X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fclear%2Ffwd.ma;h=1b05b45a89c3d9caa3264c5f5d9868c38ad01396;hp=5d602754db201e079e83600273ecf122b7147152;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hpb=d795687ffe924872a5e36122c2bd3069d6409454

diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma
index 5d602754d..1b05b45a8 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma
@@ -18,255 +18,175 @@ include "basic_1/clear/defs.ma".
 
 include "basic_1/C/fwd.ma".
 
-let rec clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b: B).(\forall (e: 
-C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b) u)))))) (f0: 
-(\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to (\forall (f0: 
-F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C) (c0: C) (c1: 
-clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u) \Rightarrow 
-(f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (let TMP_1 \def ((clear_ind 
-P f f0) e c2 c3) in (f0 e c2 c3 TMP_1 f1 u))].
+implied rec lemma clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b: 
+B).(\forall (e: C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b) 
+u)))))) (f0: (\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to 
+(\forall (f0: F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C) 
+(c0: C) (c1: clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u) 
+\Rightarrow (f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (f0 e c2 c3 
+((clear_ind P f f0) e c2 c3) f1 u)].
 
-theorem clear_gen_sort:
+lemma clear_gen_sort:
  \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: 
 Prop).P)))
 \def
  \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda 
-(P: Prop).(let TMP_1 \def (CSort n) in (let TMP_2 \def (\lambda (c: C).(clear 
-c x)) in (let TMP_3 \def (\lambda (_: C).P) in (let TMP_15 \def (\lambda (y: 
-C).(\lambda (H0: (clear y x)).(let TMP_4 \def (\lambda (c: C).(\lambda (_: 
-C).((eq C c (CSort n)) \to P))) in (let TMP_9 \def (\lambda (b: B).(\lambda 
-(e: C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort 
-n))).(let TMP_5 \def (Bind b) in (let TMP_6 \def (CHead e TMP_5 u) in (let 
-TMP_7 \def (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
-(CHead _ _ _) \Rightarrow True])) in (let TMP_8 \def (CSort n) in (let H2 
-\def (eq_ind C TMP_6 TMP_7 I TMP_8 H1) in (False_ind P H2)))))))))) in (let 
-TMP_14 \def (\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e 
-c)).(\lambda (_: (((eq C e (CSort n)) \to P))).(\lambda (f: F).(\lambda (u: 
-T).(\lambda (H3: (eq C (CHead e (Flat f) u) (CSort n))).(let TMP_10 \def 
-(Flat f) in (let TMP_11 \def (CHead e TMP_10 u) in (let TMP_12 \def (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) in (let TMP_13 \def (CSort n) in (let H4 \def (eq_ind C 
-TMP_11 TMP_12 I TMP_13 H3) in (False_ind P H4))))))))))))) in (clear_ind 
-TMP_4 TMP_9 TMP_14 y x H0)))))) in (insert_eq C TMP_1 TMP_2 TMP_3 TMP_15 
-H)))))))).
+(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_: 
+C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c: 
+C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let 
+H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) 
+H1) in (False_ind P H2)))))) (\lambda (e: C).(\lambda (c: C).(\lambda (_: 
+(clear e c)).(\lambda (_: (((eq C e (CSort n)) \to P))).(\lambda (f: 
+F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e (Flat f) u) (CSort n))).(let 
+H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) 
+H3) in (False_ind P H4))))))))) y x H0))) H)))).
