X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Flegacy_1%2Fcoq%2Fprops.ma;h=b5069fdf73511272af8581577af1cd7420ac0b2b;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=79b6a373344a460c3035a947dd3271a2b31d966d;hpb=9c954a9a843ebb1bf189536df4e14f77132ed1cf;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma b/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma
index 79b6a3733..b5069fdf7 100644
--- a/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma
+++ b/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma
@@ -16,17 +16,15 @@
 
 include "legacy_1/coq/fwd.ma".
 
-theorem f_equal:
+lemma f_equal:
  \forall (A: Type[0]).(\forall (B: Type[0]).(\forall (f: ((A \to 
 B))).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y)))))))
 \def
  \lambda (A: Type[0]).(\lambda (B: Type[0]).(\lambda (f: ((A \to 
-B))).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x y)).(let TMP_12 
-\def (\lambda (a: A).(let TMP_11 \def (f x) in (let TMP_10 \def (f a) in (eq 
-B TMP_11 TMP_10)))) in (let TMP_8 \def (f x) in (let TMP_9 \def (refl_equal B 
-TMP_8) in (eq_ind A x TMP_12 TMP_9 y H))))))))).
+B))).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x 
+(\lambda (a: A).(eq B (f x) (f a))) (refl_equal B (f x)) y H)))))).
 
-theorem f_equal2:
+lemma f_equal2:
  \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (B: Type[0]).(\forall 
 (f: ((A1 \to (A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: 
 A2).(\forall (y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) 
@@ -34,15 +32,12 @@ A2).(\forall (y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2)
 \def
  \lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda (B: Type[0]).(\lambda 
 (f: ((A1 \to (A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2: 
-A2).(\lambda (y2: A2).(\lambda (H: (eq A1 x1 y1)).(let TMP_21 \def (\lambda 
-(a: A1).((eq A2 x2 y2) \to (let TMP_20 \def (f x1 x2) in (let TMP_19 \def (f 
-a y2) in (eq B TMP_20 TMP_19))))) in (let TMP_18 \def (\lambda (H0: (eq A2 x2 
-y2)).(let TMP_17 \def (\lambda (a: A2).(let TMP_16 \def (f x1 x2) in (let 
-TMP_15 \def (f x1 a) in (eq B TMP_16 TMP_15)))) in (let TMP_13 \def (f x1 x2) 
-in (let TMP_14 \def (refl_equal B TMP_13) in (eq_ind A2 x2 TMP_17 TMP_14 y2 
-H0))))) in (eq_ind A1 x1 TMP_21 TMP_18 y1 H))))))))))).
-
-theorem f_equal3:
+A2).(\lambda (y2: A2).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: 
+A1).((eq A2 x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 
+y2)).(eq_ind A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B 
+(f x1 x2)) y2 H0)) y1 H))))))))).
+
+lemma f_equal3:
  \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (A3: Type[0]).(\forall 
 (B: Type[0]).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: 
 A1).(\forall (y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: 
@@ -52,52 +47,45 @@ A3).(\forall (y3: A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3)
  \lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda (A3: Type[0]).(\lambda 
 (B: Type[0]).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1: 
 A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3: 
-A3).(\lambda (y3: A3).(\lambda (H: (eq A1 x1 y1)).(let TMP_34 \def (\lambda 
-(a: A1).((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (let TMP_33 \def (f x1 x2 x3) 
-in (let TMP_32 \def (f a y2 y3) in (eq B TMP_33 TMP_32)))))) in (let TMP_31 
-\def (\lambda (H0: (eq A2 x2 y2)).(let TMP_30 \def (\lambda (a: A2).((eq A3 
-x3 y3) \to (let TMP_29 \def (f x1 x2 x3) in (let TMP_28 \def (f x1 a y3) in 
-(eq B TMP_29 TMP_28))))) in (let TMP_27 \def (\lambda (H1: (eq A3 x3 
-y3)).(let TMP_26 \def (\lambda (a: A3).(let TMP_25 \def (f x1 x2 x3) in (let 
-TMP_24 \def (f x1 x2 a) in (eq B TMP_25 TMP_24)))) in (let TMP_22 \def (f x1 
-x2 x3) in (let TMP_23 \def (refl_equal B TMP_22) in (eq_ind A3 x3 TMP_26 
-TMP_23 y3 H1))))) in (eq_ind A2 x2 TMP_30 TMP_27 y2 H0)))) in (eq_ind A1 x1 
-TMP_34 TMP_31 y1 H)))))))))))))).
-
-theorem sym_eq:
+A3).(\lambda (y3: A3).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: 
+A1).((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3))))) 
+(\lambda (H0: (eq A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) 
+\to (eq B (f x1 x2 x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 
+x3 (\lambda (a: A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 
+x3)) y3 H1)) y2 H0)) y1 H)))))))))))).
+
+lemma sym_eq:
  \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y 
 x))))
 \def
  \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x 
-y)).(let TMP_36 \def (\lambda (a: A).(eq A a x)) in (let TMP_35 \def 
-(refl_equal A x) in (eq_ind A x TMP_36 TMP_35 y H)))))).
+y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))).
 
-theorem eq_ind_r:
+lemma eq_ind_r:
  \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to 
 (\forall (y: A).((eq A y x) \to (P y))))))
 \def
  \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda 
 (H: (P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) 
-in eq with [refl_equal \Rightarrow H])))))).
+with [refl_equal \Rightarrow H])))))).
 
-theorem trans_eq:
+lemma trans_eq:
  \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A 
 x y) \to ((eq A y z) \to (eq A x z))))))
 \def
  \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (z: 
-A).(\lambda (H: (eq A x y)).(\lambda (H0: (eq A y z)).(let TMP_37 \def 
-(\lambda (a: A).(eq A x a)) in (eq_ind A y TMP_37 H z H0))))))).
+A).(\lambda (H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda 
+(a: A).(eq A x a)) H z H0)))))).
 
-theorem sym_not_eq:
+lemma sym_not_eq:
  \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to 
 (not (eq A y x)))))
 \def
  \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq 
-A x y))).(\lambda (h2: (eq A y x)).(let TMP_39 \def (\lambda (a: A).(eq A a 
-y)) in (let TMP_38 \def (refl_equal A y) in (let TMP_40 \def (eq_ind A y 
-TMP_39 TMP_38 x h2) in (h1 TMP_40)))))))).
+A x y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a 
+y)) (refl_equal A y) x h2)))))).
 
