include "ground_1/preamble.ma".
-theorem nat_dec:
+lemma nat_dec:
\forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to
(\forall (P: Prop).P))))
\def
- \lambda (n1: nat).(let TMP_3 \def (\lambda (n: nat).(\forall (n2: nat).(let
-TMP_1 \def (eq nat n n2) in (let TMP_2 \def ((eq nat n n2) \to (\forall (P:
-Prop).P)) in (or TMP_1 TMP_2))))) in (let TMP_18 \def (\lambda (n2: nat).(let
-TMP_6 \def (\lambda (n: nat).(let TMP_4 \def (eq nat O n) in (let TMP_5 \def
-((eq nat O n) \to (\forall (P: Prop).P)) in (or TMP_4 TMP_5)))) in (let TMP_7
-\def (eq nat O O) in (let TMP_8 \def ((eq nat O O) \to (\forall (P: Prop).P))
-in (let TMP_9 \def (refl_equal nat O) in (let TMP_10 \def (or_introl TMP_7
-TMP_8 TMP_9) in (let TMP_17 \def (\lambda (n: nat).(\lambda (_: (or (eq nat O
-n) ((eq nat O n) \to (\forall (P: Prop).P)))).(let TMP_11 \def (S n) in (let
-TMP_12 \def (eq nat O TMP_11) in (let TMP_13 \def ((eq nat O (S n)) \to
-(\forall (P: Prop).P)) in (let TMP_16 \def (\lambda (H0: (eq nat O (S
-n))).(\lambda (P: Prop).(let TMP_14 \def (\lambda (ee: nat).(match ee with [O
-\Rightarrow True | (S _) \Rightarrow False])) in (let TMP_15 \def (S n) in
-(let H1 \def (eq_ind nat O TMP_14 I TMP_15 H0) in (False_ind P H1)))))) in
-(or_intror TMP_12 TMP_13 TMP_16))))))) in (nat_ind TMP_6 TMP_10 TMP_17
-n2)))))))) in (let TMP_71 \def (\lambda (n: nat).(\lambda (H: ((\forall (n2:
+ \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat
+n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2:
+nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall
+(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P:
+Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n)
+((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq
+nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S
+n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match
+ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H0) in
+(False_ind P H1))))))) n2)) (\lambda (n: nat).(\lambda (H: ((\forall (n2:
nat).(or (eq nat n n2) ((eq nat n n2) \to (\forall (P: Prop).P)))))).(\lambda
-(n2: nat).(let TMP_22 \def (\lambda (n0: nat).(let TMP_19 \def (S n) in (let
-TMP_20 \def (eq nat TMP_19 n0) in (let TMP_21 \def ((eq nat (S n) n0) \to
-(\forall (P: Prop).P)) in (or TMP_20 TMP_21))))) in (let TMP_23 \def (S n) in
-(let TMP_24 \def (eq nat TMP_23 O) in (let TMP_25 \def ((eq nat (S n) O) \to
-(\forall (P: Prop).P)) in (let TMP_28 \def (\lambda (H0: (eq nat (S n)
-O)).(\lambda (P: Prop).(let TMP_26 \def (S n) in (let TMP_27 \def (\lambda
-(ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) in
-(let H1 \def (eq_ind nat TMP_26 TMP_27 I O H0) in (False_ind P H1)))))) in
-(let TMP_29 \def (or_intror TMP_24 TMP_25 TMP_28) in (let TMP_70 \def
+(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)))).(let TMP_30 \def (eq nat n n0) in (let TMP_31 \def
-((eq nat n n0) \to (\forall (P: Prop).P)) in (let TMP_32 \def (S n) in (let
-TMP_33 \def (S n0) in (let TMP_34 \def (eq nat TMP_32 TMP_33) in (let TMP_35
-\def ((eq nat (S n) (S n0)) \to (\forall (P: Prop).P)) in (let TMP_36 \def
-(or TMP_34 TMP_35) in (let TMP_53 \def (\lambda (H1: (eq nat n n0)).(let
-TMP_40 \def (\lambda (n3: nat).(let TMP_37 \def (S n) in (let TMP_38 \def (eq
-nat TMP_37 n3) in (let TMP_39 \def ((eq nat (S n) n3) \to (\forall (P:
-Prop).P)) in (or TMP_38 TMP_39))))) in (let H2 \def (eq_ind_r nat n0 TMP_40
-H0 n H1) in (let TMP_45 \def (\lambda (n3: nat).(let TMP_41 \def (S n) in
-(let TMP_42 \def (S n3) in (let TMP_43 \def (eq nat TMP_41 TMP_42) in (let
-TMP_44 \def ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)) in (or TMP_43
-TMP_44)))))) in (let TMP_46 \def (S n) in (let TMP_47 \def (S n) in (let
-TMP_48 \def (eq nat TMP_46 TMP_47) in (let TMP_49 \def ((eq nat (S n) (S n))
-\to (\forall (P: Prop).P)) in (let TMP_50 \def (S n) in (let TMP_51 \def
-(refl_equal nat TMP_50) in (let TMP_52 \def (or_introl TMP_48 TMP_49 TMP_51)
-in (eq_ind nat n TMP_45 TMP_52 n0 H1)))))))))))) in (let TMP_68 \def (\lambda
-(H1: (((eq nat n n0) \to (\forall (P: Prop).P)))).(let TMP_54 \def (S n) in
-(let TMP_55 \def (S n0) in (let TMP_56 \def (eq nat TMP_54 TMP_55) in (let
-TMP_57 \def ((eq nat (S n) (S n0)) \to (\forall (P: Prop).P)) in (let TMP_67
-\def (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: Prop).(let TMP_58 \def
-(\lambda (e: nat).(match e with [O \Rightarrow n | (S n3) \Rightarrow n3]))
-in (let TMP_59 \def (S n) in (let TMP_60 \def (S n0) in (let H3 \def (f_equal
-nat nat TMP_58 TMP_59 TMP_60 H2) in (let TMP_61 \def (\lambda (n3: nat).((eq
-nat n n3) \to (\forall (P0: Prop).P0))) in (let H4 \def (eq_ind_r nat n0
-TMP_61 H1 n H3) in (let TMP_65 \def (\lambda (n3: nat).(let TMP_62 \def (S n)
-in (let TMP_63 \def (eq nat TMP_62 n3) in (let TMP_64 \def ((eq nat (S n) n3)
-\to (\forall (P0: Prop).P0)) in (or TMP_63 TMP_64))))) in (let H5 \def
-(eq_ind_r nat n0 TMP_65 H0 n H3) in (let TMP_66 \def (refl_equal nat n) in
-(H4 TMP_66 P)))))))))))) in (or_intror TMP_56 TMP_57 TMP_67))))))) in (let
-TMP_69 \def (H n0) in (or_ind TMP_30 TMP_31 TMP_36 TMP_53 TMP_68
-TMP_69))))))))))))) in (nat_ind TMP_22 TMP_29 TMP_70 n2))))))))))) in
-(nat_ind TMP_3 TMP_18 TMP_71 n1)))).