 
-theorem clear_gen_bind:
+lemma clear_gen_bind:
  \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear 
 (CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
 \def
  \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: 
-(clear (CHead e (Bind b) u) x)).(let TMP_1 \def (Bind b) in (let TMP_2 \def 
-(CHead e TMP_1 u) in (let TMP_3 \def (\lambda (c: C).(clear c x)) in (let 
-TMP_4 \def (\lambda (c: C).(eq C x c)) in (let TMP_53 \def (\lambda (y: 
-C).(\lambda (H0: (clear y x)).(let TMP_5 \def (\lambda (c: C).(\lambda (c0: 
-C).((eq C c (CHead e (Bind b) u)) \to (eq C c0 c)))) in (let TMP_43 \def 
-(\lambda (b0: B).(\lambda (e0: C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead 
-e0 (Bind b0) u0) (CHead e (Bind b) u))).(let TMP_6 \def (\lambda (e1: 
-C).(match e1 with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) 
-in (let TMP_7 \def (Bind b0) in (let TMP_8 \def (CHead e0 TMP_7 u0) in (let 
-TMP_9 \def (Bind b) in (let TMP_10 \def (CHead e TMP_9 u) in (let H2 \def 
-(f_equal C C TMP_6 TMP_8 TMP_10 H1) in (let TMP_11 \def (\lambda (e1: 
+(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda 
+(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: 
+(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e 
+(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) 
+u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 with [(CSort _) 
+\Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind b0) u0) 
+(CHead e (Bind b) u) H1) in ((let H3 \def (f_equal C B (\lambda (e1: 
 C).(match e1 with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) in 
-(let TMP_12 \def (Bind b0) in (let TMP_13 \def (CHead e0 TMP_12 u0) in (let 
-TMP_14 \def (Bind b) in (let TMP_15 \def (CHead e TMP_14 u) in (let H3 \def 
-(f_equal C B TMP_11 TMP_13 TMP_15 H1) in (let TMP_16 \def (\lambda (e1: 
-C).(match e1 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
-in (let TMP_17 \def (Bind b0) in (let TMP_18 \def (CHead e0 TMP_17 u0) in 
-(let TMP_19 \def (Bind b) in (let TMP_20 \def (CHead e TMP_19 u) in (let H4 
-\def (f_equal C T TMP_16 TMP_18 TMP_20 H1) in (let TMP_41 \def (\lambda (H5: 
-(eq B b0 b)).(\lambda (H6: (eq C e0 e)).(let TMP_25 \def (\lambda (t: T).(let 
-TMP_21 \def (Bind b0) in (let TMP_22 \def (CHead e0 TMP_21 t) in (let TMP_23 
-\def (Bind b0) in (let TMP_24 \def (CHead e0 TMP_23 t) in (eq C TMP_22 
-TMP_24)))))) in (let TMP_30 \def (\lambda (c: C).(let TMP_26 \def (Bind b0) 
-in (let TMP_27 \def (CHead c TMP_26 u) in (let TMP_28 \def (Bind b0) in (let 
-TMP_29 \def (CHead c TMP_28 u) in (eq C TMP_27 TMP_29)))))) in (let TMP_35 
-\def (\lambda (b1: B).(let TMP_31 \def (Bind b1) in (let TMP_32 \def (CHead e 
-TMP_31 u) in (let TMP_33 \def (Bind b1) in (let TMP_34 \def (CHead e TMP_33 
-u) in (eq C TMP_32 TMP_34)))))) in (let TMP_36 \def (Bind b) in (let TMP_37 
-\def (CHead e TMP_36 u) in (let TMP_38 \def (refl_equal C TMP_37) in (let 
-TMP_39 \def (eq_ind_r B b TMP_35 TMP_38 b0 H5) in (let TMP_40 \def (eq_ind_r 
-C e TMP_30 TMP_39 e0 H6) in (eq_ind_r T u TMP_25 TMP_40 u0 H4))))))))))) in 
-(let TMP_42 \def (TMP_41 H3) in (TMP_42 H2))))))))))))))))))))))))) in (let 
-TMP_52 \def (\lambda (e0: C).(\lambda (c: C).(\lambda (_: (clear e0 
-c)).(\lambda (_: (((eq C e0 (CHead e (Bind b) u)) \to (eq C c e0)))).(\lambda 
-(f: F).(\lambda (u0: T).(\lambda (H3: (eq C (CHead e0 (Flat f) u0) (CHead e 
-(Bind b) u))).(let TMP_44 \def (Flat f) in (let TMP_45 \def (CHead e0 TMP_44 
-u0) in (let TMP_46 \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_47 \def (Bind 
-b) in (let TMP_48 \def (CHead e TMP_47 u) in (let H4 \def (eq_ind C TMP_45 
-TMP_46 I TMP_48 H3) in (let TMP_49 \def (Flat f) in (let TMP_50 \def (CHead 
-e0 TMP_49 u0) in (let TMP_51 \def (eq C c TMP_50) in (False_ind TMP_51 
-H4))))))))))))))))) in (clear_ind TMP_5 TMP_43 TMP_52 y x H0)))))) in 
-(insert_eq C TMP_2 TMP_3 TMP_4 TMP_53 H)))))))))).
