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=a9223f1b14f63fea039caaf1cfbf3f9d7ec2a245;hpb=24b593925bae7964bdc61e28408d389e5a74bd7a;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 a9223f1b1..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_3 \def 
-(\lambda (a: A).(let TMP_1 \def (f x) in (let TMP_2 \def (f a) in (eq B TMP_1 
-TMP_2)))) in (let TMP_4 \def (f x) in (let TMP_5 \def (refl_equal B TMP_4) in 
-(eq_ind A x TMP_3 TMP_5 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_3 \def (\lambda 
-(a: A1).((eq A2 x2 y2) \to (let TMP_1 \def (f x1 x2) in (let TMP_2 \def (f a 
-y2) in (eq B TMP_1 TMP_2))))) in (let TMP_9 \def (\lambda (H0: (eq A2 x2 
-y2)).(let TMP_6 \def (\lambda (a: A2).(let TMP_4 \def (f x1 x2) in (let TMP_5 
-\def (f x1 a) in (eq B TMP_4 TMP_5)))) in (let TMP_7 \def (f x1 x2) in (let 
-TMP_8 \def (refl_equal B TMP_7) in (eq_ind A2 x2 TMP_6 TMP_8 y2 H0))))) in 
-(eq_ind A1 x1 TMP_3 TMP_9 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_3 \def (\lambda 
-(a: A1).((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (let TMP_1 \def (f x1 x2 x3) in 
-(let TMP_2 \def (f a y2 y3) in (eq B TMP_1 TMP_2)))))) in (let TMP_13 \def 
-(\lambda (H0: (eq A2 x2 y2)).(let TMP_6 \def (\lambda (a: A2).((eq A3 x3 y3) 
-\to (let TMP_4 \def (f x1 x2 x3) in (let TMP_5 \def (f x1 a y3) in (eq B 
-TMP_4 TMP_5))))) in (let TMP_12 \def (\lambda (H1: (eq A3 x3 y3)).(let TMP_9 
-\def (\lambda (a: A3).(let TMP_7 \def (f x1 x2 x3) in (let TMP_8 \def (f x1 
-x2 a) in (eq B TMP_7 TMP_8)))) in (let TMP_10 \def (f x1 x2 x3) in (let 
-TMP_11 \def (refl_equal B TMP_10) in (eq_ind A3 x3 TMP_9 TMP_11 y3 H1))))) in 
-(eq_ind A2 x2 TMP_6 TMP_12 y2 H0)))) in (eq_ind A1 x1 TMP_3 TMP_13 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_1 \def (\lambda (a: A).(eq A a x)) in (let TMP_2 \def 
-(refl_equal A x) in (eq_ind A x TMP_1 TMP_2 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)).(let TMP_1 \def (sym_eq 
-A y x H0) in (match TMP_1 with [refl_equal \Rightarrow H]))))))).
+(H: (P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) 
+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_1 \def 
-(\lambda (a: A).(eq A x a)) in (eq_ind A y TMP_1 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_1 \def (\lambda (a: A).(eq A a 
-y)) in (let TMP_2 \def (refl_equal A y) in (let TMP_3 \def (eq_ind A y TMP_1 
-TMP_2 x h2) in (h1 TMP_3)))))))).
+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: 
@@ -106,643 +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_1 \def (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) in (let 
-TMP_7 \def (\lambda (n0: nat).(\lambda (H2: ((\forall (m: nat).(R n0 
-m)))).(\lambda (m: nat).(let TMP_3 \def (\lambda (n1: nat).(let TMP_2 \def (S 
-n0) in (R TMP_2 n1))) in (let TMP_4 \def (H0 n0) in (let TMP_6 \def (\lambda 
-(n1: nat).(\lambda (_: (R (S n0) n1)).(let TMP_5 \def (H2 n1) in (H1 n0 n1 
-TMP_5)))) in (nat_ind TMP_3 TMP_4 TMP_6 m))))))) in (nat_ind TMP_1 H TMP_7 
-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_1 \def (S n) in (let TMP_2 \def (S m) in (f_equal nat nat pred TMP_1 
-TMP_2 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_1 \def (S n) in 
-(let TMP_2 \def (\lambda (n0: nat).(IsSucc n0)) in (let TMP_3 \def (S n) in 
-(let TMP_4 \def (sym_eq nat O TMP_3 H) in (eq_ind nat TMP_1 TMP_2 I O 
-TMP_4)))))).
