X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fpr2%2Fpr2.ma;h=3151fc53cc7dcb6c90e4e03a76c940d2c8335ef9;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=406d8bcd2c3daccea3ec9a4834326eeb58c3c51b;hpb=40e0eed5dc1a5f56f089f491841a5077723b891a;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma
index 406d8bcd2..3151fc53c 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma
@@ -20,24 +20,20 @@ include "basic_1/pr0/pr0.ma".
 
 include "basic_1/getl/fwd.ma".
 
-theorem pr2_confluence__pr2_free_free:
+fact pr2_confluence__pr2_free_free:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 
 t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
 T).(pr2 c t2 t))))))))
 \def
  \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(let TMP_1 \def (\lambda (t: 
-T).(pr0 t2 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t1 t)) in (let TMP_3 
-\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t: T).(pr2 c 
-t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_10 \def (\lambda 
-(x: T).(\lambda (H1: (pr0 t2 x)).(\lambda (H2: (pr0 t1 x)).(let TMP_6 \def 
-(\lambda (t: T).(pr2 c t1 t)) in (let TMP_7 \def (\lambda (t: T).(pr2 c t2 
-t)) in (let TMP_8 \def (pr2_free c t1 x H2) in (let TMP_9 \def (pr2_free c t2 
-x H1) in (ex_intro2 T TMP_6 TMP_7 x TMP_8 TMP_9)))))))) in (let TMP_11 \def 
-(pr0_confluence t0 t2 H0 t1 H) in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_10 
-TMP_11))))))))))))).
+(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 
+t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) 
+(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 
+x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) 
+(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) 
+(pr0_confluence t0 t2 H0 t1 H))))))).
 
-theorem pr2_confluence__pr2_free_delta:
+fact pr2_confluence__pr2_free_delta:
  \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall 
 (t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to 
 ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) 
@@ -47,35 +43,23 @@ t))))))))))))))
  \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda 
 (t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 
 t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 
-t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(let TMP_1 \def (\lambda (t: 
-T).(pr0 t4 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t1 t)) in (let TMP_3 
-\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t: T).(pr2 c 
-t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_31 \def (\lambda 
-(x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: (pr0 t1 x)).(let TMP_6 \def 
-(pr0 t2 x) in (let TMP_7 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_8 
-\def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_9 \def (ex2 T TMP_7 
-TMP_8) in (let TMP_10 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_11 \def 
-(\lambda (t: T).(pr2 c t2 t)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) in 
-(let TMP_17 \def (\lambda (H5: (pr0 t2 x)).(let TMP_13 \def (\lambda (t: 
-T).(pr2 c t1 t)) in (let TMP_14 \def (\lambda (t: T).(pr2 c t2 t)) in (let 
-TMP_15 \def (pr2_free c t1 x H4) in (let TMP_16 \def (pr2_free c t2 x H5) in 
-(ex_intro2 T TMP_13 TMP_14 x TMP_15 TMP_16)))))) in (let TMP_28 \def (\lambda 
-(H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i u x 
-w2)))).(let TMP_18 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_19 \def 
-(\lambda (w2: T).(subst0 i u x w2)) in (let TMP_20 \def (\lambda (t: T).(pr2 
-c t1 t)) in (let TMP_21 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_22 
-\def (ex2 T TMP_20 TMP_21) in (let TMP_27 \def (\lambda (x0: T).(\lambda (H6: 
-(pr0 t2 x0)).(\lambda (H7: (subst0 i u x x0)).(let TMP_23 \def (\lambda (t: 
-T).(pr2 c t1 t)) in (let TMP_24 \def (\lambda (t: T).(pr2 c t2 t)) in (let 
-TMP_25 \def (pr2_delta c d u i H0 t1 x H4 x0 H7) in (let TMP_26 \def 
-(pr2_free c t2 x0 H6) in (ex_intro2 T TMP_23 TMP_24 x0 TMP_25 TMP_26)))))))) 
-in (ex2_ind T TMP_18 TMP_19 TMP_22 TMP_27 H5)))))))) in (let TMP_29 \def 
-(pr0_refl u) in (let TMP_30 \def (pr0_subst0 t4 x H3 u t2 i H2 u TMP_29) in 
-(or_ind TMP_6 TMP_9 TMP_12 TMP_17 TMP_28 TMP_30))))))))))))))) in (let TMP_32 
-\def (pr0_confluence t0 t4 H1 t1 H) in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_31 
-TMP_32))))))))))))))))))).
