X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_1%2Fext%2Farith.ma;h=724a34747372f80cc491fbc63a85f22a9d59db06;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=393440c2591179487cddcf5afbc2b6c2b55a7ed6;hpb=9c954a9a843ebb1bf189536df4e14f77132ed1cf;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma b/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma
index 393440c25..724a34747 100644
--- a/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma
@@ -16,883 +16,577 @@
 
 include "ground_1/preamble.ma".
 
-theorem nat_dec:
+lemma nat_dec:
  \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to 
 (\forall (P: Prop).P))))
 \def
- \lambda (n1: nat).(let TMP_81 \def (\lambda (n: nat).(\forall (n2: nat).(let 
-TMP_80 \def (eq nat n n2) in (let TMP_79 \def ((eq nat n n2) \to (\forall (P: 
-Prop).P)) in (or TMP_80 TMP_79))))) in (let TMP_78 \def (\lambda (n2: 
-nat).(let TMP_77 \def (\lambda (n: nat).(let TMP_76 \def (eq nat O n) in (let 
-TMP_75 \def ((eq nat O n) \to (\forall (P: Prop).P)) in (or TMP_76 TMP_75)))) 
-in (let TMP_73 \def (eq nat O O) in (let TMP_72 \def ((eq nat O O) \to 
-(\forall (P: Prop).P)) in (let TMP_71 \def (refl_equal nat O) in (let TMP_74 
-\def (or_introl TMP_73 TMP_72 TMP_71) in (let TMP_70 \def (\lambda (n: 
-nat).(\lambda (_: (or (eq nat O n) ((eq nat O n) \to (\forall (P: 
-Prop).P)))).(let TMP_68 \def (S n) in (let TMP_69 \def (eq nat O TMP_68) in 
-(let TMP_67 \def ((eq nat O (S n)) \to (\forall (P: Prop).P)) in (let TMP_66 
-\def (\lambda (H0: (eq nat O (S n))).(\lambda (P: Prop).(let TMP_65 \def 
-(\lambda (ee: nat).(match ee in nat with [O \Rightarrow True | (S _) 
-\Rightarrow False])) in (let TMP_64 \def (S n) in (let H1 \def (eq_ind nat O 
-TMP_65 I TMP_64 H0) in (False_ind P H1)))))) in (or_intror TMP_69 TMP_67 
-TMP_66))))))) in (nat_ind TMP_77 TMP_74 TMP_70 n2)))))))) in (let TMP_63 \def 
-(\lambda (n: nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat 
-n n2) \to (\forall (P: Prop).P)))))).(\lambda (n2: nat).(let TMP_62 \def 
-(\lambda (n0: nat).(let TMP_60 \def (S n) in (let TMP_61 \def (eq nat TMP_60 
-n0) in (let TMP_59 \def ((eq nat (S n) n0) \to (\forall (P: Prop).P)) in (or 
-TMP_61 TMP_59))))) in (let TMP_56 \def (S n) in (let TMP_57 \def (eq nat 
-TMP_56 O) in (let TMP_55 \def ((eq nat (S n) O) \to (\forall (P: Prop).P)) in 
-(let TMP_54 \def (\lambda (H0: (eq nat (S n) O)).(\lambda (P: Prop).(let 
-TMP_53 \def (S n) in (let TMP_52 \def (\lambda (ee: nat).(match ee in nat 
-with [O \Rightarrow False | (S _) \Rightarrow True])) in (let H1 \def (eq_ind 
-nat TMP_53 TMP_52 I O H0) in (False_ind P H1)))))) in (let TMP_58 \def 
-(or_intror TMP_57 TMP_55 TMP_54) in (let TMP_51 \def (\lambda (n0: 
-nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: 
-Prop).P)))).(let TMP_50 \def (eq nat n n0) in (let TMP_49 \def ((eq nat n n0) 
-\to (\forall (P: Prop).P)) in (let TMP_46 \def (S n) in (let TMP_45 \def (S 
-n0) in (let TMP_47 \def (eq nat TMP_46 TMP_45) in (let TMP_44 \def ((eq nat 
-(S n) (S n0)) \to (\forall (P: Prop).P)) in (let TMP_48 \def (or TMP_47 
-TMP_44) in (let TMP_43 \def (\lambda (H1: (eq nat n n0)).(let TMP_30 \def 
-(\lambda (n3: nat).(let TMP_28 \def (S n) in (let TMP_29 \def (eq nat TMP_28 
-n3) in (let TMP_27 \def ((eq nat (S n) n3) \to (\forall (P: Prop).P)) in (or 
-TMP_29 TMP_27))))) in (let H2 \def (eq_ind_r nat n0 TMP_30 H0 n H1) in (let 
-TMP_42 \def (\lambda (n3: nat).(let TMP_40 \def (S n) in (let TMP_39 \def (S 
-n3) in (let TMP_41 \def (eq nat TMP_40 TMP_39) in (let TMP_38 \def ((eq nat 
-(S n) (S n3)) \to (\forall (P: Prop).P)) in (or TMP_41 TMP_38)))))) in (let 
-TMP_35 \def (S n) in (let TMP_34 \def (S n) in (let TMP_36 \def (eq nat 
-TMP_35 TMP_34) in (let TMP_33 \def ((eq nat (S n) (S n)) \to (\forall (P: 
-Prop).P)) in (let TMP_31 \def (S n) in (let TMP_32 \def (refl_equal nat 
-TMP_31) in (let TMP_37 \def (or_introl TMP_36 TMP_33 TMP_32) in (eq_ind nat n 
-TMP_42 TMP_37 n0 H1)))))))))))) in (let TMP_26 \def (\lambda (H1: (((eq nat n 
-n0) \to (\forall (P: Prop).P)))).(let TMP_24 \def (S n) in (let TMP_23 \def 
-(S n0) in (let TMP_25 \def (eq nat TMP_24 TMP_23) in (let TMP_22 \def ((eq 
-nat (S n) (S n0)) \to (\forall (P: Prop).P)) in (let TMP_21 \def (\lambda 
-(H2: (eq nat (S n) (S n0))).(\lambda (P: Prop).(let TMP_14 \def (\lambda (e: 
-nat).(match e in nat with [O \Rightarrow n | (S n3) \Rightarrow n3])) in (let 
-TMP_13 \def (S n) in (let TMP_12 \def (S n0) in (let H3 \def (f_equal nat nat 
-TMP_14 TMP_13 TMP_12 H2) in (let TMP_15 \def (\lambda (n3: nat).((eq nat n 
-n3) \to (\forall (P0: Prop).P0))) in (let H4 \def (eq_ind_r nat n0 TMP_15 H1 
-n H3) in (let TMP_19 \def (\lambda (n3: nat).(let TMP_17 \def (S n) in (let 
-TMP_18 \def (eq nat TMP_17 n3) in (let TMP_16 \def ((eq nat (S n) n3) \to 
-(\forall (P0: Prop).P0)) in (or TMP_18 TMP_16))))) in (let H5 \def (eq_ind_r 
-nat n0 TMP_19 H0 n H3) in (let TMP_20 \def (refl_equal nat n) in (H4 TMP_20 
-P)))))))))))) in (or_intror TMP_25 TMP_22 TMP_21))))))) in (let TMP_11 \def 
-(H n0) in (or_ind TMP_50 TMP_49 TMP_48 TMP_43 TMP_26 TMP_11))))))))))))) in 
-(nat_ind TMP_62 TMP_58 TMP_51 n2))))))))))) in (nat_ind TMP_81 TMP_78 TMP_63 
-n1)))).
-
-theorem simpl_plus_r:
+ \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat 
+n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2: 
+nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall 
+(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P: 
+Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n) 
+((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq 
+nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S 
+n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match 
+ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H0) in 
+(False_ind P H1))))))) n2)) (\lambda (n: nat).(\lambda (H: ((\forall (n2: 
+nat).(or (eq nat n n2) ((eq nat n n2) \to (\forall (P: Prop).P)))))).(\lambda 
+(n2: nat).(nat_ind (\lambda (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) 
+n0) \to (\forall (P: Prop).P)))) (or_intror (eq nat (S n) O) ((eq nat (S n) 
+O) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat (S n) O)).(\lambda (P: 
+Prop).(let H1 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) 
+(\lambda (n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to 
+(\forall (P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall 
+(P: Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall 
+(P: Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 
+(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: 
+Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S 
+n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat 
+(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat 
+(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P: 
+Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to 
+(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: 
+Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e with [O 
+\Rightarrow n | (S n3) \Rightarrow n3])) (S n) (S n0) H2) in (let H4 \def 
+(eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) \to (\forall (P0: 
+Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 (\lambda (n3: nat).(or 
+(eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: Prop).P0)))) H0 n H3) 
+in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) n1).
+
+lemma simpl_plus_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) 
 (plus p n)) \to (eq nat m p))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat 
-(plus m n) (plus p n))).(let TMP_92 \def (plus m n) in (let TMP_91 \def 
-(\lambda (n0: nat).(let TMP_90 \def (plus n p) in (eq nat n0 TMP_90))) in 
-(let TMP_88 \def (plus p n) in (let TMP_87 \def (\lambda (n0: nat).(let 
-TMP_86 \def (plus n p) in (eq nat n0 TMP_86))) in (let TMP_85 \def (plus_sym 
-p n) in (let TMP_84 \def (plus m n) in (let TMP_89 \def (eq_ind_r nat TMP_88 
-TMP_87 TMP_85 TMP_84 H) in (let TMP_83 \def (plus n m) in (let TMP_82 \def 
-(plus_sym n m) in (let TMP_93 \def (eq_ind_r nat TMP_92 TMP_91 TMP_89 TMP_83 
-TMP_82) in (simpl_plus_l n m p TMP_93)))))))))))))).
-
-theorem minus_Sx_Sy:
+(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda 
+(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0: 
+nat).(eq nat n0 (plus n p))) (plus_sym p n) (plus m n) H) (plus n m) 
+(plus_sym n m)))))).
+
+lemma minus_Sx_Sy:
  \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
 \def
- \lambda (x: nat).(\lambda (y: nat).(let TMP_94 \def (minus x y) in 
-(refl_equal nat TMP_94))).
+ \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
 
-theorem minus_plus_r:
+lemma minus_plus_r:
  \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
 \def
- \lambda (m: nat).(\lambda (n: nat).(let TMP_100 \def (plus n m) in (let 
-TMP_99 \def (\lambda (n0: nat).(let TMP_98 \def (minus n0 n) in (eq nat 
-TMP_98 m))) in (let TMP_97 \def (minus_plus n m) in (let TMP_96 \def (plus m 
-n) in (let TMP_95 \def (plus_sym m n) in (eq_ind_r nat TMP_100 TMP_99 TMP_97 
-TMP_96 TMP_95))))))).
+ \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: 
+nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
 