-
-theorem simpl_plus_r:
+(\forall (P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall
+(P: Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall
+(P: Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0
+(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P:
+Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S
+n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat
+(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat
+(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P:
+Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to
+(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P:
+Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e with [O
+\Rightarrow n | (S n3) \Rightarrow n3])) (S n) (S n0) H2) in (let H4 \def
+(eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) \to (\forall (P0:
+Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 (\lambda (n3: nat).(or
+(eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: Prop).P0)))) H0 n H3)
+in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) n1).
+
+lemma simpl_plus_r:
\forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n)
(plus p n)) \to (eq nat m p))))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat
-(plus m n) (plus p n))).(let TMP_1 \def (plus m n) in (let TMP_3 \def
-(\lambda (n0: nat).(let TMP_2 \def (plus n p) in (eq nat n0 TMP_2))) in (let
-TMP_4 \def (plus p n) in (let TMP_6 \def (\lambda (n0: nat).(let TMP_5 \def
-(plus n p) in (eq nat n0 TMP_5))) in (let TMP_7 \def (plus_sym p n) in (let
-TMP_8 \def (plus m n) in (let TMP_9 \def (eq_ind_r nat TMP_4 TMP_6 TMP_7
-TMP_8 H) in (let TMP_10 \def (plus n m) in (let TMP_11 \def (plus_sym n m) in
-(let TMP_12 \def (eq_ind_r nat TMP_1 TMP_3 TMP_9 TMP_10 TMP_11) in
-(simpl_plus_l n m p TMP_12)))))))))))))).
-
-theorem minus_Sx_Sy:
+(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda
+(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0:
+nat).(eq nat n0 (plus n p))) (plus_sym p n) (plus m n) H) (plus n m)
+(plus_sym n m)))))).
+
+lemma minus_Sx_Sy:
\forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
\def
- \lambda (x: nat).(\lambda (y: nat).(let TMP_1 \def (minus x y) in
-(refl_equal nat TMP_1))).
+ \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
-theorem minus_plus_r:
+lemma minus_plus_r:
\forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
\def
- \lambda (m: nat).(\lambda (n: nat).(let TMP_1 \def (plus n m) in (let TMP_3
-\def (\lambda (n0: nat).(let TMP_2 \def (minus n0 n) in (eq nat TMP_2 m))) in
-(let TMP_4 \def (minus_plus n m) in (let TMP_5 \def (plus m n) in (let TMP_6
-\def (plus_sym m n) in (eq_ind_r nat TMP_1 TMP_3 TMP_4 TMP_5 TMP_6))))))).
+ \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0:
+nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
-theorem plus_permute_2_in_3:
+lemma plus_permute_2_in_3:
\forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x
y) z) (plus (plus x z) y))))
\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(let TMP_1 \def (plus y
-z) in (let TMP_2 \def (plus x TMP_1) in (let TMP_5 \def (\lambda (n:
-nat).(let TMP_3 \def (plus x z) in (let TMP_4 \def (plus TMP_3 y) in (eq nat
-n TMP_4)))) in (let TMP_6 \def (plus z y) in (let TMP_10 \def (\lambda (n:
-nat).(let TMP_7 \def (plus x n) in (let TMP_8 \def (plus x z) in (let TMP_9
-\def (plus TMP_8 y) in (eq nat TMP_7 TMP_9))))) in (let TMP_11 \def (plus x
-z) in (let TMP_12 \def (plus TMP_11 y) in (let TMP_15 \def (\lambda (n:
-nat).(let TMP_13 \def (plus x z) in (let TMP_14 \def (plus TMP_13 y) in (eq
-nat n TMP_14)))) in (let TMP_16 \def (plus x z) in (let TMP_17 \def (plus
-TMP_16 y) in (let TMP_18 \def (refl_equal nat TMP_17) in (let TMP_19 \def
-(plus z y) in (let TMP_20 \def (plus x TMP_19) in (let TMP_21 \def
-(plus_assoc_r x z y) in (let TMP_22 \def (eq_ind nat TMP_12 TMP_15 TMP_18
-TMP_20 TMP_21) in (let TMP_23 \def (plus y z) in (let TMP_24 \def (plus_sym y
-z) in (let TMP_25 \def (eq_ind_r nat TMP_6 TMP_10 TMP_22 TMP_23 TMP_24) in
-(let TMP_26 \def (plus x y) in (let TMP_27 \def (plus TMP_26 z) in (let
-TMP_28 \def (plus_assoc_r x y z) in (eq_ind_r nat TMP_2 TMP_5 TMP_25 TMP_27
-TMP_28)))))))))))))))))))))))).
