr2_gen_lref c t2 n H5) in (let TMP_47 \def (TLRef n) in (let TMP_48 
+\def (eq T t2 TMP_47) in (let TMP_51 \def (\lambda (d: C).(\lambda (u: 
+T).(let TMP_49 \def (Bind Abbr) in (let TMP_50 \def (CHead d TMP_49 u) in 
+(getl n c TMP_50))))) in (let TMP_54 \def (\lambda (_: C).(\lambda (u: 
+T).(let TMP_52 \def (S n) in (let TMP_53 \def (lift TMP_52 O u) in (eq T t2 
+TMP_53))))) in (let TMP_55 \def (ex2_2 C T TMP_51 TMP_54) in (let TMP_56 \def 
+(TLRef n) in (let TMP_57 \def (eq T t3 TMP_56) in (let TMP_60 \def (\lambda 
+(d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_58 \def (Bind Abbr) in (let 
+TMP_59 \def (CHead d TMP_58 u) in (getl n c TMP_59)))))) in (let TMP_61 \def 
+(\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_64 
+\def (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_62 \def (S n) 
+in (let TMP_63 \def (lift TMP_62 O v) in (eq T t3 TMP_63)))))) in (let TMP_65 
+\def (ex3_3 C T T TMP_60 TMP_61 TMP_64) in (let TMP_66 \def (or TMP_57 
+TMP_65) in (let TMP_82 \def (\lambda (H7: (eq T t2 (TLRef n))).(let TMP_76 
+\def (\lambda (t: T).((eq T t (TLRef n)) \to (let TMP_67 \def (eq T t3 t) in 
+(let TMP_70 \def (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_68 
+\def (Bind Abbr) in (let TMP_69 \def (CHead d TMP_68 u) in (getl n c 
+TMP_69)))))) in (let TMP_71 \def (\lambda (d: C).(\lambda (u: T).(\lambda (v: 
+T).(pr3 d u v)))) in (let TMP_74 \def (\lambda (_: C).(\lambda (_: 
+T).(\lambda (v: T).(let TMP_72 \def (S n) in (let TMP_73 \def (lift TMP_72 O 
+v) in (eq T t3 TMP_73)))))) in (let TMP_75 \def (ex3_3 C T T TMP_70 TMP_71 
+TMP_74) in (or TMP_67 TMP_75)))))))) in (let TMP_77 \def (TLRef n) in (let H8 
+\def (eq_ind T t2 TMP_76 H3 TMP_77 H7) in (let TMP_78 \def (\lambda (t: 
+T).(pr3 c t t3)) in (let TMP_79 \def (TLRef n) in (let H9 \def (eq_ind T t2 
+TMP_78 H2 TMP_79 H7) in (let TMP_80 \def (TLRef n) in (let TMP_81 \def 
+(refl_equal T TMP_80) in (H8 TMP_81)))))))))) in (let TMP_152 \def (\lambda 
+(H7: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind 
+Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O 
+u)))))).(let TMP_85 \def (\lambda (d: C).(\lambda (u: T).(let TMP_83 \def 
+(Bind Abbr) in (let TMP_84 \def (CHead d TMP_83 u) in (getl n c TMP_84))))) 
+in (let TMP_88 \def (\lambda (_: C).(\lambda (u: T).(let TMP_86 \def (S n) in 
+(let TMP_87 \def (lift TMP_86 O u) in (eq T t2 TMP_87))))) in (let TMP_89 
+\def (TLRef n) in (let TMP_90 \def (eq T t3 TMP_89) in (let TMP_93 \def 
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_91 \def (Bind Abbr) 
+in (let TMP_92 \def (CHead d TMP_91 u) in (getl n c TMP_92)))))) in (let 
+TMP_94 \def (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in 
+(let TMP_97 \def (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_95 
+\def (S n) in (let TMP_96 \def (lift TMP_95 O v) in (eq T t3 TMP_96)))))) in 
+(let TMP_98 \def (ex3_3 C T T TMP_93 TMP_94 TMP_97) in (let TMP_99 \def (or 
+TMP_90 TMP_98) in (let TMP_151 \def (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H8: (getl n c (CHead x0 (Bind Abbr) x1))).(\lambda (H9: (eq T t2 
+(lift (S n) O x1))).(let TMP_109 \def (\lambda (t: T).((eq T t (TLRef n)) \to 
+(let TMP_100 \def (eq T t3 t) in (let TMP_103 \def (\lambda (d: C).(\lambda 
+(u: T).(\lambda (_: T).(let TMP_101 \def (Bind Abbr) in (let TMP_102 \def 
+(CHead d TMP_101 u) in (getl n c TMP_102)))))) in (let TMP_104 \def (\lambda 
+(d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_107 \def 
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_105 \def (S n) in 
+(let TMP_106 \def (lift TMP_105 O v) in (eq T t3 TMP_106)))))) in (let 
+TMP_108 \def (ex3_3 C T T TMP_103 TMP_104 TMP_107) in (or TMP_100 
+TMP_108)))))))) in (let TMP_110 \def (S n) in (let TMP_111 \def (lift TMP_110 
+O x1) in (let H10 \def (eq_ind T t2 TMP_109 H3 TMP_111 H9) in (let TMP_112 
+\def (\lambda (t: T).(pr3 c t t3)) in (let TMP_113 \def (S n) in (let TMP_114 
+\def (lift TMP_113 O x1) in (let H11 \def (eq_ind T t2 TMP_112 H2 TMP_114 H9) 
+in (let TMP_115 \def (S n) in (let TMP_116 \def (getl_drop Abbr c x0 x1 n H8) 
+in (let H12 \def (pr3_gen_lift c x1 t3 TMP_115 O H11 x0 TMP_116) in (let 
+TMP_119 \def (\lambda (t4: T).(let TMP_117 \def (S n) in (let TMP_118 \def 
+(lift TMP_117 O t4) in (eq T t3 TMP_118)))) in (let TMP_120 \def (\lambda 
+(t4: T).(pr3 x0 x1 t4)) in (let TMP_121 \def (TLRef n) in (let TMP_122 \def 
+(eq T t3 TMP_121) in (let TMP_125 \def (\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: T).(let TMP_123 \def (Bind Abbr) in (let TMP_124 \def (CHead 
+d TMP_123 u) in (getl n c TMP_124)))))) in (let TMP_126 \def (\lambda (d: 
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_129 \def 
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_127 \def (S n) in 
+(let TMP_128 \def (lift TMP_127 O v) in (eq T t3 TMP_128)))))) in (let 
+TMP_130 \def (ex3_3 C T T TMP_125 TMP_126 TMP_129) in (let TMP_131 \def (or 
+TMP_122 TMP_130) in (let TMP_150 \def (\lambda (x2: T).(\lambda (H13: (eq T 
+t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(let TMP_132 \def 
+(TLRef n) in (let TMP_133 \def (eq T t3 TMP_132) in (let TMP_136 \def 
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_134 \def (Bind Abbr) 
+in (let TMP_135 \def (CHead d TMP_134 u) in (getl n c TMP_135)))))) in (let 
+TMP_137 \def (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) 
+in (let TMP_140 \def (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let 
+TMP_138 \def (S n) in (let TMP_139 \def (lift TMP_138 O v) in (eq T t3 
+TMP_139)))))) in (let TMP_141 \def (ex3_3 C T T TMP_136 TMP_137 TMP_140) in 
+(let TMP_144 \def (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let 
+TMP_142 \def (Bind Abbr) in (let TMP_143 \def (CHead d TMP_142 u) in (getl n 
+c TMP_143)))))) in (let TMP_145 \def (\lambda (d: C).(\lambda (u: T).(\lambda 
+(v: T).(pr3 d u v)))) in (let TMP_148 \def (\lambda (_: C).(\lambda (_: 
+T).(\lambda (v: T).(let TMP_146 \def (S n) in (let TMP_147 \def (lift TMP_146 
+O v) in (eq T t3 TMP_147)))))) in (let TMP_149 \def (ex3_3_intro C T T 
+TMP_144 TMP_145 TMP_148 x0 x1 x2 H8 H14 H13) in (or_intror TMP_133 TMP_141 
+TMP_149)))))))))))))) in (ex2_ind T TMP_119 TMP_120 TMP_131 TMP_150 
+H12)))))))))))))))))))))))))) in (ex2_2_ind C T TMP_85 TMP_88 TMP_99 TMP_151 
+H7)))))))))))) in (let TMP_153 \def (or_ind TMP_48 TMP_55 TMP_66 TMP_82 
+TMP_152 H6) in (eq_ind_r T TMP_36 TMP_46 TMP_153 t1 
+H4))))))))))))))))))))))))))))) in (pr3_ind c TMP_22 TMP_33 TMP_154 y x 
+H0)))))) in (insert_eq T TMP_1 TMP_2 TMP_12 TMP_155 H)))))))).
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma
index c7b2fb7aa60fb4575c2daf66be96966fcb621c54..7ff6beaf4bdbf370c98c1a976e2fab153717ea1c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma
@@ -14,11 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/fwd.ma".
+include "basic_1/pr3/props.ma".
 
