(* This file was automatically generated: do not edit *********************)
-include "Ground-1/preamble.ma".
+include "ground_1/preamble.ma".
-theorem nat_dec:
+lemma nat_dec:
\forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to
(\forall (P: Prop).P))))
\def
((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq
nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S
n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match
-ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
-\Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n:
-nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to
-(\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or
-(eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror
-(eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0:
-(eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda
-(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
-False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda
-(n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall
-(P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
-Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P:
-Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0
+ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H0) in
+(False_ind P H1))))))) n2)) (\lambda (n: nat).(\lambda (H: ((\forall (n2:
+nat).(or (eq nat n n2) ((eq nat n n2) \to (\forall (P: Prop).P)))))).(\lambda
+(n2: nat).(nat_ind (\lambda (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n)
+n0) \to (\forall (P: Prop).P)))) (or_intror (eq nat (S n) O) ((eq nat (S n)
+O) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat (S n) O)).(\lambda (P:
+Prop).(let H1 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1)))))
+(\lambda (n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to
+(\forall (P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall
+(P: Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall
+(P: Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0
(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P:
Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S
n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat
(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P:
Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to
(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P:
-Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
-(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n)
-(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3)
-\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0
-(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0:
-Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2))))
-n1).
-(* COMMENTS
-Initial nodes: 676
-END *)
-
-theorem simpl_plus_r:
+Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e with [O
+\Rightarrow n | (S n3) \Rightarrow n3])) (S n) (S n0) H2) in (let H4 \def
+(eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) \to (\forall (P0:
+Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 (\lambda (n3: nat).(or
+(eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: Prop).P0)))) H0 n H3)
+in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) n1).
+
+lemma simpl_plus_r:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n)
(plus p n)) \to (eq nat m p))))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat
(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda
(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0:
-nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n
-p)) (plus m n) H) (plus n m) (plus_sym n m)))))).
-(* COMMENTS
-Initial nodes: 119
-END *)
+nat).(eq nat n0 (plus n p))) (plus_sym p n) (plus m n) H) (plus n m)
+(plus_sym n m)))))).
-theorem minus_Sx_Sy:
+lemma minus_Sx_Sy:
\forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
\def
\lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
-(* COMMENTS
-Initial nodes: 13
-END *)
-theorem minus_plus_r:
+lemma minus_plus_r:
\forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
\def
\lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0:
nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
-(* COMMENTS
-Initial nodes: 45
-END *)
-theorem plus_permute_2_in_3:
+lemma plus_permute_2_in_3:
\forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x
y) z) (plus (plus x z) y))))
\def
nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z
y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
-(* COMMENTS
-Initial nodes: 163
-END *)
-theorem plus_permute_2_in_3_assoc:
+lemma plus_permute_2_in_3_assoc:
\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n
h) k) (plus n (plus k h)))))
\def
nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k
h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
-(* COMMENTS
-Initial nodes: 119
-END *)
-theorem plus_O:
+lemma plus_O:
\forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat
x O) (eq nat y O))))
\def
(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O)
H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O)
\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq
-nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0:
-nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq
-nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n)
-y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in
-nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y
-O)) H2)))]) in (H1 (refl_equal nat O))))))) x).
-(* COMMENTS
-Initial nodes: 233
-END *)
-
-theorem minus_Sx_SO:
+nat (plus (S n) y) O)).(let H1 \def (match H0 with [refl_equal \Rightarrow
+(\lambda (H1: (eq nat (plus (S n) y) O)).(let H2 \def (eq_ind nat (plus (S n)
+y) (\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y O)) H2)))]) in
+(H1 (refl_equal nat O))))))) x).
+
+lemma minus_Sx_SO:
\forall (x: nat).(eq nat (minus (S x) (S O)) x)
\def
\lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal
nat x) (minus x O) (minus_n_O x)).
-(* COMMENTS
-Initial nodes: 33
-END *)
-theorem eq_nat_dec:
+lemma nat_dec_neg:
\forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
\def
\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq
n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not
(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H
n0)))) j)))) i).