+t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 
+t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda 
+(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: 
+(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda 
+(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 
+c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 
+x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: 
+T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda 
+(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: 
+(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 
+H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) 
+(pr0_confluence t0 t4 H1 t1 H))))))))))))).
 
-theorem pr2_confluence__pr2_delta_delta:
+fact pr2_confluence__pr2_delta_delta:
  \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall 
 (t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: 
 T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d 
@@ -89,143 +73,80 @@ t))))))))))))))))))))
 T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i 
 c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 
 i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda 
-(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(let TMP_1 \def 
-(\lambda (t: T).(pr0 t4 t)) in (let TMP_2 \def (\lambda (t: T).(pr0 t3 t)) in 
-(let TMP_3 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_4 \def (\lambda (t: 
-T).(pr2 c t2 t)) in (let TMP_5 \def (ex2 T TMP_3 TMP_4) in (let TMP_165 \def 
-(\lambda (x: T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (pr0 t3 x)).(let 
-TMP_6 \def (pr0 t1 x) in (let TMP_7 \def (\lambda (w2: T).(pr0 t1 w2)) in 
-(let TMP_8 \def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_9 \def (ex2 T 
-TMP_7 TMP_8) in (let TMP_10 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_11 
-\def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) 
-in (let TMP_38 \def (\lambda (H7: (pr0 t1 x)).(let TMP_13 \def (pr0 t2 x) in 
-(let TMP_14 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_15 \def (\lambda 
-(w2: T).(subst0 i0 u0 x w2)) in (let TMP_16 \def (ex2 T TMP_14 TMP_15) in 
-(let TMP_17 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_18 \def (\lambda 
-(t: T).(pr2 c t2 t)) in (let TMP_19 \def (ex2 T TMP_17 TMP_18) in (let TMP_24 
-\def (\lambda (H8: (pr0 t2 x)).(let TMP_20 \def (\lambda (t: T).(pr2 c t1 t)) 
-in (let TMP_21 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_22 \def 
-(pr2_free c t1 x H7) in (let TMP_23 \def (pr2_free c t2 x H8) in (ex_intro2 T 
-TMP_20 TMP_21 x TMP_22 TMP_23)))))) in (let TMP_35 \def (\lambda (H8: (ex2 T 
-(\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)))).(let 
-TMP_25 \def (\lambda (w2: T).(pr0 t2 w2)) in (let TMP_26 \def (\lambda (w2: 
-T).(subst0 i0 u0 x w2)) in (let TMP_27 \def (\lambda (t: T).(pr2 c t1 t)) in 
-(let TMP_28 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_29 \def (ex2 T 
-TMP_27 TMP_28) in (let TMP_34 \def (\lambda (x0: T).(\lambda (H9: (pr0 t2 
-x0)).(\lambda (H10: (subst0 i0 u0 x x0)).(let TMP_30 \def (\lambda (t: 
-T).(pr2 c t1 t)) in (let TMP_31 \def (\lambda (t: T).(pr2 c t2 t)) in (let 
-TMP_32 \def (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) in (let TMP_33 \def 
-(pr2_free c t2 x0 H9) in (ex_intro2 T TMP_30 TMP_31 x0 TMP_32 TMP_33)))))))) 
-in (ex2_ind T TMP_25 TMP_26 TMP_29 TMP_34 H8)))))))) in (let TMP_36 \def 
-(pr0_refl u0) in (let TMP_37 \def (pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 TMP_36) 
-in (or_ind TMP_13 TMP_16 TMP_19 TMP_24 TMP_35 TMP_37))))))))))))) in (let 
-TMP_162 \def (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: 
-T).(subst0 i u x w2)))).(let TMP_39 \def (\lambda (w2: T).(pr0 t1 w2)) in 
-(let TMP_40 \def (\lambda (w2: T).(subst0 i u x w2)) in (let TMP_41 \def 
-(\lambda (t: T).(pr2 c t1 t)) in (let TMP_42 \def (\lambda (t: T).(pr2 c t2 
-t)) in (let TMP_43 \def (ex2 T TMP_41 TMP_42) in (let TMP_161 \def (\lambda 
-(x0: T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(let 