-include "Basic-1/iso/props.ma".
+include "basic_1/iso/props.ma".
 
-include "Basic-1/tlist/props.ma".
+include "basic_1/tlist/fwd.ma".
 
 theorem pr3_iso_appls_abbr:
  \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
@@ -56,20 +56,19 @@ T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
 i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C 
 (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind 
 Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) 
-H4)) in (let H9 \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) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d 
-(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal 
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead 
-x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind 
-Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 
-(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 
-\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def 
-(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d 
-H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d 
-H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) 
-H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: 
+H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) w) 
+(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 
+(Bind Abbr) x1) H4)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
+Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H 
+(CHead x0 (Bind Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def 
+(eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) 
+in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in 
+(let H14 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) 
+w))) H12 d H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w 
+x2)) H13 d H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 
+H15))))))) H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: 
 TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef 
 i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: 
 Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) 
@@ -199,9 +198,6 @@ u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))
 c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat 
 Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 
 (lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
-(* COMMENTS
-Initial nodes: 3759
-END *)
 
 theorem pr3_iso_appls_cast:
  \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 
@@ -362,9 +358,6 @@ O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
 (pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
 (THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
 x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
-(* COMMENTS
-Initial nodes: 3297
-END *)
 
 theorem pr3_iso_appl_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
@@ -450,59 +443,44 @@ T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
 (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda 
 (x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
 x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) 
-t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return 
-(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow 
-Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) 
-x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 
-| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind 
-Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T 
-x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda 
-(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def 
-(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def 
-(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst 
-H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H 
-Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 
-\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: 
-False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) 
-(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) 
-H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind 
-Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O 
-x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift 
-(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead 
-(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O 
-v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop 
-(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) 
-x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead 
-(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead 
-(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 
-x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) 
-x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 
-(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead 
-(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta 
-x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) 
-(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead 
-(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) 
-(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) 
-H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) 
+(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) x0 
+x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
+x5) H10) in (\lambda (H16: (eq T x0 x4)).(\lambda (H17: (eq B Abst b)).(let 
+H18 \def (eq_ind_r T x5 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) 
+H12 x1 H15) in (let H19 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) 
+H11 x0 H16) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c 
+(Bind b0) v2) t x1)) H18 Abst H17) in (let H21 \def (eq_ind_r B b (\lambda 
+(b0: B).(not (eq B b0 Abst))) H Abst H17) in (eq_ind B Abst (\lambda (b0: 
+B).(pr3 c (THead (Bind b0) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
+(Bind Abbr) x2 x3))) (let H22 \def (match (H21 (refl_equal B Abst)) in False 
+with []) in H22) b H17)))))))) H14)) H13))))))) H9)) (\lambda (H9: (pr3 
+(CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1)))).(pr3_t 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead 
+(Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 
+t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat 
+Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) (Bind b) 
+(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x2) (pr3_lift 
+(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) 
+v1 x2 H5) t (lift (S O) O (THead (Bind Abst) x0 x1)) H9 Appl)) (THead (Bind 
+Abbr) x2 x3) (eq_ind T (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) 
+x0 x1))) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 t0) (THead (Bind Abbr) x2 
+x3))) (pr3_sing c (THead (Bind Abbr) x2 x1) (THead (Bind b) v2 (lift (S O) O 
+(THead (Flat Appl) x2 (THead (Bind Abst) x0 x1)))) (pr2_free c (THead (Bind 
+b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) x0 x1)))) (THead 
+(Bind Abbr) x2 x1) (pr0_zeta b H (THead (Flat Appl) x2 (THead (Bind Abst) x0 
+x1)) (THead (Bind Abbr) x2 x1) (pr0_beta x0 x2 x2 (pr0_refl x2) x1 x1 
+(pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) (pr3_head_12 c x2 x2 (pr3_refl 
+c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead (Flat Appl) (lift (S O) O x2) 
+(lift (S O) O (THead (Bind Abst) x0 x1))) (lift_flat Appl x2 (THead (Bind 
+Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) H3)) (\lambda (H3: (ex6_6 B T T 
+T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda 
@@ -513,54 +491,64 @@ O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) 
-y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O 
-v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 
-Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 
-x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) 
-O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 
-x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind 
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t 
-(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind 
-b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat 
-Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) 
-(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq 
-T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 
-x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal 
-T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) 
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match 
-k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 
-_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
-T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in 
-(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def 
-(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) 
-in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) 
-in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 
-x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 
-Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat 
-Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) 
-(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S 
-O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O 
-x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c 
-(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: 
+B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 Abst))).(\lambda (H5: (pr3 c 
+(THead (Bind b) v2 t) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead 
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: 
+(pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind 
+x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
+x4) x3)) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let 
+H_x \def (pr3_gen_bind b H c v2 t (THead (Bind x0) x1 x2) H5) in (let H10 
+\def H_x in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead 
+(Bind x0) x1 x2) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) 
+v2) t t2)))) (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 
+x2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) (\lambda (H11: 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) 
+(THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t 
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c 
+v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2))) 
+(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) (\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H12: (eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) x6 x7))).(\lambda (H13: (pr3 c v2 x6)).(\lambda (H14: (pr3 (CHead c (Bind 
+b) v2) t x7)).(let H15 \def (f_equal T B (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) 
+\Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
+x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in ((let H16 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x1 | (TLRef 
+_) \Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) 
+(THead (Bind b) x6 x7) H12) in ((let H17 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 
+x7) H12) in (\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 
+\def (eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 
+H17) in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 
+H18) in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) 
+x5) x2 x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq 
+B b0 Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 
+x5 H8) (Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) 
+(lift (S O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift 
+(S O) O x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c 
+c (drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 
 (pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 
 (Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: 
 (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t 
@@ -587,9 +575,6 @@ H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl)
 O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
 (Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) 
 H10))) u2 H6))))))))))))) H3)) H2)))))))))).
-(* COMMENTS
-Initial nodes: 4805
-END *)
 
 theorem pr3_iso_appls_appl_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: 
@@ -758,9 +743,6 @@ Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat
 Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) 
 (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 
 H7)))))))))))))) H4)) H3))))))))) vs)))))).
-(* COMMENTS
-Initial nodes: 3571
-END *)
 
 theorem pr3_iso_appls_bind:
  \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: 
@@ -841,9 +823,6 @@ t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
 (Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat 
 Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) 
 (lifts_tapp (S O) O t ts))))))))))) vs))).
-(* COMMENTS
-Initial nodes: 1681
-END *)
 
 theorem pr3_iso_beta:
  \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat 
@@ -915,16 +894,15 @@ u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
 (x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead 
 (Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: 
 B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal 
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) 
-\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in 
-(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 
-H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 
-H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) x4 x5) H8) in ((let H12 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) 
+x4 x5) H8) in (\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t 
+t0)))) H10 x1 H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w 
+t0)) H9 x0 H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c 
 (pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) 
 (H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) 
 (\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
@@ -964,30 +942,25 @@ u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
 (x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead 
 (Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall 
 (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def 
-(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with 
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) 
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead k _ _) \Rightarrow (match k with [(Bind b) 
 \Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead 
 (Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) 
-\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) 
-(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | 
-(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) 
-x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 
-x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda 
-(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) 
-H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) 
-H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind 
-b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: 
+e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in 
+((let H15 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq 
+T x1 x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t 
+t0)))) H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w 
+t0)) H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c 
+(Bind b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: 
 B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) 
 H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b 
 Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in 
-False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) 
-in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))).
-(* COMMENTS
-Initial nodes: 2459
-END *)
+False with []) in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) 
+H1)))))))).
 
 theorem pr3_iso_appls_beta:
  \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 
@@ -1149,7 +1122,4 @@ O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
 (pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
 (THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
 x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
-(* COMMENTS
-Initial nodes: 3345
-END *)
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma
index e428daf3715bbd1f98806ff10642965bfd7c7081..3501531aadb3759684f29519caf0eae2b260386d 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma
@@ -14,21 +14,20 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/defs.ma".
+include "basic_1/pr3/defs.ma".
 
-include "Basic-1/pr1/defs.ma".
+include "basic_1/pr1/fwd.ma".
 
 theorem pr3_pr1:
  \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 
 t2))))
 \def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: 
-T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 
-t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: 
-C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))).
-(* COMMENTS
-Initial nodes: 95
-END *)
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(let TMP_1 \def 
+(\lambda (t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) in (let 
+TMP_2 \def (\lambda (t: T).(\lambda (c: C).(pr3_refl c t))) in (let TMP_5 
+\def (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda 
+(t4: T).(\lambda (_: (pr1 t0 t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 
+t4)))).(\lambda (c: C).(let TMP_3 \def (pr2_free c t3 t0 H0) in (let TMP_4 
+\def (H2 c) in (pr3_sing c t0 t3 TMP_3 t4 TMP_4)))))))))) in (pr1_ind TMP_1 
+TMP_2 TMP_5 t1 t2 H)))))).
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma
index 935850b1c200c6e47e48e3fd98b8165327f4a80d..b50ab1cbb8c3fdb86ef4939055adf287dbbf7d44 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/props.ma".
+include "basic_1/pr3/props.ma".
 