-
-theorem plus_permute_2_in_3_assoc:
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x
+(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat
+(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind
+nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
+(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z
+y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
+
+lemma plus_permute_2_in_3_assoc:
\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n
h) k) (plus n (plus k h)))))
\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(let TMP_1 \def (plus n
-k) in (let TMP_2 \def (plus TMP_1 h) in (let TMP_5 \def (\lambda (n0:
-nat).(let TMP_3 \def (plus k h) in (let TMP_4 \def (plus n TMP_3) in (eq nat
-n0 TMP_4)))) in (let TMP_6 \def (plus n k) in (let TMP_7 \def (plus TMP_6 h)
-in (let TMP_10 \def (\lambda (n0: nat).(let TMP_8 \def (plus n k) in (let
-TMP_9 \def (plus TMP_8 h) in (eq nat TMP_9 n0)))) in (let TMP_11 \def (plus n
-k) in (let TMP_12 \def (plus TMP_11 h) in (let TMP_13 \def (refl_equal nat
-TMP_12) in (let TMP_14 \def (plus k h) in (let TMP_15 \def (plus n TMP_14) in
-(let TMP_16 \def (plus_assoc_l n k h) in (let TMP_17 \def (eq_ind_r nat TMP_7
-TMP_10 TMP_13 TMP_15 TMP_16) in (let TMP_18 \def (plus n h) in (let TMP_19
-\def (plus TMP_18 k) in (let TMP_20 \def (plus_permute_2_in_3 n h k) in
-(eq_ind_r nat TMP_2 TMP_5 TMP_17 TMP_19 TMP_20))))))))))))))))))).
-
-theorem plus_O:
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus
+(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r
+nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
+(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k
+h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
+
+lemma plus_O:
\forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat
x O) (eq nat y O))))
\def
- \lambda (x: nat).(let TMP_3 \def (\lambda (n: nat).(\forall (y: nat).((eq
-nat (plus n y) O) \to (let TMP_1 \def (eq nat n O) in (let TMP_2 \def (eq nat
-y O) in (land TMP_1 TMP_2)))))) in (let TMP_7 \def (\lambda (y: nat).(\lambda
-(H: (eq nat (plus O y) O)).(let TMP_4 \def (eq nat O O) in (let TMP_5 \def
-(eq nat y O) in (let TMP_6 \def (refl_equal nat O) in (conj TMP_4 TMP_5 TMP_6
-H)))))) in (let TMP_16 \def (\lambda (n: nat).(\lambda (_: ((\forall (y:
-nat).((eq nat (plus n y) O) \to (land (eq nat n O) (eq nat y O)))))).(\lambda
-(y: nat).(\lambda (H0: (eq nat (plus (S n) y) O)).(let H1 \def (match H0 with
-[refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n) y) O)).(let TMP_8
-\def (S n) in (let TMP_9 \def (plus TMP_8 y) in (let TMP_10 \def (\lambda (e:
-nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) in (let
-H2 \def (eq_ind nat TMP_9 TMP_10 I O H1) in (let TMP_11 \def (S n) in (let
-TMP_12 \def (eq nat TMP_11 O) in (let TMP_13 \def (eq nat y O) in (let TMP_14
-\def (land TMP_12 TMP_13) in (False_ind TMP_14 H2))))))))))]) in (let TMP_15
-\def (refl_equal nat O) in (H1 TMP_15))))))) in (nat_ind TMP_3 TMP_7 TMP_16
-x)))).
-
-theorem minus_Sx_SO:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus
+n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda
+(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O)
+H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O)
+\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq
+nat (plus (S n) y) O)).(let H1 \def (match H0 with [refl_equal \Rightarrow
+(\lambda (H1: (eq nat (plus (S n) y) O)).(let H2 \def (eq_ind nat (plus (S n)
+y) (\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y O)) H2)))]) in
+(H1 (refl_equal nat O))))))) x).
+
+lemma minus_Sx_SO:
\forall (x: nat).(eq nat (minus (S x) (S O)) x)
\def
- \lambda (x: nat).(let TMP_1 \def (\lambda (n: nat).(eq nat n x)) in (let
-TMP_2 \def (refl_equal nat x) in (let TMP_3 \def (minus x O) in (let TMP_4
-\def (minus_n_O x) in (eq_ind nat x TMP_1 TMP_2 TMP_3 TMP_4))))).
+ \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal
+nat x) (minus x O) (minus_n_O x)).