-TMP_44 \def (pr0 t2 x) in (let TMP_45 \def (\lambda (w2: T).(pr0 t2 w2)) in 
-(let TMP_46 \def (\lambda (w2: T).(subst0 i0 u0 x w2)) in (let TMP_47 \def 
-(ex2 T TMP_45 TMP_46) in (let TMP_48 \def (\lambda (t: T).(pr2 c t1 t)) in 
-(let TMP_49 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_50 \def (ex2 T 
-TMP_48 TMP_49) in (let TMP_55 \def (\lambda (H10: (pr0 t2 x)).(let TMP_51 
-\def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_52 \def (\lambda (t: T).(pr2 c 
-t2 t)) in (let TMP_53 \def (pr2_free c t1 x0 H8) in (let TMP_54 \def 
-(pr2_delta c d u i H t2 x H10 x0 H9) in (ex_intro2 T TMP_51 TMP_52 x0 TMP_53 
-TMP_54)))))) in (let TMP_158 \def (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 
-t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)))).(let TMP_56 \def (\lambda 
-(w2: T).(pr0 t2 w2)) in (let TMP_57 \def (\lambda (w2: T).(subst0 i0 u0 x 
-w2)) in (let TMP_58 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_59 \def 
-(\lambda (t: T).(pr2 c t2 t)) in (let TMP_60 \def (ex2 T TMP_58 TMP_59) in 
-(let TMP_157 \def (\lambda (x1: T).(\lambda (H11: (pr0 t2 x1)).(\lambda (H12: 
-(subst0 i0 u0 x x1)).(let TMP_61 \def (\lambda (t: T).(pr2 c t1 t)) in (let 
-TMP_62 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_63 \def (ex2 T TMP_61 
-TMP_62) in (let TMP_76 \def (\lambda (H13: (not (eq nat i i0))).(let TMP_64 
-\def (\lambda (t: T).(subst0 i u x1 t)) in (let TMP_65 \def (\lambda (t: 
-T).(subst0 i0 u0 x0 t)) in (let TMP_66 \def (\lambda (t: T).(pr2 c t1 t)) in 
-(let TMP_67 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_68 \def (ex2 T 
-TMP_66 TMP_67) in (let TMP_73 \def (\lambda (x2: T).(\lambda (H14: (subst0 i 
-u x1 x2)).(\lambda (H15: (subst0 i0 u0 x0 x2)).(let TMP_69 \def (\lambda (t: 
-T).(pr2 c t1 t)) in (let TMP_70 \def (\lambda (t: T).(pr2 c t2 t)) in (let 
-TMP_71 \def (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) in (let TMP_72 \def 
-(pr2_delta c d u i H t2 x1 H11 x2 H14) in (ex_intro2 T TMP_69 TMP_70 x2 
-TMP_71 TMP_72)))))))) in (let TMP_74 \def (sym_not_eq nat i i0 H13) in (let 
-TMP_75 \def (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 TMP_74) in 
-(ex2_ind T TMP_64 TMP_65 TMP_68 TMP_73 TMP_75)))))))))) in (let TMP_156 \def 
-(\lambda (H13: (eq nat i i0)).(let TMP_77 \def (\lambda (n: nat).(subst0 n u0 
-x x1)) in (let H14 \def (eq_ind_r nat i0 TMP_77 H12 i H13) in (let TMP_80 
-\def (\lambda (n: nat).(let TMP_78 \def (Bind Abbr) in (let TMP_79 \def 
-(CHead d0 TMP_78 u0) in (getl n c TMP_79)))) in (let H15 \def (eq_ind_r nat 
-i0 TMP_80 H2 i H13) in (let TMP_81 \def (Bind Abbr) in (let TMP_82 \def 
-(CHead d TMP_81 u) in (let TMP_83 \def (\lambda (c0: C).(getl i c c0)) in 
-(let TMP_84 \def (Bind Abbr) in (let TMP_85 \def (CHead d0 TMP_84 u0) in (let 
-TMP_86 \def (Bind Abbr) in (let TMP_87 \def (CHead d TMP_86 u) in (let TMP_88 
-\def (Bind Abbr) in (let TMP_89 \def (CHead d0 TMP_88 u0) in (let TMP_90 \def 
-(getl_mono c TMP_87 i H TMP_89 H15) in (let H16 \def (eq_ind C TMP_82 TMP_83 
-H TMP_85 TMP_90) in (let TMP_91 \def (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) in (let TMP_92 \def (Bind 
-Abbr) in (let TMP_93 \def (CHead d TMP_92 u) in (let TMP_94 \def (Bind Abbr) 
-in (let TMP_95 \def (CHead d0 TMP_94 u0) in (let TMP_96 \def (Bind Abbr) in 
-(let TMP_97 \def (CHead d TMP_96 u) in (let TMP_98 \def (Bind Abbr) in (let 
-TMP_99 \def (CHead d0 TMP_98 u0) in (let TMP_100 \def (getl_mono c TMP_97 i H 
-TMP_99 H15) in (let H17 \def (f_equal C C TMP_91 TMP_93 TMP_95 TMP_100) in 
-(let TMP_101 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) in (let TMP_102 \def (Bind Abbr) in (let 
-TMP_103 \def (CHead d TMP_102 u) in (let TMP_104 \def (Bind Abbr) in (let 
-TMP_105 \def (CHead d0 TMP_104 u0) in (let TMP_106 \def (Bind Abbr) in (let 
-TMP_107 \def (CHead d TMP_106 u) in (let TMP_108 \def (Bind Abbr) in (let 
-TMP_109 \def (CHead d0 TMP_108 u0) in (let TMP_110 \def (getl_mono c TMP_107 