+(match k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead 
+e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def (f_equal C T 
+(\lambda (e1: C).(match e1 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
+\Rightarrow t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in (\lambda 
+(H5: (eq B b0 b)).(\lambda (H6: (eq C e0 e)).(eq_ind_r T u (\lambda (t: 
+T).(eq C (CHead e0 (Bind b0) t) (CHead e0 (Bind b0) t))) (eq_ind_r C e 
+(\lambda (c: C).(eq C (CHead c (Bind b0) u) (CHead c (Bind b0) u))) (eq_ind_r 
+B b (\lambda (b1: B).(eq C (CHead e (Bind b1) u) (CHead e (Bind b1) u))) 
+(refl_equal C (CHead e (Bind b) u)) b0 H5) e0 H6) u0 H4)))) H3)) H2)))))) 
+(\lambda (e0: C).(\lambda (c: C).(\lambda (_: (clear e0 c)).(\lambda (_: 
+(((eq C e0 (CHead e (Bind b) u)) \to (eq C c e0)))).(\lambda (f: F).(\lambda 
+(u0: T).(\lambda (H3: (eq C (CHead e0 (Flat f) u0) (CHead e (Bind b) 
+u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead e (Bind 
+b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) u0)) H4))))))))) y x H0))) 
+H))))).
 
-theorem clear_gen_flat:
+lemma clear_gen_flat:
  \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear 
 (CHead e (Flat f) u) x) \to (clear e x)))))
 \def
  \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: 
-(clear (CHead e (Flat f) u) x)).(let TMP_1 \def (Flat f) in (let TMP_2 \def 
-(CHead e TMP_1 u) in (let TMP_3 \def (\lambda (c: C).(clear c x)) in (let 
-TMP_4 \def (\lambda (_: C).(clear e x)) in (let TMP_35 \def (\lambda (y: 
-C).(\lambda (H0: (clear y x)).(let TMP_5 \def (\lambda (c: C).(\lambda (c0: 
-C).((eq C c (CHead e (Flat f) u)) \to (clear e c0)))) in (let TMP_14 \def 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead 
-e0 (Bind b) u0) (CHead e (Flat f) u))).(let TMP_6 \def (Bind b) in (let TMP_7 
-\def (CHead e0 TMP_6 u0) in (let TMP_8 \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_9 \def 
-(Flat f) in (let TMP_10 \def (CHead e TMP_9 u) in (let H2 \def (eq_ind C 
-TMP_7 TMP_8 I TMP_10 H1) in (let TMP_11 \def (Bind b) in (let TMP_12 \def 
-(CHead e0 TMP_11 u0) in (let TMP_13 \def (clear e TMP_12) in (False_ind 
-TMP_13 H2)))))))))))))) in (let TMP_34 \def (\lambda (e0: C).(\lambda (c: 
-C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0 (CHead e (Flat f) u)) 
-\to (clear e c)))).(\lambda (f0: F).(\lambda (u0: T).(\lambda (H3: (eq C 
-(CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let TMP_15 \def (\lambda (e1: 
-C).(match e1 with [(CSort _) \Rightarrow e0 | (CHead c0 _ _) \Rightarrow 
-c0])) in (let TMP_16 \def (Flat f0) in (let TMP_17 \def (CHead e0 TMP_16 u0) 
-in (let TMP_18 \def (Flat f) in (let TMP_19 \def (CHead e TMP_18 u) in (let 
-H4 \def (f_equal C C TMP_15 TMP_17 TMP_19 H3) in (let TMP_20 \def (\lambda 
-(e1: C).(match e1 with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) in 
-(let TMP_21 \def (Flat f0) in (let TMP_22 \def (CHead e0 TMP_21 u0) in (let 
-TMP_23 \def (Flat f) in (let TMP_24 \def (CHead e TMP_23 u) in (let H5 \def 
-(f_equal C F TMP_20 TMP_22 TMP_24 H3) in (let TMP_25 \def (\lambda (e1: 
-C).(match e1 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
-in (let TMP_26 \def (Flat f0) in (let TMP_27 \def (CHead e0 TMP_26 u0) in 
-(let TMP_28 \def (Flat f) in (let TMP_29 \def (CHead e TMP_28 u) in (let H6 
-\def (f_equal C T TMP_25 TMP_27 TMP_29 H3) in (let TMP_32 \def (\lambda (_: 
-(eq F f0 f)).(\lambda (H8: (eq C e0 e)).(let TMP_30 \def (\lambda (c0: 
-C).((eq C c0 (CHead e (Flat f) u)) \to (clear e c))) in (let H9 \def (eq_ind 
-C e0 TMP_30 H2 e H8) in (let TMP_31 \def (\lambda (c0: C).(clear c0 c)) in 
-(let H10 \def (eq_ind C e0 TMP_31 H1 e H8) in H10)))))) in (let TMP_33 \def 
-(TMP_32 H5) in (TMP_33 H4)))))))))))))))))))))))))))) in (clear_ind TMP_5 
-TMP_14 TMP_34 y x H0)))))) in (insert_eq C TMP_2 TMP_3 TMP_4 TMP_35 
-H)))))))))).