-theorem plus_permute_2_in_3:
+lemma plus_permute_2_in_3:
  \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x 
 y) z) (plus (plus x z) y))))
 \def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(let TMP_127 \def (plus 
-y z) in (let TMP_128 \def (plus x TMP_127) in (let TMP_126 \def (\lambda (n: 
-nat).(let TMP_124 \def (plus x z) in (let TMP_125 \def (plus TMP_124 y) in 
-(eq nat n TMP_125)))) in (let TMP_122 \def (plus z y) in (let TMP_121 \def 
-(\lambda (n: nat).(let TMP_120 \def (plus x n) in (let TMP_118 \def (plus x 
-z) in (let TMP_119 \def (plus TMP_118 y) in (eq nat TMP_120 TMP_119))))) in 
-(let TMP_115 \def (plus x z) in (let TMP_116 \def (plus TMP_115 y) in (let 
-TMP_114 \def (\lambda (n: nat).(let TMP_112 \def (plus x z) in (let TMP_113 
-\def (plus TMP_112 y) in (eq nat n TMP_113)))) in (let TMP_109 \def (plus x 
-z) in (let TMP_110 \def (plus TMP_109 y) in (let TMP_111 \def (refl_equal nat 
-TMP_110) in (let TMP_107 \def (plus z y) in (let TMP_108 \def (plus x 
-TMP_107) in (let TMP_106 \def (plus_assoc_r x z y) in (let TMP_117 \def 
-(eq_ind nat TMP_116 TMP_114 TMP_111 TMP_108 TMP_106) in (let TMP_105 \def 
-(plus y z) in (let TMP_104 \def (plus_sym y z) in (let TMP_123 \def (eq_ind_r 
-nat TMP_122 TMP_121 TMP_117 TMP_105 TMP_104) in (let TMP_102 \def (plus x y) 
-in (let TMP_103 \def (plus TMP_102 z) in (let TMP_101 \def (plus_assoc_r x y 
-z) in (eq_ind_r nat TMP_128 TMP_126 TMP_123 TMP_103 
-TMP_101)))))))))))))))))))))))).
-
-theorem plus_permute_2_in_3_assoc:
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x 
+(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat 
+(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind 
+nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) 
+(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z 
+y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
+
+lemma plus_permute_2_in_3_assoc:
  \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n 
 h) k) (plus n (plus k h)))))
 \def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(let TMP_147 \def (plus 
-n k) in (let TMP_148 \def (plus TMP_147 h) in (let TMP_146 \def (\lambda (n0: 
-nat).(let TMP_144 \def (plus k h) in (let TMP_145 \def (plus n TMP_144) in 
-(eq nat n0 TMP_145)))) in (let TMP_141 \def (plus n k) in (let TMP_142 \def 
-(plus TMP_141 h) in (let TMP_140 \def (\lambda (n0: nat).(let TMP_138 \def 
-(plus n k) in (let TMP_139 \def (plus TMP_138 h) in (eq nat TMP_139 n0)))) in 
-(let TMP_135 \def (plus n k) in (let TMP_136 \def (plus TMP_135 h) in (let 
-TMP_137 \def (refl_equal nat TMP_136) in (let TMP_133 \def (plus k h) in (let 
-TMP_134 \def (plus n TMP_133) in (let TMP_132 \def (plus_assoc_l n k h) in 
-(let TMP_143 \def (eq_ind_r nat TMP_142 TMP_140 TMP_137 TMP_134 TMP_132) in 
-(let TMP_130 \def (plus n h) in (let TMP_131 \def (plus TMP_130 k) in (let 
-TMP_129 \def (plus_permute_2_in_3 n h k) in (eq_ind_r nat TMP_148 TMP_146 
-TMP_143 TMP_131 TMP_129))))))))))))))))))).
-
-theorem plus_O:
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus 
+(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r 
+nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0)) 
+(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k 
+h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
+
+lemma plus_O:
  \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat 
 x O) (eq nat y O))))
 \def
- \lambda (x: nat).(let TMP_164 \def (\lambda (n: nat).(\forall (y: nat).((eq 
-nat (plus n y) O) \to (let TMP_163 \def (eq nat n O) in (let TMP_162 \def (eq 
-nat y O) in (land TMP_163 TMP_162)))))) in (let TMP_161 \def (\lambda (y: 
-nat).(\lambda (H: (eq nat (plus O y) O)).(let TMP_160 \def (eq nat O O) in 
-(let TMP_159 \def (eq nat y O) in (let TMP_158 \def (refl_equal nat O) in 
-(conj TMP_160 TMP_159 TMP_158 H)))))) in (let TMP_157 \def (\lambda (n: 
-nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) \to (land (eq nat 
-n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq nat (plus (S n) y) 
-O)).(let H1 \def (match H0 in eq with [refl_equal \Rightarrow (\lambda (H1: 
-(eq nat (plus (S n) y) O)).(let TMP_150 \def (S n) in (let TMP_151 \def (plus 
-TMP_150 y) in (let TMP_149 \def (\lambda (e: nat).(match e in nat with [O 
-\Rightarrow False | (S _) \Rightarrow True])) in (let H2 \def (eq_ind nat 
-TMP_151 TMP_149 I O H1) in (let TMP_153 \def (S n) in (let TMP_154 \def (eq 
-nat TMP_153 O) in (let TMP_152 \def (eq nat y O) in (let TMP_155 \def (land 
-TMP_154 TMP_152) in (False_ind TMP_155 H2))))))))))]) in (let TMP_156 \def 
-(refl_equal nat O) in (H1 TMP_156))))))) in (nat_ind TMP_164 TMP_161 TMP_157 
-x)))).
-
-theorem minus_Sx_SO:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus 
+n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda 
+(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) 
+H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) 
+\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq 
+nat (plus (S n) y) O)).(let H1 \def (match H0 with [refl_equal \Rightarrow 
+(\lambda (H1: (eq nat (plus (S n) y) O)).(let H2 \def (eq_ind nat (plus (S n) 
+y) (\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y O)) H2)))]) in 
+(H1 (refl_equal nat O))))))) x).
+
+lemma minus_Sx_SO:
  \forall (x: nat).(eq nat (minus (S x) (S O)) x)
 \def
- \lambda (x: nat).(let TMP_168 \def (\lambda (n: nat).(eq nat n x)) in (let 
-TMP_167 \def (refl_equal nat x) in (let TMP_166 \def (minus x O) in (let 
-TMP_165 \def (minus_n_O x) in (eq_ind nat x TMP_168 TMP_167 TMP_166 
-TMP_165))))).
+ \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal 
+nat x) (minus x O) (minus_n_O x)).
 
-theorem nat_dec_neg:
+lemma nat_dec_neg:
  \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
 \def
- \lambda (i: nat).(let TMP_236 \def (\lambda (n: nat).(\forall (j: nat).(let 
-TMP_234 \def (eq nat n j) in (let TMP_235 \def (not TMP_234) in (let TMP_233 
-\def (eq nat n j) in (or TMP_235 TMP_233)))))) in (let TMP_232 \def (\lambda 
-(j: nat).(let TMP_231 \def (\lambda (n: nat).(let TMP_229 \def (eq nat O n) 
-in (let TMP_230 \def (not TMP_229) in (let TMP_228 \def (eq nat O n) in (or 
-TMP_230 TMP_228))))) in (let TMP_225 \def (eq nat O O) in (let TMP_226 \def 
-(not TMP_225) in (let TMP_224 \def (eq nat O O) in (let TMP_223 \def 
-(refl_equal nat O) in (let TMP_227 \def (or_intror TMP_226 TMP_224 TMP_223) 
-in (let TMP_222 \def (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) 
-(eq nat O n))).(let TMP_219 \def (S n) in (let TMP_220 \def (eq nat O 
-TMP_219) in (let TMP_221 \def (not TMP_220) in (let TMP_217 \def (S n) in 
-(let TMP_218 \def (eq nat O TMP_217) in (let TMP_216 \def (O_S n) in 
-(or_introl TMP_221 TMP_218 TMP_216))))))))) in (nat_ind TMP_231 TMP_227 
-TMP_222 j))))))))) in (let TMP_215 \def (\lambda (n: nat).(\lambda (H: 
-((\forall (j: nat).(or (not (eq nat n j)) (eq nat n j))))).(\lambda (j: 
-nat).(let TMP_214 \def (\lambda (n0: nat).(let TMP_211 \def (S n) in (let 
-TMP_212 \def (eq nat TMP_211 n0) in (let TMP_213 \def (not TMP_212) in (let 
-TMP_209 \def (S n) in (let TMP_210 \def (eq nat TMP_209 n0) in (or TMP_213 
-TMP_210))))))) in (let TMP_205 \def (S n) in (let TMP_206 \def (eq nat 
-TMP_205 O) in (let TMP_207 \def (not TMP_206) in (let TMP_203 \def (S n) in 
-(let TMP_204 \def (eq nat TMP_203 O) in (let TMP_201 \def (S n) in (let 
-TMP_200 \def (O_S n) in (let TMP_202 \def (sym_not_eq nat O TMP_201 TMP_200) 
-in (let TMP_208 \def (or_introl TMP_207 TMP_204 TMP_202) in (let TMP_199 \def 
-(\lambda (n0: nat).(\lambda (_: (or (not (eq nat (S n) n0)) (eq nat (S n) 
-n0))).(let TMP_197 \def (eq nat n n0) in (let TMP_198 \def (not TMP_197) in 
-(let TMP_196 \def (eq nat n n0) in (let TMP_192 \def (S n) in (let TMP_191 
-\def (S n0) in (let TMP_193 \def (eq nat TMP_192 TMP_191) in (let TMP_194 
-\def (not TMP_193) in (let TMP_189 \def (S n) in (let TMP_188 \def (S n0) in 
-(let TMP_190 \def (eq nat TMP_189 TMP_188) in (let TMP_195 \def (or TMP_194 
-TMP_190) in (let TMP_187 \def (\lambda (H1: (not (eq nat n n0))).(let TMP_184 
-\def (S n) in (let TMP_183 \def (S n0) in (let TMP_185 \def (eq nat TMP_184 
-TMP_183) in (let TMP_186 \def (not TMP_185) in (let TMP_181 \def (S n) in 
-(let TMP_180 \def (S n0) in (let TMP_182 \def (eq nat TMP_181 TMP_180) in 
-(let TMP_179 \def (not_eq_S n n0 H1) in (or_introl TMP_186 TMP_182 
-TMP_179)))))))))) in (let TMP_178 \def (\lambda (H1: (eq nat n n0)).(let 
-TMP_175 \def (S n) in (let TMP_174 \def (S n0) in (let TMP_176 \def (eq nat 
-TMP_175 TMP_174) in (let TMP_177 \def (not TMP_176) in (let TMP_172 \def (S 
-n) in (let TMP_171 \def (S n0) in (let TMP_173 \def (eq nat TMP_172 TMP_171) 
-in (let TMP_170 \def (f_equal nat nat S n n0 H1) in (or_intror TMP_177 
-TMP_173 TMP_170)))))))))) in (let TMP_169 \def (H n0) in (or_ind TMP_198 
-TMP_196 TMP_195 TMP_187 TMP_178 TMP_169))))))))))))))))) in (nat_ind TMP_214 
-TMP_208 TMP_199 j))))))))))))))) in (nat_ind TMP_236 TMP_232 TMP_215 i)))).
-
-theorem neq_eq_e:
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq 
+nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or 
+(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) 
+(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq 
+nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) 
+(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq 
+nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat 
+(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S 
+n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or 
+(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq 
+nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda 
+(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S 
+n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not 
+(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H 
+n0)))) j)))) i).
+
+lemma neq_eq_e:
  \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) 
 \to P)) \to ((((eq nat i j) \to P)) \to P))))
 \def
  \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not 
 (eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def 
-(nat_dec_neg i j) in (let TMP_238 \def (eq nat i j) in (let TMP_239 \def (not 
-TMP_238) in (let TMP_237 \def (eq nat i j) in (or_ind TMP_239 TMP_237 P H H0 
-o))))))))).
+(nat_dec_neg i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
 