-include "Basic-1/pr2/pr2.ma".
+include "basic_1/pr2/pr2.ma".
 
 theorem pr3_strip:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
@@ -24,26 +24,32 @@ theorem pr3_strip:
 T).(pr3 c t2 t))))))))
 \def
  \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 
-t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t 
-t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 
-t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t 
-t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 
-t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda 
-(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 
-t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: 
-T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda 
-(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: 
-T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c 
-t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 
-x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) 
-(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda 
-(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T 
-(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c 
-x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) 
-t0 t1 H)))).
-(* COMMENTS
-Initial nodes: 375
-END *)
+t1)).(let TMP_3 \def (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 
+c t t3) \to (let TMP_1 \def (\lambda (t4: T).(pr3 c t2 t4)) in (let TMP_2 
+\def (\lambda (t4: T).(pr3 c t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let 
+TMP_8 \def (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let 
+TMP_4 \def (\lambda (t3: T).(pr3 c t t3)) in (let TMP_5 \def (\lambda (t3: 
+T).(pr3 c t2 t3)) in (let TMP_6 \def (pr3_pr2 c t t2 H0) in (let TMP_7 \def 
+(pr3_refl c 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: (pr2 c t3 
+t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H2: ((\forall 
+(t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: 
+T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr2 c t3 t5)).(let 
+TMP_9 \def (\lambda (t: T).(pr2 c t5 t)) in (let TMP_10 \def (\lambda (t: 
+T).(pr2 c t2 t)) in (let TMP_11 \def (\lambda (t: T).(pr3 c t4 t)) in (let 
+TMP_12 \def (\lambda (t: T).(pr3 c t5 t)) in (let TMP_13 \def (ex2 T TMP_11 
+TMP_12) in (let TMP_24 \def (\lambda (x: T).(\lambda (H4: (pr2 c t5 
+x)).(\lambda (H5: (pr2 c t2 x)).(let TMP_14 \def (\lambda (t: T).(pr3 c t4 
+t)) in (let TMP_15 \def (\lambda (t: T).(pr3 c x t)) in (let TMP_16 \def 
+(\lambda (t: T).(pr3 c t4 t)) in (let TMP_17 \def (\lambda (t: T).(pr3 c t5 
+t)) in (let TMP_18 \def (ex2 T TMP_16 TMP_17) in (let TMP_22 \def (\lambda 
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(let TMP_19 
+\def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_20 \def (\lambda (t: T).(pr3 c 
+t5 t)) in (let TMP_21 \def (pr3_sing c x t5 H4 x0 H7) in (ex_intro2 T TMP_19 
+TMP_20 x0 H6 TMP_21))))))) in (let TMP_23 \def (H2 x H5) in (ex2_ind T TMP_14 
+TMP_15 TMP_18 TMP_22 TMP_23))))))))))) in (let TMP_25 \def (pr2_confluence c 
+t3 t5 H3 t2 H0) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_24 
+TMP_25)))))))))))))))) in (pr3_ind c TMP_3 TMP_8 TMP_26 t0 t1 H))))))).
 
 theorem pr3_confluence:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
@@ -51,24 +57,29 @@ theorem pr3_confluence:
 T).(pr3 c t2 t))))))))
 \def
  \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 
-t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t 
-t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 
-t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t 
-t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 
-t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda 
-(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda 
-(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 
-t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c 
-t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 
-t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) 
-(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 
-x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) 
-(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda 
-(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T 
-(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 
-c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 
-H)))).
-(* COMMENTS
-Initial nodes: 367
-END *)
+t1)).(let TMP_3 \def (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 
+c t t3) \to (let TMP_1 \def (\lambda (t4: T).(pr3 c t2 t4)) in (let TMP_2 
+\def (\lambda (t4: T).(pr3 c t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let 
+TMP_7 \def (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(let 
+TMP_4 \def (\lambda (t3: T).(pr3 c t t3)) in (let TMP_5 \def (\lambda (t3: 
+T).(pr3 c t2 t3)) in (let TMP_6 \def (pr3_refl c t2) in (ex_intro2 T TMP_4 
+TMP_5 t2 H0 TMP_6))))))) in (let TMP_25 \def (\lambda (t2: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 
+t4)).(\lambda (H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: 
+T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda 
+(H3: (pr3 c t3 t5)).(let TMP_8 \def (\lambda (t: T).(pr3 c t5 t)) in (let 
+TMP_9 \def (\lambda (t: T).(pr3 c t2 t)) in (let TMP_10 \def (\lambda (t: 
+T).(pr3 c t4 t)) in (let TMP_11 \def (\lambda (t: T).(pr3 c t5 t)) in (let 
+TMP_12 \def (ex2 T TMP_10 TMP_11) in (let TMP_23 \def (\lambda (x: 
+T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 x)).(let TMP_13 \def 
+(\lambda (t: T).(pr3 c t4 t)) in (let TMP_14 \def (\lambda (t: T).(pr3 c x 
+t)) in (let TMP_15 \def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_16 \def 
+(\lambda (t: T).(pr3 c t5 t)) in (let TMP_17 \def (ex2 T TMP_15 TMP_16) in 
+(let TMP_21 \def (\lambda (x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: 
+(pr3 c x x0)).(let TMP_18 \def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_19 
+\def (\lambda (t: T).(pr3 c t5 t)) in (let TMP_20 \def (pr3_t x t5 c H4 x0 
+H7) in (ex_intro2 T TMP_18 TMP_19 x0 H6 TMP_20))))))) in (let TMP_22 \def (H2 
+x H5) in (ex2_ind T TMP_13 TMP_14 TMP_17 TMP_21 TMP_22))))))))))) in (let 
+TMP_24 \def (pr3_strip c t3 t5 H3 t2 H0) in (ex2_ind T TMP_8 TMP_9 TMP_12 
+TMP_23 TMP_24)))))))))))))))) in (pr3_ind c TMP_3 TMP_7 TMP_25 t0 t1 H))))))).
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
index 777f5d85d886636069e512ed5bd96c6b36196f26..299b08b9a21d30de77e693dfd1de7d8e506b691a 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
@@ -14,11 +14,13 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/pr1.ma".
+include "basic_1/pr3/fwd.ma".
 
-include "Basic-1/pr2/props.ma".
+include "basic_1/pr3/pr1.ma".
 