-theorem nat_dec_neg:
+lemma nat_dec_neg:
\forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
\def
- \lambda (i: nat).(let TMP_4 \def (\lambda (n: nat).(\forall (j: nat).(let
-TMP_1 \def (eq nat n j) in (let TMP_2 \def (not TMP_1) in (let TMP_3 \def (eq
-nat n j) in (or TMP_2 TMP_3)))))) in (let TMP_21 \def (\lambda (j: nat).(let
-TMP_8 \def (\lambda (n: nat).(let TMP_5 \def (eq nat O n) in (let TMP_6 \def
-(not TMP_5) in (let TMP_7 \def (eq nat O n) in (or TMP_6 TMP_7))))) in (let
-TMP_9 \def (eq nat O O) in (let TMP_10 \def (not TMP_9) in (let TMP_11 \def
-(eq nat O O) in (let TMP_12 \def (refl_equal nat O) in (let TMP_13 \def
-(or_intror TMP_10 TMP_11 TMP_12) in (let TMP_20 \def (\lambda (n:
-nat).(\lambda (_: (or (not (eq nat O n)) (eq nat O n))).(let TMP_14 \def (S
-n) in (let TMP_15 \def (eq nat O TMP_14) in (let TMP_16 \def (not TMP_15) in
-(let TMP_17 \def (S n) in (let TMP_18 \def (eq nat O TMP_17) in (let TMP_19
-\def (O_S n) in (or_introl TMP_16 TMP_18 TMP_19))))))))) in (nat_ind TMP_8
-TMP_13 TMP_20 j))))))))) in (let TMP_68 \def (\lambda (n: nat).(\lambda (H:
-((\forall (j: nat).(or (not (eq nat n j)) (eq nat n j))))).(\lambda (j:
-nat).(let TMP_27 \def (\lambda (n0: nat).(let TMP_22 \def (S n) in (let
-TMP_23 \def (eq nat TMP_22 n0) in (let TMP_24 \def (not TMP_23) in (let
-TMP_25 \def (S n) in (let TMP_26 \def (eq nat TMP_25 n0) in (or TMP_24
-TMP_26))))))) in (let TMP_28 \def (S n) in (let TMP_29 \def (eq nat TMP_28 O)
-in (let TMP_30 \def (not TMP_29) in (let TMP_31 \def (S n) in (let TMP_32
-\def (eq nat TMP_31 O) in (let TMP_33 \def (S n) in (let TMP_34 \def (O_S n)
-in (let TMP_35 \def (sym_not_eq nat O TMP_33 TMP_34) in (let TMP_36 \def
-(or_introl TMP_30 TMP_32 TMP_35) in (let TMP_67 \def (\lambda (n0:
-nat).(\lambda (_: (or (not (eq nat (S n) n0)) (eq nat (S n) n0))).(let TMP_37
-\def (eq nat n n0) in (let TMP_38 \def (not TMP_37) in (let TMP_39 \def (eq
-nat n n0) in (let TMP_40 \def (S n) in (let TMP_41 \def (S n0) in (let TMP_42
-\def (eq nat TMP_40 TMP_41) in (let TMP_43 \def (not TMP_42) in (let TMP_44
-\def (S n) in (let TMP_45 \def (S n0) in (let TMP_46 \def (eq nat TMP_44
-TMP_45) in (let TMP_47 \def (or TMP_43 TMP_46) in (let TMP_56 \def (\lambda
-(H1: (not (eq nat n n0))).(let TMP_48 \def (S n) in (let TMP_49 \def (S n0)
-in (let TMP_50 \def (eq nat TMP_48 TMP_49) in (let TMP_51 \def (not TMP_50)
-in (let TMP_52 \def (S n) in (let TMP_53 \def (S n0) in (let TMP_54 \def (eq
-nat TMP_52 TMP_53) in (let TMP_55 \def (not_eq_S n n0 H1) in (or_introl
-TMP_51 TMP_54 TMP_55)))))))))) in (let TMP_65 \def (\lambda (H1: (eq nat n
-n0)).(let TMP_57 \def (S n) in (let TMP_58 \def (S n0) in (let TMP_59 \def
-(eq nat TMP_57 TMP_58) in (let TMP_60 \def (not TMP_59) in (let TMP_61 \def
-(S n) in (let TMP_62 \def (S n0) in (let TMP_63 \def (eq nat TMP_61 TMP_62)
-in (let TMP_64 \def (f_equal nat nat S n n0 H1) in (or_intror TMP_60 TMP_63
-TMP_64)))))))))) in (let TMP_66 \def (H n0) in (or_ind TMP_38 TMP_39 TMP_47
-TMP_56 TMP_65 TMP_66))))))))))))))))) in (nat_ind TMP_27 TMP_36 TMP_67
-j))))))))))))))) in (nat_ind TMP_4 TMP_21 TMP_68 i)))).
-
-theorem neq_eq_e:
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq
+nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or
+(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O)
+(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq
+nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j))
+(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq
+nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat
+(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S
+n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or
+(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq
+nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda
+(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S
+n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not
+(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H
+n0)))) j)))) i).
+
+lemma neq_eq_e:
\forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j))
\to P)) \to ((((eq nat i j) \to P)) \to P))))
\def
\lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not
(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def
-(nat_dec_neg i j) in (let TMP_1 \def (eq nat i j) in (let TMP_2 \def (not
-TMP_1) in (let TMP_3 \def (eq nat i j) in (or_ind TMP_2 TMP_3 P H H0
-o))))))))).