-theorem le_false:
+lemma le_false:
  \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S 
 n) m) \to P))))
 \def
- \lambda (m: nat).(let TMP_262 \def (\lambda (n: nat).(\forall (n0: 
-nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P))))) in (let 
-TMP_261 \def (\lambda (n: nat).(\lambda (P: Prop).(\lambda (_: (le O 
-n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le with [le_n 
-\Rightarrow (\lambda (H1: (eq nat (S n) O)).(let TMP_259 \def (S n) in (let 
-TMP_258 \def (\lambda (e: nat).(match e in nat with [O \Rightarrow False | (S 
-_) \Rightarrow True])) in (let H2 \def (eq_ind nat TMP_259 TMP_258 I O H1) in 
-(False_ind P H2))))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0) 
-O)).(let TMP_255 \def (S m0) in (let TMP_254 \def (\lambda (e: nat).(match e 
-in nat with [O \Rightarrow False | (S _) \Rightarrow True])) in (let H3 \def 
-(eq_ind nat TMP_255 TMP_254 I O H2) in (let TMP_256 \def ((le (S n) m0) \to 
-P) in (let TMP_257 \def (False_ind TMP_256 H3) in (TMP_257 H1)))))))]) in 
-(let TMP_260 \def (refl_equal nat O) in (H1 TMP_260))))))) in (let TMP_253 
-\def (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: 
-Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(let 
-TMP_252 \def (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S 
-n1) (S n)) \to P)))) in (let TMP_251 \def (\lambda (P: Prop).(\lambda (H0: 
-(le (S n) O)).(\lambda (_: (le (S O) (S n))).(let H2 \def (match H0 in le 
-with [le_n \Rightarrow (\lambda (H2: (eq nat (S n) O)).(let TMP_249 \def (S 
-n) in (let TMP_248 \def (\lambda (e: nat).(match e in nat with [O \Rightarrow 
-False | (S _) \Rightarrow True])) in (let H3 \def (eq_ind nat TMP_249 TMP_248 
-I O H2) in (False_ind P H3))))) | (le_S m0 H2) \Rightarrow (\lambda (H3: (eq 
-nat (S m0) O)).(let TMP_245 \def (S m0) in (let TMP_244 \def (\lambda (e: 
-nat).(match e in nat with [O \Rightarrow False | (S _) \Rightarrow True])) in 
-(let H4 \def (eq_ind nat TMP_245 TMP_244 I O H3) in (let TMP_246 \def ((le (S 
-n) m0) \to P) in (let TMP_247 \def (False_ind TMP_246 H4) in (TMP_247 
-H2)))))))]) in (let TMP_250 \def (refl_equal nat O) in (H2 TMP_250)))))) in 
-(let TMP_243 \def (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S 
-n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda (P: Prop).(\lambda (H1: (le 
-(S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(let TMP_242 \def (le_S_n 
-n n1 H1) in (let TMP_240 \def (S n1) in (let TMP_241 \def (le_S_n TMP_240 n 
-H2) in (H n1 P TMP_242 TMP_241))))))))) in (nat_ind TMP_252 TMP_251 TMP_243 
-n0))))))) in (nat_ind TMP_262 TMP_261 TMP_253 m)))).
-
-theorem le_Sx_x:
+ \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: 
+Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: 
+Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match 
+H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def 
+(eq_ind nat (S n) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
+_) \Rightarrow True])) I O H1) in (False_ind P H2))) | (le_S m0 H1) 
+\Rightarrow (\lambda (H2: (eq nat (S m0) O)).((let H3 \def (eq_ind nat (S m0) 
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 
+(refl_equal nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: 
+nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda 
+(n0: nat).(nat_ind (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to 
+((le (S n1) (S n)) \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) 
+O)).(\lambda (_: (le (S O) (S n))).(let H2 \def (match H0 with [le_n 
+\Rightarrow (\lambda (H2: (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) 
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow (\lambda 
+(H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) 
+in (False_ind ((le (S n) m0) \to P) H4)) H2))]) in (H2 (refl_equal nat 
+O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S n) n1) 
+\to ((le (S n1) (S n)) \to P))))).(\lambda (P: Prop).(\lambda (H1: (le (S n) 
+(S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(H n1 P (le_S_n n n1 H1) 
+(le_S_n (S n1) n H2))))))) n0)))) m).
+
+lemma le_Sx_x:
  \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
 \def
  \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def 
-le_Sn_n in (let TMP_263 \def (H0 x H) in (False_ind P TMP_263))))).
+le_Sn_n in (False_ind P (H0 x H))))).
 
-theorem le_n_pred:
+lemma le_n_pred:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_272 \def 
-(\lambda (n0: nat).(let TMP_271 \def (pred n) in (let TMP_270 \def (pred n0) 
-in (le TMP_271 TMP_270)))) in (let TMP_268 \def (pred n) in (let TMP_269 \def 
-(le_n TMP_268) in (let TMP_267 \def (\lambda (m0: nat).(\lambda (_: (le n 
-m0)).(\lambda (H1: (le (pred n) (pred m0))).(let TMP_266 \def (pred n) in 
-(let TMP_265 \def (pred m0) in (let TMP_264 \def (le_pred_n m0) in (le_trans 
-TMP_266 TMP_265 m0 H1 TMP_264))))))) in (le_ind n TMP_272 TMP_269 TMP_267 m 
-H))))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
+(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0: 
+nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans 
+(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
 
-theorem minus_le:
+lemma minus_le:
  \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
 \def
- \lambda (x: nat).(let TMP_285 \def (\lambda (n: nat).(\forall (y: nat).(let 
-TMP_284 \def (minus n y) in (le TMP_284 n)))) in (let TMP_283 \def (\lambda 
-(_: nat).(le_O_n O)) in (let TMP_282 \def (\lambda (n: nat).(\lambda (H: 
-((\forall (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(let TMP_281 \def 
-(\lambda (n0: nat).(let TMP_279 \def (S n) in (let TMP_280 \def (minus 
-TMP_279 n0) in (let TMP_278 \def (S n) in (le TMP_280 TMP_278))))) in (let 
-TMP_276 \def (S n) in (let TMP_277 \def (le_n TMP_276) in (let TMP_275 \def 
-(\lambda (n0: nat).(\lambda (_: (le (match n0 with [O \Rightarrow (S n) | (S 
-l) \Rightarrow (minus n l)]) (S n))).(let TMP_274 \def (minus n n0) in (let 
-TMP_273 \def (H n0) in (le_S TMP_274 n TMP_273))))) in (nat_ind TMP_281 
-TMP_277 TMP_275 y)))))))) in (nat_ind TMP_285 TMP_283 TMP_282 x)))).
-
-theorem le_plus_minus_sym:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n 
+y) n))) (\lambda (_: nat).(le_O_n O)) (\lambda (n: nat).(\lambda (H: 
+((\forall (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda 
+(n0: nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: 
+nat).(\lambda (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow 
+(minus n l)]) (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
+
+lemma le_plus_minus_sym:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) 
 n))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_292 \def 
-(minus m n) in (let TMP_293 \def (plus n TMP_292) in (let TMP_291 \def 
-(\lambda (n0: nat).(eq nat m n0)) in (let TMP_290 \def (le_plus_minus n m H) 
-in (let TMP_288 \def (minus m n) in (let TMP_289 \def (plus TMP_288 n) in 
-(let TMP_286 \def (minus m n) in (let TMP_287 \def (plus_sym TMP_286 n) in 
-(eq_ind_r nat TMP_293 TMP_291 TMP_290 TMP_289 TMP_287))))))))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat 
+(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) 
+(plus (minus m n) n) (plus_sym (minus m n) n)))).
 