-include "Basic-1/pr1/props.ma".
+include "basic_1/pr2/props.ma".
+
+include "basic_1/pr1/props.ma".
 
 theorem clear_pr3_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to 
@@ -30,9 +32,6 @@ T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda
 (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: 
 T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 
 t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 107
-END *)
 
 theorem pr3_pr2:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c 
@@ -40,9 +39,6 @@ t1 t2))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
 t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
-(* COMMENTS
-Initial nodes: 31
-END *)
 
 theorem pr3_t:
  \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall 
@@ -55,9 +51,6 @@ c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall 
 (t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: 
 (pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 127
-END *)
 
 theorem pr3_thin_dx:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
@@ -72,9 +65,6 @@ T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0
 t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u 
 t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c 
 t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 167
-END *)
 
 theorem pr3_head_1:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -89,9 +79,6 @@ T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
 (THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing 
 c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) 
 (H2 k t)))))))))) u1 u2 H)))).
-(* COMMENTS
-Initial nodes: 167
-END *)
 
 theorem pr3_head_2:
  \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
@@ -106,9 +93,6 @@ T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_:
 (pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u 
 t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) 
 (THead k u t4) H2))))))) t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 175
-END *)
 
 theorem pr3_head_21:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -119,9 +103,6 @@ c (THead k u1 t1) (THead k u2 t2)))))))))
 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
 (CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c 
 u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
-(* COMMENTS
-Initial nodes: 89
-END *)
 
 theorem pr3_head_12:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -132,9 +113,6 @@ c (THead k u1 t1) (THead k u2 t2)))))))))
 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
 (CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c 
 u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
-(* COMMENTS
-Initial nodes: 89
-END *)
 
 theorem pr3_cflat:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
@@ -148,9 +126,6 @@ T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
 (_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead 
 c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c 
 (Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 151
-END *)
 
 theorem pr3_flat:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -161,9 +136,6 @@ theorem pr3_flat:
 u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda 
 (f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f 
 u2))))))))).
-(* COMMENTS
-Initial nodes: 59
-END *)
 
 theorem pr3_pr0_pr2_t:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall 
@@ -189,28 +161,26 @@ c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d
 (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d 
 (Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda 
 (H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(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) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
-(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) 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 _ k0 _) \Rightarrow (match k0 in K 
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead 
+c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) 
 (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) 
-u2) (CHead d (Bind Abbr) u) 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) 
-u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind 
-b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr 
-b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0: 
-T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3 
-(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 
-t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t) 
-(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x 
-t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr) 
-u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind 
-Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) 
+u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 \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) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d 
+(Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) 
+H9))) in ((let H12 \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) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
+(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda 
+(H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u 
+(\lambda (t0: T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda 
+(b0: B).(pr3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: 
+T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind 
+Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda 
+(H17: (pr0 x t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c 
+(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 
+(CHead c (Bind Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) 
 (pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: 
 F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) 
 u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O 
@@ -235,9 +205,6 @@ u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))
 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) 
 (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f 
 u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
-(* COMMENTS
-Initial nodes: 1557
-END *)
 
 theorem pr3_pr2_pr2_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
@@ -272,56 +239,52 @@ t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t)
 (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 
 (CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 
 (Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda 
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 
-| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind 
-b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 
-\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 d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) 
-(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind 
-Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) 
-u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let 
-H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in 
-(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind 
-T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i 
-O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda 
-(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let 
-H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O 
-i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n 
-u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7)) 
-(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda 
-(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0 
-x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind 
-Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead 
-c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i) 
-(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1) 
-(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x 
-O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14)) 
-H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0 
-(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 
-(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0 
-(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5 
-H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0) 
-H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0 
-(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4 
-t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr) 
-u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl 
-(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) 
-t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t) 
-(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta 
-(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind 
-Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4 
-t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat 
-f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 
-(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) 
-c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10))))) 
-i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))).
-(* COMMENTS
-Initial nodes: 1697
-END *)
+(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow 
+c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 
+(CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 \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 d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead 
+d0 (Bind Abbr) u0) t H12)) in ((let H15 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) 
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead 
+d0 (Bind Abbr) u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq 
+C d0 c0)).(let H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) 
+H11 t H15) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 
+t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: 
+T).(subst0 (S (plus i O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) 
+(\lambda (x: T).(\lambda (H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S 
+(plus i O)) u x t6)).(let H21 \def (f_equal nat nat S (plus i O) i (sym_eq 
+nat i (plus i O) (plus_n_O i))) in (let H22 \def (eq_ind nat (S (plus i O)) 
+(\lambda (n: nat).(subst0 n u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda 
+(t7: T).(subst0 O t1 t5 t7)) (\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 
+(Bind Abbr) t1) t4 t6) (\lambda (x0: T).(\lambda (H23: (subst0 O t1 t5 
+x0)).(\lambda (H24: (pr0 x0 x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 
+(pr2_delta (CHead c0 (Bind Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 
+x0 H23) t6 (pr3_pr2 (CHead c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 
+(Bind Abbr) t1) d u (S i) (getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 
+t1 (clear_bind Abbr c0 t1) (CHead d (Bind Abbr) u) i H0) x0 x H24 t6 
+H22)))))) (pr0_subst0_back t2 t5 x O H19 t1 H1))))))) (subst0_subst0 t5 t6 t 
+O H18 t2 u i H2)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: 
+(clear (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 
+(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 O (getl_intro O c0 
+(CHead d0 (Bind Abbr) u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead d0 
+(Bind Abbr) u0) t H12)) t4 t5 H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k 
+t) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda (_: (((getl 
+i1 (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to 
+(pr3 (CHead c0 k t1) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) 
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind 
+(\lambda (k0: K).((getl (S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to 
+(pr3 (CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) 
+(CHead c0 (Bind b) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) 
+t1) t4 t6 (pr2_delta (CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) 
+i1 c0 (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) 
+u0) t i1 H12) t1) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S 
+i1) (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 
+(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) 
+(getl_gen_S (Flat f) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) 
+f t1)))) k H10))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c 
+u1 u2 H)))).
 
