X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fwcpr0%2Ffwd.ma;h=8d4ccb52e5bc01e2271b3ac9c5f9bd79ed43916c;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=deda207baf6bf3e0ed2d7c80e8864c5e4e8ee5a4;hpb=40e0eed5dc1a5f56f089f491841a5077723b891a;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma
index deda207ba..8d4ccb52e 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma
@@ -16,139 +16,90 @@
 
 include "basic_1/wcpr0/defs.ma".
 
-let rec wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P c c))) 
-(f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) \to 
-(\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P (CHead 
-c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on w: P c 
-c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp c1 c2 
-w0 u1 u2 p k) \Rightarrow (let TMP_1 \def ((wcpr0_ind P f f0) c1 c2 w0) in 
-(f0 c1 c2 w0 TMP_1 u1 u2 p k))].
+implied rec lemma wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P 
+c c))) (f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) 
+\to (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P 
+(CHead c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on 
+w: P c c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp 
+c1 c2 w0 u1 u2 p k) \Rightarrow (f0 c1 c2 w0 ((wcpr0_ind P f f0) c1 c2 w0) u1 
+u2 p k)].
 
-theorem wcpr0_gen_sort:
+lemma wcpr0_gen_sort:
  \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort 
 n))))
 \def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let 
-TMP_1 \def (CSort n) in (let TMP_2 \def (\lambda (c: C).(wcpr0 c x)) in (let 
-TMP_3 \def (\lambda (c: C).(eq C x c)) in (let TMP_19 \def (\lambda (y: 
-C).(\lambda (H0: (wcpr0 y x)).(let TMP_4 \def (\lambda (c: C).(\lambda (c0: 
-C).((eq C c (CSort n)) \to (eq C c0 c)))) in (let TMP_11 \def (\lambda (c: 
-C).(\lambda (H1: (eq C c (CSort n))).(let TMP_5 \def (\lambda (e: C).e) in 
-(let TMP_6 \def (CSort n) in (let H2 \def (f_equal C C TMP_5 c TMP_6 H1) in 
-(let TMP_7 \def (CSort n) in (let TMP_8 \def (\lambda (c0: C).(eq C c0 c0)) 
-in (let TMP_9 \def (CSort n) in (let TMP_10 \def (refl_equal C TMP_9) in 
-(eq_ind_r C TMP_7 TMP_8 TMP_10 c H2)))))))))) in (let TMP_18 \def (\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 
-(CSort n)) \to (eq C c2 c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: 
-(pr0 u1 u2)).(\lambda (k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort 
-n))).(let TMP_12 \def (CHead c1 k u1) in (let TMP_13 \def (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
-True])) in (let TMP_14 \def (CSort n) in (let H5 \def (eq_ind C TMP_12 TMP_13 
-I TMP_14 H4) in (let TMP_15 \def (CHead c2 k u2) in (let TMP_16 \def (CHead 
-c1 k u1) in (let TMP_17 \def (eq C TMP_15 TMP_16) in (False_ind TMP_17 
-H5))))))))))))))))) in (wcpr0_ind TMP_4 TMP_11 TMP_18 y x H0)))))) in 
-(insert_eq C TMP_1 TMP_2 TMP_3 TMP_19 H))))))).
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) 
+x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C 
+x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: 
+C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: 
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: 
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) 
+(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 
+c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C 
+(CHead c1 k u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False 
+| (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C 
+(CHead c2 k u2) (CHead c1 k u1)) H5))))))))))) y x H0))) H))).