-theorem le_minus_minus:
+lemma le_minus_minus:
  \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) 
 \to (le (minus y x) (minus z x))))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z: 
-nat).(\lambda (H0: (le y z)).(let TMP_308 \def (minus y x) in (let TMP_307 
-\def (minus z x) in (let TMP_305 \def (\lambda (n: nat).(let TMP_303 \def 
-(minus z x) in (let TMP_304 \def (plus x TMP_303) in (le n TMP_304)))) in 
-(let TMP_301 \def (\lambda (n: nat).(le y n)) in (let TMP_299 \def (minus z 
-x) in (let TMP_300 \def (plus x TMP_299) in (let TMP_297 \def (le_trans x y z 
-H H0) in (let TMP_298 \def (le_plus_minus_r x z TMP_297) in (let TMP_302 \def 
-(eq_ind_r nat z TMP_301 H0 TMP_300 TMP_298) in (let TMP_295 \def (minus y x) 
-in (let TMP_296 \def (plus x TMP_295) in (let TMP_294 \def (le_plus_minus_r x 
-y H) in (let TMP_306 \def (eq_ind_r nat y TMP_305 TMP_302 TMP_296 TMP_294) in 
-(simpl_le_plus_l x TMP_308 TMP_307 TMP_306)))))))))))))))))).
-
-theorem le_minus_plus:
+nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x) 
+(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat 
+z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z 
+(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
+
+lemma le_minus_plus:
  \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat 
 (minus (plus x y) z) (plus (minus x z) y)))))
 \def
- \lambda (z: nat).(let TMP_368 \def (\lambda (n: nat).(\forall (x: nat).((le 
-n x) \to (\forall (y: nat).(let TMP_366 \def (plus x y) in (let TMP_367 \def 
-(minus TMP_366 n) in (let TMP_364 \def (minus x n) in (let TMP_365 \def (plus 
-TMP_364 y) in (eq nat TMP_367 TMP_365))))))))) in (let TMP_363 \def (\lambda 
-(x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le with [le_n 
-\Rightarrow (\lambda (H0: (eq nat O x)).(let TMP_361 \def (\lambda (n: 
-nat).(\forall (y: nat).(let TMP_359 \def (plus n y) in (let TMP_360 \def 
-(minus TMP_359 O) in (let TMP_357 \def (minus n O) in (let TMP_358 \def (plus 
-TMP_357 y) in (eq nat TMP_360 TMP_358))))))) in (let TMP_356 \def (\lambda 
-(y: nat).(let TMP_354 \def (minus O O) in (let TMP_355 \def (plus TMP_354 y) 
-in (let TMP_352 \def (plus O y) in (let TMP_353 \def (minus TMP_352 O) in 
-(let TMP_350 \def (plus O y) in (let TMP_351 \def (minus_n_O TMP_350) in 
-(sym_eq nat TMP_355 TMP_353 TMP_351)))))))) in (eq_ind nat O TMP_361 TMP_356 
-x H0)))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) x)).(let 
-TMP_349 \def (S m) in (let TMP_348 \def (\lambda (n: nat).((le O m) \to 
-(\forall (y: nat).(let TMP_346 \def (plus n y) in (let TMP_347 \def (minus 
-TMP_346 O) in (let TMP_344 \def (minus n O) in (let TMP_345 \def (plus 
-TMP_344 y) in (eq nat TMP_347 TMP_345)))))))) in (let TMP_343 \def (\lambda 
-(_: (le O m)).(\lambda (y: nat).(let TMP_340 \def (S m) in (let TMP_341 \def 
-(minus TMP_340 O) in (let TMP_342 \def (plus TMP_341 y) in (refl_equal nat 
-TMP_342)))))) in (eq_ind nat TMP_349 TMP_348 TMP_343 x H1 H0)))))]) in (let 
-TMP_362 \def (refl_equal nat x) in (H0 TMP_362))))) in (let TMP_339 \def 
-(\lambda (z0: nat).(\lambda (H: ((\forall (x: nat).((le z0 x) \to (\forall 
-(y: nat).(eq nat (minus (plus x y) z0) (plus (minus x z0) y))))))).(\lambda 
-(x: nat).(let TMP_338 \def (\lambda (n: nat).((le (S z0) n) \to (\forall (y: 
-nat).(let TMP_336 \def (plus n y) in (let TMP_335 \def (S z0) in (let TMP_337 
-\def (minus TMP_336 TMP_335) in (let TMP_332 \def (S z0) in (let TMP_333 \def 
-(minus n TMP_332) in (let TMP_334 \def (plus TMP_333 y) in (eq nat TMP_337 
-TMP_334)))))))))) in (let TMP_331 \def (\lambda (H0: (le (S z0) O)).(\lambda 
-(y: nat).(let H1 \def (match H0 in le with [le_n \Rightarrow (\lambda (H1: 
-(eq nat (S z0) O)).(let TMP_322 \def (S z0) in (let TMP_321 \def (\lambda (e: 
-nat).(match e in nat with [O \Rightarrow False | (S _) \Rightarrow True])) in 
-(let H2 \def (eq_ind nat TMP_322 TMP_321 I O H1) in (let TMP_327 \def (plus O 
-y) in (let TMP_326 \def (S z0) in (let TMP_328 \def (minus TMP_327 TMP_326) 
-in (let TMP_323 \def (S z0) in (let TMP_324 \def (minus O TMP_323) in (let 
-TMP_325 \def (plus TMP_324 y) in (let TMP_329 \def (eq nat TMP_328 TMP_325) 
-in (False_ind TMP_329 H2)))))))))))) | (le_S m H1) \Rightarrow (\lambda (H2: 
-(eq nat (S m) O)).(let TMP_312 \def (S m) in (let TMP_311 \def (\lambda (e: 
-nat).(match e in nat with [O \Rightarrow False | (S _) \Rightarrow True])) in 
-(let H3 \def (eq_ind nat TMP_312 TMP_311 I O H2) in (let TMP_319 \def ((le (S 
-z0) m) \to (let TMP_317 \def (plus O y) in (let TMP_316 \def (S z0) in (let 
-TMP_318 \def (minus TMP_317 TMP_316) in (let TMP_313 \def (S z0) in (let 
-TMP_314 \def (minus O TMP_313) in (let TMP_315 \def (plus TMP_314 y) in (eq 
-nat TMP_318 TMP_315)))))))) in (let TMP_320 \def (False_ind TMP_319 H3) in 
-(TMP_320 H1)))))))]) in (let TMP_330 \def (refl_equal nat O) in (H1 
-TMP_330))))) in (let TMP_310 \def (\lambda (n: nat).(\lambda (_: (((le (S z0) 
-n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n (S 
-z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(let TMP_309 
-\def (le_S_n z0 n H1) in (H n TMP_309 y)))))) in (nat_ind TMP_338 TMP_331 
-TMP_310 x))))))) in (nat_ind TMP_368 TMP_363 TMP_339 z)))).
-
-theorem le_minus:
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to 
+(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) 
+(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H with [le_n 
+\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: 
+nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) 
+(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) 
+(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq 
+nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: 
+nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O 
+m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) 
+in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: 
+nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus 
+(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S 
+z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n 
+(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def 
+(match H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 
+\def (eq_ind nat (S z0) (\lambda (e: nat).(match e with [O \Rightarrow False 
+| (S _) \Rightarrow True])) I O H1) in (False_ind (eq nat (minus (plus O y) 
+(S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) \Rightarrow (\lambda 
+(H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) 
+in (False_ind ((le (S z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus 
+(minus O (S z0)) y))) H3)) H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: 
+nat).(\lambda (_: (((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus 
+n y) (S z0)) (plus (minus n (S z0)) y)))))).(\lambda (H1: (le (S z0) (S 
+n))).(\lambda (y: nat).(H n (le_S_n z0 n H1) y))))) x)))) z).
+
+lemma le_minus:
  \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to 
 (le x (minus z y)))))
 \def
  \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus 
-x y) z)).(let TMP_375 \def (plus x y) in (let TMP_376 \def (minus TMP_375 y) 
-in (let TMP_374 \def (\lambda (n: nat).(let TMP_373 \def (minus z y) in (le n 
-TMP_373))) in (let TMP_371 \def (plus x y) in (let TMP_370 \def (le_plus_r x 
-y) in (let TMP_372 \def (le_minus_minus y TMP_371 TMP_370 z H) in (let 
-TMP_369 \def (minus_plus_r x y) in (eq_ind nat TMP_376 TMP_374 TMP_372 x 
-TMP_369))))))))))).
-
-theorem le_trans_plus_r:
+x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z 
+y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x 
+y))))).
+
+lemma le_trans_plus_r:
  \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to 
 (le y z))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus 
-x y) z)).(let TMP_378 \def (plus x y) in (let TMP_377 \def (le_plus_r x y) in 
-(le_trans y TMP_378 z TMP_377 H)))))).
+x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
 
-theorem lt_x_O:
+lemma lt_x_O:
  \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
 \def
- \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let TMP_379 
-\def (S x) in (let H_y \def (le_n_O_eq TMP_379 H) in (let TMP_381 \def 
-(\lambda (ee: nat).(match ee in nat with [O \Rightarrow True | (S _) 
-\Rightarrow False])) in (let TMP_380 \def (S x) in (let H0 \def (eq_ind nat O 
-TMP_381 I TMP_380 H_y) in (False_ind P H0)))))))).
+ \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def 
+(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match 
+ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) H_y) in 
+(False_ind P H0))))).
 