+ \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_1 \def (eq_add_S n m H0) in (H TMP_1))))).
+(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_1 \def (S m) in (let TMP_2 \def (pred TMP_1) in 
-(refl_equal nat TMP_2))).
+ \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_1 \def (S 
-m) in (let TMP_4 \def (\lambda (n0: nat).(let TMP_2 \def (pred n0) in (let 
-TMP_3 \def (S TMP_2) in (eq nat n0 TMP_3)))) in (let TMP_5 \def (S m) in (let 
-TMP_6 \def (pred TMP_5) in (let TMP_7 \def (S TMP_6) in (let TMP_8 \def 
-(refl_equal nat TMP_7) in (let TMP_12 \def (\lambda (m0: nat).(\lambda (_: 
-(le (S m) m0)).(\lambda (_: (eq nat m0 (S (pred m0)))).(let TMP_9 \def (S m0) 
-in (let TMP_10 \def (pred TMP_9) in (let TMP_11 \def (S TMP_10) in 
-(refl_equal nat TMP_11))))))) in (le_ind TMP_1 TMP_4 TMP_8 TMP_12 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_1 \def (\lambda (n0: nat).(le n n0)) in 
-(let TMP_2 \def (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: 
-(le n m0)).(le_S n m0 IHle)))) in (le_ind m TMP_1 H TMP_2 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_1 
-\def (S n) in (let TMP_2 \def (le_n n) in (let TMP_3 \def (le_S n n TMP_2) in 
-(le_trans n TMP_1 m TMP_3 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_3 \def 
-(\lambda (n0: nat).(let TMP_1 \def (S n) in (let TMP_2 \def (S n0) in (le 
-TMP_1 TMP_2)))) in (let TMP_4 \def (S n) in (let TMP_5 \def (le_n TMP_4) in 
-(let TMP_8 \def (\lambda (m0: nat).(\lambda (_: (le n m0)).(\lambda (IHle: 
-(le (S n) (S m0))).(let TMP_6 \def (S n) in (let TMP_7 \def (S m0) in (le_S 
-TMP_6 TMP_7 IHle)))))) in (le_ind n TMP_3 TMP_5 TMP_8 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_1 \def (\lambda (n0: nat).(le O n0)) in (let TMP_2 
-\def (le_n O) in (let TMP_3 \def (\lambda (n0: nat).(\lambda (IHn: (le O 
-n0)).(le_S O n0 IHn))) in (nat_ind TMP_1 TMP_2 TMP_3 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_1 
-\def (S n) in (let TMP_5 \def (\lambda (n0: nat).(let TMP_2 \def (S n) in 
-(let TMP_3 \def (pred TMP_2) in (let TMP_4 \def (pred n0) in (le TMP_3 
-TMP_4))))) in (let TMP_6 \def (le_n n) in (let TMP_7 \def (\lambda (m0: 
+ \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)))) in (let TMP_8 \def (S m) in (le_ind TMP_1 TMP_5 TMP_6 TMP_7 TMP_8 
-H)))))))).
+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_1 \def (S n) in (let 
-TMP_2 \def (\lambda (n0: nat).(IsSucc n0)) in (let TMP_3 \def (\lambda (m: 
-nat).(\lambda (_: (le (S n) m)).(\lambda (_: (IsSucc m)).I))) in (le_ind 
-TMP_1 TMP_2 I TMP_3 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_3 \def (\lambda (n0: nat).(let TMP_1 \def (S n0) 
-in (let TMP_2 \def (le TMP_1 n0) in (not TMP_2)))) in (let TMP_4 \def 
-(le_Sn_O O) in (let TMP_7 \def (\lambda (n0: nat).(\lambda (IHn: (not (le (S 
-n0) n0))).(\lambda (H: (le (S (S n0)) (S n0))).(let TMP_5 \def (S n0) in (let 
-TMP_6 \def (le_S_n TMP_5 n0 H) in (IHn TMP_6)))))) in (nat_ind TMP_3 TMP_4 
-TMP_7 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_1 \def 
-(\lambda (n0: nat).((le n0 n) \to (eq nat n n0))) in (let TMP_2 \def (\lambda 
-(_: (le n n)).(refl_equal nat n)) in (let TMP_8 \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_3 \def (S m0) in (let TMP_4 \def 