-i H TMP_109 H15) in (let H18 \def (f_equal C T TMP_101 TMP_103 TMP_105 
-TMP_110) in (let TMP_155 \def (\lambda (H19: (eq C d d0)).(let TMP_111 \def 
-(\lambda (t: T).(subst0 i t x x1)) in (let H20 \def (eq_ind_r T u0 TMP_111 
-H14 u H18) in (let TMP_114 \def (\lambda (t: T).(let TMP_112 \def (Bind Abbr) 
-in (let TMP_113 \def (CHead d0 TMP_112 t) in (getl i c TMP_113)))) in (let 
-H21 \def (eq_ind_r T u0 TMP_114 H16 u H18) in (let TMP_117 \def (\lambda (c0: 
-C).(let TMP_115 \def (Bind Abbr) in (let TMP_116 \def (CHead c0 TMP_115 u) in 
-(getl i c TMP_116)))) in (let H22 \def (eq_ind_r C d0 TMP_117 H21 d H19) in 
-(let TMP_118 \def (eq T x1 x0) in (let TMP_119 \def (\lambda (t: T).(subst0 i 
-u x1 t)) in (let TMP_120 \def (\lambda (t: T).(subst0 i u x0 t)) in (let 
-TMP_121 \def (ex2 T TMP_119 TMP_120) in (let TMP_122 \def (subst0 i u x1 x0) 
-in (let TMP_123 \def (subst0 i u x0 x1) in (let TMP_124 \def (\lambda (t: 
-T).(pr2 c t1 t)) in (let TMP_125 \def (\lambda (t: T).(pr2 c t2 t)) in (let 
-TMP_126 \def (ex2 T TMP_124 TMP_125) in (let TMP_132 \def (\lambda (H23: (eq 
-T x1 x0)).(let TMP_127 \def (\lambda (t: T).(pr0 t2 t)) in (let H24 \def 
-(eq_ind T x1 TMP_127 H11 x0 H23) in (let TMP_128 \def (\lambda (t: T).(pr2 c 
-t1 t)) in (let TMP_129 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_130 
-\def (pr2_free c t1 x0 H8) in (let TMP_131 \def (pr2_free c t2 x0 H24) in 
-(ex_intro2 T TMP_128 TMP_129 x0 TMP_130 TMP_131)))))))) in (let TMP_143 \def 
-(\lambda (H23: (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: 
-T).(subst0 i u x0 t)))).(let TMP_133 \def (\lambda (t: T).(subst0 i u x1 t)) 
-in (let TMP_134 \def (\lambda (t: T).(subst0 i u x0 t)) in (let TMP_135 \def 
-(\lambda (t: T).(pr2 c t1 t)) in (let TMP_136 \def (\lambda (t: T).(pr2 c t2 
-t)) in (let TMP_137 \def (ex2 T TMP_135 TMP_136) in (let TMP_142 \def 
-(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i 
-u x0 x2)).(let TMP_138 \def (\lambda (t: T).(pr2 c t1 t)) in (let TMP_139 
-\def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_140 \def (pr2_delta c d u i 
-H22 t1 x0 H8 x2 H25) in (let TMP_141 \def (pr2_delta c d u i H22 t2 x1 H11 x2 
-H24) in (ex_intro2 T TMP_138 TMP_139 x2 TMP_140 TMP_141)))))))) in (ex2_ind T 
-TMP_133 TMP_134 TMP_137 TMP_142 H23)))))))) in (let TMP_148 \def (\lambda 
-(H23: (subst0 i u x1 x0)).(let TMP_144 \def (\lambda (t: T).(pr2 c t1 t)) in 
-(let TMP_145 \def (\lambda (t: T).(pr2 c t2 t)) in (let TMP_146 \def 
-(pr2_free c t1 x0 H8) in (let TMP_147 \def (pr2_delta c d u i H22 t2 x1 H11 
-x0 H23) in (ex_intro2 T TMP_144 TMP_145 x0 TMP_146 TMP_147)))))) in (let 
-TMP_153 \def (\lambda (H23: (subst0 i u x0 x1)).(let TMP_149 \def (\lambda 
-(t: T).(pr2 c t1 t)) in (let TMP_150 \def (\lambda (t: T).(pr2 c t2 t)) in 
-(let TMP_151 \def (pr2_delta c d u i H22 t1 x0 H8 x1 H23) in (let TMP_152 
-\def (pr2_free c t2 x1 H11) in (ex_intro2 T TMP_149 TMP_150 x1 TMP_151 
-TMP_152)))))) in (let TMP_154 \def (subst0_confluence_eq x x1 u i H20 x0 H9) 
-in (or4_ind TMP_118 TMP_121 TMP_122 TMP_123 TMP_126 TMP_132 TMP_143 TMP_148 
-TMP_153 TMP_154)))))))))))))))))))))) in (TMP_155 
-H17)))))))))))))))))))))))))))))))))))))))) in (neq_eq_e i i0 TMP_63 TMP_76 
-TMP_156))))))))) in (ex2_ind T TMP_56 TMP_57 TMP_60 TMP_157 H10)))))))) in 
-(let TMP_159 \def (pr0_refl u0) in (let TMP_160 \def (pr0_subst0 t4 x H5 u0 
-t2 i0 H4 u0 TMP_159) in (or_ind TMP_44 TMP_47 TMP_50 TMP_55 TMP_158 
-TMP_160))))))))))))))) in (ex2_ind T TMP_39 TMP_40 TMP_43 TMP_161 H7)))))))) 
-in (let TMP_163 \def (pr0_refl u) in (let TMP_164 \def (pr0_subst0 t3 x H6 u 
-t1 i H1 u TMP_163) in (or_ind TMP_6 TMP_9 TMP_12 TMP_38 TMP_162 
-TMP_164))))))))))))))) in (let TMP_166 \def (pr0_confluence t0 t4 H3 t3 H0) 
-in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_165 TMP_166))))))))))))))))))))))))).
+(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: 
+T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 