-theorem le_gen_S:
+lemma le_gen_S:
  \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: 
 nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
 \def
  \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def 
-(match H in le with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(let 
-TMP_409 \def (S m) in (let TMP_408 \def (\lambda (n: nat).(let TMP_407 \def 
-(\lambda (n0: nat).(let TMP_406 \def (S n0) in (eq nat n TMP_406))) in (let 
-TMP_405 \def (\lambda (n0: nat).(le m n0)) in (ex2 nat TMP_407 TMP_405)))) in 
-(let TMP_403 \def (\lambda (n: nat).(let TMP_402 \def (S m) in (let TMP_401 
-\def (S n) in (eq nat TMP_402 TMP_401)))) in (let TMP_400 \def (\lambda (n: 
-nat).(le m n)) in (let TMP_398 \def (S m) in (let TMP_399 \def (refl_equal 
-nat TMP_398) in (let TMP_397 \def (le_n m) in (let TMP_404 \def (ex_intro2 
-nat TMP_403 TMP_400 m TMP_399 TMP_397) in (eq_ind nat TMP_409 TMP_408 TMP_404 
-x H0)))))))))) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) 
-x)).(let TMP_396 \def (S m0) in (let TMP_395 \def (\lambda (n: nat).((le (S 
-m) m0) \to (let TMP_394 \def (\lambda (n0: nat).(let TMP_393 \def (S n0) in 
-(eq nat n TMP_393))) in (let TMP_392 \def (\lambda (n0: nat).(le m n0)) in 
-(ex2 nat TMP_394 TMP_392))))) in (let TMP_391 \def (\lambda (H2: (le (S m) 
-m0)).(let TMP_390 \def (\lambda (n: nat).(let TMP_389 \def (S m0) in (let 
-TMP_388 \def (S n) in (eq nat TMP_389 TMP_388)))) in (let TMP_387 \def 
-(\lambda (n: nat).(le m n)) in (let TMP_385 \def (S m0) in (let TMP_386 \def 
-(refl_equal nat TMP_385) in (let TMP_382 \def (S m) in (let TMP_383 \def 
-(le_S TMP_382 m0 H2) in (let TMP_384 \def (le_S_n m m0 TMP_383) in (ex_intro2 
-nat TMP_390 TMP_387 m0 TMP_386 TMP_384))))))))) in (eq_ind nat TMP_396 
-TMP_395 TMP_391 x H1 H0)))))]) in (let TMP_410 \def (refl_equal nat x) in (H0 
-TMP_410))))).
-
-theorem lt_x_plus_x_Sy:
+(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat 
+(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) 
+(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S 
+m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x 
+H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat 
+(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq 
+nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) 
+m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: 
+nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) 
+x H1 H0))]) in (H0 (refl_equal nat x))))).
+
+lemma lt_x_plus_x_Sy:
  \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
 \def
- \lambda (x: nat).(\lambda (y: nat).(let TMP_427 \def (S y) in (let TMP_428 
-\def (plus TMP_427 x) in (let TMP_426 \def (\lambda (n: nat).(lt x n)) in 
-(let TMP_424 \def (S x) in (let TMP_422 \def (plus y x) in (let TMP_423 \def 
-(S TMP_422) in (let TMP_420 \def (S x) in (let TMP_418 \def (plus y x) in 
-(let TMP_419 \def (S TMP_418) in (let TMP_416 \def (plus y x) in (let TMP_415 
-\def (le_plus_r y x) in (let TMP_417 \def (le_n_S x TMP_416 TMP_415) in (let 
-TMP_421 \def (le_n_S TMP_420 TMP_419 TMP_417) in (let TMP_425 \def (le_S_n 
-TMP_424 TMP_423 TMP_421) in (let TMP_413 \def (S y) in (let TMP_414 \def 
-(plus x TMP_413) in (let TMP_411 \def (S y) in (let TMP_412 \def (plus_sym x 
-TMP_411) in (eq_ind_r nat TMP_428 TMP_426 TMP_425 TMP_414 
-TMP_412)))))))))))))))))))).
-
-theorem simpl_lt_plus_r:
+ \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: 
+nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) 
+(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
+
+lemma simpl_lt_plus_r:
  \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m 
 p)) \to (lt n m))))
 \def
  \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus 
-n p) (plus m p))).(let TMP_433 \def (plus n p) in (let TMP_432 \def (\lambda 
-(n0: nat).(let TMP_431 \def (plus m p) in (lt n0 TMP_431))) in (let TMP_430 
-\def (plus p n) in (let TMP_429 \def (plus_sym n p) in (let H0 \def (eq_ind 
-nat TMP_433 TMP_432 H TMP_430 TMP_429) in (let TMP_438 \def (plus m p) in 
-(let TMP_437 \def (\lambda (n0: nat).(let TMP_436 \def (plus p n) in (lt 
-TMP_436 n0))) in (let TMP_435 \def (plus p m) in (let TMP_434 \def (plus_sym 
-m p) in (let H1 \def (eq_ind nat TMP_438 TMP_437 H0 TMP_435 TMP_434) in 
-(simpl_lt_plus_l n m p H1)))))))))))))).
-
-theorem minus_x_Sy:
+n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p) 
+(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let 
+H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 
+(plus p m) (plus_sym m p)) in H1)))))).
+
+lemma minus_x_Sy:
  \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S 
 (minus x (S y))))))
 \def
- \lambda (x: nat).(let TMP_478 \def (\lambda (n: nat).(\forall (y: nat).((lt 
-y n) \to (let TMP_477 \def (minus n y) in (let TMP_474 \def (S y) in (let 
-TMP_475 \def (minus n TMP_474) in (let TMP_476 \def (S TMP_475) in (eq nat 
-TMP_477 TMP_476)))))))) in (let TMP_473 \def (\lambda (y: nat).(\lambda (H: 
-(lt y O)).(let H0 \def (match H in le with [le_n \Rightarrow (\lambda (H0: 
-(eq nat (S y) O)).(let TMP_466 \def (S y) in (let TMP_465 \def (\lambda (e: 
-nat).(match e in nat with [O \Rightarrow False | (S _) \Rightarrow True])) in 
-(let H1 \def (eq_ind nat TMP_466 TMP_465 I O H0) in (let TMP_470 \def (minus 
-O y) in (let TMP_467 \def (S y) in (let TMP_468 \def (minus O TMP_467) in 
-(let TMP_469 \def (S TMP_468) in (let TMP_471 \def (eq nat TMP_470 TMP_469) 
-in (False_ind TMP_471 H1)))))))))) | (le_S m H0) \Rightarrow (\lambda (H1: 
-(eq nat (S m) O)).(let TMP_458 \def (S m) in (let TMP_457 \def (\lambda (e: 
-nat).(match e in nat with [O \Rightarrow False | (S _) \Rightarrow True])) in 
-(let H2 \def (eq_ind nat TMP_458 TMP_457 I O H1) in (let TMP_463 \def ((le (S 
-y) m) \to (let TMP_462 \def (minus O y) in (let TMP_459 \def (S y) in (let 
-TMP_460 \def (minus O TMP_459) in (let TMP_461 \def (S TMP_460) in (eq nat 
-TMP_462 TMP_461)))))) in (let TMP_464 \def (False_ind TMP_463 H2) in (TMP_464 
-H0)))))))]) in (let TMP_472 \def (refl_equal nat O) in (H0 TMP_472))))) in 
-(let TMP_456 \def (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) 
-\to (eq nat (minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(let 
-TMP_455 \def (\lambda (n0: nat).((lt n0 (S n)) \to (let TMP_453 \def (S n) in 
-(let TMP_454 \def (minus TMP_453 n0) in (let TMP_450 \def (S n) in (let 
-TMP_449 \def (S n0) in (let TMP_451 \def (minus TMP_450 TMP_449) in (let 
-TMP_452 \def (S TMP_451) in (eq nat TMP_454 TMP_452))))))))) in (let TMP_448 
-\def (\lambda (_: (lt O (S n))).(let TMP_447 \def (\lambda (n0: nat).(let 
-TMP_446 \def (S n) in (let TMP_445 \def (S n0) in (eq nat TMP_446 TMP_445)))) 
-in (let TMP_443 \def (S n) in (let TMP_444 \def (refl_equal nat TMP_443) in 
-(let TMP_442 \def (minus n O) in (let TMP_441 \def (minus_n_O n) in (eq_ind 
-nat n TMP_447 TMP_444 TMP_442 TMP_441))))))) in (let TMP_440 \def (\lambda 
-(n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus 
-(S n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let TMP_439 \def (S n0) 
-in (let H2 \def (le_S_n TMP_439 n H1) in (H n0 H2)))))) in (nat_ind TMP_455 
-TMP_448 TMP_440 y))))))) in (nat_ind TMP_478 TMP_473 TMP_456 x)))).
-
-theorem lt_plus_minus:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to 
+(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: 
+(lt y O)).(let H0 \def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat 
+(S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat 
+(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda 
+(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) 
+in (False_ind ((le (S y) m) \to (eq nat (minus O y) (S (minus O (S y))))) 
+H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H: 
+((\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S 
+y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to 
+(eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda (_: (lt O (S 
+n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S n0))) (refl_equal nat 
+(S n)) (minus n O) (minus_n_O n))) (\lambda (n0: nat).(\lambda (_: (((lt n0 
+(S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))).(\lambda 
+(H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) n H1) in (H n0 H2))))) 
+y)))) x).
+
+lemma lt_plus_minus:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus 
 y (S x)))))))
 \def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(let TMP_479 \def 
-(S x) in (le_plus_minus TMP_479 y H)))).
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S 
+x) y H))).
 
-theorem lt_plus_minus_r:
+lemma lt_plus_minus_r:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y 
 (S x)) x)))))
 \def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(let TMP_489 \def 
-(S x) in (let TMP_490 \def (minus y TMP_489) in (let TMP_491 \def (plus x 
-TMP_490) in (let TMP_488 \def (\lambda (n: nat).(let TMP_487 \def (S n) in 
-(eq nat y TMP_487))) in (let TMP_486 \def (lt_plus_minus x y H) in (let 
-TMP_483 \def (S x) in (let TMP_484 \def (minus y TMP_483) in (let TMP_485 
-\def (plus TMP_484 x) in (let TMP_480 \def (S x) in (let TMP_481 \def (minus 
-y TMP_480) in (let TMP_482 \def (plus_sym TMP_481 x) in (eq_ind_r nat TMP_491 
-TMP_488 TMP_486 TMP_485 TMP_482)))))))))))))).
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat 
+(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x 
+y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
 
-theorem minus_x_SO:
+lemma minus_x_SO:
  \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
 \def
- \lambda (x: nat).(\lambda (H: (lt O x)).(let TMP_502 \def (minus x O) in 
-(let TMP_501 \def (\lambda (n: nat).(eq nat x n)) in (let TMP_499 \def 
-(\lambda (n: nat).(eq nat x n)) in (let TMP_498 \def (refl_equal nat x) in 
-(let TMP_497 \def (minus x O) in (let TMP_496 \def (minus_n_O x) in (let 
-TMP_500 \def (eq_ind nat x TMP_499 TMP_498 TMP_497 TMP_496) in (let TMP_493 
-\def (S O) in (let TMP_494 \def (minus x TMP_493) in (let TMP_495 \def (S 
-TMP_494) in (let TMP_492 \def (minus_x_Sy x O H) in (eq_ind nat TMP_502 
-TMP_501 TMP_500 TMP_495 TMP_492))))))))))))).
+ \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: 
+nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal 
+nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
 
-theorem le_x_pred_y:
+lemma le_x_pred_y:
  \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
 \def
- \lambda (y: nat).(let TMP_514 \def (\lambda (n: nat).(\forall (x: nat).((lt 
-x n) \to (let TMP_513 \def (pred n) in (le x TMP_513))))) in (let TMP_512 
-\def (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def (match H in le 
-with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let TMP_509 \def (S 
-x) in (let TMP_508 \def (\lambda (e: nat).(match e in nat with [O \Rightarrow 
-False | (S _) \Rightarrow True])) in (let H1 \def (eq_ind nat TMP_509 TMP_508 
-I O H0) in (let TMP_510 \def (le x O) in (False_ind TMP_510 H1)))))) | (le_S 
-m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).(let TMP_505 \def (S m) in 
-(let TMP_504 \def (\lambda (e: nat).(match e in nat with [O \Rightarrow False 
-| (S _) \Rightarrow True])) in (let H2 \def (eq_ind nat TMP_505 TMP_504 I O 
-H1) in (let TMP_506 \def ((le (S x) m) \to (le x O)) in (let TMP_507 \def 
-(False_ind TMP_506 H2) in (TMP_507 H0)))))))]) in (let TMP_511 \def 
-(refl_equal nat O) in (H0 TMP_511))))) in (let TMP_503 \def (\lambda (n: 
-nat).(\lambda (_: ((\forall (x: nat).((lt x n) \to (le x (pred 
-n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S n))).(le_S_n x n H0))))) in 
-(nat_ind TMP_514 TMP_512 TMP_503 y)))).
-
-theorem lt_le_minus:
+ \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to 
+(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def 
+(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let H1 \def 
+(eq_ind nat (S x) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
+_) \Rightarrow True])) I O H0) in (False_ind (le x O) H1))) | (le_S m H0) 
+\Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) 
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 
+(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt 
+x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S 
+n))).(le_S_n x n H0))))) y).
+
+lemma lt_le_minus:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
 \def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(let TMP_523 \def 
-(S O) in (let TMP_520 \def (S O) in (let TMP_521 \def (plus TMP_520 x) in 
-(let TMP_519 \def (\lambda (n: nat).(le n y)) in (let TMP_517 \def (S O) in 
-(let TMP_518 \def (plus x TMP_517) in (let TMP_515 \def (S O) in (let TMP_516 
-\def (plus_sym x TMP_515) in (let TMP_522 \def (eq_ind_r nat TMP_521 TMP_519 
-H TMP_518 TMP_516) in (le_minus x y TMP_523 TMP_522)))))))))))).
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S 
+O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) 
+(plus_sym x (S O)))))).
 