-theorem nat_double_ind:
+lemma nat_double_ind:
  \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to 
 (((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m: 
 nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m: 
@@ -106,664 +94,504 @@ nat).(R n m))))))
  \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O 
 n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall 
 (n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n: 
-nat).(let TMP_47 \def (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) in (let 
-TMP_46 \def (\lambda (n0: nat).(\lambda (H2: ((\forall (m: nat).(R n0 
-m)))).(\lambda (m: nat).(let TMP_45 \def (\lambda (n1: nat).(let TMP_44 \def 
-(S n0) in (R TMP_44 n1))) in (let TMP_43 \def (H0 n0) in (let TMP_42 \def 
-(\lambda (n1: nat).(\lambda (_: (R (S n0) n1)).(let TMP_41 \def (H2 n1) in 
-(H1 n0 n1 TMP_41)))) in (nat_ind TMP_45 TMP_43 TMP_42 m))))))) in (nat_ind 
-TMP_47 H TMP_46 n))))))).
-
-theorem eq_add_S:
+nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0: 
+nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind 
+(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R 
+(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))).
+
+lemma eq_add_S:
  \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m)))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S m))).(let 
-TMP_49 \def (S n) in (let TMP_48 \def (S m) in (f_equal nat nat pred TMP_49 
-TMP_48 H))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S 
+m))).(f_equal nat nat pred (S n) (S m) H))).
 
-theorem O_S:
+lemma O_S:
  \forall (n: nat).(not (eq nat O (S n)))
 \def
- \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(let TMP_53 \def (S n) in 
-(let TMP_52 \def (\lambda (n0: nat).(IsSucc n0)) in (let TMP_50 \def (S n) in 
-(let TMP_51 \def (sym_eq nat O TMP_50 H) in (eq_ind nat TMP_53 TMP_52 I O 
-TMP_51)))))).
+ \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda 
+(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))).
 
-theorem not_eq_S:
+lemma not_eq_S:
  \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S 
 n) (S m)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda 
-(H0: (eq nat (S n) (S m))).(let TMP_54 \def (eq_add_S n m H0) in (H 
-TMP_54))))).
+(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))).
 
-theorem pred_Sn:
+lemma pred_Sn:
  \forall (m: nat).(eq nat m (pred (S m)))
 \def
- \lambda (m: nat).(let TMP_55 \def (S m) in (let TMP_56 \def (pred TMP_55) in 
-(refl_equal nat TMP_56))).
+ \lambda (m: nat).(refl_equal nat (pred (S m))).
 
-theorem S_pred:
+lemma S_pred:
  \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n)))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(let TMP_68 \def 
-(S m) in (let TMP_67 \def (\lambda (n0: nat).(let TMP_65 \def (pred n0) in 
-(let TMP_66 \def (S TMP_65) in (eq nat n0 TMP_66)))) in (let TMP_61 \def (S 
-m) in (let TMP_62 \def (pred TMP_61) in (let TMP_63 \def (S TMP_62) in (let 
-TMP_64 \def (refl_equal nat TMP_63) in (let TMP_60 \def (\lambda (m0: 
-nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 (S (pred m0)))).(let 
-TMP_57 \def (S m0) in (let TMP_58 \def (pred TMP_57) in (let TMP_59 \def (S 
-TMP_58) in (refl_equal nat TMP_59))))))) in (le_ind TMP_68 TMP_67 TMP_64 
-TMP_60 n H)))))))))).
-
-theorem le_trans:
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m) 
+(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S 
+m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 
+(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))).
+
+lemma le_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) 
 \to (le n p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
-m)).(\lambda (H0: (le m p)).(let TMP_70 \def (\lambda (n0: nat).(le n n0)) in 
-(let TMP_69 \def (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: 
-(le n m0)).(le_S n m0 IHle)))) in (le_ind m TMP_70 H TMP_69 p H0))))))).
+m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H 
+(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n 
+m0 IHle)))) p H0))))).
 
-theorem le_trans_S:
+lemma le_trans_S:
  \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m)))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(let TMP_73 
-\def (S n) in (let TMP_71 \def (le_n n) in (let TMP_72 \def (le_S n n TMP_71) 
-in (le_trans n TMP_73 m TMP_72 H)))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S 
+n) m (le_S n n (le_n n)) H))).
 
-theorem le_n_S:
+lemma le_n_S:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_81 \def 
-(\lambda (n0: nat).(let TMP_80 \def (S n) in (let TMP_79 \def (S n0) in (le 
-TMP_80 TMP_79)))) in (let TMP_77 \def (S n) in (let TMP_78 \def (le_n TMP_77) 
-in (let TMP_76 \def (\lambda (m0: nat).(\lambda (_: (le n m0)).(\lambda 
-(IHle: (le (S n) (S m0))).(let TMP_75 \def (S n) in (let TMP_74 \def (S m0) 
-in (le_S TMP_75 TMP_74 IHle)))))) in (le_ind n TMP_81 TMP_78 TMP_76 m 
-H))))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
+(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le 
+n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))).
 
-theorem le_O_n:
+lemma le_O_n:
  \forall (n: nat).(le O n)
 \def
- \lambda (n: nat).(let TMP_84 \def (\lambda (n0: nat).(le O n0)) in (let 
-TMP_83 \def (le_n O) in (let TMP_82 \def (\lambda (n0: nat).(\lambda (IHn: 
-(le O n0)).(le_S O n0 IHn))) in (nat_ind TMP_84 TMP_83 TMP_82 n)))).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda 
+(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n).
 
-theorem le_S_n:
+lemma le_S_n:
  \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m)))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(let 
-TMP_92 \def (S n) in (let TMP_91 \def (\lambda (n0: nat).(let TMP_89 \def (S 
-n) in (let TMP_90 \def (pred TMP_89) in (let TMP_88 \def (pred n0) in (le 
-TMP_90 TMP_88))))) in (let TMP_87 \def (le_n n) in (let TMP_86 \def (\lambda 
-(m0: nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred 
-m0))).(le_trans_S n m0 H0)))) in (let TMP_85 \def (S m) in (le_ind TMP_92 
-TMP_91 TMP_87 TMP_86 TMP_85 H)))))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S 
+n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0: 
+nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S 
+n m0 H0)))) (S m) H))).
 
-theorem le_Sn_O:
+lemma le_Sn_O:
  \forall (n: nat).(not (le (S n) O))
 \def
- \lambda (n: nat).(\lambda (H: (le (S n) O)).(let TMP_95 \def (S n) in (let 
-TMP_94 \def (\lambda (n0: nat).(IsSucc n0)) in (let TMP_93 \def (\lambda (m: 
-nat).(\lambda (_: (le (S n) m)).(\lambda (_: (IsSucc m)).I))) in (le_ind 
-TMP_95 TMP_94 I TMP_93 O H))))).
+ \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: 
+nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: 
+(IsSucc m)).I))) O H)).
 
-theorem le_Sn_n:
+lemma le_Sn_n:
  \forall (n: nat).(not (le (S n) n))
 \def
- \lambda (n: nat).(let TMP_102 \def (\lambda (n0: nat).(let TMP_100 \def (S 
-n0) in (let TMP_101 \def (le TMP_100 n0) in (not TMP_101)))) in (let TMP_99 
-\def (le_Sn_O O) in (let TMP_98 \def (\lambda (n0: nat).(\lambda (IHn: (not 
-(le (S n0) n0))).(\lambda (H: (le (S (S n0)) (S n0))).(let TMP_96 \def (S n0) 
-in (let TMP_97 \def (le_S_n TMP_96 n0 H) in (IHn TMP_97)))))) in (nat_ind 
-TMP_102 TMP_99 TMP_98 n)))).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O 
+O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le 
+(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n).
 