+x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 
+t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 
+x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
+w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
+(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda 
+(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda 
+(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 
+u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
+(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x 
+x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) 
+x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) 
+(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T 
+(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind 
+T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T 
+(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: 
+T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 
+t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
+w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
+(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) 
+(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H 
+t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) 
+(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 
+w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 
+x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: 
+T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i 
+i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 
+i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
+t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: 
+(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d 
+u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 
+(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def 
+(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 
+\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) 
+H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: 
+C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
+Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) 
+(getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in 
+((let H18 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead 
+d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind 
+Abbr) u0) H15)) in (\lambda (H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 
+(\lambda (t: T).(subst0 i t x x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 
+(\lambda (t: T).(getl i c (CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 
+\def (eq_ind_r C d0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 
+d H19) in (or4_ind (eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) 
+(\lambda (t: T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) 
+(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda 
+(H23: (eq T x1 x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) 
+H11 x0 H23) in (ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda 
+(H23: (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u 
+x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: 
+T).(subst0 i u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t))) (\lambda (x2: T).(\lambda (H24: (subst0 i u x1 
+x2)).(\lambda (H25: (subst0 i u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c 
+t1 t)) (\lambda (t: T).(pr2 c t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 
+H25) (pr2_delta c d u i H22 t2 x1 H11 x2 H24))))) H23)) (\lambda (H23: 
+(subst0 i u x1 x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 
+H23))) (\lambda (H23: (subst0 i u x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c 
+t1 t)) (\lambda (t: T).(pr2 c t2 t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 
+H23) (pr2_free c t2 x1 H11))) (subst0_confluence_eq x x1 u i H20 x0 H9))))))) 
+H17)))))))))) H10)) (pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) 
+H7)) (pr0_subst0 t3 x H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 
+H3 t3 H0))))))))))))))))))).
