X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fwf3%2Ffwd.ma;h=d4718ec6e378e060fa4f67bebbe16ee8daa7b049;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=62ecb8cc0e5c45111809b4cc716fdff50020b1cf;hpb=7efd72a7efa0e709d6395b2020a7995f7d3a2f89;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma
index 62ecb8cc0..d4718ec6e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma
@@ -16,9 +16,9 @@
 
 include "basic_1/wf3/defs.ma".
 
-let rec wf3_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (m: nat).(P 
-(CSort m) (CSort m)))) (f0: (\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) 
-\to ((P c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to 
+implied rec lemma wf3_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (m: 
+nat).(P (CSort m) (CSort m)))) (f0: (\forall (c1: C).(\forall (c2: C).((wf3 g 
+c1 c2) \to ((P c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to 
 (\forall (b: B).(P (CHead c1 (Bind b) u) (CHead c2 (Bind b) u))))))))))) (f1: 
 (\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to ((P c1 c2) \to (\forall 
 (u: T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(P 
@@ -26,61 +26,44 @@ let rec wf3_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (m: nat).(P
 (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to ((P c1 c2) \to (\forall (u: 
 T).(\forall (f2: F).(P (CHead c1 (Flat f2) u) c2)))))))) (c: C) (c0: C) (w: 
 wf3 g c c0) on w: P c c0 \def match w with [(wf3_sort m) \Rightarrow (f m) | 
-(wf3_bind c1 c2 w0 u t t0 b) \Rightarrow (let TMP_3 \def ((wf3_ind g P f f0 
-f1 f2) c1 c2 w0) in (f0 c1 c2 w0 TMP_3 u t t0 b)) | (wf3_void c1 c2 w0 u f3 
-b) \Rightarrow (let TMP_2 \def ((wf3_ind g P f f0 f1 f2) c1 c2 w0) in (f1 c1 
-c2 w0 TMP_2 u f3 b)) | (wf3_flat c1 c2 w0 u f3) \Rightarrow (let TMP_1 \def 
-((wf3_ind g P f f0 f1 f2) c1 c2 w0) in (f2 c1 c2 w0 TMP_1 u f3))].
+(wf3_bind c1 c2 w0 u t t0 b) \Rightarrow (f0 c1 c2 w0 ((wf3_ind g P f f0 f1 
+f2) c1 c2 w0) u t t0 b) | (wf3_void c1 c2 w0 u f3 b) \Rightarrow (f1 c1 c2 w0 
+((wf3_ind g P f f0 f1 f2) c1 c2 w0) u f3 b) | (wf3_flat c1 c2 w0 u f3) 
+\Rightarrow (f2 c1 c2 w0 ((wf3_ind g P f f0 f1 f2) c1 c2 w0) u f3)].
 
-theorem wf3_gen_sort1:
+lemma wf3_gen_sort1:
  \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to 
 (eq C x (CSort m)))))
 \def
  \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort 
-m) x)).(let TMP_1 \def (CSort m) in (let TMP_2 \def (\lambda (c: C).(wf3 g c 
-x)) in (let TMP_3 \def (\lambda (c: C).(eq C x c)) in (let TMP_43 \def 
-(\lambda (y: C).(\lambda (H0: (wf3 g y x)).(let TMP_4 \def (\lambda (c: 
-C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) in (let TMP_13 
-\def (\lambda (m0: nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let TMP_5 
-\def (\lambda (e: C).(match e with [(CSort n) \Rightarrow n | (CHead _ _ _) 
-\Rightarrow m0])) in (let TMP_6 \def (CSort m0) in (let TMP_7 \def (CSort m) 
-in (let H2 \def (f_equal C nat TMP_5 TMP_6 TMP_7 H1) in (let TMP_10 \def 
-(\lambda (n: nat).(let TMP_8 \def (CSort n) in (let TMP_9 \def (CSort n) in 
-(eq C TMP_8 TMP_9)))) in (let TMP_11 \def (CSort m) in (let TMP_12 \def 
-(refl_equal C TMP_11) in (eq_ind_r nat m TMP_10 TMP_12 m0 H2)))))))))) in 
-(let TMP_23 \def (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 
-c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: 
-(eq C (CHead c1 (Bind b) u) (CSort m))).(let TMP_14 \def (Bind b) in (let 
-TMP_15 \def (CHead c1 TMP_14 u) in (let TMP_16 \def (\lambda (ee: C).(match 
-ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) in 
-(let TMP_17 \def (CSort m) in (let H5 \def (eq_ind C TMP_15 TMP_16 I TMP_17 
-H4) in (let TMP_18 \def (Bind b) in (let TMP_19 \def (CHead c2 TMP_18 u) in 
-(let TMP_20 \def (Bind b) in (let TMP_21 \def (CHead c1 TMP_20 u) in (let 
-TMP_22 \def (eq C TMP_19 TMP_21) in (False_ind TMP_22 H5)))))))))))))))))))) 
-in (let TMP_34 \def (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 
-c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: 
-T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b: 
-B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let TMP_24 \def 
-(Bind b) in (let TMP_25 \def (CHead c1 TMP_24 u) in (let TMP_26 \def (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) in (let TMP_27 \def (CSort m) in (let H5 \def (eq_ind C 
-TMP_25 TMP_26 I TMP_27 H4) in (let TMP_28 \def (Bind Void) in (let TMP_29 
-\def (TSort O) in (let TMP_30 \def (CHead c2 TMP_28 TMP_29) in (let TMP_31 
-\def (Bind b) in (let TMP_32 \def (CHead c1 TMP_31 u) in (let TMP_33 \def (eq 
-C TMP_30 TMP_32) in (False_ind TMP_33 H5)))))))))))))))))))) in (let TMP_42 
-\def (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 c2)).(\lambda 
-(_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: T).(\lambda (f: 
-F).(\lambda (H3: (eq C (CHead c1 (Flat f) u) (CSort m))).(let TMP_35 \def 
-(Flat f) in (let TMP_36 \def (CHead c1 TMP_35 u) in (let TMP_37 \def (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) in (let TMP_38 \def (CSort m) in (let H4 \def (eq_ind C 
-TMP_36 TMP_37 I TMP_38 H3) in (let TMP_39 \def (Flat f) in (let TMP_40 \def 
-(CHead c1 TMP_39 u) in (let TMP_41 \def (eq C c2 TMP_40) in (False_ind TMP_41 
-H4)))))))))))))))) in (wf3_ind g TMP_4 TMP_13 TMP_23 TMP_34 TMP_42 y x 
-H0)))))))) in (insert_eq C TMP_1 TMP_2 TMP_3 TMP_43 H)))))))).