-(* COMMENTS
-Initial nodes: 401
-END *)
-theorem neq_eq_e:
+lemma neq_eq_e:
\forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j))
\to P)) \to ((((eq nat i j) \to P)) \to P))))
\def
\lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not
(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def
-(eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
-(* COMMENTS
-Initial nodes: 61
-END *)
+(nat_dec_neg i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
-theorem le_false:
+lemma le_false:
\forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S
n) m) \to P))))
\def
\lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P:
Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P:
Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match
-H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to
-P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
-(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in
-(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0)
-O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
-I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal
-nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P:
-Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind
-(\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n))
-\to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S
-O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda
-(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2:
-(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e
-in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow
-(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda
-(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
-False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P)
-H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_:
-((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda
-(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S
-n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
-(* COMMENTS
-Initial nodes: 409
-END *)
-
-theorem le_Sx_x:
+H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
+(eq_ind nat (S n) (\lambda (e: nat).(match e with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H1) in (False_ind P H2))) | (le_S m0 H1)
+\Rightarrow (\lambda (H2: (eq nat (S m0) O)).((let H3 \def (eq_ind nat (S m0)
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1
+(refl_equal nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0:
+nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda
+(n0: nat).(nat_ind (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to
+((le (S n1) (S n)) \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n)
+O)).(\lambda (_: (le (S O) (S n))).(let H2 \def (match H0 with [le_n
+\Rightarrow (\lambda (H2: (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n)
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow (\lambda
+(H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3)
+in (False_ind ((le (S n) m0) \to P) H4)) H2))]) in (H2 (refl_equal nat
+O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S n) n1)
+\to ((le (S n1) (S n)) \to P))))).(\lambda (P: Prop).(\lambda (H1: (le (S n)
+(S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(H n1 P (le_S_n n n1 H1)
+(le_S_n (S n1) n H2))))))) n0)))) m).
+
+lemma le_Sx_x:
\forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
\def
\lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def
le_Sn_n in (False_ind P (H0 x H))))).
-(* COMMENTS
-Initial nodes: 23
-END *)
-theorem le_n_pred:
+lemma le_n_pred:
\forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0:
nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans
(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
-(* COMMENTS
-Initial nodes: 71
-END *)
-theorem minus_le:
+lemma minus_le:
\forall (x: nat).(\forall (y: nat).(le (minus x y) x))
\def
\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n
-y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall
-(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0:
-nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda
-(_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)])
-(S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
-(* COMMENTS
-Initial nodes: 101
-END *)
-
-theorem le_plus_minus_sym:
+y) n))) (\lambda (_: nat).(le_O_n O)) (\lambda (n: nat).(\lambda (H:
+((\forall (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda
+(n0: nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0:
+nat).(\lambda (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow
+(minus n l)]) (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
+
+lemma le_plus_minus_sym:
\forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n)
n))))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat
(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H)
(plus (minus m n) n) (plus_sym (minus m n) n)))).
-(* COMMENTS
-Initial nodes: 61
-END *)
-theorem le_minus_minus:
+lemma le_minus_minus:
\forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z)
\to (le (minus y x) (minus z x))))))
\def
(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat
z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z
(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
-(* COMMENTS
-Initial nodes: 117
-END *)
-theorem le_minus_plus:
+lemma le_minus_plus:
\forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat
(minus (plus x y) z) (plus (minus x z) y)))))
\def
\lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to
(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y))))))
-(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return
-(\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y:
-nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n
+(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H with [le_n
\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n:
nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))
(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O)
(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S
z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def
-(match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
-\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with
-[le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat
-(S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with
-[O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq
-nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1)
-\Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m)
-(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S
-z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3))
-H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S
-z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
-(S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n
-(le_S_n z0 n H1) y))))) x)))) z).
-(* COMMENTS
-Initial nodes: 603
-END *)
-
-theorem le_minus:
+(match H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2
+\def (eq_ind nat (S z0) (\lambda (e: nat).(match e with [O \Rightarrow False
+| (S _) \Rightarrow True])) I O H1) in (False_ind (eq nat (minus (plus O y)
+(S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) \Rightarrow (\lambda
+(H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2)
+in (False_ind ((le (S z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus
+(minus O (S z0)) y))) H3)) H1))]) in (H1 (refl_equal nat O))))) (\lambda (n:
+nat).(\lambda (_: (((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus
+n y) (S z0)) (plus (minus n (S z0)) y)))))).(\lambda (H1: (le (S z0) (S
+n))).(\lambda (y: nat).(H n (le_S_n z0 n H1) y))))) x)))) z).