 
 theorem pr2_confluence:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall 
@@ -235,118 +156,81 @@ T).(pr2 c t2 t))))))))
  \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 
 t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H with 
 [(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: 
-(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(let TMP_54 \def (\lambda (_: 
-C).((eq T t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (let TMP_52 \def 
-(\lambda (t: T).(pr2 c t1 t)) in (let TMP_53 \def (\lambda (t: T).(pr2 c t2 
-t)) in (ex2 T TMP_52 TMP_53))))))) in (let TMP_98 \def (\lambda (H5: (eq T t3 
-t0)).(let TMP_57 \def (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (let 
-TMP_55 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_56 \def (\lambda (t5: 
-T).(pr2 c t2 t5)) in (ex2 T TMP_55 TMP_56)))))) in (let TMP_96 \def (\lambda 
-(H6: (eq T t4 t1)).(let TMP_60 \def (\lambda (t: T).((pr0 t0 t) \to (let 
-TMP_58 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_59 \def (\lambda (t5: 
-T).(pr2 c t2 t5)) in (ex2 T TMP_58 TMP_59))))) in (let TMP_94 \def (\lambda 
-(H7: (pr0 t0 t1)).(let H8 \def (match H0 with [(pr2_free c1 t5 t6 H8) 
-\Rightarrow (\lambda (H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda 
-(H11: (eq T t6 t2)).(let TMP_78 \def (\lambda (_: C).((eq T t5 t0) \to ((eq T 
-t6 t2) \to ((pr0 t5 t6) \to (let TMP_76 \def (\lambda (t: T).(pr2 c t1 t)) in 
-(let TMP_77 \def (\lambda (t: T).(pr2 c t2 t)) in (ex2 T TMP_76 TMP_77))))))) 
-in (let TMP_89 \def (\lambda (H12: (eq T t5 t0)).(let TMP_81 \def (\lambda 
-(t: T).((eq T t6 t2) \to ((pr0 t t6) \to (let TMP_79 \def (\lambda (t7: 
-T).(pr2 c t1 t7)) in (let TMP_80 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2 
-T TMP_79 TMP_80)))))) in (let TMP_87 \def (\lambda (H13: (eq T t6 t2)).(let 
-TMP_84 \def (\lambda (t: T).((pr0 t0 t) \to (let TMP_82 \def (\lambda (t7: 
-T).(pr2 c t1 t7)) in (let TMP_83 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2 
-T TMP_82 TMP_83))))) in (let TMP_85 \def (\lambda (H14: (pr0 t0 
-t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) in (let TMP_86 \def 
-(sym_eq T t6 t2 H13) in (eq_ind T t2 TMP_84 TMP_85 t6 TMP_86))))) in (let 
-TMP_88 \def (sym_eq T t5 t0 H12) in (eq_ind T t0 TMP_81 TMP_87 t5 TMP_88))))) 
-in (let TMP_90 \def (sym_eq C c1 c H9) in (eq_ind C c TMP_78 TMP_89 c1 TMP_90 
-H10 H11 H8))))))) | (pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow 
-(\lambda (H11: (eq C c1 c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T 
-t t2)).(let TMP_63 \def (\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to 
-((getl i c2 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) 
-\to (let TMP_61 \def (\lambda (t7: T).(pr2 c t1 t7)) in (let TMP_62 \def 
-(\lambda (t7: T).(pr2 c t2 t7)) in (ex2 T TMP_61 TMP_62))))))))) in (let 
-TMP_74 \def (\lambda (H14: (eq T t5 t0)).(let TMP_66 \def (\lambda (t7: 
-T).((eq T t t2) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to 
-((subst0 i u t6 t) \to (let TMP_64 \def (\lambda (t8: T).(pr2 c t1 t8)) in 
-(let TMP_65 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T TMP_64 
-TMP_65)))))))) in (let TMP_72 \def (\lambda (H15: (eq T t t2)).(let TMP_69 
-\def (\lambda (t7: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) 
-\to ((subst0 i u t6 t7) \to (let TMP_67 \def (\lambda (t8: T).(pr2 c t1 t8)) 
-in (let TMP_68 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T TMP_67 
-TMP_68))))))) in (let TMP_70 \def (\lambda (H16: (getl i c (CHead d (Bind 
-Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda (H18: (subst0 i u t6 
-t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i H7 H16 H17 H18)))) 
-in (let TMP_71 \def (sym_eq T t t2 H15) in (eq_ind T t2 TMP_69 TMP_70 t 
-TMP_71))))) in (let TMP_73 \def (sym_eq T t5 t0 H14) in (eq_ind T t0 TMP_66 
-TMP_72 t5 TMP_73))))) in (let TMP_75 \def (sym_eq C c1 c H11) in (eq_ind C c 
-TMP_63 TMP_74 c1 TMP_75 H12 H13 H8 H9 H10)))))))]) in (let TMP_91 \def 
-(refl_equal C c) in (let TMP_92 \def (refl_equal T t0) in (let TMP_93 \def 
-(refl_equal T t2) in (H8 TMP_91 TMP_92 TMP_93)))))) in (let TMP_95 \def 
-(sym_eq T t4 t1 H6) in (eq_ind T t1 TMP_60 TMP_94 t4 TMP_95))))) in (let 
-TMP_97 \def (sym_eq T t3 t0 H5) in (eq_ind T t0 TMP_57 TMP_96 t3 TMP_97))))) 
-in (let TMP_99 \def (sym_eq C c0 c H2) in (eq_ind C c TMP_54 TMP_98 c0 TMP_99 
-H3 H4 H1))))))) | (pr2_delta c0 d u i H1 t3 t4 H2 t H3) \Rightarrow (\lambda 
-(H4: (eq C c0 c)).(\lambda (H5: (eq T t3 t0)).(\lambda (H6: (eq T t t1)).(let 