-theorem lt_le_e:
+lemma lt_le_e:
  \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) 
 \to ((((le d n) \to P)) \to P))))
 \def
  \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n 
 d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in 
-(let TMP_525 \def (le d n) in (let TMP_524 \def (lt n d) in (or_ind TMP_525 
-TMP_524 P H0 H H1)))))))).
+(or_ind (le d n) (lt n d) P H0 H H1)))))).
 
-theorem lt_eq_e:
+lemma lt_eq_e:
  \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) 
 \to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x 
 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x 
-y)).(let TMP_528 \def (lt x y) in (let TMP_527 \def (eq nat x y) in (let 
-TMP_526 \def (le_lt_or_eq x y H1) in (or_ind TMP_528 TMP_527 P H H0 
-TMP_526))))))))).
+y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
 
-theorem lt_eq_gt_e:
+lemma lt_eq_gt_e:
  \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) 
 \to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x 
 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x) 
-\to P))).(let TMP_531 \def (\lambda (H2: (le y x)).(let TMP_530 \def (\lambda 
-(H3: (eq nat y x)).(let TMP_529 \def (sym_eq nat y x H3) in (H0 TMP_529))) in 
-(lt_eq_e y x P H1 TMP_530 H2))) in (lt_le_e x y P H TMP_531))))))).
+\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda 
+(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
 
-theorem lt_gen_xS:
+lemma lt_gen_xS:
  \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 
 nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
 \def
- \lambda (x: nat).(let TMP_561 \def (\lambda (n: nat).(\forall (n0: nat).((lt 
-n (S n0)) \to (let TMP_560 \def (eq nat n O) in (let TMP_558 \def (\lambda 
-(m: nat).(let TMP_557 \def (S m) in (eq nat n TMP_557))) in (let TMP_556 \def 
-(\lambda (m: nat).(lt m n0)) in (let TMP_559 \def (ex2 nat TMP_558 TMP_556) 
-in (or TMP_560 TMP_559)))))))) in (let TMP_555 \def (\lambda (n: 
-nat).(\lambda (_: (lt O (S n))).(let TMP_554 \def (eq nat O O) in (let 
-TMP_552 \def (\lambda (m: nat).(let TMP_551 \def (S m) in (eq nat O 
-TMP_551))) in (let TMP_550 \def (\lambda (m: nat).(lt m n)) in (let TMP_553 
-\def (ex2 nat TMP_552 TMP_550) in (let TMP_549 \def (refl_equal nat O) in 
-(or_introl TMP_554 TMP_553 TMP_549)))))))) in (let TMP_548 \def (\lambda (n: 
-nat).(\lambda (_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) 
-(ex2 nat (\lambda (m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m 
-n0)))))))).(\lambda (n0: nat).(\lambda (H0: (lt (S n) (S n0))).(let TMP_546 
-\def (S n) in (let TMP_547 \def (eq nat TMP_546 O) in (let TMP_544 \def 
-(\lambda (m: nat).(let TMP_543 \def (S n) in (let TMP_542 \def (S m) in (eq 
-nat TMP_543 TMP_542)))) in (let TMP_541 \def (\lambda (m: nat).(lt m n0)) in 
-(let TMP_545 \def (ex2 nat TMP_544 TMP_541) in (let TMP_539 \def (\lambda (m: 
-nat).(let TMP_538 \def (S n) in (let TMP_537 \def (S m) in (eq nat TMP_538 
-TMP_537)))) in (let TMP_536 \def (\lambda (m: nat).(lt m n0)) in (let TMP_534 
-\def (S n) in (let TMP_535 \def (refl_equal nat TMP_534) in (let TMP_532 \def 
-(S n) in (let TMP_533 \def (le_S_n TMP_532 n0 H0) in (let TMP_540 \def 
-(ex_intro2 nat TMP_539 TMP_536 n TMP_535 TMP_533) in (or_intror TMP_547 
-TMP_545 TMP_540))))))))))))))))) in (nat_ind TMP_561 TMP_555 TMP_548 x)))).
-
-theorem le_lt_false:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S 
+n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) 
+(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S 
+n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m))) 
+(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda 
+(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda 
+(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0: 
+nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat 
+(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0))) 
+(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt 
+m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
+
+lemma le_lt_false:
  \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: 
 Prop).P))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt 
-y x)).(\lambda (P: Prop).(let TMP_562 \def (le_not_lt x y H H0) in (False_ind 
-P TMP_562)))))).
+y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
 
-theorem lt_neq:
+lemma lt_neq:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
 \def
  \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq 
-nat x y)).(let TMP_563 \def (\lambda (n: nat).(lt n y)) in (let H1 \def 
-(eq_ind nat x TMP_563 H y H0) in (lt_n_n y H1)))))).
+nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in 
+(lt_n_n y H1))))).
 