+m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c: 
+C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda 
+(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0: 
+nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat 
+(\lambda (e: C).(match e with [(CSort n) \Rightarrow n | (CHead _ _ _) 
+\Rightarrow m0])) (CSort m0) (CSort m) H1) in (eq_ind_r nat m (\lambda (n: 
+nat).(eq C (CSort n) (CSort n))) (refl_equal C (CSort m)) m0 H2)))) (\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 
+(CSort m)) \to (eq C c2 c1)))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
+(ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) 
+(CSort m))).(let H5 \def (eq_ind C (CHead c1 (Bind b) u) (\lambda (ee: 
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
+True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 
+(Bind b) u)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
+(wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 
+c1)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to 
+False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort 
+m))).(let H5 \def (eq_ind C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I 
+(CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O)) (CHead c1 
+(Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 
+g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: 
+T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat f) u) (CSort m))).(let 
+H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m) 
+H3) in (False_ind (eq C c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
 
-theorem wf3_gen_bind1:
+lemma wf3_gen_bind1:
  \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b: 
 B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2: 
 C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda 
@@ -90,509 +73,305 @@ C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
 False))))))))))
 \def
  \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b: 
-B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(let TMP_1 \def (Bind b) in 
-(let TMP_2 \def (CHead c1 TMP_1 v) in (let TMP_3 \def (\lambda (c: C).(wf3 g 
-c x)) in (let TMP_17 \def (\lambda (_: C).(let TMP_6 \def (\lambda (c2: 
-C).(\lambda (_: T).(let TMP_4 \def (Bind b) in (let TMP_5 \def (CHead c2 
-TMP_4 v) in (eq C x TMP_5))))) in (let TMP_7 \def (\lambda (c2: C).(\lambda 
-(_: T).(wf3 g c1 c2))) in (let TMP_8 \def (\lambda (_: C).(\lambda (w: 
-T).(ty3 g c1 v w))) in (let TMP_9 \def (ex3_2 C T TMP_6 TMP_7 TMP_8) in (let 
-TMP_13 \def (\lambda (c2: C).(let TMP_10 \def (Bind Void) in (let TMP_11 \def 
-(TSort O) in (let TMP_12 \def (CHead c2 TMP_10 TMP_11) in (eq C x TMP_12))))) 
-in (let TMP_14 \def (\lambda (c2: C).(wf3 g c1 c2)) in (let TMP_15 \def 
-(\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) in (let TMP_16 
-\def (ex3 C TMP_13 TMP_14 TMP_15) in (or TMP_9 TMP_16)))))))))) in (let 
-TMP_242 \def (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(let TMP_31 \def 
-(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (let 
-TMP_20 \def (\lambda (c2: C).(\lambda (_: T).(let TMP_18 \def (Bind b) in 
-(let TMP_19 \def (CHead c2 TMP_18 v) in (eq C c0 TMP_19))))) in (let TMP_21 
-\def (\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) in (let TMP_22 \def 
-(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) in (let TMP_23 \def (ex3_2 C 
-T TMP_20 TMP_21 TMP_22) in (let TMP_27 \def (\lambda (c2: C).(let TMP_24 \def 
-(Bind Void) in (let TMP_25 \def (TSort O) in (let TMP_26 \def (CHead c2 
-TMP_24 TMP_25) in (eq C c0 TMP_26))))) in (let TMP_28 \def (\lambda (c2: 
-C).(wf3 g c1 c2)) in (let TMP_29 \def (\lambda (_: C).(\forall (w: T).((ty3 g 
-c1 v w) \to False))) in (let TMP_30 \def (ex3 C TMP_27 TMP_28 TMP_29) in (or 
-TMP_23 TMP_30)))))))))))) in (let TMP_52 \def (\lambda (m: nat).(\lambda (H1: 
-(eq C (CSort m) (CHead c1 (Bind b) v))).(let TMP_32 \def (CSort m) in (let 
-TMP_33 \def (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | 
-(CHead _ _ _) \Rightarrow False])) in (let TMP_34 \def (Bind b) in (let 
-TMP_35 \def (CHead c1 TMP_34 v) in (let H2 \def (eq_ind C TMP_32 TMP_33 I 
-TMP_35 H1) in (let TMP_39 \def (\lambda (c2: C).(\lambda (_: T).(let TMP_36 
-\def (CSort m) in (let TMP_37 \def (Bind b) in (let TMP_38 \def (CHead c2 
-TMP_37 v) in (eq C TMP_36 TMP_38)))))) in (let TMP_40 \def (\lambda (c2: 
-C).(\lambda (_: T).(wf3 g c1 c2))) in (let TMP_41 \def (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c1 v w))) in (let TMP_42 \def (ex3_2 C T TMP_39 
-TMP_40 TMP_41) in (let TMP_47 \def (\lambda (c2: C).(let TMP_43 \def (CSort 
-m) in (let TMP_44 \def (Bind Void) in (let TMP_45 \def (TSort O) in (let 
-TMP_46 \def (CHead c2 TMP_44 TMP_45) in (eq C TMP_43 TMP_46)))))) in (let 
-TMP_48 \def (\lambda (c2: C).(wf3 g c1 c2)) in (let TMP_49 \def (\lambda (_: 
-C).(\forall (w: T).((ty3 g c1 v w) \to False))) in (let TMP_50 \def (ex3 C 
-TMP_47 TMP_48 TMP_49) in (let TMP_51 \def (or TMP_42 TMP_50) in (False_ind 
-TMP_51 H2))))))))))))))))) in (let TMP_153 \def (\lambda (c0: C).(\lambda 
-(c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Bind 
-b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 
+B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind 
+b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda 
+(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: 
+C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
+v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) 
+(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
+w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or 
+(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v)))) 
+(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void) 
+(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C 
+(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda 
+(ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead c1 (Bind b) v) H1) in (False_ind (or (ex3_2 C T 
+(\lambda (c2: C).(\lambda (_: T).(eq C (CSort m) (CHead c2 (Bind b) v)))) 
+(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C (CSort m) (CHead c2 (Bind 
+Void) (TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall 
+(w: T).((ty3 g c1 v w) \to False))))) H2)))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Bind b) 
+v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 
 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: 
 C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead 
 c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: 
 C).(\forall (w: T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda 
 (t: T).(\lambda (H3: (ty3 g c0 u t)).(\lambda (b0: B).(\lambda (H4: (eq C 
-(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v))).(let TMP_53 \def (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) 
-in (let TMP_54 \def (Bind b0) in (let TMP_55 \def (CHead c0 TMP_54 u) in (let 
-TMP_56 \def (Bind b) in (let TMP_57 \def (CHead c1 TMP_56 v) in (let H5 \def 
-(f_equal C C TMP_53 TMP_55 TMP_57 H4) in (let TMP_58 \def (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match 
-k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) in (let 
-TMP_59 \def (Bind b0) in (let TMP_60 \def (CHead c0 TMP_59 u) in (let TMP_61 
-\def (Bind b) in (let TMP_62 \def (CHead c1 TMP_61 v) in (let H6 \def 
-(f_equal C B TMP_58 TMP_60 TMP_62 H4) in (let TMP_63 \def (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
-in (let TMP_64 \def (Bind b0) in (let TMP_65 \def (CHead c0 TMP_64 u) in (let 
-TMP_66 \def (Bind b) in (let TMP_67 \def (CHead c1 TMP_66 v) in (let H7 \def 
-(f_equal C T TMP_63 TMP_65 TMP_67 H4) in (let TMP_151 \def (\lambda (H8: (eq 
-B b0 b)).(\lambda (H9: (eq C c0 c1)).(let TMP_85 \def (\lambda (b1: B).(let 
-TMP_72 \def (\lambda (c3: C).(\lambda (_: T).(let TMP_68 \def (Bind b1) in 
-(let TMP_69 \def (CHead c2 TMP_68 u) in (let TMP_70 \def (Bind b) in (let 
-TMP_71 \def (CHead c3 TMP_70 v) in (eq C TMP_69 TMP_71))))))) in (let TMP_73 