 
-theorem wcpr0_gen_head:
+lemma wcpr0_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 
 (CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: 
 C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
 T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
 \def
  \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
-(H: (wcpr0 (CHead c1 k u1) x)).(let TMP_1 \def (CHead c1 k u1) in (let TMP_2 
-\def (\lambda (c: C).(wcpr0 c x)) in (let TMP_9 \def (\lambda (c: C).(let 
-TMP_3 \def (eq C x c) in (let TMP_5 \def (\lambda (c2: C).(\lambda (u2: 
-T).(let TMP_4 \def (CHead c2 k u2) in (eq C x TMP_4)))) in (let TMP_6 \def 
-(\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) in (let TMP_7 \def (\lambda 
-(_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_8 \def (ex3_2 C T TMP_5 
-TMP_6 TMP_7) in (or TMP_3 TMP_8))))))) in (let TMP_111 \def (\lambda (y: 
-C).(\lambda (H0: (wcpr0 y x)).(let TMP_16 \def (\lambda (c: C).(\lambda (c0: 
-C).((eq C c (CHead c1 k u1)) \to (let TMP_10 \def (eq C c0 c) in (let TMP_12 
-\def (\lambda (c2: C).(\lambda (u2: T).(let TMP_11 \def (CHead c2 k u2) in 
-(eq C c0 TMP_11)))) in (let TMP_13 \def (\lambda (c2: C).(\lambda (_: 
-T).(wcpr0 c1 c2))) in (let TMP_14 \def (\lambda (_: C).(\lambda (u2: T).(pr0 
-u1 u2))) in (let TMP_15 \def (ex3_2 C T TMP_12 TMP_13 TMP_14) in (or TMP_10 
-TMP_15))))))))) in (let TMP_39 \def (\lambda (c: C).(\lambda (H1: (eq C c 
-(CHead c1 k u1))).(let TMP_17 \def (\lambda (e: C).e) in (let TMP_18 \def 
-(CHead c1 k u1) in (let H2 \def (f_equal C C TMP_17 c TMP_18 H1) in (let 
-TMP_19 \def (CHead c1 k u1) in (let TMP_26 \def (\lambda (c0: C).(let TMP_20 
-\def (eq C c0 c0) in (let TMP_22 \def (\lambda (c2: C).(\lambda (u2: T).(let 
-TMP_21 \def (CHead c2 k u2) in (eq C c0 TMP_21)))) in (let TMP_23 \def 
-(\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) in (let TMP_24 \def (\lambda 
-(_: C).(\lambda (u2: T).(pr0 u1 u2))) in (let TMP_25 \def (ex3_2 C T TMP_22 
-TMP_23 TMP_24) in (or TMP_20 TMP_25))))))) in (let TMP_27 \def (CHead c1 k 
-u1) in (let TMP_28 \def (CHead c1 k u1) in (let TMP_29 \def (eq C TMP_27 
-TMP_28) in (let TMP_32 \def (\lambda (c2: C).(\lambda (u2: T).(let TMP_30 
-\def (CHead c1 k u1) in (let TMP_31 \def (CHead c2 k u2) in (eq C TMP_30 
-TMP_31))))) in (let TMP_33 \def (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 
-c2))) in (let TMP_34 \def (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) in 
-(let TMP_35 \def (ex3_2 C T TMP_32 TMP_33 TMP_34) in (let TMP_36 \def (CHead 
-c1 k u1) in (let TMP_37 \def (refl_equal C TMP_36) in (let TMP_38 \def 
-(or_introl TMP_29 TMP_35 TMP_37) in (eq_ind_r C TMP_19 TMP_26 TMP_38 c 
-H2)))))))))))))))))) in (let TMP_110 \def (\lambda (c0: C).(\lambda (c2: 
+(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: 
+C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: 
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda 
+(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: 
+C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: 
+C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) 
+(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal 
+C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) 
+(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: 
+T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 
+c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C 
+(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: 
+T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) 
+(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: 
 C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to 
 (or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead 
 c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: 
 C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: 
 T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 
-k0 u0) (CHead c1 k u1))).(let TMP_40 \def (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) in (let TMP_41 
-\def (CHead c0 k0 u0) in (let TMP_42 \def (CHead c1 k u1) in (let H5 \def 
-(f_equal C C TMP_40 TMP_41 TMP_42 H4) in (let TMP_43 \def (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) 
-in (let TMP_44 \def (CHead c0 k0 u0) in (let TMP_45 \def (CHead c1 k u1) in 
-(let H6 \def (f_equal C K TMP_43 TMP_44 TMP_45 H4) in (let TMP_46 \def 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) in (let TMP_47 \def (CHead c0 k0 u0) in (let TMP_48 \def 
-(CHead c1 k u1) in (let H7 \def (f_equal C T TMP_46 TMP_47 TMP_48 H4) in (let 
-TMP_108 \def (\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 c1)).(let 
-TMP_58 \def (\lambda (k1: K).(let TMP_49 \def (CHead c2 k1 u2) in (let TMP_50 
-\def (CHead c0 k1 u0) in (let TMP_51 \def (eq C TMP_49 TMP_50) in (let TMP_54 
-\def (\lambda (c3: C).(\lambda (u3: T).(let TMP_52 \def (CHead c2 k1 u2) in 
-(let TMP_53 \def (CHead c3 k u3) in (eq C TMP_52 TMP_53))))) in (let TMP_55 