-TMP_3 \def (\lambda (c1: C).((eq T t3 t0) \to ((eq T t t1) \to ((getl i c1 
-(CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) \to ((subst0 i u t4 t) \to (let 
-TMP_1 \def (\lambda (t5: T).(pr2 c t1 t5)) in (let TMP_2 \def (\lambda (t5: 
-T).(pr2 c t2 t5)) in (ex2 T TMP_1 TMP_2))))))))) in (let TMP_50 \def (\lambda 
-(H7: (eq T t3 t0)).(let TMP_6 \def (\lambda (t5: T).((eq T t t1) \to ((getl i 
-c (CHead d (Bind Abbr) u)) \to ((pr0 t5 t4) \to ((subst0 i u t4 t) \to (let 
-TMP_4 \def (\lambda (t6: T).(pr2 c t1 t6)) in (let TMP_5 \def (\lambda (t6: 
-T).(pr2 c t2 t6)) in (ex2 T TMP_4 TMP_5)))))))) in (let TMP_48 \def (\lambda 
-(H8: (eq T t t1)).(let TMP_9 \def (\lambda (t5: T).((getl i c (CHead d (Bind 
-Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t5) \to (let TMP_7 \def 
-(\lambda (t6: T).(pr2 c t1 t6)) in (let TMP_8 \def (\lambda (t6: T).(pr2 c t2 
-t6)) in (ex2 T TMP_7 TMP_8))))))) in (let TMP_46 \def (\lambda (H9: (getl i c 
-(CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 
-i u t4 t1)).(let H12 \def (match H0 with [(pr2_free c1 t5 t6 H12) \Rightarrow 
-(\lambda (H13: (eq C c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T 
-t6 t2)).(let TMP_27 \def (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to 
-((pr0 t5 t6) \to (let TMP_25 \def (\lambda (t7: T).(pr2 c t1 t7)) in (let 
-TMP_26 \def (\lambda (t7: T).(pr2 c t2 t7)) in (ex2 T TMP_25 TMP_26))))))) in 
-(let TMP_41 \def (\lambda (H16: (eq T t5 t0)).(let TMP_30 \def (\lambda (t7: 
-T).((eq T t6 t2) \to ((pr0 t7 t6) \to (let TMP_28 \def (\lambda (t8: T).(pr2 
-c t1 t8)) in (let TMP_29 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 T 
-TMP_28 TMP_29)))))) in (let TMP_39 \def (\lambda (H17: (eq T t6 t2)).(let 
-TMP_33 \def (\lambda (t7: T).((pr0 t0 t7) \to (let TMP_31 \def (\lambda (t8: 
-T).(pr2 c t1 t8)) in (let TMP_32 \def (\lambda (t8: T).(pr2 c t2 t8)) in (ex2 
-T TMP_31 TMP_32))))) in (let TMP_37 \def (\lambda (H18: (pr0 t0 t2)).(let 
-TMP_34 \def (pr2 c t2) in (let TMP_35 \def (pr2 c t1) in (let TMP_36 \def 
-(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11) in 
-(ex2_sym T TMP_34 TMP_35 TMP_36))))) in (let TMP_38 \def (sym_eq T t6 t2 H17) 
-in (eq_ind T t2 TMP_33 TMP_37 t6 TMP_38))))) in (let TMP_40 \def (sym_eq T t5 
-t0 H16) in (eq_ind T t0 TMP_30 TMP_39 t5 TMP_40))))) in (let TMP_42 \def 
-(sym_eq C c1 c H13) in (eq_ind C c TMP_27 TMP_41 c1 TMP_42 H14 H15 H12))))))) 
-| (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow (\lambda (H15: (eq 
-C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T t7 t2)).(let 
-TMP_12 \def (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to ((getl i0 c2 
-(CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 t6 t7) \to 
-(let TMP_10 \def (\lambda (t8: T).(pr2 c t1 t8)) in (let TMP_11 \def (\lambda 
-(t8: T).(pr2 c t2 t8)) in (ex2 T TMP_10 TMP_11))))))))) in (let TMP_23 \def 
-(\lambda (H18: (eq T t5 t0)).(let TMP_15 \def (\lambda (t8: T).((eq T t7 t2) 
-\to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to ((subst0 i0 
-u0 t6 t7) \to (let TMP_13 \def (\lambda (t9: T).(pr2 c t1 t9)) in (let TMP_14 
-\def (\lambda (t9: T).(pr2 c t2 t9)) in (ex2 T TMP_13 TMP_14)))))))) in (let 
-TMP_21 \def (\lambda (H19: (eq T t7 t2)).(let TMP_18 \def (\lambda (t8: 
-T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to ((subst0 i0 u0 
-t6 t8) \to (let TMP_16 \def (\lambda (t9: T).(pr2 c t1 t9)) in (let TMP_17 
-\def (\lambda (t9: T).(pr2 c t2 t9)) in (ex2 T TMP_16 TMP_17))))))) in (let 
-TMP_19 \def (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda 
-(H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 
+(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T 
+t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c 
+t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind 
+T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: 
+T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 
+t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: 
+T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 
+t1)).(let H8 \def (match H0 with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda 
+(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 
+t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 
+t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
+t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 
+t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: 
+T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: 