-(eq nat n TMP_3) in (let TMP_5 \def (S m0) in (let H2 \def (le_trans TMP_5 n 
-m0 H1 H) in (let H3 \def (le_Sn_n m0) in (let TMP_6 \def (\lambda (H4: (le (S 
-m0) m0)).(H3 H4)) in (let TMP_7 \def (TMP_6 H2) in (False_ind TMP_4 
-TMP_7)))))))))))) in (le_ind n TMP_1 TMP_2 TMP_8 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_1 \def (le_O_n n) in 
-(le_antisym O n TMP_1 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_1 \def (\lambda (n0: 
-nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) in (let TMP_2 \def (\lambda 
-(m: nat).(\lambda (_: (le O m)).(H m))) in (let TMP_14 \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_3 \def (S n0) in (let TMP_5 
-\def (\lambda (n1: nat).(let TMP_4 \def (S n0) in (P TMP_4 n1))) in (let 
-TMP_6 \def (le_n n0) in (let TMP_7 \def (le_n n0) in (let TMP_8 \def (IHn n0 
-TMP_7) in (let TMP_9 \def (H0 n0 n0 TMP_6 TMP_8) in (let TMP_13 \def (\lambda 
-(m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: (P (S n0) m0)).(let 
-TMP_10 \def (le_trans_S n0 m0 H1) in (let TMP_11 \def (le_trans_S n0 m0 H1) 
-in (let TMP_12 \def (IHn m0 TMP_11) in (H0 n0 m0 TMP_10 TMP_12))))))) in 
-(le_ind TMP_3 TMP_5 TMP_9 TMP_13 m Le)))))))))))) in (nat_ind TMP_1 TMP_2 
-TMP_14 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_1 \def (S 
-n) in (le_n_S TMP_1 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_1 \def (S n) in (le_n TMP_1)).
+ \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_1 
-\def (S n) in (le_S_n TMP_1 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_1 \def (S m) in (let TMP_2 \def (\lambda 
-(n0: nat).(lt n n0)) in (let TMP_3 \def (S n) in (let TMP_4 \def (le_S TMP_3 
-m H) in (let TMP_6 \def (\lambda (m0: nat).(\lambda (_: (le (S m) 
-m0)).(\lambda (IHle: (lt n m0)).(let TMP_5 \def (S n) in (le_S TMP_5 m0 
-IHle))))) in (le_ind TMP_1 TMP_2 TMP_4 TMP_6 p H0)))))))))).
+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))))).
 
-theorem lt_O_Sn:
+lemma lt_O_Sn:
  \forall (n: nat).(lt O (S n))
 \def
- \lambda (n: nat).(let TMP_1 \def (le_O_n n) in (le_n_S O n TMP_1)).
+ \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_2 \def 
-(\lambda (n0: nat).(let TMP_1 \def (lt n0 n) in (not TMP_1))) in (let TMP_3 
-\def (lt_n_n n) in (let TMP_6 \def (\lambda (m0: nat).(\lambda (_: (le n 
-m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(let TMP_4 
-\def (S m0) in (let TMP_5 \def (le_trans_S TMP_4 n H1) in (IHle TMP_5))))))) 
-in (le_ind n TMP_2 TMP_3 TMP_6 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_1 \def (S m) in (let TMP_2 \def (\lambda 
-(n0: nat).(lt n n0)) in (let TMP_3 \def (le_n_S n m H) in (let TMP_5 \def 
-(\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt n 
-m0)).(let TMP_4 \def (S n) in (le_S TMP_4 m0 IHle))))) in (le_ind TMP_1 TMP_2 
-TMP_3 TMP_5 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_1 \def (\lambda (n0: nat).(lt n n0)) in 
-(let TMP_3 \def (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: 
-(lt n m0)).(let TMP_2 \def (S n) in (le_S TMP_2 m0 IHle))))) in (le_ind m 
-TMP_1 H TMP_3 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_3 \def 
-(\lambda (n0: nat).(let TMP_1 \def (lt n n0) in (let TMP_2 \def (eq nat n n0) 
-in (or TMP_1 TMP_2)))) in (let TMP_4 \def (lt n n) in (let TMP_5 \def (eq nat 