 theorem pr3_pr2_pr3_t:
  \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
@@ -338,9 +301,6 @@ T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
 \to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 
 u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
 u1 H3)))))))))) t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 199
-END *)
 
 theorem pr3_pr3_pr3_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -357,9 +317,6 @@ t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3
 (CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: 
 T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 
 t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
-(* COMMENTS
-Initial nodes: 187
-END *)
 
 theorem pr3_lift:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
@@ -374,9 +331,6 @@ d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda
 t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d 
 t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 
 H0)))))))).
-(* COMMENTS
-Initial nodes: 167
-END *)
 
 theorem pr3_eta:
  \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind 
@@ -397,11 +351,1254 @@ 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 
+(lift (S O) O u) (subst1_lift_S u O O (le_O_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))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead 
 (Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).
-(* COMMENTS
-Initial nodes: 523
-END *)
+
+theorem pr3_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
+t1) (\lambda (t: T).(pr3 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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y 
+x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \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).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 
+(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \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).(pr3 c t u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c 
+(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
+(THead (Bind Void) x0 x1)) \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).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
+(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind 
+Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
+(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) 
+(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) 
+u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c 
+t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let 
+H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 
+x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) 
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
+t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) 
+O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind 
+Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O 
+t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 
+(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c 
+(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 
+x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 
+(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead 
+c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: 
+(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind 
+Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift 
+(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
+t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) 
+(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S 
+O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) 
+O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
+t1) (\lambda (t: T).(pr3 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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda 
+(t: T).((eq T y (THead (Bind Abbr) u1 t)) \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).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) 
+O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind 
+Abbr) t x0)) \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).(pr3 c t u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 
+(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: 
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind 
+Abbr) x0 x1)) \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).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 
+x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 
+x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) 
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
+T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 
+t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 
+(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
+b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead 
+c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z 
+t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 
+c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
+(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 
+(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: 
+T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
+(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 
+(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 
+x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) 
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c 
+(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c 
+x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T 
+(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 
+x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 
+(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 
+H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
+x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) 
+O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: 
+T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 
+x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
+(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
+T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) 
+x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall 
+(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S 
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 
+x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) 
+x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 
+x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) 
+(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 
+x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c 
+(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 
+x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O 
+t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T 
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) 
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
+T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
+(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 
+x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) 
+x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall 
+(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S 
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
+(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 
+x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) 
+x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 
+x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 
+H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind 
+Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 
+(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) 
+(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
+x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 
+(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing 
+(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 
+(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) 
+H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing 
+(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) 
+(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c 
+(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
+c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 
+t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (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).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 
+(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 c t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda 
+(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (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).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall 
+(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) 
+(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t 
+(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda 
+(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) 
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat 
+Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 
+x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda 
+(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 
+t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat 
+Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 
+t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (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 x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 
+x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t 
+(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: 
+T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 
+(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4 
+x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T 
+(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4 
+x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5)))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) 
+x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
+x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T 
+(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 
+x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: 
+(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda 
+(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro 
+T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: 
+T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
+t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 
+(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind 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).(pr3 c x3 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: 
+T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 
+c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 
+(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 
+x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) 
+x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (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).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) 
+x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 
+(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) 
+(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq 
+T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 
+x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 
+x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t 
+(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: 
+T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (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).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind 
+Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) 
+x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (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 x1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+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 x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (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 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T 
+x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 
+(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 
+x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) 
+x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 c t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 
+(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) 
+x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (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).(pr3 
+c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let 
+H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 
+(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (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).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (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).(pr3 c 
+(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) 
+H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) 
+x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
+T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind 
+b) u1) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
+(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind 
+b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 
+(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B 
+Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: 
+T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def 
+(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
+Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x 
+(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 
+(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 
+H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
+(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: 
+(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 
+\def (match (H (refl_equal B Abst)) in False with []) in H1))))))) (\lambda 
+(_: (not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 
+\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda 
+(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) 
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
+u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 
+x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S 
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 
+H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
index 3bcbd4965141cafa18bf97783ad72f7d572ab063..048fc9bebf363085918831e8dd13ddf80186f85a 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/defs.ma".
+include "basic_1/pr3/fwd.ma".
 
-include "Basic-1/pr2/subst1.ma".
+include "basic_1/pr2/subst1.ma".
 
 theorem pr3_subst1:
  \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
@@ -26,27 +26,33 @@ w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
 \def
  \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
 (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: 
-T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 
-w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: 
-T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 
-w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda 
-(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: 
-T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v 
-t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i 
-v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T 
-(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T 
-(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) 
-(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 
-x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 
-w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 
-w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v 
-t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 
-i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c 
-e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
-(* COMMENTS
-Initial nodes: 425
-END *)
+T).(\lambda (H0: (pr3 c t1 t2)).(let TMP_3 \def (\lambda (t: T).(\lambda (t0: 
+T).(\forall (w1: T).((subst1 i v t w1) \to (let TMP_1 \def (\lambda (w2: 
+T).(pr3 c w1 w2)) in (let TMP_2 \def (\lambda (w2: T).(subst1 i v t0 w2)) in 
+(ex2 T TMP_1 TMP_2))))))) in (let TMP_7 \def (\lambda (t: T).(\lambda (w1: 
+T).(\lambda (H1: (subst1 i v t w1)).(let TMP_4 \def (\lambda (w2: T).(pr3 c 
+w1 w2)) in (let TMP_5 \def (\lambda (w2: T).(subst1 i v t w2)) in (let TMP_6 
+\def (pr3_refl c w1) in (ex_intro2 T TMP_4 TMP_5 w1 TMP_6 H1))))))) in (let 
+TMP_25 \def (\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 
+t3)).(\lambda (t5: T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall 
+(w1: T).((subst1 i v t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) 
+(\lambda (w2: T).(subst1 i v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: 
+(subst1 i v t4 w1)).(let TMP_8 \def (\lambda (w2: T).(pr2 c w1 w2)) in (let 
+TMP_9 \def (\lambda (w2: T).(subst1 i v t3 w2)) in (let TMP_10 \def (\lambda 
+(w2: T).(pr3 c w1 w2)) in (let TMP_11 \def (\lambda (w2: T).(subst1 i v t5 
+w2)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) in (let TMP_23 \def (\lambda 
+(x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 x)).(let 
+TMP_13 \def (\lambda (w2: T).(pr3 c x w2)) in (let TMP_14 \def (\lambda (w2: 
+T).(subst1 i v t5 w2)) in (let TMP_15 \def (\lambda (w2: T).(pr3 c w1 w2)) in 
+(let TMP_16 \def (\lambda (w2: T).(subst1 i v t5 w2)) in (let TMP_17 \def 
+(ex2 T TMP_15 TMP_16) in (let TMP_21 \def (\lambda (x0: T).(\lambda (H7: (pr3 
+c x x0)).(\lambda (H8: (subst1 i v t5 x0)).(let TMP_18 \def (\lambda (w2: 
+T).(pr3 c w1 w2)) in (let TMP_19 \def (\lambda (w2: T).(subst1 i v t5 w2)) in 
+(let TMP_20 \def (pr3_sing c x w1 H5 x0 H7) in (ex_intro2 T TMP_18 TMP_19 x0 
+TMP_20 H8))))))) in (let TMP_22 \def (H3 x H6) in (ex2_ind T TMP_13 TMP_14 
+TMP_17 TMP_21 TMP_22))))))))))) in (let TMP_24 \def (pr2_subst1 c e v i H t4 
+t3 H1 w1 H4) in (ex2_ind T TMP_8 TMP_9 TMP_12 TMP_23 TMP_24)))))))))))))))) 
+in (pr3_ind c TMP_3 TMP_7 TMP_25 t1 t2 H0))))))))))).
 