-\def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_56 \def 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_57 \def (ex3_2 C T 
-TMP_54 TMP_55 TMP_56) in (or TMP_51 TMP_57))))))))) in (let TMP_59 \def 
-(\lambda (t: T).(pr0 t u2)) in (let H10 \def (eq_ind T u0 TMP_59 H3 u1 H7) in 
-(let TMP_69 \def (\lambda (t: T).(let TMP_60 \def (CHead c2 k u2) in (let 
-TMP_61 \def (CHead c0 k t) in (let TMP_62 \def (eq C TMP_60 TMP_61) in (let 
-TMP_65 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_63 \def (CHead c2 k 
-u2) in (let TMP_64 \def (CHead c3 k u3) in (eq C TMP_63 TMP_64))))) in (let 
-TMP_66 \def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_67 
-\def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_68 \def 
-(ex3_2 C T TMP_65 TMP_66 TMP_67) in (or TMP_62 TMP_68))))))))) in (let TMP_76 
-\def (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (let TMP_70 \def (eq C c2 
-c) in (let TMP_72 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_71 \def 
-(CHead c3 k u3) in (eq C c2 TMP_71)))) in (let TMP_73 \def (\lambda (c3: 
-C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_74 \def (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_75 \def (ex3_2 C T TMP_72 
-TMP_73 TMP_74) in (or TMP_70 TMP_75)))))))) in (let H11 \def (eq_ind C c0 
-TMP_76 H2 c1 H9) in (let TMP_77 \def (\lambda (c: C).(wcpr0 c c2)) in (let 
-H12 \def (eq_ind C c0 TMP_77 H1 c1 H9) in (let TMP_87 \def (\lambda (c: 
-C).(let TMP_78 \def (CHead c2 k u2) in (let TMP_79 \def (CHead c k u1) in 
-(let TMP_80 \def (eq C TMP_78 TMP_79) in (let TMP_83 \def (\lambda (c3: 
-C).(\lambda (u3: T).(let TMP_81 \def (CHead c2 k u2) in (let TMP_82 \def 
-(CHead c3 k u3) in (eq C TMP_81 TMP_82))))) in (let TMP_84 \def (\lambda (c3: 
-C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_85 \def (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_86 \def (ex3_2 C T TMP_83 
-TMP_84 TMP_85) in (or TMP_80 TMP_86))))))))) in (let TMP_88 \def (CHead c2 k 
-u2) in (let TMP_89 \def (CHead c1 k u1) in (let TMP_90 \def (eq C TMP_88 
-TMP_89) in (let TMP_93 \def (\lambda (c3: C).(\lambda (u3: T).(let TMP_91 
-\def (CHead c2 k u2) in (let TMP_92 \def (CHead c3 k u3) in (eq C TMP_91 
-TMP_92))))) in (let TMP_94 \def (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 
-c3))) in (let TMP_95 \def (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in 
-(let TMP_96 \def (ex3_2 C T TMP_93 TMP_94 TMP_95) in (let TMP_99 \def 
-(\lambda (c3: C).(\lambda (u3: T).(let TMP_97 \def (CHead c2 k u2) in (let 
-TMP_98 \def (CHead c3 k u3) in (eq C TMP_97 TMP_98))))) in (let TMP_100 \def 
-(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) in (let TMP_101 \def 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) in (let TMP_102 \def (CHead c2 
-k u2) in (let TMP_103 \def (refl_equal C TMP_102) in (let TMP_104 \def 
-(ex3_2_intro C T TMP_99 TMP_100 TMP_101 c2 u2 TMP_103 H12 H10) in (let 
-TMP_105 \def (or_intror TMP_90 TMP_96 TMP_104) in (let TMP_106 \def (eq_ind_r 
-C c1 TMP_87 TMP_105 c0 H9) in (let TMP_107 \def (eq_ind_r T u1 TMP_69 TMP_106 
-u0 H7) in (eq_ind_r K k TMP_58 TMP_107 k0 H8)))))))))))))))))))))))))))) in 
-(let TMP_109 \def (TMP_108 H6) in (TMP_109 H5)))))))))))))))))))))))) in 
-(wcpr0_ind TMP_16 TMP_39 TMP_110 y x H0)))))) in (insert_eq C TMP_1 TMP_2 
-TMP_9 TMP_111 H))))))))).
+k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 
+u0) (CHead c1 k u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 
+k0 u0) (CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
+(CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda 
+(H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C (CHead c2 k1 u2) 
+(CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead 
+c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 \def (eq_ind T u0 
+(\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 (\lambda (t: T).(or 
+(eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda (c3: C).(\lambda 
+(u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda 
+(_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let 
+H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or (eq C 
+c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c2 (CHead c3 k 
+u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def (eq_ind C c0 
+(\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 (\lambda (c: C).(or 
+(eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda 
+(u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda 
+(_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) 
+(or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: 
+C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: 
+C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 
+u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k 
+u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C (CHead c2 
+k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x H0))) 
+H))))).