-n n) in (let TMP_6 \def (refl_equal nat n) in (let TMP_7 \def (or_intror 
-TMP_4 TMP_5 TMP_6) in (let TMP_13 \def (\lambda (m0: nat).(\lambda (H0: (le n 
-m0)).(\lambda (_: (or (lt n m0) (eq nat n m0))).(let TMP_8 \def (S m0) in 
-(let TMP_9 \def (lt n TMP_8) in (let TMP_10 \def (S m0) in (let TMP_11 \def 
-(eq nat n TMP_10) in (let TMP_12 \def (le_n_S n m0 H0) in (or_introl TMP_9 
-TMP_11 TMP_12))))))))) in (le_ind n TMP_3 TMP_7 TMP_13 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_3 \def (\lambda (n0: 
-nat).(\lambda (n1: nat).(let TMP_1 \def (le n0 n1) in (let TMP_2 \def (lt n1 
-n0) in (or TMP_1 TMP_2))))) in (let TMP_7 \def (\lambda (n0: nat).(let TMP_4 
-\def (le O n0) in (let TMP_5 \def (lt n0 O) in (let TMP_6 \def (le_O_n n0) in 
-(or_introl TMP_4 TMP_5 TMP_6))))) in (let TMP_15 \def (\lambda (n0: nat).(let 
-TMP_8 \def (S n0) in (let TMP_9 \def (le TMP_8 O) in (let TMP_10 \def (S n0) 
-in (let TMP_11 \def (lt O TMP_10) in (let TMP_12 \def (S n0) in (let TMP_13 
-\def (lt_O_Sn n0) in (let TMP_14 \def (lt_le_S O TMP_12 TMP_13) in (or_intror 
-TMP_9 TMP_11 TMP_14))))))))) in (let TMP_42 \def (\lambda (n0: nat).(\lambda 
-(m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 n0))).(let TMP_16 \def (le n0 
-m0) in (let TMP_17 \def (lt m0 n0) in (let TMP_18 \def (S n0) in (let TMP_19 
-\def (S m0) in (let TMP_20 \def (le TMP_18 TMP_19) in (let TMP_21 \def (S m0) 
-in (let TMP_22 \def (S n0) in (let TMP_23 \def (lt TMP_21 TMP_22) in (let 
-TMP_24 \def (or TMP_20 TMP_23) in (let TMP_32 \def (\lambda (H0: (le n0 
-m0)).(let TMP_25 \def (S n0) in (let TMP_26 \def (S m0) in (let TMP_27 \def 
-(le TMP_25 TMP_26) in (let TMP_28 \def (S m0) in (let TMP_29 \def (S n0) in 
-(let TMP_30 \def (lt TMP_28 TMP_29) in (let TMP_31 \def (le_n_S n0 m0 H0) in 
-(or_introl TMP_27 TMP_30 TMP_31))))))))) in (let TMP_41 \def (\lambda (H0: 
-(lt m0 n0)).(let TMP_33 \def (S n0) in (let TMP_34 \def (S m0) in (let TMP_35 
-\def (le TMP_33 TMP_34) in (let TMP_36 \def (S m0) in (let TMP_37 \def (S n0) 
-in (let TMP_38 \def (lt TMP_36 TMP_37) in (let TMP_39 \def (S m0) in (let 
-TMP_40 \def (le_n_S TMP_39 n0 H0) in (or_intror TMP_35 TMP_38 
-TMP_40)))))))))) in (or_ind TMP_16 TMP_17 TMP_24 TMP_32 TMP_41 
-H))))))))))))))) in (nat_double_ind TMP_3 TMP_7 TMP_15 TMP_42 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_2 \def (\lambda (n0: nat).(let TMP_1 \def (plus n0 
-O) in (eq nat n0 TMP_1))) in (let TMP_3 \def (refl_equal nat O) in (let TMP_5 
-\def (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 O))).(let TMP_4 \def 
-(plus n0 O) in (f_equal nat nat S n0 TMP_4 H)))) in (nat_ind TMP_2 TMP_3 
-TMP_5 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_5 \def (\lambda (n0: nat).(let 
-TMP_1 \def (plus n0 n) in (let TMP_2 \def (S TMP_1) in (let TMP_3 \def (S n) 
-in (let TMP_4 \def (plus n0 TMP_3) in (eq nat TMP_2 TMP_4)))))) in (let TMP_6 
-\def (S n) in (let TMP_7 \def (refl_equal nat TMP_6) in (let TMP_12 \def 
-(\lambda (n0: nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S 
-n)))).(let TMP_8 \def (plus n0 n) in (let TMP_9 \def (S TMP_8) in (let TMP_10 
-\def (S n) in (let TMP_11 \def (plus n0 TMP_10) in (f_equal nat nat S TMP_9 
-TMP_11 H))))))) in (nat_ind TMP_5 TMP_7 TMP_12 m)))))).