-\def (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) in (let TMP_74 \def 
-(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) in (let TMP_75 \def (ex3_2 C 
-T TMP_72 TMP_73 TMP_74) in (let TMP_81 \def (\lambda (c3: C).(let TMP_76 \def 
-(Bind b1) in (let TMP_77 \def (CHead c2 TMP_76 u) in (let TMP_78 \def (Bind 
-Void) in (let TMP_79 \def (TSort O) in (let TMP_80 \def (CHead c3 TMP_78 
-TMP_79) in (eq C TMP_77 TMP_80))))))) in (let TMP_82 \def (\lambda (c3: 
-C).(wf3 g c1 c3)) in (let TMP_83 \def (\lambda (_: C).(\forall (w: T).((ty3 g 
-c1 v w) \to False))) in (let TMP_84 \def (ex3 C TMP_81 TMP_82 TMP_83) in (or 
-TMP_75 TMP_84)))))))))) in (let TMP_86 \def (\lambda (t0: T).(ty3 g c0 t0 t)) 
-in (let H10 \def (eq_ind T u TMP_86 H3 v H7) in (let TMP_104 \def (\lambda 
-(t0: T).(let TMP_91 \def (\lambda (c3: C).(\lambda (_: T).(let TMP_87 \def 
-(Bind b) in (let TMP_88 \def (CHead c2 TMP_87 t0) in (let TMP_89 \def (Bind 
-b) in (let TMP_90 \def (CHead c3 TMP_89 v) in (eq C TMP_88 TMP_90))))))) in 
-(let TMP_92 \def (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) in (let 
-TMP_93 \def (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) in (let TMP_94 
-\def (ex3_2 C T TMP_91 TMP_92 TMP_93) in (let TMP_100 \def (\lambda (c3: 
-C).(let TMP_95 \def (Bind b) in (let TMP_96 \def (CHead c2 TMP_95 t0) in (let 
-TMP_97 \def (Bind Void) in (let TMP_98 \def (TSort O) in (let TMP_99 \def 
-(CHead c3 TMP_97 TMP_98) in (eq C TMP_96 TMP_99))))))) in (let TMP_101 \def 
-(\lambda (c3: C).(wf3 g c1 c3)) in (let TMP_102 \def (\lambda (_: C).(\forall 
-(w: T).((ty3 g c1 v w) \to False))) in (let TMP_103 \def (ex3 C TMP_100 
-TMP_101 TMP_102) in (or TMP_94 TMP_103)))))))))) in (let TMP_105 \def 
-(\lambda (c: C).(ty3 g c v t)) in (let H11 \def (eq_ind C c0 TMP_105 H10 c1 
-H9) in (let TMP_119 \def (\lambda (c: C).((eq C c (CHead c1 (Bind b) v)) \to 
-(let TMP_108 \def (\lambda (c3: C).(\lambda (_: T).(let TMP_106 \def (Bind b) 
-in (let TMP_107 \def (CHead c3 TMP_106 v) in (eq C c2 TMP_107))))) in (let 
-TMP_109 \def (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) in (let TMP_110 
-\def (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) in (let TMP_111 \def 
-(ex3_2 C T TMP_108 TMP_109 TMP_110) in (let TMP_115 \def (\lambda (c3: 
-C).(let TMP_112 \def (Bind Void) in (let TMP_113 \def (TSort O) in (let 
-TMP_114 \def (CHead c3 TMP_112 TMP_113) in (eq C c2 TMP_114))))) in (let 
-TMP_116 \def (\lambda (c3: C).(wf3 g c1 c3)) in (let TMP_117 \def (\lambda 
-(_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) in (let TMP_118 \def (ex3 
-C TMP_115 TMP_116 TMP_117) in (or TMP_111 TMP_118))))))))))) in (let H12 \def 
-(eq_ind C c0 TMP_119 H2 c1 H9) in (let TMP_120 \def (\lambda (c: C).(wf3 g c 
-c2)) in (let H13 \def (eq_ind C c0 TMP_120 H1 c1 H9) in (let TMP_125 \def 
-(\lambda (c3: C).(\lambda (_: T).(let TMP_121 \def (Bind b) in (let TMP_122 
-\def (CHead c2 TMP_121 v) in (let TMP_123 \def (Bind b) in (let TMP_124 \def 
-(CHead c3 TMP_123 v) in (eq C TMP_122 TMP_124))))))) in (let TMP_126 \def 
-(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) in (let TMP_127 \def 
-(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) in (let TMP_128 \def (ex3_2 
-C T TMP_125 TMP_126 TMP_127) in (let TMP_134 \def (\lambda (c3: C).(let 
-TMP_129 \def (Bind b) in (let TMP_130 \def (CHead c2 TMP_129 v) in (let 
-TMP_131 \def (Bind Void) in (let TMP_132 \def (TSort O) in (let TMP_133 \def 
-(CHead c3 TMP_131 TMP_132) in (eq C TMP_130 TMP_133))))))) in (let TMP_135 
-\def (\lambda (c3: C).(wf3 g c1 c3)) in (let TMP_136 \def (\lambda (_: 
-C).(\forall (w: T).((ty3 g c1 v w) \to False))) in (let TMP_137 \def (ex3 C 
-TMP_134 TMP_135 TMP_136) in (let TMP_142 \def (\lambda (c3: C).(\lambda (_: 
-T).(let TMP_138 \def (Bind b) in (let TMP_139 \def (CHead c2 TMP_138 v) in 
-(let TMP_140 \def (Bind b) in (let TMP_141 \def (CHead c3 TMP_140 v) in (eq C 
-TMP_139 TMP_141))))))) in (let TMP_143 \def (\lambda (c3: C).(\lambda (_: 
-T).(wf3 g c1 c3))) in (let TMP_144 \def (\lambda (_: C).(\lambda (w: T).(ty3 
-g c1 v w))) in (let TMP_145 \def (Bind b) in (let TMP_146 \def (CHead c2 
-TMP_145 v) in (let TMP_147 \def (refl_equal C TMP_146) in (let TMP_148 \def 
-(ex3_2_intro C T TMP_142 TMP_143 TMP_144 c2 t TMP_147 H13 H11) in (let 
-TMP_149 \def (or_introl TMP_128 TMP_137 TMP_148) in (let TMP_150 \def 
-(eq_ind_r T v TMP_104 TMP_149 u H7) in (eq_ind_r B b TMP_85 TMP_150 b0 
-H8)))))))))))))))))))))))))))))) in (let TMP_152 \def (TMP_151 H6) in 
-(TMP_152 H5)))))))))))))))))))))))))))))) in (let TMP_221 \def (\lambda (c0: 
-C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 
-(CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: 
-T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g 
-c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda 
-(c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g 
-c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to 
-False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g c0 u t) 
-\to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) 
-(CHead c1 (Bind b) v))).(let TMP_154 \def (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) in (let TMP_155 
-\def (Bind b0) in (let TMP_156 \def (CHead c0 TMP_155 u) in (let TMP_157 \def 
-(Bind b) in (let TMP_158 \def (CHead c1 TMP_157 v) in (let H5 \def (f_equal C 
-C TMP_154 TMP_156 TMP_158 H4) in (let TMP_159 \def (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k with 
-[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) in (let TMP_160 \def 
-(Bind b0) in (let TMP_161 \def (CHead c0 TMP_160 u) in (let TMP_162 \def 
-(Bind b) in (let TMP_163 \def (CHead c1 TMP_162 v) in (let H6 \def (f_equal C 
-B TMP_159 TMP_161 TMP_163 H4) in (let TMP_164 \def (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) in (let 
-TMP_165 \def (Bind b0) in (let TMP_166 \def (CHead c0 TMP_165 u) in (let 
-TMP_167 \def (Bind b) in (let TMP_168 \def (CHead c1 TMP_167 v) in (let H7 
-\def (f_equal C T TMP_164 TMP_166 TMP_168 H4) in (let TMP_219 \def (\lambda 
-(_: (eq B b0 b)).(\lambda (H9: (eq C c0 c1)).(let TMP_169 \def (\lambda (t: 
-T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) in (let H10 \def (eq_ind T 
-u TMP_169 H3 v H7) in (let TMP_170 \def (\lambda (c: C).(\forall (t: T).((ty3 
-g c v t) \to False))) in (let H11 \def (eq_ind C c0 TMP_170 H10 c1 H9) in 
-(let TMP_184 \def (\lambda (c: C).((eq C c (CHead c1 (Bind b) v)) \to (let 
-TMP_173 \def (\lambda (c3: C).(\lambda (_: T).(let TMP_171 \def (Bind b) in 
-(let TMP_172 \def (CHead c3 TMP_171 v) in (eq C c2 TMP_172))))) in (let 
-TMP_174 \def (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) in (let TMP_175 
-\def (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) in (let TMP_176 \def 
-(ex3_2 C T TMP_173 TMP_174 TMP_175) in (let TMP_180 \def (\lambda (c3: 
-C).(let TMP_177 \def (Bind Void) in (let TMP_178 \def (TSort O) in (let 
-TMP_179 \def (CHead c3 TMP_177 TMP_178) in (eq C c2 TMP_179))))) in (let 
-TMP_181 \def (\lambda (c3: C).(wf3 g c1 c3)) in (let TMP_182 \def (\lambda 
-(_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) in (let TMP_183 \def (ex3 
-C TMP_180 TMP_181 TMP_182) in (or TMP_176 TMP_183))))))))))) in (let H12 \def 
-(eq_ind C c0 TMP_184 H2 c1 H9) in (let TMP_185 \def (\lambda (c: C).(wf3 g c 
-c2)) in (let H13 \def (eq_ind C c0 TMP_185 H1 c1 H9) in (let TMP_191 \def 
-(\lambda (c3: C).(\lambda (_: T).(let TMP_186 \def (Bind Void) in (let 
-TMP_187 \def (TSort O) in (let TMP_188 \def (CHead c2 TMP_186 TMP_187) in 
-(let TMP_189 \def (Bind b) in (let TMP_190 \def (CHead c3 TMP_189 v) in (eq C 
-TMP_188 TMP_190)))))))) in (let TMP_192 \def (\lambda (c3: C).(\lambda (_: 
-T).(wf3 g c1 c3))) in (let TMP_193 \def (\lambda (_: C).(\lambda (w: T).(ty3 
-g c1 v w))) in (let TMP_194 \def (ex3_2 C T TMP_191 TMP_192 TMP_193) in (let 