-theorem le_antisym:
+lemma le_antisym:
  \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n 
 m))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(let TMP_110 \def 
-(\lambda (n0: nat).((le n0 n) \to (eq nat n n0))) in (let TMP_109 \def 
-(\lambda (_: (le n n)).(refl_equal nat n)) in (let TMP_108 \def (\lambda (m0: 
-nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) \to (eq nat n 
-m0)))).(\lambda (H1: (le (S m0) n)).(let TMP_106 \def (S m0) in (let TMP_107 
-\def (eq nat n TMP_106) in (let TMP_103 \def (S m0) in (let H2 \def (le_trans 
-TMP_103 n m0 H1 H) in (let H3 \def (le_Sn_n m0) in (let TMP_104 \def (\lambda 
-(H4: (le (S m0) m0)).(H3 H4)) in (let TMP_105 \def (TMP_104 H2) in (False_ind 
-TMP_107 TMP_105)))))))))))) in (le_ind n TMP_110 TMP_109 TMP_108 m h)))))).
-
-theorem le_n_O_eq:
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(le_ind n (\lambda 
+(n0: nat).((le n0 n) \to (eq nat n n0))) (\lambda (_: (le n n)).(refl_equal 
+nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) 
+\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S 
+m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) 
+in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))).
+
+lemma le_n_O_eq:
  \forall (n: nat).((le n O) \to (eq nat O n))
 \def
- \lambda (n: nat).(\lambda (H: (le n O)).(let TMP_111 \def (le_O_n n) in 
-(le_antisym O n TMP_111 H))).
+ \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)).
 
-theorem le_elim_rel:
+lemma le_elim_rel:
  \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to 
 (((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S 
 q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m))))))
 \def
  \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O 
 p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p 
-q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(let TMP_125 \def (\lambda 
-(n0: nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) in (let TMP_124 \def 
-(\lambda (m: nat).(\lambda (_: (le O m)).(H m))) in (let TMP_123 \def 
-(\lambda (n0: nat).(\lambda (IHn: ((\forall (m: nat).((le n0 m) \to (P n0 
-m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0) m)).(let TMP_122 \def (S 
-n0) in (let TMP_121 \def (\lambda (n1: nat).(let TMP_120 \def (S n0) in (P 
-TMP_120 n1))) in (let TMP_118 \def (le_n n0) in (let TMP_116 \def (le_n n0) 
-in (let TMP_117 \def (IHn n0 TMP_116) in (let TMP_119 \def (H0 n0 n0 TMP_118 
-TMP_117) in (let TMP_115 \def (\lambda (m0: nat).(\lambda (H1: (le (S n0) 
-m0)).(\lambda (_: (P (S n0) m0)).(let TMP_114 \def (le_trans_S n0 m0 H1) in 
-(let TMP_112 \def (le_trans_S n0 m0 H1) in (let TMP_113 \def (IHn m0 TMP_112) 
-in (H0 n0 m0 TMP_114 TMP_113))))))) in (le_ind TMP_122 TMP_121 TMP_119 
-TMP_115 m Le)))))))))))) in (nat_ind TMP_125 TMP_124 TMP_123 n))))))).
-
-theorem lt_n_n:
+q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(nat_ind (\lambda (n0: 
+nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) (\lambda (m: nat).(\lambda 
+(_: (le O m)).(H m))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (m: 
+nat).((le n0 m) \to (P n0 m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0) 
+m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn 
+n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: 
+(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 
+H1)))))) m Le))))) n)))).
+
+lemma lt_n_n:
  \forall (n: nat).(not (lt n n))
 \def
  le_Sn_n.
 
-theorem lt_n_S:
+lemma lt_n_S:
  \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(let TMP_126 \def 
-(S n) in (le_n_S TMP_126 m H)))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m 
+H))).
 
-theorem lt_n_Sn:
+lemma lt_n_Sn:
  \forall (n: nat).(lt n (S n))
 \def
- \lambda (n: nat).(let TMP_127 \def (S n) in (le_n TMP_127)).
+ \lambda (n: nat).(le_n (S n)).
 
-theorem lt_S_n:
+lemma lt_S_n:
  \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m)))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(let 
-TMP_128 \def (S n) in (le_S_n TMP_128 m H)))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S 
+n) m H))).
 
-theorem lt_n_O:
+lemma lt_n_O:
  \forall (n: nat).(not (lt n O))
 \def
  le_Sn_O.
 
-theorem lt_trans:
+lemma lt_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) 
 \to (lt n p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
-m)).(\lambda (H0: (lt m p)).(let TMP_134 \def (S m) in (let TMP_133 \def 
-(\lambda (n0: nat).(lt n n0)) in (let TMP_131 \def (S n) in (let TMP_132 \def 
-(le_S TMP_131 m H) in (let TMP_130 \def (\lambda (m0: nat).(\lambda (_: (le 
-(S m) m0)).(\lambda (IHle: (lt n m0)).(let TMP_129 \def (S n) in (le_S 
-TMP_129 m0 IHle))))) in (le_ind TMP_134 TMP_133 TMP_132 TMP_130 p 
-H0)))))))))).
-
-theorem lt_O_Sn:
+m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S 
+(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt 
+n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+
+lemma lt_O_Sn:
  \forall (n: nat).(lt O (S n))
 \def
- \lambda (n: nat).(let TMP_135 \def (le_O_n n) in (le_n_S O n TMP_135)).
+ \lambda (n: nat).(le_n_S O n (le_O_n n)).
 
-theorem lt_le_S:
+lemma lt_le_S:
  \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p)))
 \def
  \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)).
 
-theorem le_not_lt:
+lemma le_not_lt:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_141 \def 
-(\lambda (n0: nat).(let TMP_140 \def (lt n0 n) in (not TMP_140))) in (let 
-TMP_139 \def (lt_n_n n) in (let TMP_138 \def (\lambda (m0: nat).(\lambda (_: 
-(le n m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(let 
-TMP_136 \def (S m0) in (let TMP_137 \def (le_trans_S TMP_136 n H1) in (IHle 
-TMP_137))))))) in (le_ind n TMP_141 TMP_139 TMP_138 m H)))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
+(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n 
+m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle 
+(le_trans_S (S m0) n H1)))))) m H))).
 
-theorem le_lt_n_Sm:
+lemma le_lt_n_Sm:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))).
 
-theorem le_lt_trans:
+lemma le_lt_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) 
 \to (lt n p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
-m)).(\lambda (H0: (lt m p)).(let TMP_146 \def (S m) in (let TMP_145 \def 
-(\lambda (n0: nat).(lt n n0)) in (let TMP_144 \def (le_n_S n m H) in (let 
-TMP_143 \def (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: 
-(lt n m0)).(let TMP_142 \def (S n) in (le_S TMP_142 m0 IHle))))) in (le_ind 
-TMP_146 TMP_145 TMP_144 TMP_143 p H0))))))))).
+m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) 
+(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: 
+(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))).
 