+(nat_dec_neg i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
-theorem le_false:
+lemma le_false:
\forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S
n) m) \to P))))
\def
- \lambda (m: nat).(let TMP_1 \def (\lambda (n: nat).(\forall (n0:
-nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P))))) in (let
-TMP_9 \def (\lambda (n: nat).(\lambda (P: Prop).(\lambda (_: (le O
-n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 with [le_n \Rightarrow
-(\lambda (H1: (eq nat (S n) O)).(let TMP_6 \def (S n) in (let TMP_7 \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 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])) in (let H2 \def (eq_ind nat TMP_6 TMP_7 I O H1) in (False_ind P
-H2))))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0) O)).(let
-TMP_2 \def (S m0) in (let TMP_3 \def (\lambda (e: nat).(match e with [O
-\Rightarrow False | (S _) \Rightarrow True])) in (let H3 \def (eq_ind nat
-TMP_2 TMP_3 I O H2) in (let TMP_4 \def ((le (S n) m0) \to P) in (let TMP_5
-\def (False_ind TMP_4 H3) in (TMP_5 H1)))))))]) in (let TMP_8 \def
-(refl_equal nat O) in (H1 TMP_8))))))) in (let TMP_23 \def (\lambda (n:
-nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: Prop).((le n n0) \to ((le
-(S n0) n) \to P)))))).(\lambda (n0: nat).(let TMP_10 \def (\lambda (n1:
-nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P)))) in
-(let TMP_18 \def (\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 TMP_15 \def (S n) in (let TMP_16 \def (\lambda (e:
-nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) in (let
-H3 \def (eq_ind nat TMP_15 TMP_16 I O H2) in (False_ind P H3))))) | (le_S m0
-H2) \Rightarrow (\lambda (H3: (eq nat (S m0) O)).(let TMP_11 \def (S m0) in
-(let TMP_12 \def (\lambda (e: nat).(match e with [O \Rightarrow False | (S _)
-\Rightarrow True])) in (let H4 \def (eq_ind nat TMP_11 TMP_12 I O H3) in (let
-TMP_13 \def ((le (S n) m0) \to P) in (let TMP_14 \def (False_ind TMP_13 H4)
-in (TMP_14 H2)))))))]) in (let TMP_17 \def (refl_equal nat O) in (H2
-TMP_17)))))) in (let TMP_22 \def (\lambda (n1: nat).(\lambda (_: ((\forall
-(P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda (P:
-Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S
-n))).(let TMP_19 \def (le_S_n n n1 H1) in (let TMP_20 \def (S n1) in (let
-TMP_21 \def (le_S_n TMP_20 n H2) in (H n1 P TMP_19 TMP_21))))))))) in
-(nat_ind TMP_10 TMP_18 TMP_22 n0))))))) in (nat_ind TMP_1 TMP_9 TMP_23 m)))).
-
-theorem le_Sx_x:
+True])) I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1
+(refl_equal nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0:
+nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda
+(n0: nat).(nat_ind (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to
+((le (S n1) (S n)) \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n)
+O)).(\lambda (_: (le (S O) (S n))).(let H2 \def (match H0 with [le_n
+\Rightarrow (\lambda (H2: (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n)
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow (\lambda
+(H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3)
+in (False_ind ((le (S n) m0) \to P) H4)) H2))]) in (H2 (refl_equal nat
+O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S n) n1)
+\to ((le (S n1) (S n)) \to P))))).(\lambda (P: Prop).(\lambda (H1: (le (S n)
+(S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(H n1 P (le_S_n n n1 H1)
+(le_S_n (S n1) n H2))))))) n0)))) m).
+
+lemma le_Sx_x:
\forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
\def
\lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def
-le_Sn_n in (let TMP_1 \def (H0 x H) in (False_ind P TMP_1))))).
+le_Sn_n in (False_ind P (H0 x H))))).
-theorem le_n_pred:
+lemma le_n_pred:
\forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_3 \def
-(\lambda (n0: nat).(let TMP_1 \def (pred n) in (let TMP_2 \def (pred n0) in
-(le TMP_1 TMP_2)))) in (let TMP_4 \def (pred n) in (let TMP_5 \def (le_n
-TMP_4) in (let TMP_9 \def (\lambda (m0: nat).(\lambda (_: (le n m0)).(\lambda
-(H1: (le (pred n) (pred m0))).(let TMP_6 \def (pred n) in (let TMP_7 \def
-(pred m0) in (let TMP_8 \def (le_pred_n m0) in (le_trans TMP_6 TMP_7 m0 H1
-TMP_8))))))) in (le_ind n TMP_3 TMP_5 TMP_9 m H))))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
+(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0:
+nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans
+(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
-theorem minus_le:
+lemma minus_le:
\forall (x: nat).(\forall (y: nat).(le (minus x y) x))
\def
- \lambda (x: nat).(let TMP_2 \def (\lambda (n: nat).(\forall (y: nat).(let
-TMP_1 \def (minus n y) in (le TMP_1 n)))) in (let TMP_3 \def (\lambda (_:
-nat).(le_O_n O)) in (let TMP_13 \def (\lambda (n: nat).(\lambda (H: ((\forall
-(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(let TMP_7 \def (\lambda
-(n0: nat).(let TMP_4 \def (S n) in (let TMP_5 \def (minus TMP_4 n0) in (let
-TMP_6 \def (S n) in (le TMP_5 TMP_6))))) in (let TMP_8 \def (S n) in (let
-TMP_9 \def (le_n TMP_8) in (let TMP_12 \def (\lambda (n0: nat).(\lambda (_:
-(le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)]) (S
-n))).(let TMP_10 \def (minus n n0) in (let TMP_11 \def (H n0) in (le_S TMP_10
-n TMP_11))))) in (nat_ind TMP_7 TMP_9 TMP_12 y)))))))) in (nat_ind TMP_2
-TMP_3 TMP_13 x)))).