+T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: 
+T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 
+t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 
+H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | 
+(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 
+c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c 
+(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d 
+(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda 
+(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: 
+(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c 
+(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T 
+(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) 
+(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d 
+(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda 
+(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: 
+(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda 
+(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i 
+H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 
+(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) 
+(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T 
+t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 
+H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 
+t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) 
+\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) 
+\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda 
+(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 
+(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to 
+((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 
+t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind 
+T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) 
+\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda 
+(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) 
+u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 
+\def (match H0 with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C 
+c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c 
+(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda 
+(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 
+t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
+t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 
+t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
+t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) 
+(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 
+(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 
+H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow 
+(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T 
+t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to 
+((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 
+t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
+t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T 
+t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to 
+((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda 
+(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 
+(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to 
+((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda 
+(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) 
+u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 
 t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 
-H11 H20 H21 H22)))) in (let TMP_20 \def (sym_eq T t7 t2 H19) in (eq_ind T t2 
-TMP_18 TMP_19 t7 TMP_20))))) in (let TMP_22 \def (sym_eq T t5 t0 H18) in 
-(eq_ind T t0 TMP_15 TMP_21 t5 TMP_22))))) in (let TMP_24 \def (sym_eq C c1 c 
-H15) in (eq_ind C c TMP_12 TMP_23 c1 TMP_24 H16 H17 H12 H13 H14)))))))]) in 
-(let TMP_43 \def (refl_equal C c) in (let TMP_44 \def (refl_equal T t0) in 
-(let TMP_45 \def (refl_equal T t2) in (H12 TMP_43 TMP_44 TMP_45)))))))) in 
-(let TMP_47 \def (sym_eq T t t1 H8) in (eq_ind T t1 TMP_9 TMP_46 t 
-TMP_47))))) in (let TMP_49 \def (sym_eq T t3 t0 H7) in (eq_ind T t0 TMP_6 
-TMP_48 t3 TMP_49))))) in (let TMP_51 \def (sym_eq C c0 c H4) in (eq_ind C c 
-TMP_3 TMP_50 c0 TMP_51 H5 H6 H1 H2 H3)))))))]) in (let TMP_100 \def 
-(refl_equal C c) in (let TMP_101 \def (refl_equal T t0) in (let TMP_102 \def 
-(refl_equal T t1) in (H1 TMP_100 TMP_101 TMP_102)))))))))).
+H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 
+(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) 
+(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T 
+t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C 
+c) (refl_equal T t0) (refl_equal T t1)))))))).