-theorem lt_le_trans:
+lemma lt_le_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) 
 \to (lt n p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
-m)).(\lambda (H0: (le m p)).(let TMP_149 \def (\lambda (n0: nat).(lt n n0)) 
-in (let TMP_148 \def (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda 
-(IHle: (lt n m0)).(let TMP_147 \def (S n) in (le_S TMP_147 m0 IHle))))) in 
-(le_ind m TMP_149 H TMP_148 p H0))))))).
+m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H 
+(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S 
+(S n) m0 IHle)))) p H0))))).
 
-theorem lt_le_weak:
+lemma lt_le_weak:
  \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m 
 H))).
 
-theorem lt_n_Sm_le:
+lemma lt_n_Sm_le:
  \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m 
 H))).
 
-theorem le_lt_or_eq:
+lemma le_lt_or_eq:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_162 \def 
-(\lambda (n0: nat).(let TMP_161 \def (lt n n0) in (let TMP_160 \def (eq nat n 
-n0) in (or TMP_161 TMP_160)))) in (let TMP_158 \def (lt n n) in (let TMP_157 
-\def (eq nat n n) in (let TMP_156 \def (refl_equal nat n) in (let TMP_159 
-\def (or_intror TMP_158 TMP_157 TMP_156) in (let TMP_155 \def (\lambda (m0: 
-nat).(\lambda (H0: (le n m0)).(\lambda (_: (or (lt n m0) (eq nat n m0))).(let 
-TMP_153 \def (S m0) in (let TMP_154 \def (lt n TMP_153) in (let TMP_151 \def 
-(S m0) in (let TMP_152 \def (eq nat n TMP_151) in (let TMP_150 \def (le_n_S n 
-m0 H0) in (or_introl TMP_154 TMP_152 TMP_150))))))))) in (le_ind n TMP_162 
-TMP_159 TMP_155 m H))))))))).
-
-theorem le_or_lt:
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
+(n0: nat).(or (lt n n0) (eq nat n n0))) (or_intror (lt n n) (eq nat n n) 
+(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_: 
+(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) 
+(le_n_S n m0 H0))))) m H))).
+
+lemma le_or_lt:
  \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n)))
 \def
- \lambda (n: nat).(\lambda (m: nat).(let TMP_204 \def (\lambda (n0: 
-nat).(\lambda (n1: nat).(let TMP_203 \def (le n0 n1) in (let TMP_202 \def (lt 
-n1 n0) in (or TMP_203 TMP_202))))) in (let TMP_201 \def (\lambda (n0: 
-nat).(let TMP_200 \def (le O n0) in (let TMP_199 \def (lt n0 O) in (let 
-TMP_198 \def (le_O_n n0) in (or_introl TMP_200 TMP_199 TMP_198))))) in (let 
-TMP_197 \def (\lambda (n0: nat).(let TMP_195 \def (S n0) in (let TMP_196 \def 
-(le TMP_195 O) in (let TMP_193 \def (S n0) in (let TMP_194 \def (lt O 
-TMP_193) in (let TMP_191 \def (S n0) in (let TMP_190 \def (lt_O_Sn n0) in 
-(let TMP_192 \def (lt_le_S O TMP_191 TMP_190) in (or_intror TMP_196 TMP_194 
-TMP_192))))))))) in (let TMP_189 \def (\lambda (n0: nat).(\lambda (m0: 
-nat).(\lambda (H: (or (le n0 m0) (lt m0 n0))).(let TMP_188 \def (le n0 m0) in 
-(let TMP_187 \def (lt m0 n0) in (let TMP_184 \def (S n0) in (let TMP_183 \def 
-(S m0) in (let TMP_185 \def (le TMP_184 TMP_183) in (let TMP_181 \def (S m0) 
-in (let TMP_180 \def (S n0) in (let TMP_182 \def (lt TMP_181 TMP_180) in (let 
-TMP_186 \def (or TMP_185 TMP_182) in (let TMP_179 \def (\lambda (H0: (le n0 
-m0)).(let TMP_177 \def (S n0) in (let TMP_176 \def (S m0) in (let TMP_178 
-\def (le TMP_177 TMP_176) in (let TMP_174 \def (S m0) in (let TMP_173 \def (S 
-n0) in (let TMP_175 \def (lt TMP_174 TMP_173) in (let TMP_172 \def (le_n_S n0 
-m0 H0) in (or_introl TMP_178 TMP_175 TMP_172))))))))) in (let TMP_171 \def 
-(\lambda (H0: (lt m0 n0)).(let TMP_169 \def (S n0) in (let TMP_168 \def (S 
-m0) in (let TMP_170 \def (le TMP_169 TMP_168) in (let TMP_166 \def (S m0) in 
-(let TMP_165 \def (S n0) in (let TMP_167 \def (lt TMP_166 TMP_165) in (let 
-TMP_163 \def (S m0) in (let TMP_164 \def (le_n_S TMP_163 n0 H0) in (or_intror 
-TMP_170 TMP_167 TMP_164)))))))))) in (or_ind TMP_188 TMP_187 TMP_186 TMP_179 
-TMP_171 H))))))))))))))) in (nat_double_ind TMP_204 TMP_201 TMP_197 TMP_189 n 
-m)))))).
-
-theorem plus_n_O:
+ \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0: 
+nat).(\lambda (n1: nat).(or (le n0 n1) (lt n1 n0)))) (\lambda (n0: 
+nat).(or_introl (le O n0) (lt n0 O) (le_O_n n0))) (\lambda (n0: 
+nat).(or_intror (le (S n0) O) (lt O (S n0)) (lt_le_S O (S n0) (lt_O_Sn n0)))) 
+(\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 
+n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S 
+n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S 
+n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S 
+m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)).
+
+lemma plus_n_O:
  \forall (n: nat).(eq nat n (plus n O))
 \def
- \lambda (n: nat).(let TMP_209 \def (\lambda (n0: nat).(let TMP_208 \def 
-(plus n0 O) in (eq nat n0 TMP_208))) in (let TMP_207 \def (refl_equal nat O) 
-in (let TMP_206 \def (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 
-O))).(let TMP_205 \def (plus n0 O) in (f_equal nat nat S n0 TMP_205 H)))) in 
-(nat_ind TMP_209 TMP_207 TMP_206 n)))).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) 
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 
+O))).(f_equal nat nat S n0 (plus n0 O) H))) n).
 