-theorem arith0:
+lemma arith0:
  \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) 
 \to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
 \def
  \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le 
-(plus d2 h2) n)).(\lambda (h1: nat).(let TMP_602 \def (plus d2 h1) in (let 
-TMP_603 \def (plus h2 TMP_602) in (let TMP_604 \def (minus TMP_603 h2) in 
-(let TMP_601 \def (\lambda (n0: nat).(let TMP_599 \def (plus n h1) in (let 
-TMP_600 \def (minus TMP_599 h2) in (le n0 TMP_600)))) in (let TMP_596 \def 
-(plus d2 h1) in (let TMP_597 \def (plus h2 TMP_596) in (let TMP_594 \def 
-(plus d2 h1) in (let TMP_595 \def (le_plus_l h2 TMP_594) in (let TMP_593 \def 
-(plus n h1) in (let TMP_590 \def (plus h2 d2) in (let TMP_591 \def (plus 
-TMP_590 h1) in (let TMP_589 \def (\lambda (n0: nat).(let TMP_588 \def (plus n 
-h1) in (le n0 TMP_588))) in (let TMP_586 \def (plus d2 h2) in (let TMP_585 
-\def (\lambda (n0: nat).(let TMP_584 \def (plus n0 h1) in (let TMP_583 \def 
-(plus n h1) in (le TMP_584 TMP_583)))) in (let TMP_580 \def (plus d2 h2) in 
-(let TMP_581 \def (plus TMP_580 h1) in (let TMP_579 \def (plus n h1) in (let 
-TMP_576 \def (plus d2 h2) in (let TMP_577 \def (plus TMP_576 h1) in (let 
-TMP_575 \def (plus n h1) in (let TMP_573 \def (plus d2 h2) in (let TMP_572 
-\def (le_n h1) in (let TMP_574 \def (le_plus_plus TMP_573 n h1 h1 H TMP_572) 
-in (let TMP_578 \def (le_n_S TMP_577 TMP_575 TMP_574) in (let TMP_582 \def 
-(le_S_n TMP_581 TMP_579 TMP_578) in (let TMP_571 \def (plus h2 d2) in (let 
-TMP_570 \def (plus_sym h2 d2) in (let TMP_587 \def (eq_ind_r nat TMP_586 
-TMP_585 TMP_582 TMP_571 TMP_570) in (let TMP_568 \def (plus d2 h1) in (let 
-TMP_569 \def (plus h2 TMP_568) in (let TMP_567 \def (plus_assoc_l h2 d2 h1) 
-in (let TMP_592 \def (eq_ind_r nat TMP_591 TMP_589 TMP_587 TMP_569 TMP_567) 
-in (let TMP_598 \def (le_minus_minus h2 TMP_597 TMP_595 TMP_593 TMP_592) in 
-(let TMP_566 \def (plus d2 h1) in (let TMP_564 \def (plus d2 h1) in (let 
-TMP_565 \def (minus_plus h2 TMP_564) in (eq_ind nat TMP_604 TMP_601 TMP_598 
-TMP_566 TMP_565))))))))))))))))))))))))))))))))))))))))).
-
-theorem O_minus:
+(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1)) 
+h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 
+(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat 
+(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat 
+(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus 
+(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1) 
+(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2 
+d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1) 
+(minus_plus h2 (plus d2 h1))))))).
+
+lemma O_minus:
  \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
 \def
- \lambda (x: nat).(let TMP_624 \def (\lambda (n: nat).(\forall (y: nat).((le 
-n y) \to (let TMP_623 \def (minus n y) in (eq nat TMP_623 O))))) in (let 
-TMP_622 \def (\lambda (y: nat).(\lambda (_: (le O y)).(refl_equal nat O))) in 
-(let TMP_621 \def (\lambda (x0: nat).(\lambda (H: ((\forall (y: nat).((le x0 
-y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(let TMP_620 \def 
-(\lambda (n: nat).((le (S x0) n) \to (let TMP_619 \def (match n with [O 
-\Rightarrow (S x0) | (S l) \Rightarrow (minus x0 l)]) in (eq nat TMP_619 
-O)))) in (let TMP_618 \def (\lambda (H0: (le (S x0) O)).(let TMP_617 \def 
-(\lambda (n: nat).(let TMP_616 \def (S n) in (eq nat O TMP_616))) in (let 
-TMP_615 \def (\lambda (n: nat).(le x0 n)) in (let TMP_613 \def (S x0) in (let 
-TMP_614 \def (eq nat TMP_613 O) in (let TMP_612 \def (\lambda (x1: 
-nat).(\lambda (H1: (eq nat O (S x1))).(\lambda (_: (le x0 x1)).(let TMP_609 
-\def (\lambda (ee: nat).(match ee in nat with [O \Rightarrow True | (S _) 
-\Rightarrow False])) in (let TMP_608 \def (S x1) in (let H3 \def (eq_ind nat 
-O TMP_609 I TMP_608 H1) in (let TMP_610 \def (S x0) in (let TMP_611 \def (eq 
-nat TMP_610 O) in (False_ind TMP_611 H3))))))))) in (let TMP_607 \def 
-(le_gen_S x0 O H0) in (ex2_ind nat TMP_617 TMP_615 TMP_614 TMP_612 
-TMP_607)))))))) in (let TMP_606 \def (\lambda (n: nat).(\lambda (_: (((le (S 
-x0) n) \to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow 
-(minus x0 l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(let TMP_605 \def 
-(le_S_n x0 n H1) in (H n TMP_605))))) in (nat_ind TMP_620 TMP_618 TMP_606 
-y))))))) in (nat_ind TMP_624 TMP_622 TMP_621 x)))).
-
-theorem minus_minus:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
+(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O 
+y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y: 
+nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind 
+(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S 
+x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0) 
+O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0 
+n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S 
+x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) 
+H1) in (False_ind (eq nat (S x0) O) H3))))) (le_gen_S x0 O H0))) (\lambda (n: 
+nat).(\lambda (_: (((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S 
+x0) | (S l) \Rightarrow (minus x0 l)]) O)))).(\lambda (H1: (le (S x0) (S 
+n))).(H n (le_S_n x0 n H1))))) y)))) x).
+
+lemma minus_minus:
  \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) 
 \to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
 \def
- \lambda (z: nat).(let TMP_664 \def (\lambda (n: nat).(\forall (x: 
-nat).(\forall (y: nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) 
-(minus y n)) \to (eq nat x y))))))) in (let TMP_663 \def (\lambda (x: 
-nat).(\lambda (y: nat).(\lambda (_: (le O x)).(\lambda (_: (le O y)).(\lambda 
-(H1: (eq nat (minus x O) (minus y O))).(let TMP_659 \def (minus x O) in (let 
-TMP_658 \def (\lambda (n: nat).(let TMP_657 \def (minus y O) in (eq nat n 
-TMP_657))) in (let TMP_656 \def (minus_n_O x) in (let H2 \def (eq_ind_r nat 
-TMP_659 TMP_658 H1 x TMP_656) in (let TMP_662 \def (minus y O) in (let 
-TMP_661 \def (\lambda (n: nat).(eq nat x n)) in (let TMP_660 \def (minus_n_O 
-y) in (let H3 \def (eq_ind_r nat TMP_662 TMP_661 H2 y TMP_660) in 
-H3))))))))))))) in (let TMP_655 \def (\lambda (z0: nat).(\lambda (IH: 
-((\forall (x: nat).(\forall (y: nat).((le z0 x) \to ((le z0 y) \to ((eq nat 
-(minus x z0) (minus y z0)) \to (eq nat x y)))))))).(\lambda (x: nat).(let 
-TMP_654 \def (\lambda (n: nat).(\forall (y: nat).((le (S z0) n) \to ((le (S 
-z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) \to (eq nat n y)))))) 
-in (let TMP_653 \def (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda 
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y: 
+nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq 
+nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O 
+x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y 
+O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n 
+(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O) 
+(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda 
+(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to 
+((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x 
+y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le 
+(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) 
+\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda 
 (_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S 
-z0)))).(let TMP_652 \def (\lambda (n: nat).(let TMP_651 \def (S n) in (eq nat 
-O TMP_651))) in (let TMP_650 \def (\lambda (n: nat).(le z0 n)) in (let 
-TMP_649 \def (eq nat O y) in (let TMP_648 \def (\lambda (x0: nat).(\lambda 
-(H2: (eq nat O (S x0))).(\lambda (_: (le z0 x0)).(let TMP_646 \def (\lambda 
-(ee: nat).(match ee in nat with [O \Rightarrow True | (S _) \Rightarrow 
-False])) in (let TMP_645 \def (S x0) in (let H4 \def (eq_ind nat O TMP_646 I 
-TMP_645 H2) in (let TMP_647 \def (eq nat O y) in (False_ind TMP_647 
-H4)))))))) in (let TMP_644 \def (le_gen_S z0 O H) in (ex2_ind nat TMP_652 
-TMP_650 TMP_649 TMP_648 TMP_644)))))))))) in (let TMP_643 \def (\lambda (x0: 
+z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le 
+z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S 
+x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x0) 
+H2) in (False_ind (eq nat O y) H4))))) (le_gen_S z0 O H)))))) (\lambda (x0: 
 nat).(\lambda (_: ((\forall (y: nat).((le (S z0) x0) \to ((le (S z0) y) \to 
 ((eq nat (minus x0 (S z0)) (minus y (S z0))) \to (eq nat x0 y))))))).(\lambda 
-(y: nat).(let TMP_642 \def (\lambda (n: nat).((le (S z0) (S x0)) \to ((le (S 
-z0) n) \to ((eq nat (minus (S x0) (S z0)) (minus n (S z0))) \to (let TMP_641 
-\def (S x0) in (eq nat TMP_641 n)))))) in (let TMP_640 \def (\lambda (H: (le 
-(S z0) (S x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S 
-x0) (S z0)) (minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (let TMP_639 
-\def (\lambda (n: nat).(let TMP_638 \def (S n) in (eq nat O TMP_638))) in 
-(let TMP_637 \def (\lambda (n: nat).(le z0 n)) in (let TMP_635 \def (S x0) in 
-(let TMP_636 \def (eq nat TMP_635 O) in (let TMP_634 \def (\lambda (x1: 
-nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0 x1)).(let TMP_631 
-\def (\lambda (ee: nat).(match ee in nat with [O \Rightarrow True | (S _) 
-\Rightarrow False])) in (let TMP_630 \def (S x1) in (let H4 \def (eq_ind nat 
-O TMP_631 I TMP_630 H2) in (let TMP_632 \def (S x0) in (let TMP_633 \def (eq 
-nat TMP_632 O) in (False_ind TMP_633 H4))))))))) in (let TMP_629 \def 
-(le_gen_S z0 O H0) in (ex2_ind nat TMP_639 TMP_637 TMP_636 TMP_634 
-TMP_629))))))))))) in (let TMP_628 \def (\lambda (y0: nat).(\lambda (_: (((le 
-(S z0) (S x0)) \to ((le (S z0) y0) \to ((eq nat (minus (S x0) (S z0)) (minus 
-y0 (S z0))) \to (eq nat (S x0) y0)))))).(\lambda (H: (le (S z0) (S 
-x0))).(\lambda (H0: (le (S z0) (S y0))).(\lambda (H1: (eq nat (minus (S x0) 
-(S z0)) (minus (S y0) (S z0)))).(let TMP_626 \def (le_S_n z0 x0 H) in (let 
-TMP_625 \def (le_S_n z0 y0 H0) in (let TMP_627 \def (IH x0 y0 TMP_626 TMP_625 
-H1) in (f_equal nat nat S x0 y0 TMP_627))))))))) in (nat_ind TMP_642 TMP_640 
-TMP_628 y))))))) in (nat_ind TMP_654 TMP_653 TMP_643 x))))))) in (nat_ind 
-TMP_664 TMP_663 TMP_655 z)))).
-
-theorem plus_plus:
+(y: nat).(nat_ind (\lambda (n: nat).((le (S z0) (S x0)) \to ((le (S z0) n) 
+\to ((eq nat (minus (S x0) (S z0)) (minus n (S z0))) \to (eq nat (S x0) 
+n))))) (\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) O)).(\lambda 
+(_: (eq nat (minus (S x0) (S z0)) (minus O (S z0)))).(let H_y \def (le_S_n z0 
+x0 H) in (ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: 
+nat).(le z0 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H2: (eq nat O 
+(S x1))).(\lambda (_: (le z0 x1)).(let H4 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) 
+H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) (\lambda 
+(y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to ((eq nat 
+(minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0) y0)))))).(\lambda 
+(H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S y0))).(\lambda (H1: (eq 
+nat (minus (S x0) (S z0)) (minus (S y0) (S z0)))).(f_equal nat nat S x0 y0 
+(IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z).
+
+lemma plus_plus:
  \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: 
 nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z 
 x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
 \def
- \lambda (z: nat).(let TMP_755 \def (\lambda (n: nat).(\forall (x1: 
-nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to 
-((le x2 n) \to ((eq nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to 
-(let TMP_754 \def (plus x1 y2) in (let TMP_753 \def (plus x2 y1) in (eq nat 
-TMP_754 TMP_753))))))))))) in (let TMP_752 \def (\lambda (x1: nat).(\lambda 
-(x2: nat).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le x1 
-O)).(\lambda (H0: (le x2 O)).(\lambda (H1: (eq nat y1 y2)).(let TMP_751 \def 
-(\lambda (n: nat).(let TMP_750 \def (plus x1 n) in (let TMP_749 \def (plus x2 
-y1) in (eq nat TMP_750 TMP_749)))) in (let H_y \def (le_n_O_eq x2 H0) in (let 
-TMP_747 \def (\lambda (n: nat).(let TMP_746 \def (plus x1 y1) in (let TMP_745 
-\def (plus n y1) in (eq nat TMP_746 TMP_745)))) in (let H_y0 \def (le_n_O_eq 
-x1 H) in (let TMP_743 \def (\lambda (n: nat).(let TMP_742 \def (plus n y1) in 
-(let TMP_741 \def (plus O y1) in (eq nat TMP_742 TMP_741)))) in (let TMP_739 
-\def (plus O y1) in (let TMP_740 \def (refl_equal nat TMP_739) in (let 