+(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda 
+(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda 
+(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead 
+e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) 
+u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat 
+f) u) H1) in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda 
+(e0: C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0 
+(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0: 
+T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4 
+\def (f_equal C C (\lambda (e1: C).(match e1 with [(CSort _) \Rightarrow e0 | 
+(CHead c0 _ _) \Rightarrow c0])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) 
+H3) in ((let H5 \def (f_equal C F (\lambda (e1: C).(match e1 with [(CSort _) 
+\Rightarrow f0 | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0) u0) (CHead 
+e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1: C).(match e1 
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 
+(Flat f0) u0) (CHead e (Flat f) u) H3) in (\lambda (_: (eq F f0 f)).(\lambda 
+(H8: (eq C e0 e)).(let H9 \def (eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead 
+e (Flat f) u)) \to (clear e c))) H2 e H8) in (let H10 \def (eq_ind C e0 
+(\lambda (c0: C).(clear c0 c)) H1 e H8) in H10))))) H5)) H4))))))))) y x 
+H0))) H))))).
 
-theorem clear_gen_flat_r:
+lemma clear_gen_flat_r:
  \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x 
 (CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
 \def
  \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H: 
-(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(let TMP_1 \def (Flat f) 
-in (let TMP_2 \def (CHead e TMP_1 u) in (let TMP_3 \def (\lambda (c: 
-C).(clear x c)) in (let TMP_4 \def (\lambda (_: C).P) in (let TMP_22 \def 
-(\lambda (y: C).(\lambda (H0: (clear x y)).(let TMP_5 \def (\lambda (_: 
-C).(\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to P))) in (let TMP_11 
-\def (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(\lambda (H1: (eq C 
-(CHead e0 (Bind b) u0) (CHead e (Flat f) u))).(let TMP_6 \def (Bind b) in 
-(let TMP_7 \def (CHead e0 TMP_6 u0) in (let TMP_8 \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_9 \def (Flat f) in (let TMP_10 \def (CHead e TMP_9 u) in (let H2 
-\def (eq_ind C TMP_7 TMP_8 I TMP_10 H1) in (False_ind P H2))))))))))) in (let 
-TMP_21 \def (\lambda (e0: C).(\lambda (c: C).(\lambda (H1: (clear e0 
-c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to P))).(\lambda (_: 
-F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f) u))).(let TMP_12 
-\def (\lambda (c0: C).((eq C c0 (CHead e (Flat f) u)) \to P)) in (let TMP_13 
-\def (Flat f) in (let TMP_14 \def (CHead e TMP_13 u) in (let H4 \def (eq_ind 
-C c TMP_12 H2 TMP_14 H3) in (let TMP_15 \def (\lambda (c0: C).(clear e0 c0)) 
-in (let TMP_16 \def (Flat f) in (let TMP_17 \def (CHead e TMP_16 u) in (let 
-H5 \def (eq_ind C c TMP_15 H1 TMP_17 H3) in (let TMP_18 \def (Flat f) in (let 
-TMP_19 \def (CHead e TMP_18 u) in (let TMP_20 \def (refl_equal C TMP_19) in 
-(H4 TMP_20))))))))))))))))))) in (clear_ind TMP_5 TMP_11 TMP_21 x y H0)))))) 
-in (insert_eq C TMP_2 TMP_3 TMP_4 TMP_22 H))))))))))).