-theorem plus_n_Sm:
+lemma plus_n_Sm:
  \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m))))
 \def
- \lambda (m: nat).(\lambda (n: nat).(let TMP_221 \def (\lambda (n0: nat).(let 
-TMP_219 \def (plus n0 n) in (let TMP_220 \def (S TMP_219) in (let TMP_217 
-\def (S n) in (let TMP_218 \def (plus n0 TMP_217) in (eq nat TMP_220 
-TMP_218)))))) in (let TMP_215 \def (S n) in (let TMP_216 \def (refl_equal nat 
-TMP_215) in (let TMP_214 \def (\lambda (n0: nat).(\lambda (H: (eq nat (S 
-(plus n0 n)) (plus n0 (S n)))).(let TMP_212 \def (plus n0 n) in (let TMP_213 
-\def (S TMP_212) in (let TMP_210 \def (S n) in (let TMP_211 \def (plus n0 
-TMP_210) in (f_equal nat nat S TMP_213 TMP_211 H))))))) in (nat_ind TMP_221 
-TMP_216 TMP_214 m)))))).
-
-theorem plus_sym:
+ \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S 
+(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0: 
+nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat 
+S (S (plus n0 n)) (plus n0 (S n)) H))) m)).
+
+lemma plus_sym:
  \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n)))
 \def
- \lambda (n: nat).(\lambda (m: nat).(let TMP_237 \def (\lambda (n0: nat).(let 
-TMP_236 \def (plus n0 m) in (let TMP_235 \def (plus m n0) in (eq nat TMP_236 
-TMP_235)))) in (let TMP_234 \def (plus_n_O m) in (let TMP_233 \def (\lambda 
-(y: nat).(\lambda (H: (eq nat (plus y m) (plus m y))).(let TMP_231 \def (plus 
-m y) in (let TMP_232 \def (S TMP_231) in (let TMP_230 \def (\lambda (n0: 
-nat).(let TMP_228 \def (plus y m) in (let TMP_229 \def (S TMP_228) in (eq nat 
-TMP_229 n0)))) in (let TMP_226 \def (plus y m) in (let TMP_225 \def (plus m 
-y) in (let TMP_227 \def (f_equal nat nat S TMP_226 TMP_225 H) in (let TMP_223 
-\def (S y) in (let TMP_224 \def (plus m TMP_223) in (let TMP_222 \def 
-(plus_n_Sm m y) in (eq_ind nat TMP_232 TMP_230 TMP_227 TMP_224 
-TMP_222)))))))))))) in (nat_ind TMP_237 TMP_234 TMP_233 n))))).
-
-theorem plus_Snm_nSm:
+ \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus 
+n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus 
+y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S 
+(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) 
+(plus_n_Sm m y)))) n)).
+
+lemma plus_Snm_nSm:
  \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(let TMP_257 \def (plus m n) in (let 
-TMP_256 \def (\lambda (n0: nat).(let TMP_255 \def (S n0) in (let TMP_253 \def 
-(S m) in (let TMP_254 \def (plus n TMP_253) in (eq nat TMP_255 TMP_254))))) 
-in (let TMP_250 \def (S m) in (let TMP_251 \def (plus TMP_250 n) in (let 
-TMP_249 \def (\lambda (n0: nat).(let TMP_247 \def (plus m n) in (let TMP_248 
-\def (S TMP_247) in (eq nat TMP_248 n0)))) in (let TMP_244 \def (S m) in (let 
-TMP_245 \def (plus TMP_244 n) in (let TMP_246 \def (refl_equal nat TMP_245) 
-in (let TMP_242 \def (S m) in (let TMP_243 \def (plus n TMP_242) in (let 
-TMP_240 \def (S m) in (let TMP_241 \def (plus_sym n TMP_240) in (let TMP_252 
-\def (eq_ind_r nat TMP_251 TMP_249 TMP_246 TMP_243 TMP_241) in (let TMP_239 
-\def (plus n m) in (let TMP_238 \def (plus_sym n m) in (eq_ind_r nat TMP_257 
-TMP_256 TMP_252 TMP_239 TMP_238))))))))))))))))).
-
-theorem plus_assoc_l:
+ \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0: 
+nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda 
+(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n 
+(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))).
+
+lemma plus_assoc_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m 
 p)) (plus (plus n m) p))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_269 \def 
-(\lambda (n0: nat).(let TMP_267 \def (plus m p) in (let TMP_268 \def (plus n0 
-TMP_267) in (let TMP_265 \def (plus n0 m) in (let TMP_266 \def (plus TMP_265 
-p) in (eq nat TMP_268 TMP_266)))))) in (let TMP_263 \def (plus m p) in (let 
-TMP_264 \def (refl_equal nat TMP_263) in (let TMP_262 \def (\lambda (n0: 
-nat).(\lambda (H: (eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))).(let 
-TMP_260 \def (plus m p) in (let TMP_261 \def (plus n0 TMP_260) in (let 
-TMP_258 \def (plus n0 m) in (let TMP_259 \def (plus TMP_258 p) in (f_equal 
-nat nat S TMP_261 TMP_259 H))))))) in (nat_ind TMP_269 TMP_264 TMP_262 
-n))))))).
-
-theorem plus_assoc_r:
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
+nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat 
+(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p)) 
+(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 
+m) p) H))) n))).
+
+lemma plus_assoc_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n 
 m) p) (plus n (plus m p)))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_273 \def (plus 
-m p) in (let TMP_274 \def (plus n TMP_273) in (let TMP_271 \def (plus n m) in 
-(let TMP_272 \def (plus TMP_271 p) in (let TMP_270 \def (plus_assoc_l n m p) 
-in (sym_eq nat TMP_274 TMP_272 TMP_270)))))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n 
+(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))).
 
-theorem simpl_plus_l:
+lemma simpl_plus_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) 
 (plus n p)) \to (eq nat m p))))
 \def
- \lambda (n: nat).(let TMP_287 \def (\lambda (n0: nat).(\forall (m: 
-nat).(\forall (p: nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) 
-in (let TMP_286 \def (\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat 
-m p)).H))) in (let TMP_285 \def (\lambda (n0: nat).(\lambda (IHn: ((\forall 
-(m: nat).(\forall (p: nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m 
-p)))))).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat (S (plus n0 
-m)) (S (plus n0 p)))).(let TMP_283 \def (plus n0 m) in (let TMP_282 \def 
-(plus n0 p) in (let TMP_280 \def (plus n0) in (let TMP_279 \def (plus n0 m) 
-in (let TMP_278 \def (plus n0 p) in (let TMP_276 \def (plus n0 m) in (let 
-TMP_275 \def (plus n0 p) in (let TMP_277 \def (eq_add_S TMP_276 TMP_275 H) in 
-(let TMP_281 \def (f_equal nat nat TMP_280 TMP_279 TMP_278 TMP_277) in (let 
-TMP_284 \def (IHn TMP_283 TMP_282 TMP_281) in (IHn m p 
-TMP_284)))))))))))))))) in (nat_ind TMP_287 TMP_286 TMP_285 n)))).
-
-theorem minus_n_O:
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(\forall (p: 
+nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) (\lambda (m: 
+nat).(\lambda (p: nat).(\lambda (H: (eq nat m p)).H))) (\lambda (n0: 
+nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m) 
+(plus n0 p)) \to (eq nat m p)))))).(\lambda (m: nat).(\lambda (p: 
+nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn 
+(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) 
+(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n).
+
+lemma minus_n_O:
  \forall (n: nat).(eq nat n (minus n O))
 \def
- \lambda (n: nat).(let TMP_292 \def (\lambda (n0: nat).(let TMP_291 \def 
-(minus n0 O) in (eq nat n0 TMP_291))) in (let TMP_290 \def (refl_equal nat O) 
-in (let TMP_289 \def (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 
-O))).(let TMP_288 \def (S n0) in (refl_equal nat TMP_288)))) in (nat_ind 
-TMP_292 TMP_290 TMP_289 n)))).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) 
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 
+O))).(refl_equal nat (S n0)))) n).
 