-
-theorem le_plus_minus_sym:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n
+y) n))) (\lambda (_: nat).(le_O_n O)) (\lambda (n: nat).(\lambda (H:
+((\forall (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda
+(n0: nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0:
+nat).(\lambda (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow
+(minus n l)]) (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
+
+lemma le_plus_minus_sym:
\forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n)
n))))
\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_1 \def
-(minus m n) in (let TMP_2 \def (plus n TMP_1) in (let TMP_3 \def (\lambda
-(n0: nat).(eq nat m n0)) in (let TMP_4 \def (le_plus_minus n m H) in (let
-TMP_5 \def (minus m n) in (let TMP_6 \def (plus TMP_5 n) in (let TMP_7 \def
-(minus m n) in (let TMP_8 \def (plus_sym TMP_7 n) in (eq_ind_r nat TMP_2
-TMP_3 TMP_4 TMP_6 TMP_8))))))))))).
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat
+(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H)
+(plus (minus m n) n) (plus_sym (minus m n) n)))).
-theorem le_minus_minus:
+lemma le_minus_minus:
\forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z)
\to (le (minus y x) (minus z x))))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z:
-nat).(\lambda (H0: (le y z)).(let TMP_1 \def (minus y x) in (let TMP_2 \def
-(minus z x) in (let TMP_5 \def (\lambda (n: nat).(let TMP_3 \def (minus z x)
-in (let TMP_4 \def (plus x TMP_3) in (le n TMP_4)))) in (let TMP_6 \def
-(\lambda (n: nat).(le y n)) in (let TMP_7 \def (minus z x) in (let TMP_8 \def
-(plus x TMP_7) in (let TMP_9 \def (le_trans x y z H H0) in (let TMP_10 \def
-(le_plus_minus_r x z TMP_9) in (let TMP_11 \def (eq_ind_r nat z TMP_6 H0
-TMP_8 TMP_10) in (let TMP_12 \def (minus y x) in (let TMP_13 \def (plus x
-TMP_12) in (let TMP_14 \def (le_plus_minus_r x y H) in (let TMP_15 \def
-(eq_ind_r nat y TMP_5 TMP_11 TMP_13 TMP_14) in (simpl_le_plus_l x TMP_1 TMP_2
-TMP_15)))))))))))))))))).
-
-theorem le_minus_plus:
+nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x)
+(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat
+z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z
+(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
+
+lemma le_minus_plus:
\forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat
(minus (plus x y) z) (plus (minus x z) y)))))
\def
- \lambda (z: nat).(let TMP_5 \def (\lambda (n: nat).(\forall (x: nat).((le n
-x) \to (\forall (y: nat).(let TMP_1 \def (plus x y) in (let TMP_2 \def (minus
-TMP_1 n) in (let TMP_3 \def (minus x n) in (let TMP_4 \def (plus TMP_3 y) in
-(eq nat TMP_2 TMP_4))))))))) in (let TMP_29 \def (\lambda (x: nat).(\lambda
-(H: (le O x)).(let H0 \def (match H with [le_n \Rightarrow (\lambda (H0: (eq
-nat O x)).(let TMP_20 \def (\lambda (n: nat).(\forall (y: nat).(let TMP_16
-\def (plus n y) in (let TMP_17 \def (minus TMP_16 O) in (let TMP_18 \def
-(minus n O) in (let TMP_19 \def (plus TMP_18 y) in (eq nat TMP_17
-TMP_19))))))) in (let TMP_27 \def (\lambda (y: nat).(let TMP_21 \def (minus O
-O) in (let TMP_22 \def (plus TMP_21 y) in (let TMP_23 \def (plus O y) in (let
-TMP_24 \def (minus TMP_23 O) in (let TMP_25 \def (plus O y) in (let TMP_26
-\def (minus_n_O TMP_25) in (sym_eq nat TMP_22 TMP_24 TMP_26)))))))) in
-(eq_ind nat O TMP_20 TMP_27 x H0)))) | (le_S m H0) \Rightarrow (\lambda (H1:
-(eq nat (S m) x)).(let TMP_6 \def (S m) in (let TMP_11 \def (\lambda (n:
-nat).((le O m) \to (\forall (y: nat).(let TMP_7 \def (plus n y) in (let TMP_8
-\def (minus TMP_7 O) in (let TMP_9 \def (minus n O) in (let TMP_10 \def (plus
-TMP_9 y) in (eq nat TMP_8 TMP_10)))))))) in (let TMP_15 \def (\lambda (_: (le
-O m)).(\lambda (y: nat).(let TMP_12 \def (S m) in (let TMP_13 \def (minus
-TMP_12 O) in (let TMP_14 \def (plus TMP_13 y) in (refl_equal nat TMP_14))))))
-in (eq_ind nat TMP_6 TMP_11 TMP_15 x H1 H0)))))]) in (let TMP_28 \def
-(refl_equal nat x) in (H0 TMP_28))))) in (let TMP_60 \def (\lambda (z0:
-nat).(\lambda (H: ((\forall (x: nat).((le z0 x) \to (\forall (y: nat).(eq nat
-(minus (plus x y) z0) (plus (minus x z0) y))))))).(\lambda (x: nat).(let
-TMP_36 \def (\lambda (n: nat).((le (S z0) n) \to (\forall (y: nat).(let
-TMP_30 \def (plus n y) in (let TMP_31 \def (S z0) in (let TMP_32 \def (minus
-TMP_30 TMP_31) in (let TMP_33 \def (S z0) in (let TMP_34 \def (minus n
-TMP_33) in (let TMP_35 \def (plus TMP_34 y) in (eq nat TMP_32