-TMP_201 \def (\lambda (c3: C).(let TMP_195 \def (Bind Void) in (let TMP_196 
-\def (TSort O) in (let TMP_197 \def (CHead c2 TMP_195 TMP_196) in (let 
-TMP_198 \def (Bind Void) in (let TMP_199 \def (TSort O) in (let TMP_200 \def 
-(CHead c3 TMP_198 TMP_199) in (eq C TMP_197 TMP_200)))))))) in (let TMP_202 
-\def (\lambda (c3: C).(wf3 g c1 c3)) in (let TMP_203 \def (\lambda (_: 
-C).(\forall (w: T).((ty3 g c1 v w) \to False))) in (let TMP_204 \def (ex3 C 
-TMP_201 TMP_202 TMP_203) in (let TMP_211 \def (\lambda (c3: C).(let TMP_205 
-\def (Bind Void) in (let TMP_206 \def (TSort O) in (let TMP_207 \def (CHead 
-c2 TMP_205 TMP_206) in (let TMP_208 \def (Bind Void) in (let TMP_209 \def 
-(TSort O) in (let TMP_210 \def (CHead c3 TMP_208 TMP_209) in (eq C TMP_207 
-TMP_210)))))))) in (let TMP_212 \def (\lambda (c3: C).(wf3 g c1 c3)) in (let 
-TMP_213 \def (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) in 
-(let TMP_214 \def (Bind Void) in (let TMP_215 \def (TSort O) in (let TMP_216 
-\def (CHead c2 TMP_214 TMP_215) in (let TMP_217 \def (refl_equal C TMP_216) 
-in (let TMP_218 \def (ex3_intro C TMP_211 TMP_212 TMP_213 c2 TMP_217 H13 H11) 
-in (or_intror TMP_194 TMP_204 TMP_218))))))))))))))))))))))))))) in (let 
-TMP_220 \def (TMP_219 H6) in (TMP_220 H5))))))))))))))))))))))))))))) in (let 
-TMP_241 \def (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 
-c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T 
-(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda 
-(c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 
-g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort 
+(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 
+\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b0 | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
+\Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let 
+H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) 
+H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r B b 
+(\lambda (b1: B).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead 
+c2 (Bind b1) u) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: 
+T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C 
+(\lambda (c3: C).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind Void) (TSort 
 O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g 
-c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C 
-(CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let TMP_222 \def (Flat f) in 
-(let TMP_223 \def (CHead c0 TMP_222 u) in (let TMP_224 \def (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) in 
-(let TMP_225 \def (Bind b) in (let TMP_226 \def (CHead c1 TMP_225 v) in (let 
-H4 \def (eq_ind C TMP_223 TMP_224 I TMP_226 H3) in (let TMP_229 \def (\lambda 
-(c3: C).(\lambda (_: T).(let TMP_227 \def (Bind b) in (let TMP_228 \def 
-(CHead c3 TMP_227 v) in (eq C c2 TMP_228))))) in (let TMP_230 \def (\lambda 
-(c3: C).(\lambda (_: T).(wf3 g c1 c3))) in (let TMP_231 \def (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c1 v w))) in (let TMP_232 \def (ex3_2 C T TMP_229 
-TMP_230 TMP_231) in (let TMP_236 \def (\lambda (c3: C).(let TMP_233 \def 
-(Bind Void) in (let TMP_234 \def (TSort O) in (let TMP_235 \def (CHead c3 
-TMP_233 TMP_234) in (eq C c2 TMP_235))))) in (let TMP_237 \def (\lambda (c3: 
-C).(wf3 g c1 c3)) in (let TMP_238 \def (\lambda (_: C).(\forall (w: T).((ty3 
-g c1 v w) \to False))) in (let TMP_239 \def (ex3 C TMP_236 TMP_237 TMP_238) 
-in (let TMP_240 \def (or TMP_232 TMP_239) in (False_ind TMP_240 
-H4))))))))))))))))))))))) in (wf3_ind g TMP_31 TMP_52 TMP_153 TMP_221 TMP_241 
-y x H0)))))))) in (insert_eq C TMP_2 TMP_3 TMP_17 TMP_242 H))))))))))).
+c1 v w) \to False)))))) (let H10 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 
+t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0: T).(or (ex3_2 C T (\lambda 
+(c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) t0) (CHead c3 (Bind b) v)))) 
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) t0) 
+(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda 
+(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H11 \def (eq_ind 
+C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1 H9) in (let H12 \def (eq_ind C c0 
+(\lambda (c: C).((eq C c (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda 
+(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: 
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) 
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
+w) \to False))))))) H2 c1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c: 
+C).(wf3 g c c2)) H1 c1 H9) in (or_introl (ex3_2 C T (\lambda (c3: C).(\lambda 
+(_: T).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: 
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
+v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind 
+Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall 
+(w: T).((ty3 g c1 v w) \to False)))) (ex3_2_intro C T (\lambda (c3: 
+C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind b) v)))) 
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w))) c2 t (refl_equal C (CHead c2 (Bind b) v)) H13 H11))))) u 
+H7)) b0 H8)))) H6)) H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda 
+(H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Bind b) v)) \to (or 
+(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) 
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) 
+(TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall 
+(t: T).((ty3 g c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C 
+(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 
+\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b0 | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
+\Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let 
+H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) 
+H4) in (\lambda (_: (eq B b0 b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def 
+(eq_ind T u (\lambda (t: T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 
+v H7) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c 
+v t) \to False))) H10 c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: 
+C).((eq C c (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: 
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: 
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) 
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
+w) \to False))))))) H2 c1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c: 
+C).(wf3 g c c2)) H1 c1 H9) in (or_intror (ex3_2 C T (\lambda (c3: C).(\lambda 
+(_: T).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind b) v)))) 
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind Void) 
+(TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) 
+(\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))) (ex3_intro C 
+(\lambda (c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) 
+(TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g c1 v w) \to False))) c2 (refl_equal C (CHead c2 (Bind Void) (TSort 
+O))) H13 H11))))))))) H6)) H5)))))))))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) 
+\to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind 
+b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: 
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead 
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: 
+C).(\forall (w: T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda 
+(f: F).(\lambda (H3: (eq C (CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let 
+H4 \def (eq_ind C (CHead c0 (Flat f) u) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead c1 (Bind b) v) 
+H3) in (False_ind (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 
+(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) 
+(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq 
+C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) 
+(\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H4))))))))) y 
+x H0))) H)))))).