-theorem minus_n_n:
+lemma minus_n_n:
  \forall (n: nat).(eq nat O (minus n n))
 \def
- \lambda (n: nat).(let TMP_296 \def (\lambda (n0: nat).(let TMP_295 \def 
-(minus n0 n0) in (eq nat O TMP_295))) in (let TMP_294 \def (refl_equal nat O) 
-in (let TMP_293 \def (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 
-n0))).IHn)) in (nat_ind TMP_296 TMP_294 TMP_293 n)))).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) 
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 
+n0))).IHn)) n).
 
-theorem minus_Sn_m:
+lemma minus_Sn_m:
  \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) 
 (minus (S n) m))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(let TMP_307 \def 
-(\lambda (n0: nat).(\lambda (n1: nat).(let TMP_305 \def (minus n1 n0) in (let 
-TMP_306 \def (S TMP_305) in (let TMP_303 \def (S n1) in (let TMP_304 \def 
-(minus TMP_303 n0) in (eq nat TMP_306 TMP_304))))))) in (let TMP_302 \def 
-(\lambda (p: nat).(let TMP_301 \def (minus p O) in (let TMP_299 \def (minus p 
-O) in (let TMP_298 \def (minus_n_O p) in (let TMP_300 \def (sym_eq nat p 
-TMP_299 TMP_298) in (f_equal nat nat S TMP_301 p TMP_300)))))) in (let 
-TMP_297 \def (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: (le p 
-q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow (S q) 
-| (S l) \Rightarrow (minus q l)]))).H0)))) in (le_elim_rel TMP_307 TMP_302 
-TMP_297 m n Le)))))).
-
-theorem plus_minus:
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(le_elim_rel 
+(\lambda (n0: nat).(\lambda (n1: nat).(eq nat (S (minus n1 n0)) (minus (S n1) 
+n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p 
+(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: 
+(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow 
+(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))).
+
+lemma plus_minus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) 
 \to (eq nat p (minus n m)))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_322 \def 
-(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (let 
-TMP_321 \def (minus n1 n0) in (eq nat p TMP_321))))) in (let TMP_320 \def 
-(\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(let TMP_319 \def (\lambda 
-(n1: nat).(eq nat p n1)) in (let TMP_318 \def (sym_eq nat n0 p H) in (let 
-TMP_317 \def (minus n0 O) in (let TMP_316 \def (minus_n_O n0) in (eq_ind nat 
-n0 TMP_319 TMP_318 TMP_317 TMP_316))))))) in (let TMP_315 \def (\lambda (n0: 
-nat).(\lambda (H: (eq nat O (S (plus n0 p)))).(let TMP_314 \def (eq nat p O) 
-in (let H0 \def H in (let TMP_311 \def (plus n0 p) in (let H1 \def (O_S 
-TMP_311) in (let TMP_312 \def (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 
-H2)) in (let TMP_313 \def (TMP_312 H0) in (False_ind TMP_314 TMP_313))))))))) 
-in (let TMP_310 \def (\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq 
-nat m0 (plus n0 p)) \to (eq nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S 
-m0) (S (plus n0 p)))).(let TMP_308 \def (plus n0 p) in (let TMP_309 \def 
-(eq_add_S m0 TMP_308 H0) in (H TMP_309))))))) in (nat_double_ind TMP_322 
-TMP_320 TMP_315 TMP_310 m n))))))).
-
-theorem minus_plus:
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_double_ind 
+(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (eq nat p 
+(minus n1 n0))))) (\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(eq_ind nat 
+n0 (\lambda (n1: nat).(eq nat p n1)) (sym_eq nat n0 p H) (minus n0 O) 
+(minus_n_O n0)))) (\lambda (n0: nat).(\lambda (H: (eq nat O (S (plus n0 
+p)))).(False_ind (eq nat p O) (let H0 \def H in ((let H1 \def (O_S (plus n0 
+p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda 
+(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq 
+nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H 
+(eq_add_S m0 (plus n0 p) H0)))))) m n))).
+
+lemma minus_plus:
  \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m))
 \def
- \lambda (n: nat).(\lambda (m: nat).(let TMP_327 \def (plus n m) in (let 
-TMP_328 \def (minus TMP_327 n) in (let TMP_325 \def (plus n m) in (let 
-TMP_323 \def (plus n m) in (let TMP_324 \def (refl_equal nat TMP_323) in (let 
-TMP_326 \def (plus_minus TMP_325 n m TMP_324) in (sym_eq nat m TMP_328 
-TMP_326)))))))).
+ \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) 
+(plus_minus (plus n m) n m (refl_equal nat (plus n m))))).
 
-theorem le_pred_n:
+lemma le_pred_n:
  \forall (n: nat).(le (pred n) n)
 \def
- \lambda (n: nat).(let TMP_335 \def (\lambda (n0: nat).(let TMP_334 \def 
-(pred n0) in (le TMP_334 n0))) in (let TMP_333 \def (le_n O) in (let TMP_332 
-\def (\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(let TMP_330 \def (S 
-n0) in (let TMP_331 \def (pred TMP_330) in (let TMP_329 \def (le_n n0) in 
-(le_S TMP_331 n0 TMP_329)))))) in (nat_ind TMP_335 TMP_333 TMP_332 n)))).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) 
+(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 
+(le_n n0)))) n).
 
-theorem le_plus_l:
+lemma le_plus_l:
  \forall (n: nat).(\forall (m: nat).(le n (plus n m)))
 \def
- \lambda (n: nat).(let TMP_341 \def (\lambda (n0: nat).(\forall (m: nat).(let 
-TMP_340 \def (plus n0 m) in (le n0 TMP_340)))) in (let TMP_339 \def (\lambda 
-(m: nat).(le_O_n m)) in (let TMP_338 \def (\lambda (n0: nat).(\lambda (IHn: 
-((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(let TMP_337 \def 
-(plus n0 m) in (let TMP_336 \def (IHn m) in (le_n_S n0 TMP_337 TMP_336)))))) 
-in (nat_ind TMP_341 TMP_339 TMP_338 n)))).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus 
+n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn: 
+((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus 
+n0 m) (IHn m))))) n).
 