-TMP_35)))))))))) in (let TMP_57 \def (\lambda (H0: (le (S z0) O)).(\lambda
-(y: nat).(let H1 \def (match H0 with [le_n \Rightarrow (\lambda (H1: (eq nat
-(S z0) O)).(let TMP_47 \def (S z0) in (let TMP_48 \def (\lambda (e:
-nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) in (let
-H2 \def (eq_ind nat TMP_47 TMP_48 I O H1) in (let TMP_49 \def (plus O y) in
-(let TMP_50 \def (S z0) in (let TMP_51 \def (minus TMP_49 TMP_50) in (let
-TMP_52 \def (S z0) in (let TMP_53 \def (minus O TMP_52) in (let TMP_54 \def
-(plus TMP_53 y) in (let TMP_55 \def (eq nat TMP_51 TMP_54) in (False_ind
-TMP_55 H2)))))))))))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m)
-O)).(let TMP_37 \def (S m) in (let TMP_38 \def (\lambda (e: nat).(match e
-with [O \Rightarrow False | (S _) \Rightarrow True])) in (let H3 \def (eq_ind
-nat TMP_37 TMP_38 I O H2) in (let TMP_45 \def ((le (S z0) m) \to (let TMP_39
-\def (plus O y) in (let TMP_40 \def (S z0) in (let TMP_41 \def (minus TMP_39
-TMP_40) in (let TMP_42 \def (S z0) in (let TMP_43 \def (minus O TMP_42) in
-(let TMP_44 \def (plus TMP_43 y) in (eq nat TMP_41 TMP_44)))))))) in (let
-TMP_46 \def (False_ind TMP_45 H3) in (TMP_46 H1)))))))]) in (let TMP_56 \def
-(refl_equal nat O) in (H1 TMP_56))))) in (let TMP_59 \def (\lambda (n:
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to
+(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y))))))
+(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H with [le_n
+\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n:
+nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))
+(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O)
+(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq
+nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y:
+nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O
+m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))])
+in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x:
+nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus
+(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S
+z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
+(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def
+(match H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2
+\def (eq_ind nat (S z0) (\lambda (e: nat).(match e with [O \Rightarrow False
+| (S _) \Rightarrow True])) I O H1) in (False_ind (eq nat (minus (plus O y)
+(S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) \Rightarrow (\lambda
+(H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2)
+in (False_ind ((le (S z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus
+(minus O (S z0)) y))) H3)) H1))]) in (H1 (refl_equal nat O))))) (\lambda (n:
nat).(\lambda (_: (((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus
n y) (S z0)) (plus (minus n (S z0)) y)))))).(\lambda (H1: (le (S z0) (S
-n))).(\lambda (y: nat).(let TMP_58 \def (le_S_n z0 n H1) in (H n TMP_58
-y)))))) in (nat_ind TMP_36 TMP_57 TMP_59 x))))))) in (nat_ind TMP_5 TMP_29
-TMP_60 z)))).
+n))).(\lambda (y: nat).(H n (le_S_n z0 n H1) y))))) x)))) z).
-theorem le_minus:
+lemma le_minus:
\forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to
(le x (minus z y)))))
\def
\lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus
-x y) z)).(let TMP_1 \def (plus x y) in (let TMP_2 \def (minus TMP_1 y) in
-(let TMP_4 \def (\lambda (n: nat).(let TMP_3 \def (minus z y) in (le n
-TMP_3))) in (let TMP_5 \def (plus x y) in (let TMP_6 \def (le_plus_r x y) in
-(let TMP_7 \def (le_minus_minus y TMP_5 TMP_6 z H) in (let TMP_8 \def
-(minus_plus_r x y) in (eq_ind nat TMP_2 TMP_4 TMP_7 x TMP_8))))))))))).
+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))))).
-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)).(let TMP_1 \def (plus x y) in (let TMP_2 \def (le_plus_r x y) in
-(le_trans y TMP_1 z TMP_2 H)))))).
+x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
-theorem lt_x_O:
+lemma lt_x_O:
\forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
\def
- \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let TMP_1
-\def (S x) in (let H_y \def (le_n_O_eq TMP_1 H) in (let TMP_2 \def (\lambda
-(ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) in
-(let TMP_3 \def (S x) in (let H0 \def (eq_ind nat O TMP_2 I TMP_3 H_y) in
-(False_ind P H0)))))))).
+ \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def
+(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match
+ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) H_y) in
+(False_ind P H0))))).