 
-theorem wf3_gen_flat1:
+lemma wf3_gen_flat1:
  \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f: 
 F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
 \def
  \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f: 
-F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(let TMP_1 \def (Flat f) in 
-(let TMP_2 \def (CHead c1 TMP_1 v) in (let TMP_3 \def (\lambda (c: C).(wf3 g 
-c x)) in (let TMP_4 \def (\lambda (_: C).(wf3 g c1 x)) in (let TMP_52 \def 
-(\lambda (y: C).(\lambda (H0: (wf3 g y x)).(let TMP_5 \def (\lambda (c: 
-C).(\lambda (c0: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) in 
-(let TMP_12 \def (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c1 
-(Flat f) v))).(let TMP_6 \def (CSort m) in (let TMP_7 \def (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow 
-False])) in (let TMP_8 \def (Flat f) in (let TMP_9 \def (CHead c1 TMP_8 v) in 
-(let H2 \def (eq_ind C TMP_6 TMP_7 I TMP_9 H1) in (let TMP_10 \def (CSort m) 
-in (let TMP_11 \def (wf3 g c1 TMP_10) in (False_ind TMP_11 H2)))))))))) in 
-(let TMP_21 \def (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 
-c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 
-c2)))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda 
-(b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let 
-TMP_13 \def (Bind b) in (let TMP_14 \def (CHead c0 TMP_13 u) in (let TMP_15 
-\def (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ 
-k _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) in (let TMP_16 \def (Flat f) in (let TMP_17 \def 
-(CHead c1 TMP_16 v) in (let H5 \def (eq_ind C TMP_14 TMP_15 I TMP_17 H4) in 
-(let TMP_18 \def (Bind b) in (let TMP_19 \def (CHead c2 TMP_18 u) in (let 
-TMP_20 \def (wf3 g c1 TMP_19) in (False_ind TMP_20 H5))))))))))))))))))) in 
-(let TMP_31 \def (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 
-c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 
-c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0 u t) \to 
-False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) (CHead c1 
-(Flat f) v))).(let TMP_22 \def (Bind b) in (let TMP_23 \def (CHead c0 TMP_22 
-u) in (let TMP_24 \def (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
-False | (CHead _ k _) \Rightarrow (match k with [(Bind _) \Rightarrow True | 
-(Flat _) \Rightarrow False])])) in (let TMP_25 \def (Flat f) in (let TMP_26 
-\def (CHead c1 TMP_25 v) in (let H5 \def (eq_ind C TMP_23 TMP_24 I TMP_26 H4) 
-in (let TMP_27 \def (Bind Void) in (let TMP_28 \def (TSort O) in (let TMP_29 
-\def (CHead c2 TMP_27 TMP_28) in (let TMP_30 \def (wf3 g c1 TMP_29) in 
-(False_ind TMP_30 H5))))))))))))))))))) in (let TMP_51 \def (\lambda (c0: 
-C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 
-(CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: 
-F).(\lambda (H3: (eq C (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let 
-TMP_32 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead 
-c _ _) \Rightarrow c])) in (let TMP_33 \def (Flat f0) in (let TMP_34 \def 
-(CHead c0 TMP_33 u) in (let TMP_35 \def (Flat f) in (let TMP_36 \def (CHead 
-c1 TMP_35 v) in (let H4 \def (f_equal C C TMP_32 TMP_34 TMP_36 H3) in (let 
-TMP_37 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow f0 | (CHead 
-_ k _) \Rightarrow (match k with [(Bind _) \Rightarrow f0 | (Flat f1) 
-\Rightarrow f1])])) in (let TMP_38 \def (Flat f0) in (let TMP_39 \def (CHead 
-c0 TMP_38 u) in (let TMP_40 \def (Flat f) in (let TMP_41 \def (CHead c1 
-TMP_40 v) in (let H5 \def (f_equal C F TMP_37 TMP_39 TMP_41 H3) in (let 
-TMP_42 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ 
-_ t) \Rightarrow t])) in (let TMP_43 \def (Flat f0) in (let TMP_44 \def 
-(CHead c0 TMP_43 u) in (let TMP_45 \def (Flat f) in (let TMP_46 \def (CHead 
-c1 TMP_45 v) in (let H6 \def (f_equal C T TMP_42 TMP_44 TMP_46 H3) in (let 
-TMP_49 \def (\lambda (_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let TMP_47 
-\def (\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) in 
-(let H9 \def (eq_ind C c0 TMP_47 H2 c1 H8) in (let TMP_48 \def (\lambda (c: 
-C).(wf3 g c c2)) in (let H10 \def (eq_ind C c0 TMP_48 H1 c1 H8) in H10)))))) 
-in (let TMP_50 \def (TMP_49 H5) in (TMP_50 H4)))))))))))))))))))))))))))) in 
-(wf3_ind g TMP_5 TMP_12 TMP_21 TMP_31 TMP_51 y x H0)))))))) in (insert_eq C 
-TMP_2 TMP_3 TMP_4 TMP_52 H))))))))))).
+F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat 
+f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y: 
+C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0: 
+C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m: 
+nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def 
+(eq_ind C (CSort m) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
+True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Flat f) v) H1) in 
+(False_ind (wf3 g c1 (CSort m)) H2)))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) 
+\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u 
+t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat 
+f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda (ee: C).(match 
+ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
+with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead c1 
+(Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2 (Bind b) u)) H5))))))))))) 
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: 
+(((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: 
+T).(\lambda (_: ((\forall (t: T).((ty3 g c0 u t) \to False)))).(\lambda (b: 
+B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 
+\def (eq_ind C (CHead c0 (Bind b) u) (\lambda (ee: C).(match ee with [(CSort 
+_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead c1 (Flat f) v) 
+H4) in (False_ind (wf3 g c1 (CHead c2 (Bind Void) (TSort O))) H5)))))))))) 
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: 
+(((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: 
+T).(\lambda (f0: F).(\lambda (H3: (eq C (CHead c0 (Flat f0) u) (CHead c1 
+(Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort 
+_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) 
+(CHead c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow (match 
+k with [(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 
+(Flat f0) u) (CHead c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow 
+t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda (_: (eq F 
+f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0 (\lambda (c: 
+C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8) in (let H10 
+\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in H10))))) H5)) 
+H4))))))))) y x H0))) H)))))).