+ \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)).
 
-theorem plus_sym:
+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_3 \def (\lambda (n0: nat).(let 
-TMP_1 \def (plus n0 m) in (let TMP_2 \def (plus m n0) in (eq nat TMP_1 
-TMP_2)))) in (let TMP_4 \def (plus_n_O m) in (let TMP_16 \def (\lambda (y: 
-nat).(\lambda (H: (eq nat (plus y m) (plus m y))).(let TMP_5 \def (plus m y) 
-in (let TMP_6 \def (S TMP_5) in (let TMP_9 \def (\lambda (n0: nat).(let TMP_7 
-\def (plus y m) in (let TMP_8 \def (S TMP_7) in (eq nat TMP_8 n0)))) in (let 
-TMP_10 \def (plus y m) in (let TMP_11 \def (plus m y) in (let TMP_12 \def 
-(f_equal nat nat S TMP_10 TMP_11 H) in (let TMP_13 \def (S y) in (let TMP_14 
-\def (plus m TMP_13) in (let TMP_15 \def (plus_n_Sm m y) in (eq_ind nat TMP_6 
-TMP_9 TMP_12 TMP_14 TMP_15)))))))))))) in (nat_ind TMP_3 TMP_4 TMP_16 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_1 \def (plus m n) in (let TMP_5 
-\def (\lambda (n0: nat).(let TMP_2 \def (S n0) in (let TMP_3 \def (S m) in 
-(let TMP_4 \def (plus n TMP_3) in (eq nat TMP_2 TMP_4))))) in (let TMP_6 \def 
-(S m) in (let TMP_7 \def (plus TMP_6 n) in (let TMP_10 \def (\lambda (n0: 
-nat).(let TMP_8 \def (plus m n) in (let TMP_9 \def (S TMP_8) in (eq nat TMP_9 
-n0)))) in (let TMP_11 \def (S m) in (let TMP_12 \def (plus TMP_11 n) in (let 
-TMP_13 \def (refl_equal nat TMP_12) in (let TMP_14 \def (S m) in (let TMP_15 
-\def (plus n TMP_14) in (let TMP_16 \def (S m) in (let TMP_17 \def (plus_sym 
-n TMP_16) in (let TMP_18 \def (eq_ind_r nat TMP_7 TMP_10 TMP_13 TMP_15 
-TMP_17) in (let TMP_19 \def (plus n m) in (let TMP_20 \def (plus_sym n m) in 
-(eq_ind_r nat TMP_1 TMP_5 TMP_18 TMP_19 TMP_20))))))))))))))))).
-
-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_5 \def 
-(\lambda (n0: nat).(let TMP_1 \def (plus m p) in (let TMP_2 \def (plus n0 
-TMP_1) in (let TMP_3 \def (plus n0 m) in (let TMP_4 \def (plus TMP_3 p) in 
-(eq nat TMP_2 TMP_4)))))) in (let TMP_6 \def (plus m p) in (let TMP_7 \def 
-(refl_equal nat TMP_6) in (let TMP_12 \def (\lambda (n0: nat).(\lambda (H: 
-(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))).(let TMP_8 \def (plus m 
-p) in (let TMP_9 \def (plus n0 TMP_8) in (let TMP_10 \def (plus n0 m) in (let 
-TMP_11 \def (plus TMP_10 p) in (f_equal nat nat S TMP_9 TMP_11 H))))))) in 
-(nat_ind TMP_5 TMP_7 TMP_12 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_1 \def (plus m 
-p) in (let TMP_2 \def (plus n TMP_1) in (let TMP_3 \def (plus n m) in (let 
-TMP_4 \def (plus TMP_3 p) in (let TMP_5 \def (plus_assoc_l n m p) in (sym_eq 
-nat TMP_2 TMP_4 TMP_5)))))))).