-TMP_744 \def (eq_ind nat O TMP_743 TMP_740 x1 H_y0) in (let TMP_748 \def 
-(eq_ind nat O TMP_747 TMP_744 x2 H_y) in (eq_ind nat y1 TMP_751 TMP_748 y2 
-H1))))))))))))))))) in (let TMP_738 \def (\lambda (z0: nat).(\lambda (IH: 
-((\forall (x1: nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: 
-nat).((le x1 z0) \to ((le x2 z0) \to ((eq nat (plus (minus z0 x1) y1) (plus 
-(minus z0 x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1))))))))))).(\lambda 
-(x1: nat).(let TMP_737 \def (\lambda (n: nat).(\forall (x2: nat).(\forall 
-(y1: nat).(\forall (y2: nat).((le n (S z0)) \to ((le x2 (S z0)) \to ((eq nat 
-(plus (minus (S z0) n) y1) (plus (minus (S z0) x2) y2)) \to (let TMP_736 \def 
-(plus n y2) in (let TMP_735 \def (plus x2 y1) in (eq nat TMP_736 
-TMP_735)))))))))) in (let TMP_734 \def (\lambda (x2: nat).(let TMP_733 \def 
-(\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O (S z0)) \to 
-((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus (minus (S z0) n) 
-y2)) \to (let TMP_732 \def (plus O y2) in (let TMP_731 \def (plus n y1) in 
-(eq nat TMP_732 TMP_731))))))))) in (let TMP_730 \def (\lambda (y1: 
-nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O (S 
-z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y \def 
-(IH O O) in (let TMP_724 \def (minus z0 O) in (let TMP_723 \def (\lambda (n: 
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2: 
+nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq 
+nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2) 
+(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1: 
+nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2 
+O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat 
+(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O 
+(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def 
+(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus 
+O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1)))))))) 
+(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2: 
+nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to 
+((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus 
+x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n: 
+nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0)) 
+\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S 
+z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2: 
+nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O 
+(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus 
+(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda 
+(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O 
+(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y 
+\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: 
 nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq 
-nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) in (let TMP_722 \def 
-(minus_n_O z0) in (let H2 \def (eq_ind_r nat TMP_724 TMP_723 H_y z0 TMP_722) 
-in (let TMP_729 \def (le_O_n z0) in (let TMP_728 \def (le_O_n z0) in (let 
-TMP_726 \def (plus z0 y1) in (let TMP_725 \def (plus z0 y2) in (let TMP_727 
-\def (eq_add_S TMP_726 TMP_725 H1) in (H2 y1 y2 TMP_729 TMP_728 
-TMP_727)))))))))))))))) in (let TMP_721 \def (\lambda (x3: nat).(\lambda (_: 
-((\forall (y1: nat).(\forall (y2: nat).((le O (S z0)) \to ((le x3 (S z0)) \to 
-((eq nat (S (plus z0 y1)) (plus (match x3 with [O \Rightarrow (S z0) | (S l) 
-\Rightarrow (minus z0 l)]) y2)) \to (eq nat y2 (plus x3 y1))))))))).(\lambda 
-(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S 
-x3) (S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) 
-y2))).(let TMP_699 \def (S y1) in (let H_y \def (IH O x3 TMP_699) in (let 
-TMP_704 \def (minus z0 O) in (let TMP_703 \def (\lambda (n: nat).(\forall 
-(y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S y1)) (plus 
-(minus z0 x3) y3)) \to (let TMP_701 \def (S y1) in (let TMP_702 \def (plus x3 
-TMP_701) in (eq nat y3 TMP_702)))))))) in (let TMP_700 \def (minus_n_O z0) in 
-(let H2 \def (eq_ind_r nat TMP_704 TMP_703 H_y z0 TMP_700) in (let TMP_711 
-\def (S y1) in (let TMP_712 \def (plus z0 TMP_711) in (let TMP_710 \def 
-(\lambda (n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat 
-n (plus (minus z0 x3) y3)) \to (let TMP_708 \def (S y1) in (let TMP_709 \def 
-(plus x3 TMP_708) in (eq nat y3 TMP_709)))))))) in (let TMP_706 \def (plus z0 
-y1) in (let TMP_707 \def (S TMP_706) in (let TMP_705 \def (plus_n_Sm z0 y1) 
-in (let H3 \def (eq_ind_r nat TMP_712 TMP_710 H2 TMP_707 TMP_705) in (let 
-TMP_717 \def (S y1) in (let TMP_718 \def (plus x3 TMP_717) in (let TMP_716 
-\def (\lambda (n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq 
-nat (S (plus z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) in (let 
-TMP_714 \def (plus x3 y1) in (let TMP_715 \def (S TMP_714) in (let TMP_713 
-\def (plus_n_Sm x3 y1) in (let H4 \def (eq_ind_r nat TMP_718 TMP_716 H3 
-TMP_715 TMP_713) in (let TMP_720 \def (le_O_n z0) in (let TMP_719 \def 
-(le_S_n x3 z0 H0) in (H4 y2 TMP_720 TMP_719 H1)))))))))))))))))))))))))))))) 
-in (nat_ind TMP_733 TMP_730 TMP_721 x2))))) in (let TMP_698 \def (\lambda 
-(x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: nat).(\forall (y2: 
-nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat (plus (minus (S z0) x2) 
-y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 y2) (plus x3 
-y1)))))))))).(\lambda (x3: nat).(let TMP_697 \def (\lambda (n: nat).(\forall 
-(y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S z0)) \to ((eq 
-nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) \to (let 
-TMP_695 \def (S x2) in (let TMP_696 \def (plus TMP_695 y2) in (let TMP_694 
-\def (plus n y1) in (eq nat TMP_696 TMP_694)))))))))) in (let TMP_693 \def 
-(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S 
-z0))).(\lambda (_: (le O (S z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) 
-y1) (S (plus z0 y2)))).(let TMP_671 \def (S y2) in (let H_y \def (IH x2 O y1 
-TMP_671) in (let TMP_676 \def (minus z0 O) in (let TMP_675 \def (\lambda (n: 
-nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) (plus n 
-(S y2))) \to (let TMP_673 \def (S y2) in (let TMP_674 \def (plus x2 TMP_673) 
-in (eq nat TMP_674 y1))))))) in (let TMP_672 \def (minus_n_O z0) in (let H2 
-\def (eq_ind_r nat TMP_676 TMP_675 H_y z0 TMP_672) in (let TMP_683 \def (S 
-y2) in (let TMP_684 \def (plus z0 TMP_683) in (let TMP_682 \def (\lambda (n: 
-nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to 
-(let TMP_680 \def (S y2) in (let TMP_681 \def (plus x2 TMP_680) in (eq nat 
-TMP_681 y1))))))) in (let TMP_678 \def (plus z0 y2) in (let TMP_679 \def (S 
-TMP_678) in (let TMP_677 \def (plus_n_Sm z0 y2) in (let H3 \def (eq_ind_r nat 
-TMP_684 TMP_682 H2 TMP_679 TMP_677) in (let TMP_689 \def (S y2) in (let 
-TMP_690 \def (plus x2 TMP_689) in (let TMP_688 \def (\lambda (n: nat).((le x2 
-z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to 
-(eq nat n y1))))) in (let TMP_686 \def (plus x2 y2) in (let TMP_687 \def (S 
-TMP_686) in (let TMP_685 \def (plus_n_Sm x2 y2) in (let H4 \def (eq_ind_r nat 
-TMP_690 TMP_688 H3 TMP_687 TMP_685) in (let TMP_692 \def (le_S_n x2 z0 H) in 
-(let TMP_691 \def (le_O_n z0) in (H4 TMP_692 TMP_691 
-H1)))))))))))))))))))))))))))) in (let TMP_670 \def (\lambda (x4: 
-nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) 
-\to ((le x4 (S z0)) \to ((eq nat (plus (minus z0 x2) y1) (plus (match x4 with 
-[O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2)) \to (eq nat (S 
-(plus x2 y2)) (plus x4 y1))))))))).(\lambda (y1: nat).(\lambda (y2: 
-nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (H0: (le (S x4) (S 
-z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (plus (minus z0 x4) 
-y2))).(let TMP_669 \def (plus x2 y2) in (let TMP_668 \def (plus x4 y1) in 
-(let TMP_666 \def (le_S_n x2 z0 H) in (let TMP_665 \def (le_S_n x4 z0 H0) in 
-(let TMP_667 \def (IH x2 x4 y1 y2 TMP_666 TMP_665 H1) in (f_equal nat nat S 
-TMP_669 TMP_668 TMP_667))))))))))))) in (nat_ind TMP_697 TMP_693 TMP_670 
-x3))))))) in (nat_ind TMP_737 TMP_734 TMP_698 x1))))))) in (nat_ind TMP_755 
-TMP_752 TMP_738 z)))).
-
-theorem le_S_minus:
+nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) 
+in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2) 
+H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: 
+nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus 
+(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2)) 
+\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2: 
+nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda 
+(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O 
+x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: 
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S 
+y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 
+(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: 
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus 
+(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) 
+(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda 
+(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus 
+z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) 
+(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) 
+x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: 
+nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat 
+(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 
+y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n: 
+nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S 
+z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) 
+\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda 
+(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S 
+z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let 
+H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O) 
+(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) 
+y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O 
+z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2 
+z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus 
+x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def 
+(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0) 
+\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) 
+H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0) 
+H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall 
+(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus 
+z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow 
+(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4 
+y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S 
+z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0 
+x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4 
+y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) 
+x1)))) z).
+
+lemma le_S_minus:
  \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to 
 (le d (S (minus n h))))))
 \def
  \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus 
-d h) n)).(let TMP_757 \def (plus d h) in (let TMP_756 \def (le_plus_l d h) in 
-(let H0 \def (le_trans d TMP_757 n TMP_756 H) in (let TMP_764 \def (\lambda 
-(n0: nat).(le d n0)) in (let TMP_762 \def (minus n h) in (let TMP_763 \def 
-(plus TMP_762 h) in (let TMP_759 \def (plus d h) in (let TMP_758 \def 
-(le_plus_r d h) in (let TMP_760 \def (le_trans h TMP_759 n TMP_758 H) in (let 
-TMP_761 \def (le_plus_minus_sym h n TMP_760) in (let H1 \def (eq_ind nat n 
-TMP_764 H0 TMP_763 TMP_761) in (let TMP_766 \def (minus n h) in (let TMP_765 
-\def (le_minus d n h H) in (le_S d TMP_766 TMP_765))))))))))))))))).
-
-theorem lt_x_pred_y:
+d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 
+\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) 
+(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S 
+d (minus n h) (le_minus d n h H))))))).
+
+lemma lt_x_pred_y:
  \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
 \def
- \lambda (x: nat).(\lambda (y: nat).(let TMP_772 \def (\lambda (n: nat).((lt 
-x (pred n)) \to (let TMP_771 \def (S x) in (lt TMP_771 n)))) in (let TMP_770 
-\def (\lambda (H: (lt x O)).(let TMP_768 \def (S x) in (let TMP_769 \def (lt 
-TMP_768 O) in (lt_x_O x H TMP_769)))) in (let TMP_767 \def (\lambda (n: 
-nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda (H0: (lt x 
-n)).(lt_n_S x n H0)))) in (nat_ind TMP_772 TMP_770 TMP_767 y))))).
+ \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred 
+n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O))) 
+(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda 
+(H0: (lt x n)).(lt_n_S x n H0)))) y)).