 
-theorem wf3_gen_head2:
+lemma wf3_gen_head2:
  \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k: 
 K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
 \def
  \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k: 
-K).(\lambda (H: (wf3 g x (CHead c k v))).(let TMP_1 \def (CHead c k v) in 
-(let TMP_2 \def (\lambda (c0: C).(wf3 g x c0)) in (let TMP_5 \def (\lambda 
-(_: C).(let TMP_4 \def (\lambda (b: B).(let TMP_3 \def (Bind b) in (eq K k 
-TMP_3))) in (ex B TMP_4))) in (let TMP_102 \def (\lambda (y: C).(\lambda (H0: 
-(wf3 g x y)).(let TMP_8 \def (\lambda (_: C).(\lambda (c1: C).((eq C c1 
-(CHead c k v)) \to (let TMP_7 \def (\lambda (b: B).(let TMP_6 \def (Bind b) 
-in (eq K k TMP_6))) in (ex B TMP_7))))) in (let TMP_15 \def (\lambda (m: 
-nat).(\lambda (H1: (eq C (CSort m) (CHead c k v))).(let TMP_9 \def (CSort m) 
-in (let TMP_10 \def (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
-True | (CHead _ _ _) \Rightarrow False])) in (let TMP_11 \def (CHead c k v) 
-in (let H2 \def (eq_ind C TMP_9 TMP_10 I TMP_11 H1) in (let TMP_13 \def 
-(\lambda (b: B).(let TMP_12 \def (Bind b) in (eq K k TMP_12))) in (let TMP_14 
-\def (ex B TMP_13) in (False_ind TMP_14 H2))))))))) in (let TMP_49 \def 
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: 
-(((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind 
-b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u 
-t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c k 
-v))).(let TMP_16 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 
-| (CHead c0 _ _) \Rightarrow c0])) in (let TMP_17 \def (Bind b) in (let 
-TMP_18 \def (CHead c2 TMP_17 u) in (let TMP_19 \def (CHead c k v) in (let H5 
-\def (f_equal C C TMP_16 TMP_18 TMP_19 H4) in (let TMP_20 \def (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow 
-k0])) in (let TMP_21 \def (Bind b) in (let TMP_22 \def (CHead c2 TMP_21 u) in 
-(let TMP_23 \def (CHead c k v) in (let H6 \def (f_equal C K TMP_20 TMP_22 
-TMP_23 H4) in (let TMP_24 \def (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) in (let TMP_25 \def (Bind b) 
-in (let TMP_26 \def (CHead c2 TMP_25 u) in (let TMP_27 \def (CHead c k v) in 
-(let H7 \def (f_equal C T TMP_24 TMP_26 TMP_27 H4) in (let TMP_47 \def 
-(\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let TMP_28 \def 
-(\lambda (t0: T).(ty3 g c1 t0 t)) in (let H10 \def (eq_ind T u TMP_28 H3 v 
-H7) in (let TMP_31 \def (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (let 
-TMP_30 \def (\lambda (b0: B).(let TMP_29 \def (Bind b0) in (eq K k TMP_29))) 
-in (ex B TMP_30)))) in (let H11 \def (eq_ind C c2 TMP_31 H2 c H9) in (let 
-TMP_32 \def (\lambda (c0: C).(wf3 g c1 c0)) in (let H12 \def (eq_ind C c2 
-TMP_32 H1 c H9) in (let TMP_35 \def (\lambda (k0: K).((eq C c (CHead c k0 v)) 
-\to (let TMP_34 \def (\lambda (b0: B).(let TMP_33 \def (Bind b0) in (eq K k0 
-TMP_33))) in (ex B TMP_34)))) in (let TMP_36 \def (Bind b) in (let H13 \def 
-(eq_ind_r K k TMP_35 H11 TMP_36 H8) in (let TMP_37 \def (Bind b) in (let 
-TMP_40 \def (\lambda (k0: K).(let TMP_39 \def (\lambda (b0: B).(let TMP_38 
-\def (Bind b0) in (eq K k0 TMP_38))) in (ex B TMP_39))) in (let TMP_43 \def 
-(\lambda (b0: B).(let TMP_41 \def (Bind b) in (let TMP_42 \def (Bind b0) in 
-(eq K TMP_41 TMP_42)))) in (let TMP_44 \def (Bind b) in (let TMP_45 \def 
-(refl_equal K TMP_44) in (let TMP_46 \def (ex_intro B TMP_43 b TMP_45) in 
-(eq_ind K TMP_37 TMP_40 TMP_46 k H8)))))))))))))))))) in (let TMP_48 \def 
-(TMP_47 H6) in (TMP_48 H5))))))))))))))))))))))))))) in (let TMP_90 \def 
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: 
-(((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind 
-b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to 
-False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void) (TSort 
-O)) (CHead c k v))).(let TMP_50 \def (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) in (let TMP_51 \def (Bind 
-Void) in (let TMP_52 \def (TSort O) in (let TMP_53 \def (CHead c2 TMP_51 
-TMP_52) in (let TMP_54 \def (CHead c k v) in (let H5 \def (f_equal C C TMP_50 
-TMP_53 TMP_54 H4) in (let TMP_55 \def (\lambda (e: C).(match e with [(CSort 
-_) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow k0])) in (let TMP_56 
-\def (Bind Void) in (let TMP_57 \def (TSort O) in (let TMP_58 \def (CHead c2 
-TMP_56 TMP_57) in (let TMP_59 \def (CHead c k v) in (let H6 \def (f_equal C K 
-TMP_55 TMP_58 TMP_59 H4) in (let TMP_60 \def (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) in (let 
-TMP_61 \def (Bind Void) in (let TMP_62 \def (TSort O) in (let TMP_63 \def 
-(CHead c2 TMP_61 TMP_62) in (let TMP_64 \def (CHead c k v) in (let H7 \def 
-(f_equal C T TMP_60 TMP_63 TMP_64 H4) in (let TMP_88 \def (\lambda (H8: (eq K 
-(Bind Void) k)).(\lambda (H9: (eq C c2 c)).(let TMP_67 \def (\lambda (c0: 
-C).((eq C c0 (CHead c k v)) \to (let TMP_66 \def (\lambda (b0: B).(let TMP_65 
-\def (Bind b0) in (eq K k TMP_65))) in (ex B TMP_66)))) in (let H10 \def 
-(eq_ind C c2 TMP_67 H2 c H9) in (let TMP_68 \def (\lambda (c0: C).(wf3 g c1 
-c0)) in (let H11 \def (eq_ind C c2 TMP_68 H1 c H9) in (let TMP_71 \def 
-(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (let TMP_70 \def (\lambda (b0: 
-B).(let TMP_69 \def (Bind b0) in (eq K k0 TMP_69))) in (ex B TMP_70)))) in 
-(let TMP_72 \def (Bind Void) in (let H12 \def (eq_ind_r K k TMP_71 H10 TMP_72 
-H8) in (let TMP_73 \def (Bind Void) in (let TMP_76 \def (\lambda (k0: K).(let 
-TMP_75 \def (\lambda (b0: B).(let TMP_74 \def (Bind b0) in (eq K k0 TMP_74))) 
-in (ex B TMP_75))) in (let TMP_80 \def (\lambda (t: T).((eq C c (CHead c 
-(Bind Void) t)) \to (let TMP_79 \def (\lambda (b0: B).(let TMP_77 \def (Bind 
-Void) in (let TMP_78 \def (Bind b0) in (eq K TMP_77 TMP_78)))) in (ex B 
-TMP_79)))) in (let TMP_81 \def (TSort O) in (let H13 \def (eq_ind_r T v 
-TMP_80 H12 TMP_81 H7) in (let TMP_84 \def (\lambda (b0: B).(let TMP_82 \def 
-(Bind Void) in (let TMP_83 \def (Bind b0) in (eq K TMP_82 TMP_83)))) in (let 
-TMP_85 \def (Bind Void) in (let TMP_86 \def (refl_equal K TMP_85) in (let 
-TMP_87 \def (ex_intro B TMP_84 Void TMP_86) in (eq_ind K TMP_73 TMP_76 TMP_87 
-k H8))))))))))))))))))) in (let TMP_89 \def (TMP_88 H6) in (TMP_89 
-H5))))))))))))))))))))))))))))) in (let TMP_101 \def (\lambda (c1: 
+K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda 
+(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind 
+b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_: 
+C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K 
+k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k 
+v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee with [(CSort 
+_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c k v) H1) 
+in (False_ind (ex B (\lambda (b: B).(eq K k (Bind b)))) H2)))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 
+(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda 
+(H4: (eq C (CHead c2 (Bind b) u) (CHead c k v))).(let H5 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def 
+(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in 
+((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead 
+c k v) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 
+c)).(let H10 \def (eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in 
+(let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex 
+B (\lambda (b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C 
+c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k 
+(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0 
+(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B 
+(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K 
+(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6)) 
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 
+c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K 
+k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u 
+t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void) 
+(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 
+(Bind Void) (TSort O)) (CHead c k v) H4) in ((let H6 \def (f_equal C K 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind Void) | (CHead _ 
+k0 _) \Rightarrow k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in 
+((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind Void) 
+(TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void) k)).(\lambda 
+(H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 
+(CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0)))))) H2 c H9) in 
+(let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let 
+H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B 
+(\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in (eq_ind K 
+(Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind b0))))) 
+(let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind Void) t)) 
+\to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12 (TSort O) H7) 
+in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0))) Void 
+(refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1: 
 C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 
 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_: 
-T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let TMP_91 \def 
-(\lambda (e: C).e) in (let TMP_92 \def (CHead c k v) in (let H4 \def (f_equal 
-C C TMP_91 c2 TMP_92 H3) in (let TMP_95 \def (\lambda (c0: C).((eq C c0 
-(CHead c k v)) \to (let TMP_94 \def (\lambda (b: B).(let TMP_93 \def (Bind b) 
-in (eq K k TMP_93))) in (ex B TMP_94)))) in (let TMP_96 \def (CHead c k v) in 
-(let H5 \def (eq_ind C c2 TMP_95 H2 TMP_96 H4) in (let TMP_97 \def (\lambda 
-(c0: C).(wf3 g c1 c0)) in (let TMP_98 \def (CHead c k v) in (let H6 \def 
-(eq_ind C c2 TMP_97 H1 TMP_98 H4) in (let TMP_99 \def (CHead c k v) in (let 
-TMP_100 \def (refl_equal C TMP_99) in (H5 TMP_100))))))))))))))))))) in 
-(wf3_ind g TMP_8 TMP_15 TMP_49 TMP_90 TMP_101 x y H0)))))))) in (insert_eq C 
-TMP_1 TMP_2 TMP_5 TMP_102 H)))))))))).