-theorem le_plus_r:
+lemma le_plus_r:
  \forall (n: nat).(\forall (m: nat).(le m (plus n m)))
 \def
- \lambda (n: nat).(\lambda (m: nat).(let TMP_346 \def (\lambda (n0: nat).(let 
-TMP_345 \def (plus n0 m) in (le m TMP_345))) in (let TMP_344 \def (le_n m) in 
-(let TMP_343 \def (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(let 
-TMP_342 \def (plus n0 m) in (le_S m TMP_342 H)))) in (nat_ind TMP_346 TMP_344 
-TMP_343 n))))).
+ \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus 
+n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m 
+(plus n0 m) H))) n)).
 
-theorem simpl_le_plus_l:
+lemma simpl_le_plus_l:
  \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p 
 m)) \to (le n m))))
 \def
- \lambda (p: nat).(let TMP_352 \def (\lambda (n: nat).(\forall (n0: 
-nat).(\forall (m: nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) in (let 
-TMP_351 \def (\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) 
-in (let TMP_350 \def (\lambda (p0: nat).(\lambda (IHp: ((\forall (n: 
-nat).(\forall (m: nat).((le (plus p0 n) (plus p0 m)) \to (le n 
-m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S (plus p0 n)) 
-(S (plus p0 m)))).(let TMP_348 \def (plus p0 n) in (let TMP_347 \def (plus p0 
-m) in (let TMP_349 \def (le_S_n TMP_348 TMP_347 H) in (IHp n m 
-TMP_349))))))))) in (nat_ind TMP_352 TMP_351 TMP_350 p)))).
-
-theorem le_plus_trans:
+ \lambda (p: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (m: 
+nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) (\lambda (n: 
+nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) (\lambda (p0: 
+nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n) 
+(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda 
+(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus 
+p0 m) H))))))) p).
+
+lemma le_plus_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n 
 (plus m p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
-m)).(let TMP_354 \def (plus m p) in (let TMP_353 \def (le_plus_l m p) in 
-(le_trans n m TMP_354 H TMP_353)))))).
+m)).(le_trans n m (plus m p) H (le_plus_l m p))))).
 
-theorem le_reg_l:
+lemma le_reg_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus 
 p n) (plus p m)))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_362 \def 
-(\lambda (n0: nat).((le n m) \to (let TMP_361 \def (plus n0 n) in (let 
-TMP_360 \def (plus n0 m) in (le TMP_361 TMP_360))))) in (let TMP_359 \def 
-(\lambda (H: (le n m)).H) in (let TMP_358 \def (\lambda (p0: nat).(\lambda 
-(IHp: (((le n m) \to (le (plus p0 n) (plus p0 m))))).(\lambda (H: (le n 
-m)).(let TMP_357 \def (plus p0 n) in (let TMP_356 \def (plus p0 m) in (let 
-TMP_355 \def (IHp H) in (le_n_S TMP_357 TMP_356 TMP_355))))))) in (nat_ind 
-TMP_362 TMP_359 TMP_358 p)))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
+nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H) 
+(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0 
+m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
+p))).
 
-theorem le_plus_plus:
+lemma le_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
 n m) \to ((le p q) \to (le (plus n p) (plus m q)))))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
-nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(let TMP_369 \def 
-(\lambda (n0: nat).(let TMP_368 \def (plus n p) in (let TMP_367 \def (plus n0 
-q) in (le TMP_368 TMP_367)))) in (let TMP_366 \def (le_reg_l p q n H0) in 
-(let TMP_365 \def (\lambda (m0: nat).(\lambda (_: (le n m0)).(\lambda (H2: 
-(le (plus n p) (plus m0 q))).(let TMP_364 \def (plus n p) in (let TMP_363 
-\def (plus m0 q) in (le_S TMP_364 TMP_363 H2)))))) in (le_ind n TMP_369 
-TMP_366 TMP_365 m H))))))))).
-
-theorem le_plus_minus:
+nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0: 
+nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0: 
+nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S 
+(plus n p) (plus m0 q) H2)))) m H)))))).
+
+lemma le_plus_minus:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m 
 n)))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(let TMP_376 \def 
-(\lambda (n0: nat).(\lambda (n1: nat).(let TMP_374 \def (minus n1 n0) in (let 
-TMP_375 \def (plus n0 TMP_374) in (eq nat n1 TMP_375))))) in (let TMP_373 
-\def (\lambda (p: nat).(minus_n_O p)) in (let TMP_372 \def (\lambda (p: 
-nat).(\lambda (q: nat).(\lambda (_: (le p q)).(\lambda (H0: (eq nat q (plus p 
-(minus q p)))).(let TMP_370 \def (minus q p) in (let TMP_371 \def (plus p 
-TMP_370) in (f_equal nat nat S q TMP_371 H0))))))) in (le_elim_rel TMP_376 
-TMP_373 TMP_372 n m Le)))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(le_elim_rel 
+(\lambda (n0: nat).(\lambda (n1: nat).(eq nat n1 (plus n0 (minus n1 n0))))) 
+(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda 
+(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat 
+S q (plus p (minus q p)) H0))))) n m Le))).
 
-theorem le_plus_minus_r:
+lemma le_plus_minus_r:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m 
 n)) m)))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_378 \def 
-(minus m n) in (let TMP_379 \def (plus n TMP_378) in (let TMP_377 \def 
-(le_plus_minus n m H) in (sym_eq nat m TMP_379 TMP_377)))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m 
+(plus n (minus m n)) (le_plus_minus n m H)))).
 
-theorem simpl_lt_plus_l:
+lemma simpl_lt_plus_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p 
 m)) \to (lt n m))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_386 \def 
-(\lambda (n0: nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) in (let 
-TMP_385 \def (\lambda (H: (lt n m)).H) in (let TMP_384 \def (\lambda (p0: 
-nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n m)))).(\lambda 
-(H: (lt (S (plus p0 n)) (S (plus p0 m)))).(let TMP_381 \def (plus p0 n) in 
-(let TMP_382 \def (S TMP_381) in (let TMP_380 \def (plus p0 m) in (let 
-TMP_383 \def (le_S_n TMP_382 TMP_380 H) in (IHp TMP_383)))))))) in (nat_ind 
-TMP_386 TMP_385 TMP_384 p)))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
+nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H) 
+(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n 
+m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S 
+(plus p0 n)) (plus p0 m) H))))) p))).
 
-theorem lt_reg_l:
+lemma lt_reg_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
 p n) (plus p m)))))
 \def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_394 \def 
-(\lambda (n0: nat).((lt n m) \to (let TMP_393 \def (plus n0 n) in (let 
-TMP_392 \def (plus n0 m) in (lt TMP_393 TMP_392))))) in (let TMP_391 \def 
-(\lambda (H: (lt n m)).H) in (let TMP_390 \def (\lambda (p0: nat).(\lambda 
-(IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 m))))).(\lambda (H: (lt n 
-m)).(let TMP_389 \def (plus p0 n) in (let TMP_388 \def (plus p0 m) in (let 
-TMP_387 \def (IHp H) in (lt_n_S TMP_389 TMP_388 TMP_387))))))) in (nat_ind 
-TMP_394 TMP_391 TMP_390 p)))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
+nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H) 
+(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 
+m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
+p))).
 