+ \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_1 \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_2 \def (\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat m 
-p)).H))) in (let TMP_13 \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_3 \def (plus n0 m) in (let TMP_4 \def (plus n0 
-p) in (let TMP_5 \def (plus n0) in (let TMP_6 \def (plus n0 m) in (let TMP_7 
-\def (plus n0 p) in (let TMP_8 \def (plus n0 m) in (let TMP_9 \def (plus n0 
-p) in (let TMP_10 \def (eq_add_S TMP_8 TMP_9 H) in (let TMP_11 \def (f_equal 
-nat nat TMP_5 TMP_6 TMP_7 TMP_10) in (let TMP_12 \def (IHn TMP_3 TMP_4 
-TMP_11) in (IHn m p TMP_12)))))))))))))))) in (nat_ind TMP_1 TMP_2 TMP_13 
-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_2 \def (\lambda (n0: nat).(let TMP_1 \def (minus 
-n0 O) in (eq nat n0 TMP_1))) in (let TMP_3 \def (refl_equal nat O) in (let 
-TMP_5 \def (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 O))).(let 
-TMP_4 \def (S n0) in (refl_equal nat TMP_4)))) in (nat_ind TMP_2 TMP_3 TMP_5 
-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_2 \def (\lambda (n0: nat).(let TMP_1 \def (minus 
-n0 n0) in (eq nat O TMP_1))) in (let TMP_3 \def (refl_equal nat O) in (let 
-TMP_4 \def (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 n0))).IHn)) 
-in (nat_ind TMP_2 TMP_3 TMP_4 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_5 \def 
-(\lambda (n0: nat).(\lambda (n1: nat).(let TMP_1 \def (minus n1 n0) in (let 
-TMP_2 \def (S TMP_1) in (let TMP_3 \def (S n1) in (let TMP_4 \def (minus 
-TMP_3 n0) in (eq nat TMP_2 TMP_4))))))) in (let TMP_10 \def (\lambda (p: 
-nat).(let TMP_6 \def (minus p O) in (let TMP_7 \def (minus p O) in (let TMP_8 
-\def (minus_n_O p) in (let TMP_9 \def (sym_eq nat p TMP_7 TMP_8) in (f_equal 
-nat nat S TMP_6 p TMP_9)))))) in (let TMP_11 \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_5 TMP_10 TMP_11 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_2 \def 
-(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (let TMP_1 
-\def (minus n1 n0) in (eq nat p TMP_1))))) in (let TMP_7 \def (\lambda (n0: 
-nat).(\lambda (H: (eq nat n0 p)).(let TMP_3 \def (\lambda (n1: nat).(eq nat p 
-n1)) in (let TMP_4 \def (sym_eq nat n0 p H) in (let TMP_5 \def (minus n0 O) 
-in (let TMP_6 \def (minus_n_O n0) in (eq_ind nat n0 TMP_3 TMP_4 TMP_5 
-TMP_6))))))) in (let TMP_12 \def (\lambda (n0: nat).(\lambda (H: (eq nat O (S 
-(plus n0 p)))).(let TMP_8 \def (eq nat p O) in (let H0 \def H in (let TMP_9 
-\def (plus n0 p) in (let H1 \def (O_S TMP_9) in (let TMP_10 \def (\lambda 
-(H2: (eq nat O (S (plus n0 p)))).(H1 H2)) in (let TMP_11 \def (TMP_10 H0) in 
-(False_ind TMP_8 TMP_11))))))))) in (let TMP_15 \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_13 
-\def (plus n0 p) in (let TMP_14 \def (eq_add_S m0 TMP_13 H0) in (H 
-TMP_14))))))) in (nat_double_ind TMP_2 TMP_7 TMP_12 TMP_15 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_1 \def (plus n m) in (let TMP_2 
-\def (minus TMP_1 n) in (let TMP_3 \def (plus n m) in (let TMP_4 \def (plus n 
-m) in (let TMP_5 \def (refl_equal nat TMP_4) in (let TMP_6 \def (plus_minus 
-TMP_3 n m TMP_5) in (sym_eq nat m TMP_2 TMP_6)))))))).