 theorem pr3_gen_cabbr:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
@@ -57,39 +63,50 @@ a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
 x1 x2))))))))))))))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: 
-T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
-C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
-(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 
-d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) 
-(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
-(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: 
-(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 
-a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d 
-x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) 
-(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: 
-T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: 
-T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
-C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
-(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: 
-T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 
-x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
-(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: 
-(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 
-a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d 
-x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda 
-(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
-x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 
-d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda 
-(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) 
-(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: 
-T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) 
-d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d 
-u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x 
-x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 
-e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 731
-END *)
+t2)).(let TMP_5 \def (\lambda (t: T).(\lambda (t0: T).(\forall (e: 
+C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to 
+(\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) 
+\to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (let TMP_3 \def 
+(\lambda (x2: T).(let TMP_1 \def (S O) in (let TMP_2 \def (lift TMP_1 d x2) 
+in (subst1 d u t0 TMP_2)))) in (let TMP_4 \def (\lambda (x2: T).(pr3 a x1 
+x2)) in (ex2 T TMP_3 TMP_4))))))))))))))) in (let TMP_11 \def (\lambda (t: 
+T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c 
+(CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: (csubst1 d u c 
+a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 a)).(\lambda (x1: 
+T).(\lambda (H3: (subst1 d u t (lift (S O) d x1))).(let TMP_8 \def (\lambda 
+(x2: T).(let TMP_6 \def (S O) in (let TMP_7 \def (lift TMP_6 d x2) in (subst1 
+d u t TMP_7)))) in (let TMP_9 \def (\lambda (x2: T).(pr3 a x1 x2)) in (let 
+TMP_10 \def (pr3_refl a x1) in (ex_intro2 T TMP_8 TMP_9 x1 H3 
+TMP_10))))))))))))))) in (let TMP_39 \def (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 
+t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl 
+d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to 
+(\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t0 
+(lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
+x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))).(\lambda (e: C).(\lambda 
+(u: T).(\lambda (d: nat).(\lambda (H3: (getl d c (CHead e (Bind Abbr) 
+u))).(\lambda (a0: C).(\lambda (H4: (csubst1 d u c a0)).(\lambda (a: 
+C).(\lambda (H5: (drop (S O) d a0 a)).(\lambda (x1: T).(\lambda (H6: (subst1 
+d u t3 (lift (S O) d x1))).(let TMP_14 \def (\lambda (x2: T).(let TMP_12 \def 
+(S O) in (let TMP_13 \def (lift TMP_12 d x2) in (subst1 d u t0 TMP_13)))) in 
+(let TMP_15 \def (\lambda (x2: T).(pr2 a x1 x2)) in (let TMP_18 \def (\lambda 
+(x2: T).(let TMP_16 \def (S O) in (let TMP_17 \def (lift TMP_16 d x2) in 
+(subst1 d u t4 TMP_17)))) in (let TMP_19 \def (\lambda (x2: T).(pr3 a x1 x2)) 
+in (let TMP_20 \def (ex2 T TMP_18 TMP_19) in (let TMP_37 \def (\lambda (x: 
+T).(\lambda (H7: (subst1 d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 
+x)).(let TMP_23 \def (\lambda (x2: T).(let TMP_21 \def (S O) in (let TMP_22 
+\def (lift TMP_21 d x2) in (subst1 d u t4 TMP_22)))) in (let TMP_24 \def 
+(\lambda (x2: T).(pr3 a x x2)) in (let TMP_27 \def (\lambda (x2: T).(let 
+TMP_25 \def (S O) in (let TMP_26 \def (lift TMP_25 d x2) in (subst1 d u t4 
+TMP_26)))) in (let TMP_28 \def (\lambda (x2: T).(pr3 a x1 x2)) in (let TMP_29 
+\def (ex2 T TMP_27 TMP_28) in (let TMP_35 \def (\lambda (x0: T).(\lambda (H9: 
+(subst1 d u t4 (lift (S O) d x0))).(\lambda (H10: (pr3 a x x0)).(let TMP_32 
+\def (\lambda (x2: T).(let TMP_30 \def (S O) in (let TMP_31 \def (lift TMP_30 
+d x2) in (subst1 d u t4 TMP_31)))) in (let TMP_33 \def (\lambda (x2: T).(pr3 
+a x1 x2)) in (let TMP_34 \def (pr3_sing a x x1 H8 x0 H10) in (ex_intro2 T 
+TMP_32 TMP_33 x0 H9 TMP_34))))))) in (let TMP_36 \def (H2 e u d H3 a0 H4 a H5 
+x H7) in (ex2_ind T TMP_23 TMP_24 TMP_29 TMP_35 TMP_36))))))))))) in (let 
+TMP_38 \def (pr2_gen_cabbr c t3 t0 H0 e u d H3 a0 H4 a H5 x1 H6) in (ex2_ind 
+T TMP_14 TMP_15 TMP_20 TMP_37 TMP_38)))))))))))))))))))))))) in (pr3_ind c 
+TMP_5 TMP_11 TMP_39 t1 t2 H))))))).
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma
index 052f4e5b227249edbe0ec314b03daf64bf531fe1..c7f6669d5c49f2b314047bb73e1cb5f8b3582777 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma
@@ -14,53 +14,70 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/props.ma".
+include "basic_1/pr3/props.ma".
 