+T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def 
+(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind 
+C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq 
+K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda 
+(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k 
+v))))))))))))) x y H0))) H)))))).
 
 theorem wf3_mono:
  \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall 
 (c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
 \def
  \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c 
-c1)).(let TMP_1 \def (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 
-g c0 c3) \to (eq C c2 c3))))) in (let TMP_7 \def (\lambda (m: nat).(\lambda 
-(c2: C).(\lambda (H0: (wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 
-m H0) in (let TMP_2 \def (CSort m) in (let TMP_4 \def (\lambda (c0: C).(let 
-TMP_3 \def (CSort m) in (eq C TMP_3 c0))) in (let TMP_5 \def (CSort m) in 
-(let TMP_6 \def (refl_equal C TMP_5) in (eq_ind_r C TMP_2 TMP_4 TMP_6 c2 
-H_y))))))))) in (let TMP_70 \def (\lambda (c2: C).(\lambda (c3: C).(\lambda 
-(_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C 
-c3 c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u 
+c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0 
+c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0: 
+(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C 
+(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2 
+H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 
+c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 
+c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u 
 t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b) 
 u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in 
-(let TMP_10 \def (\lambda (c4: C).(\lambda (_: T).(let TMP_8 \def (Bind b) in 
-(let TMP_9 \def (CHead c4 TMP_8 u) in (eq C c0 TMP_9))))) in (let TMP_11 \def 
-(\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) in (let TMP_12 \def (\lambda 
-(_: C).(\lambda (w: T).(ty3 g c2 u w))) in (let TMP_13 \def (ex3_2 C T TMP_10 
-TMP_11 TMP_12) in (let TMP_17 \def (\lambda (c4: C).(let TMP_14 \def (Bind 
-Void) in (let TMP_15 \def (TSort O) in (let TMP_16 \def (CHead c4 TMP_14 
-TMP_15) in (eq C c0 TMP_16))))) in (let TMP_18 \def (\lambda (c4: C).(wf3 g 
-c2 c4)) in (let TMP_19 \def (\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) 
-\to False))) in (let TMP_20 \def (ex3 C TMP_17 TMP_18 TMP_19) in (let TMP_21 
-\def (Bind b) in (let TMP_22 \def (CHead c3 TMP_21 u) in (let TMP_23 \def (eq 
-C TMP_22 c0) in (let TMP_45 \def (\lambda (H5: (ex3_2 C T (\lambda (c4: 
+(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind 
+b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: 
+C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead 
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: 
+C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u) 
+c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead 
+c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda 
+(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4: 
 C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: 
 C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 
-u w))))).(let TMP_26 \def (\lambda (c4: C).(\lambda (_: T).(let TMP_24 \def 
-(Bind b) in (let TMP_25 \def (CHead c4 TMP_24 u) in (eq C c0 TMP_25))))) in 
-(let TMP_27 \def (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) in (let 
-TMP_28 \def (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w))) in (let TMP_29 
-\def (Bind b) in (let TMP_30 \def (CHead c3 TMP_29 u) in (let TMP_31 \def (eq 
-C TMP_30 c0) in (let TMP_44 \def (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2 x0)).(\lambda 
-(_: (ty3 g c2 u x1)).(let TMP_32 \def (Bind b) in (let TMP_33 \def (CHead x0 
-TMP_32 u) in (let TMP_36 \def (\lambda (c4: C).(let TMP_34 \def (Bind b) in 
-(let TMP_35 \def (CHead c3 TMP_34 u) in (eq C TMP_35 c4)))) in (let TMP_37 
-\def (Bind b) in (let TMP_38 \def (Bind b) in (let TMP_39 \def (H1 x0 H7) in 
-(let TMP_40 \def (Bind b) in (let TMP_41 \def (refl_equal K TMP_40) in (let 
-TMP_42 \def (refl_equal T u) in (let TMP_43 \def (f_equal3 C K T C CHead c3 
-x0 TMP_37 TMP_38 u u TMP_39 TMP_41 TMP_42) in (eq_ind_r C TMP_33 TMP_36 
-TMP_43 c0 H6)))))))))))))))) in (ex3_2_ind C T TMP_26 TMP_27 TMP_28 TMP_31 
-TMP_44 H5))))))))) in (let TMP_69 \def (\lambda (H5: (ex3 C (\lambda (c4: 
-C).(eq C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 
-c4)) (\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False))))).(let 
-TMP_49 \def (\lambda (c4: C).(let TMP_46 \def (Bind Void) in (let TMP_47 \def 
-(TSort O) in (let TMP_48 \def (CHead c4 TMP_46 TMP_47) in (eq C c0 
-TMP_48))))) in (let TMP_50 \def (\lambda (c4: C).(wf3 g c2 c4)) in (let 
-TMP_51 \def (\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) in 
-(let TMP_52 \def (Bind b) in (let TMP_53 \def (CHead c3 TMP_52 u) in (let 
-TMP_54 \def (eq C TMP_53 c0) in (let TMP_68 \def (\lambda (x0: C).(\lambda 
-(H6: (eq C c0 (CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g c2 
-x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u w) \to False)))).(let TMP_55 
-\def (Bind Void) in (let TMP_56 \def (TSort O) in (let TMP_57 \def (CHead x0 
-TMP_55 TMP_56) in (let TMP_60 \def (\lambda (c4: C).(let TMP_58 \def (Bind b) 
-in (let TMP_59 \def (CHead c3 TMP_58 u) in (eq C TMP_59 c4)))) in (let H_x0 
-\def (H8 t H2) in (let H9 \def H_x0 in (let TMP_61 \def (Bind b) in (let 
-TMP_62 \def (CHead c3 TMP_61 u) in (let TMP_63 \def (Bind Void) in (let 
-TMP_64 \def (TSort O) in (let TMP_65 \def (CHead x0 TMP_63 TMP_64) in (let 
-TMP_66 \def (eq C TMP_62 TMP_65) in (let TMP_67 \def (False_ind TMP_66 H9) in 
-(eq_ind_r C TMP_57 TMP_60 TMP_67 c0 H6)))))))))))))))))) in (ex3_ind C TMP_49 
-TMP_50 TMP_51 TMP_54 TMP_68 H5))))))))) in (or_ind TMP_13 TMP_20 TMP_23 
-TMP_45 TMP_69 H4)))))))))))))))))))))))))) in (let TMP_141 \def (\lambda (c2: 
-C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: 
-C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: 
-((\forall (t: T).((ty3 g c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: 
-C).(\lambda (H3: (wf3 g (CHead c2 (Bind b) u) c0)).(let H_x \def 
-(wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in (let TMP_73 \def 
-(\lambda (c4: C).(\lambda (_: T).(let TMP_71 \def (Bind b) in (let TMP_72 
-\def (CHead c4 TMP_71 u) in (eq C c0 TMP_72))))) in (let TMP_74 \def (\lambda 
-(c4: C).(\lambda (_: T).(wf3 g c2 c4))) in (let TMP_75 \def (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c2 u w))) in (let TMP_76 \def (ex3_2 C T TMP_73 
-TMP_74 TMP_75) in (let TMP_80 \def (\lambda (c4: C).(let TMP_77 \def (Bind 