+
+lemma le_minus:
\forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to
(le x (minus z y)))))
\def
x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z
y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x
y))))).
-(* COMMENTS
-Initial nodes: 69
-END *)
-theorem le_trans_plus_r:
+lemma le_trans_plus_r:
\forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to
(le y z))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus
x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
-(* COMMENTS
-Initial nodes: 35
-END *)
-theorem lt_x_O:
+lemma lt_x_O:
\forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
\def
\lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def
(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match
-ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
-\Rightarrow False])) I (S x) H_y) in (False_ind P H0))))).
-(* COMMENTS
-Initial nodes: 48
-END *)
+ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) H_y) in
+(False_ind P H0))))).
-theorem le_gen_S:
+lemma le_gen_S:
\forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n:
nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
\def
\lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def
-(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x)
-\to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m
-n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat
+(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat
(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0)))
(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S
m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x
m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n:
nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2))))
x H1 H0))]) in (H0 (refl_equal nat x))))).
-(* COMMENTS
-Initial nodes: 261
-END *)
-theorem lt_x_plus_x_Sy:
+lemma lt_x_plus_x_Sy:
\forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
\def
\lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n:
nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x))
(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
-(* COMMENTS
-Initial nodes: 83
-END *)
-theorem simpl_lt_plus_r:
+lemma simpl_lt_plus_r:
\forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m
p)) \to (lt n m))))
\def
(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let
H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0
(plus p m) (plus_sym m p)) in H1)))))).
-(* COMMENTS
-Initial nodes: 101
-END *)
-theorem minus_x_Sy:
+lemma minus_x_Sy:
\forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S
(minus x (S y))))))
\def
\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H:
-(lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_:
-(le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with
-[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S
-y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
+(lt y O)).(let H0 \def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat
+(S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O
\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat
(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda
(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e:
-nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False
-| (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat
-(minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O)))))
-(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat
-(minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0:
-nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))
-(\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S
-n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0:
-nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S
-n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0)
-n H1) in (H n0 H2))))) y)))) x).
-(* COMMENTS
-Initial nodes: 383
-END *)
-
-theorem lt_plus_minus:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1)
+in (False_ind ((le (S y) m) \to (eq nat (minus O y) (S (minus O (S y)))))
+H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H:
+((\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S
+y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to
+(eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda (_: (lt O (S
+n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S n0))) (refl_equal nat
+(S n)) (minus n O) (minus_n_O n))) (\lambda (n0: nat).(\lambda (_: (((lt n0
+(S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))).(\lambda
+(H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) n H1) in (H n0 H2)))))
+y)))) x).
+
+lemma lt_plus_minus:
\forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus
y (S x)))))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S
x) y H))).
-(* COMMENTS
-Initial nodes: 19
-END *)
-theorem lt_plus_minus_r:
+lemma lt_plus_minus_r:
\forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y
(S x)) x)))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat
(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x
y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
-(* COMMENTS
-Initial nodes: 69
-END *)
-theorem minus_x_SO:
+lemma minus_x_SO:
\forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
\def
\lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n:
nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal
nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
-(* COMMENTS
-Initial nodes: 77
-END *)
-theorem le_x_pred_y:
+lemma le_x_pred_y:
\forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
\def
\lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to
(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def
-(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
-\to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let
-H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda
-(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0)
-in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat
-(S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat
-return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let H1 \def
+(eq_ind nat (S x) (\lambda (e: nat).(match e with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H0) in (False_ind (le x O) H1))) | (le_S m H0)
+\Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m)
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow
True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0
(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt
x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S
n))).(le_S_n x n H0))))) y).
-(* COMMENTS
-Initial nodes: 189
-END *)
-theorem lt_le_minus:
+lemma lt_le_minus:
\forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S
O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O))
(plus_sym x (S O)))))).
-(* COMMENTS
-Initial nodes: 57
-END *)
-theorem lt_le_e:
+lemma lt_le_e:
\forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P))
\to ((((le d n) \to P)) \to P))))
\def
\lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n
d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in
(or_ind (le d n) (lt n d) P H0 H H1)))))).