+ \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_2 \def (\lambda (n0: nat).(let TMP_1 \def (pred 
-n0) in (le TMP_1 n0))) in (let TMP_3 \def (le_n O) in (let TMP_7 \def 
-(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(let TMP_4 \def (S n0) in 
-(let TMP_5 \def (pred TMP_4) in (let TMP_6 \def (le_n n0) in (le_S TMP_5 n0 
-TMP_6)))))) in (nat_ind TMP_2 TMP_3 TMP_7 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_2 \def (\lambda (n0: nat).(\forall (m: nat).(let 
-TMP_1 \def (plus n0 m) in (le n0 TMP_1)))) in (let TMP_3 \def (\lambda (m: 
-nat).(le_O_n m)) in (let TMP_6 \def (\lambda (n0: nat).(\lambda (IHn: 
-((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(let TMP_4 \def 
-(plus n0 m) in (let TMP_5 \def (IHn m) in (le_n_S n0 TMP_4 TMP_5)))))) in 
-(nat_ind TMP_2 TMP_3 TMP_6 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_2 \def (\lambda (n0: nat).(let 
-TMP_1 \def (plus n0 m) in (le m TMP_1))) in (let TMP_3 \def (le_n m) in (let 
-TMP_5 \def (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(let TMP_4 
-\def (plus n0 m) in (le_S m TMP_4 H)))) in (nat_ind TMP_2 TMP_3 TMP_5 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_1 \def (\lambda (n: nat).(\forall (n0: 
-nat).(\forall (m: nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) in (let 
-TMP_2 \def (\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) in 
-(let TMP_6 \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_3 \def (plus p0 n) in (let TMP_4 \def (plus p0 m) in (let 
-TMP_5 \def (le_S_n TMP_3 TMP_4 H) in (IHp n m TMP_5))))))))) in (nat_ind 
-TMP_1 TMP_2 TMP_6 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_1 \def (plus m p) in (let TMP_2 \def (le_plus_l m p) in 
-(le_trans n m TMP_1 H TMP_2)))))).
+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_3 \def 
-(\lambda (n0: nat).((le n m) \to (let TMP_1 \def (plus n0 n) in (let TMP_2 
-\def (plus n0 m) in (le TMP_1 TMP_2))))) in (let TMP_4 \def (\lambda (H: (le 
-n m)).H) in (let TMP_8 \def (\lambda (p0: nat).(\lambda (IHp: (((le n m) \to 
-(le (plus p0 n) (plus p0 m))))).(\lambda (H: (le n m)).(let TMP_5 \def (plus 
-p0 n) in (let TMP_6 \def (plus p0 m) in (let TMP_7 \def (IHp H) in (le_n_S 
-TMP_5 TMP_6 TMP_7))))))) in (nat_ind TMP_3 TMP_4 TMP_8 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_3 \def (\lambda 
-(n0: nat).(let TMP_1 \def (plus n p) in (let TMP_2 \def (plus n0 q) in (le 
-TMP_1 TMP_2)))) in (let TMP_4 \def (le_reg_l p q n H0) in (let TMP_7 \def 
-(\lambda (m0: nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus 
-m0 q))).(let TMP_5 \def (plus n p) in (let TMP_6 \def (plus m0 q) in (le_S 
-TMP_5 TMP_6 H2)))))) in (le_ind n TMP_3 TMP_4 TMP_7 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_3 \def 
-(\lambda (n0: nat).(\lambda (n1: nat).(let TMP_1 \def (minus n1 n0) in (let 
-TMP_2 \def (plus n0 TMP_1) in (eq nat n1 TMP_2))))) in (let TMP_4 \def 
-(\lambda (p: nat).(minus_n_O p)) in (let TMP_7 \def (\lambda (p: 
-nat).(\lambda (q: nat).(\lambda (_: (le p q)).(\lambda (H0: (eq nat q (plus p 
-(minus q p)))).(let TMP_5 \def (minus q p) in (let TMP_6 \def (plus p TMP_5) 
-in (f_equal nat nat S q TMP_6 H0))))))) in (le_elim_rel TMP_3 TMP_4 TMP_7 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_1 \def 
-(minus m n) in (let TMP_2 \def (plus n TMP_1) in (let TMP_3 \def 
-(le_plus_minus n m H) in (sym_eq nat m TMP_2 TMP_3)))))).