+(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e 
+(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y: 
+C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: 
+C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat 
+f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match 
+ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
+with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e 
+(Flat f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: 
+C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) 
+\to P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat 
+f) u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f) 
+u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda 
+(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C 
+(CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
 
-theorem clear_gen_all:
+lemma clear_gen_all:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: 
 B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(let TMP_4 \def 
-(\lambda (_: C).(\lambda (c0: C).(let TMP_3 \def (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u: T).(let TMP_1 \def (Bind b) in (let TMP_2 \def (CHead e TMP_1 
-u) in (eq C c0 TMP_2)))))) in (ex_3 B C T TMP_3)))) in (let TMP_13 \def 
-(\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(let TMP_9 \def (\lambda (b0: 
-B).(\lambda (e0: C).(\lambda (u0: T).(let TMP_5 \def (Bind b) in (let TMP_6 
-\def (CHead e TMP_5 u) in (let TMP_7 \def (Bind b0) in (let TMP_8 \def (CHead 
-e0 TMP_7 u0) in (eq C TMP_6 TMP_8)))))))) in (let TMP_10 \def (Bind b) in 
-(let TMP_11 \def (CHead e TMP_10 u) in (let TMP_12 \def (refl_equal C TMP_11) 
-in (ex_3_intro B C T TMP_9 b e u TMP_12)))))))) in (let TMP_40 \def (\lambda 
-(e: C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T 
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind 
+(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0: 
+B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0 
+(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e: 
+C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T 
 (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b) 
-u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (let TMP_16 \def 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(let TMP_14 \def (Bind b) 
-in (let TMP_15 \def (CHead e0 TMP_14 u0) in (eq C c TMP_15)))))) in (let 
-TMP_19 \def (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(let TMP_17 
-\def (Bind b) in (let TMP_18 \def (CHead e0 TMP_17 u0) in (eq C c 
-TMP_18)))))) in (let TMP_20 \def (ex_3 B C T TMP_19) in (let TMP_39 \def 
-(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C c 
-(CHead x1 (Bind x0) x2))).(let TMP_21 \def (\lambda (c0: C).(clear e c0)) in 
-(let TMP_22 \def (Bind x0) in (let TMP_23 \def (CHead x1 TMP_22 x2) in (let 
-H4 \def (eq_ind C c TMP_21 H0 TMP_23 H3) in (let TMP_24 \def (Bind x0) in 
-(let TMP_25 \def (CHead x1 TMP_24 x2) in (let TMP_29 \def (\lambda (c0: 
-C).(let TMP_28 \def (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(let 
-TMP_26 \def (Bind b) in (let TMP_27 \def (CHead e0 TMP_26 u0) in (eq C c0 
-TMP_27)))))) in (ex_3 B C T TMP_28))) in (let TMP_34 \def (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u0: T).(let TMP_30 \def (Bind x0) in (let 
-TMP_31 \def (CHead x1 TMP_30 x2) in (let TMP_32 \def (Bind b) in (let TMP_33 
-\def (CHead e0 TMP_32 u0) in (eq C TMP_31 TMP_33)))))))) in (let TMP_35 \def 
-(Bind x0) in (let TMP_36 \def (CHead x1 TMP_35 x2) in (let TMP_37 \def 
-(refl_equal C TMP_36) in (let TMP_38 \def (ex_3_intro B C T TMP_34 x0 x1 x2 
-TMP_37) in (eq_ind_r C TMP_25 TMP_29 TMP_38 c H3))))))))))))))))) in 
-(ex_3_ind B C T TMP_16 TMP_20 TMP_39 H2)))))))))))) in (clear_ind TMP_4 
-TMP_13 TMP_40 c1 c2 H)))))).
+u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b) 
+u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c 
+(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c 
+(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C 
+(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B 
+C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind 
+x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) 
+x2))) c H3)))))) H2)))))))) c1 c2 H))).