-include "Basic-1/wcpr0/getl.ma".
+include "basic_1/wcpr0/getl.ma".
 
 theorem pr3_wcpr0_t:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: 
 T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
 \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_1 \def 
 (\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 
-t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: 
-T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) 
-(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: 
-T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
-(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead 
-c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 
-(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) 
-(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: 
-(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq 
-C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: 
-C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: 
-(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 
-k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) 
-u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda 
-(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k 
-u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind 
-Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: 
-C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: 
-T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) 
-x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T 
-(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 
-(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 
-x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta 
-(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t 
-(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 
-H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 
-H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) 
-H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 
-H2)))))))))))))) c2 c1 H))).
-(* COMMENTS
-Initial nodes: 799
-END *)
+t1 t2) \to (pr3 c t1 t2)))))) in (let TMP_2 \def (\lambda (c: C).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).H0)))) in (let TMP_57 
+\def (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda 
+(_: ((\forall (t1: T).(\forall (t2: T).((pr3 c3 t1 t2) \to (pr3 c0 t1 
+t2)))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H3: (pr3 
+(CHead c3 k u2) t1 t2)).(let TMP_3 \def (CHead c3 k u1) in (let TMP_5 \def 
+(\lambda (t: T).(\lambda (t0: T).(let TMP_4 \def (CHead c0 k u1) in (pr3 
+TMP_4 t t0)))) in (let TMP_7 \def (\lambda (t: T).(let TMP_6 \def (CHead c0 k 
+u1) in (pr3_refl TMP_6 t))) in (let TMP_54 \def (\lambda (t0: T).(\lambda 
+(t3: T).(\lambda (H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda 
+(_: (pr3 (CHead c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 
+t4)).(let TMP_8 \def (CHead c0 k u1) in (let TMP_9 \def (CHead c3 k u1) in 
+(let TMP_10 \def (\lambda (c: C).(pr2 c t3 t0)) in (let TMP_12 \def (\lambda 
+(_: C).(let TMP_11 \def (CHead c0 k u1) in (pr3 TMP_11 t3 t0))) in (let 
+TMP_52 \def (\lambda (y: C).(\lambda (H7: (pr2 y t3 t0)).(let TMP_14 \def 
+(\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq C c (CHead c3 k u1)) 
+\to (let TMP_13 \def (CHead c0 k u1) in (pr3 TMP_13 t t5)))))) in (let TMP_18 
+\def (\lambda (c: C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 
+t6)).(\lambda (_: (eq C c (CHead c3 k u1))).(let TMP_15 \def (CHead c0 k u1) 
+in (let TMP_16 \def (CHead c0 k u1) in (let TMP_17 \def (pr2_free TMP_16 t5 
+t6 H8) in (pr3_pr2 TMP_15 t5 t6 TMP_17))))))))) in (let TMP_51 \def (\lambda 
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H8: (getl 
+i c (CHead d (Bind Abbr) u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: 
+(pr0 t5 t6)).(\lambda (t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: 
+(eq C c (CHead c3 k u1))).(let TMP_21 \def (\lambda (c4: C).(let TMP_19 \def 
+(Bind Abbr) in (let TMP_20 \def (CHead d TMP_19 u) in (getl i c4 TMP_20)))) 
+in (let TMP_22 \def (CHead c3 k u1) in (let H12 \def (eq_ind C c TMP_21 H8 
+TMP_22 H11) in (let TMP_26 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_23 
+\def (CHead c0 k u1) in (let TMP_24 \def (Bind Abbr) in (let TMP_25 \def 
+(CHead e2 TMP_24 u3) in (getl i TMP_23 TMP_25)))))) in (let TMP_27 \def 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) in (let TMP_28 \def (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u3 u))) in (let TMP_29 \def (CHead c0 k u1) in 
+(let TMP_30 \def (pr3 TMP_29 t5 t) in (let TMP_44 \def (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind 
+Abbr) x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(let TMP_31 
+\def (\lambda (t7: T).(subst0 i x1 t6 t7)) in (let TMP_32 \def (\lambda (t7: 
+T).(pr0 t7 t)) in (let TMP_33 \def (CHead c0 k u1) in (let TMP_34 \def (pr3 
+TMP_33 t5 t) in (let TMP_42 \def (\lambda (x: T).(\lambda (H16: (subst0 i x1 
+t6 x)).(\lambda (H17: (pr0 x t)).(let TMP_35 \def (CHead c0 k u1) in (let 
+TMP_36 \def (CHead c0 k u1) in (let TMP_37 \def (pr2_delta TMP_36 x0 x1 i H13 
+t5 t6 H9 x H16) in (let TMP_38 \def (CHead c0 k u1) in (let TMP_39 \def 
+(CHead c0 k u1) in (let TMP_40 \def (pr2_free TMP_39 x t H17) in (let TMP_41 
+\def (pr3_pr2 TMP_38 x t TMP_40) in (pr3_sing TMP_35 x t5 TMP_37 t 
+TMP_41))))))))))) in (let TMP_43 \def (pr0_subst0_back u t6 t i H10 x1 H15) 
+in (ex2_ind T TMP_31 TMP_32 TMP_34 TMP_42 TMP_43)))))))))))) in (let TMP_45 
+\def (CHead c3 k u1) in (let TMP_46 \def (CHead c0 k u1) in (let TMP_47 \def 
+(pr0_refl u1) in (let TMP_48 \def (wcpr0_comp c0 c3 H0 u1 u1 TMP_47 k) in 
+(let TMP_49 \def (Bind Abbr) in (let TMP_50 \def (wcpr0_getl_back TMP_45 
+TMP_46 TMP_48 i d u TMP_49 H12) in (ex3_2_ind C T TMP_26 TMP_27 TMP_28 TMP_30 
+TMP_44 TMP_50))))))))))))))))))))))))))) in (pr2_ind TMP_14 TMP_18 TMP_51 y 
+t3 t0 H7)))))) in (let TMP_53 \def (insert_eq C TMP_9 TMP_10 TMP_12 TMP_52 
+H4) in (pr3_t t0 t3 TMP_8 TMP_53 t4 H6))))))))))))) in (let TMP_55 \def 
+(pr2_free c3 u1 u2 H2) in (let TMP_56 \def (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 
+TMP_55) in (pr3_ind TMP_3 TMP_5 TMP_7 TMP_54 t1 t2 TMP_56)))))))))))))))))) 
+in (wcpr0_ind TMP_1 TMP_2 TMP_57 c2 c1 H)))))).