-(* COMMENTS
-Initial nodes: 49
-END *)
-theorem lt_eq_e:
+lemma lt_eq_e:
\forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
\to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x
y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
-(* COMMENTS
-Initial nodes: 59
-END *)
-theorem lt_eq_gt_e:
+lemma lt_eq_gt_e:
\forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
\def
y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x)
\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda
(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
-(* COMMENTS
-Initial nodes: 79
-END *)
-theorem lt_gen_xS:
+lemma lt_gen_xS:
\forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2
nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
\def
(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0)))
(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt
m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
-(* COMMENTS
-Initial nodes: 243
-END *)
-theorem le_lt_false:
+lemma le_lt_false:
\forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P:
Prop).P))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt
y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
-(* COMMENTS
-Initial nodes: 31
-END *)
-theorem lt_neq:
+lemma lt_neq:
\forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq
nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in
(lt_n_n y H1))))).
-(* COMMENTS
-Initial nodes: 43
-END *)
-theorem arith0:
+lemma arith0:
\forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n)
\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
\def
(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2
d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1)
(minus_plus h2 (plus d2 h1))))))).
-(* COMMENTS
-Initial nodes: 235
-END *)
-theorem O_minus:
+lemma O_minus:
\forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
\def
\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0
n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S
x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee:
-nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
-| (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O)
-H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n)
-\to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0
-l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y))))
-x).
-(* COMMENTS
-Initial nodes: 252
-END *)
-
-theorem minus_minus:
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1)
+H1) in (False_ind (eq nat (S x0) O) H3))))) (le_gen_S x0 O H0))) (\lambda (n:
+nat).(\lambda (_: (((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
+x0) | (S l) \Rightarrow (minus x0 l)]) O)))).(\lambda (H1: (le (S x0) (S
+n))).(H n (le_S_n x0 n H1))))) y)))) x).
+
+lemma minus_minus:
\forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y)
\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
\def
z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le
z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S
x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee:
-nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
-| (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4)))))
-(le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y:
-nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus
-y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n:
-nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S
-z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S
-x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0))
-(minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda
-(n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O)
-(\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0
-x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
-I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0))))))
-(\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to
-((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0)
-y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S
-y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S
-z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0)
-H1))))))) y)))) x)))) z).
-(* COMMENTS
-Initial nodes: 751
-END *)
-
-theorem plus_plus:
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x0)
+H2) in (False_ind (eq nat O y) H4))))) (le_gen_S z0 O H)))))) (\lambda (x0:
+nat).(\lambda (_: ((\forall (y: nat).((le (S z0) x0) \to ((le (S z0) y) \to
+((eq nat (minus x0 (S z0)) (minus y (S z0))) \to (eq nat x0 y))))))).(\lambda
+(y: nat).(nat_ind (\lambda (n: nat).((le (S z0) (S x0)) \to ((le (S z0) n)
+\to ((eq nat (minus (S x0) (S z0)) (minus n (S z0))) \to (eq nat (S x0)
+n))))) (\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) O)).(\lambda
+(_: (eq nat (minus (S x0) (S z0)) (minus O (S z0)))).(let H_y \def (le_S_n z0
+x0 H) in (ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n:
+nat).(le z0 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H2: (eq nat O
+(S x1))).(\lambda (_: (le z0 x1)).(let H4 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1)
+H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) (\lambda
+(y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to ((eq nat
+(minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0) y0)))))).(\lambda
+(H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S y0))).(\lambda (H1: (eq
+nat (minus (S x0) (S z0)) (minus (S y0) (S z0)))).(f_equal nat nat S x0 y0
+(IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z).
+
+lemma plus_plus:
\forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1:
nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z
x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3))))
x1)))) z).
-(* COMMENTS
-Initial nodes: 1495
-END *)
-theorem le_S_minus:
+lemma le_S_minus:
\forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to
(le d (S (minus n h))))))
\def
\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h)
(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S
d (minus n h) (le_minus d n h H))))))).
-(* COMMENTS
-Initial nodes: 107
-END *)
-theorem lt_x_pred_y:
+lemma lt_x_pred_y:
\forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
\def
\lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred
n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O)))
(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda
-(H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x)
-n H0)))))) y)).
-(* COMMENTS
-Initial nodes: 103
-END *)
+(H0: (lt x n)).(lt_n_S x n H0)))) y)).