- \lambda (x: nat).(let TMP_2 \def (\lambda (n: nat).(\forall (y: nat).((lt y
-n) \to (let TMP_1 \def (blt y n) in (eq bool TMP_1 true))))) in (let TMP_13
-\def (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 \def (match H with
-[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let TMP_8 \def (S y) in
-(let TMP_9 \def (\lambda (e: nat).(match e with [O \Rightarrow False | (S _)
-\Rightarrow True])) in (let H1 \def (eq_ind nat TMP_8 TMP_9 I O H0) in (let
-TMP_10 \def (blt y O) in (let TMP_11 \def (eq bool TMP_10 true) in (False_ind
-TMP_11 H1))))))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m)
-O)).(let TMP_3 \def (S m) in (let TMP_4 \def (\lambda (e: nat).(match e with
-[O \Rightarrow False | (S _) \Rightarrow True])) in (let H2 \def (eq_ind nat
-TMP_3 TMP_4 I O H1) in (let TMP_6 \def ((le (S y) m) \to (let TMP_5 \def (blt
-y O) in (eq bool TMP_5 true))) in (let TMP_7 \def (False_ind TMP_6 H2) in
-(TMP_7 H0)))))))]) in (let TMP_12 \def (refl_equal nat O) in (H0 TMP_12)))))
-in (let TMP_21 \def (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y
-n) \to (eq bool (blt y n) true))))).(\lambda (y: nat).(let TMP_16 \def
-(\lambda (n0: nat).((lt n0 (S n)) \to (let TMP_14 \def (S n) in (let TMP_15
-\def (blt n0 TMP_14) in (eq bool TMP_15 true))))) in (let TMP_17 \def
-(\lambda (_: (lt O (S n))).(refl_equal bool true)) in (let TMP_20 \def
-(\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0 with
-[O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda (H1:
-(lt (S n0) (S n))).(let TMP_18 \def (S n0) in (let TMP_19 \def (le_S_n TMP_18
-n H1) in (H n0 TMP_19)))))) in (nat_ind TMP_16 TMP_17 TMP_20 y))))))) in
-(nat_ind TMP_2 TMP_13 TMP_21 x)))).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
+(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0
+\def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1
+\def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O \Rightarrow False |
+(S _) \Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true)
+H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def
+(eq_ind nat (S m) (\lambda (e: nat).(match e with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq bool (blt
+y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n:
+nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt y n)
+true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to
+(eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal bool
+true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0
+with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda
+(H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).