+ \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_1 \def 
-(\lambda (n0: nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) in (let TMP_2 
-\def (\lambda (H: (lt n m)).H) in (let TMP_7 \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_3 \def (plus p0 n) in (let TMP_4 \def 
-(S TMP_3) in (let TMP_5 \def (plus p0 m) in (let TMP_6 \def (le_S_n TMP_4 
-TMP_5 H) in (IHp TMP_6)))))))) in (nat_ind TMP_1 TMP_2 TMP_7 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_3 \def 
-(\lambda (n0: nat).((lt n m) \to (let TMP_1 \def (plus n0 n) in (let TMP_2 
-\def (plus n0 m) in (lt TMP_1 TMP_2))))) in (let TMP_4 \def (\lambda (H: (lt 
-n m)).H) in (let TMP_8 \def (\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to 
-(lt (plus p0 n) (plus p0 m))))).(\lambda (H: (lt n m)).(let TMP_5 \def (plus 
-p0 n) in (let TMP_6 \def (plus p0 m) in (let TMP_7 \def (IHp H) in (lt_n_S 
-TMP_5 TMP_6 TMP_7))))))) in (nat_ind TMP_3 TMP_4 TMP_8 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_1 \def (plus p n) in (let TMP_3 \def (\lambda (n0: nat).(let 
-TMP_2 \def (plus m p) in (lt n0 TMP_2))) in (let TMP_4 \def (plus p m) in 
-(let TMP_6 \def (\lambda (n0: nat).(let TMP_5 \def (plus p n) in (lt TMP_5 
-n0))) in (let TMP_9 \def (\lambda (n0: nat).(let TMP_7 \def (plus n0 n) in 
-(let TMP_8 \def (plus n0 m) in (lt TMP_7 TMP_8)))) in (let TMP_11 \def 
-(\lambda (n0: nat).(\lambda (_: (lt (plus n0 n) (plus n0 m))).(let TMP_10 
-\def (S n0) in (lt_reg_l n m TMP_10 H)))) in (let TMP_12 \def (nat_ind TMP_9 
-H TMP_11 p) in (let TMP_13 \def (plus m p) in (let TMP_14 \def (plus_sym m p) 
-in (let TMP_15 \def (eq_ind_r nat TMP_4 TMP_6 TMP_12 TMP_13 TMP_14) in (let 
-TMP_16 \def (plus n p) in (let TMP_17 \def (plus_sym n p) in (eq_ind_r nat 
-TMP_1 TMP_3 TMP_15 TMP_16 TMP_17)))))))))))))))).
-
-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_1 \def (S p) 
-in (let TMP_2 \def (plus n TMP_1) in (let TMP_4 \def (\lambda (n0: nat).(let 
-TMP_3 \def (plus m q) in (le n0 TMP_3))) in (let TMP_5 \def (S p) in (let 
-TMP_6 \def (le_plus_plus n m TMP_5 q H H0) in (let TMP_7 \def (S n) in (let 
-TMP_8 \def (plus TMP_7 p) in (let TMP_9 \def (plus_Snm_nSm n p) in (eq_ind_r 
-nat TMP_2 TMP_4 TMP_6 TMP_8 TMP_9)))))))))))))).
-
-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_1 \def (S n) 
-in (le_plus_plus TMP_1 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_1 \def 
-(lt_le_weak p q H0) in (lt_le_plus_plus n m p q H TMP_1))))))).
+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_2 \def (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to 
-(let TMP_1 \def (ltof A f) in (Acc A TMP_1 a))))) in (let TMP_8 \def (\lambda 
-(a: A).(\lambda (H: (lt (f a) O)).(let TMP_3 \def (ltof A f) in (let TMP_4 
-\def (Acc A TMP_3 a) in (let H0 \def H in (let TMP_5 \def (f a) in (let H1 
-\def (lt_n_O TMP_5) in (let TMP_6 \def (\lambda (H2: (lt (f a) O)).(H1 H2)) 
-in (let TMP_7 \def (TMP_6 H0) in (False_ind TMP_4 TMP_7)))))))))) in (let 
-TMP_16 \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_9 \def (ltof A f) in (let TMP_15 \def (\lambda (b: A).(\lambda 
-(ltfafb: (lt (f b) (f a))).(let TMP_10 \def (f b) in (let TMP_11 \def (f a) 
-in (let TMP_12 \def (f a) in (let TMP_13 \def (lt_n_Sm_le TMP_12 n0 ltSma) in 
-(let TMP_14 \def (lt_le_trans TMP_10 TMP_11 n0 ltfafb TMP_13) in (IHn b 
-TMP_14)))))))) in (Acc_intro A TMP_9 a TMP_15))))))) in (nat_ind TMP_2 TMP_8 
-TMP_16 n))))) in (\lambda (a: A).(let TMP_17 \def (f a) in (let TMP_18 \def 
-(S TMP_17) in (let TMP_19 \def (f a) in (let TMP_20 \def (S TMP_19) in (let 
-TMP_21 \def (le_n TMP_20) in (H TMP_18 a TMP_21))))))))).
-
-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_1 \def (\lambda (m: nat).m) in (well_founded_ltof nat TMP_1).
+ 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_1 \def 
-(\lambda (n: nat).(P n)) in (let TMP_2 \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_3 \def (lt_wf p) in 
-(Acc_ind nat lt TMP_1 TMP_2 p TMP_3)))))).
+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)))).