-Void) in (let TMP_78 \def (TSort O) in (let TMP_79 \def (CHead c4 TMP_77 
-TMP_78) in (eq C c0 TMP_79))))) in (let TMP_81 \def (\lambda (c4: C).(wf3 g 
-c2 c4)) in (let TMP_82 \def (\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) 
-\to False))) in (let TMP_83 \def (ex3 C TMP_80 TMP_81 TMP_82) in (let TMP_84 
-\def (Bind Void) in (let TMP_85 \def (TSort O) in (let TMP_86 \def (CHead c3 
-TMP_84 TMP_85) in (let TMP_87 \def (eq C TMP_86 c0) in (let TMP_111 \def 
-(\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 
+u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2 
+x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda 
+(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind 
+b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0 
+H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind 
+Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall 
+(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 
+(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda 
+(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b) 
+u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort 
+O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u 
+w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4: 
+C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def 
+H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort 
+O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3: 
+C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) 
+\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g 
+c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g 
+(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in 
+(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C 
+c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) 
+(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq 
+C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) 
+(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 
+(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda 
+(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: 
+T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u 
+w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 
 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c2 u w))))).(let TMP_90 \def (\lambda (c4: 
-C).(\lambda (_: T).(let TMP_88 \def (Bind b) in (let TMP_89 \def (CHead c4 
-TMP_88 u) in (eq C c0 TMP_89))))) in (let TMP_91 \def (\lambda (c4: 
-C).(\lambda (_: T).(wf3 g c2 c4))) in (let TMP_92 \def (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c2 u w))) in (let TMP_93 \def (Bind Void) in (let 
-TMP_94 \def (TSort O) in (let TMP_95 \def (CHead c3 TMP_93 TMP_94) in (let 
-TMP_96 \def (eq C TMP_95 c0) in (let TMP_110 \def (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (_: (wf3 g c2 
-x0)).(\lambda (H8: (ty3 g c2 u x1)).(let TMP_97 \def (Bind b) in (let TMP_98 
-\def (CHead x0 TMP_97 u) in (let TMP_102 \def (\lambda (c4: C).(let TMP_99 
-\def (Bind Void) in (let TMP_100 \def (TSort O) in (let TMP_101 \def (CHead 
-c3 TMP_99 TMP_100) in (eq C TMP_101 c4))))) in (let H_x0 \def (H2 x1 H8) in 
-(let H9 \def H_x0 in (let TMP_103 \def (Bind Void) in (let TMP_104 \def 
-(TSort O) in (let TMP_105 \def (CHead c3 TMP_103 TMP_104) in (let TMP_106 
-\def (Bind b) in (let TMP_107 \def (CHead x0 TMP_106 u) in (let TMP_108 \def 
-(eq C TMP_105 TMP_107) in (let TMP_109 \def (False_ind TMP_108 H9) in 
-(eq_ind_r C TMP_98 TMP_102 TMP_109 c0 H6)))))))))))))))))) in (ex3_2_ind C T 
-TMP_90 TMP_91 TMP_92 TMP_96 TMP_110 H5)))))))))) in (let TMP_140 \def 
-(\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void) (TSort 
-O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w: T).((ty3 g 
-c2 u w) \to False))))).(let TMP_115 \def (\lambda (c4: C).(let TMP_112 \def 
-(Bind Void) in (let TMP_113 \def (TSort O) in (let TMP_114 \def (CHead c4 
-TMP_112 TMP_113) in (eq C c0 TMP_114))))) in (let TMP_116 \def (\lambda (c4: 
-C).(wf3 g c2 c4)) in (let TMP_117 \def (\lambda (_: C).(\forall (w: T).((ty3 
-g c2 u w) \to False))) in (let TMP_118 \def (Bind Void) in (let TMP_119 \def 
-(TSort O) in (let TMP_120 \def (CHead c3 TMP_118 TMP_119) in (let TMP_121 
-\def (eq C TMP_120 c0) in (let TMP_139 \def (\lambda (x0: C).(\lambda (H6: 
-(eq C c0 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H7: (wf3 g c2 
-x0)).(\lambda (_: ((\forall (w: T).((ty3 g c2 u w) \to False)))).(let TMP_122 
-\def (Bind Void) in (let TMP_123 \def (TSort O) in (let TMP_124 \def (CHead 
-x0 TMP_122 TMP_123) in (let TMP_128 \def (\lambda (c4: C).(let TMP_125 \def 
-(Bind Void) in (let TMP_126 \def (TSort O) in (let TMP_127 \def (CHead c3 
-TMP_125 TMP_126) in (eq C TMP_127 c4))))) in (let TMP_129 \def (Bind Void) in 
-(let TMP_130 \def (Bind Void) in (let TMP_131 \def (TSort O) in (let TMP_132 
-\def (TSort O) in (let TMP_133 \def (H1 x0 H7) in (let TMP_134 \def (Bind 
-Void) in (let TMP_135 \def (refl_equal K TMP_134) in (let TMP_136 \def (TSort 
-O) in (let TMP_137 \def (refl_equal T TMP_136) in (let TMP_138 \def (f_equal3 
-C K T C CHead c3 x0 TMP_129 TMP_130 TMP_131 TMP_132 TMP_133 TMP_135 TMP_137) 
-in (eq_ind_r C TMP_124 TMP_128 TMP_138 c0 H6))))))))))))))))))) in (ex3_ind C 
-TMP_115 TMP_116 TMP_117 TMP_121 TMP_139 H5)))))))))) in (or_ind TMP_76 TMP_83 
-TMP_87 TMP_111 TMP_140 H4)))))))))))))))))))))))))) in (let TMP_142 \def 
+C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O)) 
+c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind 
+b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r 
+C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort 
+O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C 
+(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6)))))) 
+H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void) 
+(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead 
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: 
+C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void) 
+(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) 
+(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3 
+g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda 
+(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead 
+c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K 
+(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4)))))))))))) 
 (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: 
 ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u: 
 T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f) 
 u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y)))))))))) 
-in (wf3_ind g TMP_1 TMP_7 TMP_70 TMP_141 TMP_142 c c1 H))))))))).
+c c1 H)))).