-theorem lt_reg_r:
+lemma lt_reg_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
 n p) (plus m p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
-m)).(let TMP_411 \def (plus p n) in (let TMP_410 \def (\lambda (n0: nat).(let 
-TMP_409 \def (plus m p) in (lt n0 TMP_409))) in (let TMP_407 \def (plus p m) 
-in (let TMP_406 \def (\lambda (n0: nat).(let TMP_405 \def (plus p n) in (lt 
-TMP_405 n0))) in (let TMP_403 \def (\lambda (n0: nat).(let TMP_402 \def (plus 
-n0 n) in (let TMP_401 \def (plus n0 m) in (lt TMP_402 TMP_401)))) in (let 
-TMP_400 \def (\lambda (n0: nat).(\lambda (_: (lt (plus n0 n) (plus n0 
-m))).(let TMP_399 \def (S n0) in (lt_reg_l n m TMP_399 H)))) in (let TMP_404 
-\def (nat_ind TMP_403 H TMP_400 p) in (let TMP_398 \def (plus m p) in (let 
-TMP_397 \def (plus_sym m p) in (let TMP_408 \def (eq_ind_r nat TMP_407 
-TMP_406 TMP_404 TMP_398 TMP_397) in (let TMP_396 \def (plus n p) in (let 
-TMP_395 \def (plus_sym n p) in (eq_ind_r nat TMP_411 TMP_410 TMP_408 TMP_396 
-TMP_395)))))))))))))))).
-
-theorem le_lt_plus_plus:
+m)).(eq_ind_r nat (plus p n) (\lambda (n0: nat).(lt n0 (plus m p))) (eq_ind_r 
+nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0: 
+nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt 
+(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m 
+p)) (plus n p) (plus_sym n p))))).
+
+lemma le_lt_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
 n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
-nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(let TMP_419 \def (S 
-p) in (let TMP_420 \def (plus n TMP_419) in (let TMP_418 \def (\lambda (n0: 
-nat).(let TMP_417 \def (plus m q) in (le n0 TMP_417))) in (let TMP_415 \def 
-(S p) in (let TMP_416 \def (le_plus_plus n m TMP_415 q H H0) in (let TMP_413 
-\def (S n) in (let TMP_414 \def (plus TMP_413 p) in (let TMP_412 \def 
-(plus_Snm_nSm n p) in (eq_ind_r nat TMP_420 TMP_418 TMP_416 TMP_414 
-TMP_412)))))))))))))).
-
-theorem lt_le_plus_plus:
+nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n 
+(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) 
+(plus (S n) p) (plus_Snm_nSm n p))))))).
+
+lemma lt_le_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
 n m) \to ((le p q) \to (lt (plus n p) (plus m q)))))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
-nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(let TMP_421 \def (S 
-n) in (le_plus_plus TMP_421 m p q H H0))))))).
+nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m 
+p q H H0)))))).
 
-theorem lt_plus_plus:
+lemma lt_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
 n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
-nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(let TMP_422 \def 
-(lt_le_weak p q H0) in (lt_le_plus_plus n m p q H TMP_422))))))).
+nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q 
+H (lt_le_weak p q H0))))))).
 
-theorem well_founded_ltof:
+lemma well_founded_ltof:
  \forall (A: Type[0]).(\forall (f: ((A \to nat))).(well_founded A (ltof A f)))
 \def
  \lambda (A: Type[0]).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n: 
-nat).(let TMP_438 \def (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to 
-(let TMP_437 \def (ltof A f) in (Acc A TMP_437 a))))) in (let TMP_436 \def 
-(\lambda (a: A).(\lambda (H: (lt (f a) O)).(let TMP_434 \def (ltof A f) in 
-(let TMP_435 \def (Acc A TMP_434 a) in (let H0 \def H in (let TMP_431 \def (f 
-a) in (let H1 \def (lt_n_O TMP_431) in (let TMP_432 \def (\lambda (H2: (lt (f 
-a) O)).(H1 H2)) in (let TMP_433 \def (TMP_432 H0) in (False_ind TMP_435 
-TMP_433)))))))))) in (let TMP_430 \def (\lambda (n0: nat).(\lambda (IHn: 
-((\forall (a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: 
-A).(\lambda (ltSma: (lt (f a) (S n0))).(let TMP_429 \def (ltof A f) in (let 
-TMP_428 \def (\lambda (b: A).(\lambda (ltfafb: (lt (f b) (f a))).(let TMP_426 
-\def (f b) in (let TMP_425 \def (f a) in (let TMP_423 \def (f a) in (let 
-TMP_424 \def (lt_n_Sm_le TMP_423 n0 ltSma) in (let TMP_427 \def (lt_le_trans 
-TMP_426 TMP_425 n0 ltfafb TMP_424) in (IHn b TMP_427)))))))) in (Acc_intro A 
-TMP_429 a TMP_428))))))) in (nat_ind TMP_438 TMP_436 TMP_430 n))))) in 
-(\lambda (a: A).(let TMP_442 \def (f a) in (let TMP_443 \def (S TMP_442) in 
-(let TMP_439 \def (f a) in (let TMP_440 \def (S TMP_439) in (let TMP_441 \def 
-(le_n TMP_440) in (H TMP_443 a TMP_441))))))))).
-
-theorem lt_wf:
+nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A 
+(ltof A f) a)))) (\lambda (a: A).(\lambda (H: (lt (f a) O)).(False_ind (Acc A 
+(ltof A f) a) (let H0 \def H in ((let H1 \def (lt_n_O (f a)) in (\lambda (H2: 
+(lt (f a) O)).(H1 H2))) H0))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall 
+(a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda 
+(ltSma: (lt (f a) (S n0))).(Acc_intro A (ltof A f) a (\lambda (b: A).(\lambda 
+(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb 
+(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a 
+(le_n (S (f a))))))).
+
+lemma lt_wf:
  well_founded nat lt
 \def
- let TMP_444 \def (\lambda (m: nat).m) in (well_founded_ltof nat TMP_444).
+ well_founded_ltof nat (\lambda (m: nat).m).
 
-theorem lt_wf_ind:
+lemma lt_wf_ind:
  \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: 
 nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p)))
 \def
  \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n: 
-nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(let TMP_447 
-\def (\lambda (n: nat).(P n)) in (let TMP_446 \def (\lambda (x: nat).(\lambda 
-(_: ((\forall (y: nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: 
-((\forall (y: nat).((lt y x) \to (P y))))).(H x H1)))) in (let TMP_445 \def 
-(lt_wf p) in (Acc_ind nat lt TMP_447 TMP_446 p TMP_445)))))).
+nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt 
+(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y: 
+nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y 
+x) \to (P y))))).(H x H1)))) p (lt_wf p)))).