 
 theorem clear_mono:
  \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c 
 c2) \to (eq C c1 c2)))))
 \def
- \lambda (c: C).(let TMP_1 \def (\lambda (c0: C).(\forall (c1: C).((clear c0 
-c1) \to (\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) in (let TMP_3 
-\def (\lambda (n: nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) 
-c1)).(\lambda (c2: C).(\lambda (H0: (clear (CSort n) c2)).(let TMP_2 \def (eq 
-C c1 c2) in (clear_gen_sort c2 n H0 TMP_2))))))) in (let TMP_23 \def (\lambda 
-(c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to (\forall (c2: 
-C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: K).(\lambda (t: 
-T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) c1)).(\lambda (c2: 
-C).(\lambda (H1: (clear (CHead c0 k t) c2)).(let TMP_4 \def (\lambda (k0: 
-K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) \to (eq C c1 
-c2)))) in (let TMP_19 \def (\lambda (b: B).(\lambda (H2: (clear (CHead c0 
-(Bind b) t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(let TMP_5 
-\def (Bind b) in (let TMP_6 \def (CHead c0 TMP_5 t) in (let TMP_7 \def 
-(\lambda (c3: C).(eq C c1 c3)) in (let TMP_8 \def (Bind b) in (let TMP_9 \def 
-(CHead c0 TMP_8 t) in (let TMP_12 \def (\lambda (c3: C).(let TMP_10 \def 
-(Bind b) in (let TMP_11 \def (CHead c0 TMP_10 t) in (eq C c3 TMP_11)))) in 
-(let TMP_13 \def (Bind b) in (let TMP_14 \def (CHead c0 TMP_13 t) in (let 
-TMP_15 \def (refl_equal C TMP_14) in (let TMP_16 \def (clear_gen_bind b c0 c1 
-t H2) in (let TMP_17 \def (eq_ind_r C TMP_9 TMP_12 TMP_15 c1 TMP_16) in (let 
-TMP_18 \def (clear_gen_bind b c0 c2 t H3) in (eq_ind_r C TMP_6 TMP_7 TMP_17 
-c2 TMP_18)))))))))))))))) in (let TMP_22 \def (\lambda (f: F).(\lambda (H2: 
-(clear (CHead c0 (Flat f) t) c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) 
-c2)).(let TMP_20 \def (clear_gen_flat f c0 c1 t H2) in (let TMP_21 \def 
-(clear_gen_flat f c0 c2 t H3) in (H c1 TMP_20 c2 TMP_21)))))) in (K_ind TMP_4 
-TMP_19 TMP_22 k H0 H1)))))))))))) in (C_ind TMP_1 TMP_3 TMP_23 c)))).
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to 
+(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: 
+nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: 
+C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 
+c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to 
+(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) 
+c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind 
+(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) 
+\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b) 
+t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 
+(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) 
+(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 
+(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t 
+H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) 
+c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f 
+c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
 
-theorem clear_cle:
+lemma clear_cle:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
 \def
- \lambda (c1: C).(let TMP_3 \def (\lambda (c: C).(\forall (c2: C).((clear c 
-c2) \to (let TMP_1 \def (cweight c2) in (let TMP_2 \def (cweight c) in (le 
-TMP_1 TMP_2)))))) in (let TMP_6 \def (\lambda (n: nat).(\lambda (c2: 
-C).(\lambda (H: (clear (CSort n) c2)).(let TMP_4 \def (cweight c2) in (let 
-TMP_5 \def (le TMP_4 O) in (clear_gen_sort c2 n H TMP_5)))))) in (let TMP_31 
-\def (\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le 
-(cweight c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: 
-C).(\lambda (H0: (clear (CHead c k t) c2)).(let TMP_11 \def (\lambda (k0: 
-K).((clear (CHead c k0 t) c2) \to (let TMP_7 \def (cweight c2) in (let TMP_8 
-\def (cweight c) in (let TMP_9 \def (tweight t) in (let TMP_10 \def (plus 
-TMP_8 TMP_9) in (le TMP_7 TMP_10))))))) in (let TMP_24 \def (\lambda (b: 
-B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(let TMP_12 \def (Bind b) 
-in (let TMP_13 \def (CHead c TMP_12 t) in (let TMP_18 \def (\lambda (c0: 
-C).(let TMP_14 \def (cweight c0) in (let TMP_15 \def (cweight c) in (let 
-TMP_16 \def (tweight t) in (let TMP_17 \def (plus TMP_15 TMP_16) in (le 
-TMP_14 TMP_17)))))) in (let TMP_19 \def (cweight c) in (let TMP_20 \def 
-(tweight t) in (let TMP_21 \def (plus TMP_19 TMP_20) in (let TMP_22 \def 
-(le_n TMP_21) in (let TMP_23 \def (clear_gen_bind b c c2 t H1) in (eq_ind_r C 
-TMP_13 TMP_18 TMP_22 c2 TMP_23))))))))))) in (let TMP_30 \def (\lambda (f: 
-F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(let TMP_25 \def (cweight 
-c2) in (let TMP_26 \def (cweight c) in (let TMP_27 \def (tweight t) in (let 
-TMP_28 \def (clear_gen_flat f c c2 t H1) in (let TMP_29 \def (H c2 TMP_28) in 
-(le_plus_trans TMP_25 TMP_26 TMP_27 TMP_29)))))))) in (K_ind TMP_11 TMP_24 
-TMP_30 k H0)))))))))) in (C_ind TMP_3 TMP_6 TMP_31 c1)))).
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to 
+(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda 
+(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight 
+c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: 
+C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear 
+(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) 
+(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C 
+(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) 
+(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c 
+c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) 
+c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 
+(clear_gen_flat f c c2 t H1))))) k H0))))))) c1).