-theorem le_gen_S:
+lemma le_gen_S:
\forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n:
nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
\def
\lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def
-(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(let TMP_16
-\def (S m) in (let TMP_20 \def (\lambda (n: nat).(let TMP_18 \def (\lambda
-(n0: nat).(let TMP_17 \def (S n0) in (eq nat n TMP_17))) in (let TMP_19 \def
-(\lambda (n0: nat).(le m n0)) in (ex2 nat TMP_18 TMP_19)))) in (let TMP_23
-\def (\lambda (n: nat).(let TMP_21 \def (S m) in (let TMP_22 \def (S n) in
-(eq nat TMP_21 TMP_22)))) in (let TMP_24 \def (\lambda (n: nat).(le m n)) in
-(let TMP_25 \def (S m) in (let TMP_26 \def (refl_equal nat TMP_25) in (let
-TMP_27 \def (le_n m) in (let TMP_28 \def (ex_intro2 nat TMP_23 TMP_24 m
-TMP_26 TMP_27) in (eq_ind nat TMP_16 TMP_20 TMP_28 x H0)))))))))) | (le_S m0
-H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(let TMP_1 \def (S m0) in
-(let TMP_5 \def (\lambda (n: nat).((le (S m) m0) \to (let TMP_3 \def (\lambda
-(n0: nat).(let TMP_2 \def (S n0) in (eq nat n TMP_2))) in (let TMP_4 \def
-(\lambda (n0: nat).(le m n0)) in (ex2 nat TMP_3 TMP_4))))) in (let TMP_15
-\def (\lambda (H2: (le (S m) m0)).(let TMP_8 \def (\lambda (n: nat).(let
-TMP_6 \def (S m0) in (let TMP_7 \def (S n) in (eq nat TMP_6 TMP_7)))) in (let
-TMP_9 \def (\lambda (n: nat).(le m n)) in (let TMP_10 \def (S m0) in (let
-TMP_11 \def (refl_equal nat TMP_10) in (let TMP_12 \def (S m) in (let TMP_13
-\def (le_S TMP_12 m0 H2) in (let TMP_14 \def (le_S_n m m0 TMP_13) in
-(ex_intro2 nat TMP_8 TMP_9 m0 TMP_11 TMP_14))))))))) in (eq_ind nat TMP_1
-TMP_5 TMP_15 x H1 H0)))))]) in (let TMP_29 \def (refl_equal nat x) in (H0
-TMP_29))))).
-
-theorem lt_x_plus_x_Sy:
+(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat
+(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0)))
+(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S
+m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x
+H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat
+(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq
+nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m)
+m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n:
+nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2))))
+x H1 H0))]) in (H0 (refl_equal nat x))))).
+
+lemma lt_x_plus_x_Sy:
\forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
\def
- \lambda (x: nat).(\lambda (y: nat).(let TMP_1 \def (S y) in (let TMP_2 \def
-(plus TMP_1 x) in (let TMP_3 \def (\lambda (n: nat).(lt x n)) in (let TMP_4
-\def (S x) in (let TMP_5 \def (plus y x) in (let TMP_6 \def (S TMP_5) in (let
-TMP_7 \def (S x) in (let TMP_8 \def (plus y x) in (let TMP_9 \def (S TMP_8)
-in (let TMP_10 \def (plus y x) in (let TMP_11 \def (le_plus_r y x) in (let
-TMP_12 \def (le_n_S x TMP_10 TMP_11) in (let TMP_13 \def (le_n_S TMP_7 TMP_9
-TMP_12) in (let TMP_14 \def (le_S_n TMP_4 TMP_6 TMP_13) in (let TMP_15 \def
-(S y) in (let TMP_16 \def (plus x TMP_15) in (let TMP_17 \def (S y) in (let
-TMP_18 \def (plus_sym x TMP_17) in (eq_ind_r nat TMP_2 TMP_3 TMP_14 TMP_16
-TMP_18)))))))))))))))))))).
-
-theorem simpl_lt_plus_r:
+ \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n:
+nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x))
+(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
+
+lemma simpl_lt_plus_r:
\forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m
p)) \to (lt n m))))
\def
\lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus
-n p) (plus m p))).(let TMP_1 \def (plus n p) 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
-n) in (let TMP_5 \def (plus_sym n p) in (let H0 \def (eq_ind nat TMP_1 TMP_3
-H TMP_4 TMP_5) in (let TMP_6 \def (plus m p) in (let TMP_8 \def (\lambda (n0:
-nat).(let TMP_7 \def (plus p n) in (lt TMP_7 n0))) in (let TMP_9 \def (plus p
-m) in (let TMP_10 \def (plus_sym m p) in (let H1 \def (eq_ind nat TMP_6 TMP_8
-H0 TMP_9 TMP_10) in (simpl_lt_plus_l n m p H1)))))))))))))).
-
-theorem minus_x_Sy:
+n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p)
+(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let
+H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0
+(plus p m) (plus_sym m p)) in H1)))))).
+
+lemma minus_x_Sy:
\forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S
(minus x (S y))))))
\def
- \lambda (x: nat).(let TMP_5 \def (\lambda (n: nat).(\forall (y: nat).((lt y
-n) \to (let TMP_1 \def (minus n y) in (let TMP_2 \def (S y) in (let TMP_3
-\def (minus n TMP_2) in (let TMP_4 \def (S TMP_3) in (eq nat TMP_1
-TMP_4)))))))) in (let TMP_22 \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_14 \def (S y) in (let TMP_15 \def (\lambda (e: nat).(match e
-with [O \Rightarrow False | (S _) \Rightarrow True])) in (let H1 \def (eq_ind
-nat TMP_14 TMP_15 I O H0) in (let TMP_16 \def (minus O y) in (let TMP_17 \def
-(S y) in (let TMP_18 \def (minus O TMP_17) in (let TMP_19 \def (S TMP_18) in
-(let TMP_20 \def (eq nat TMP_16 TMP_19) in (False_ind TMP_20 H1)))))))))) |
-(le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).(let TMP_6 \def (S m)
-in (let TMP_7 \def (\lambda (e: nat).(match e with [O \Rightarrow False | (S
-_) \Rightarrow True])) in (let H2 \def (eq_ind nat TMP_6 TMP_7 I O H1) in
-(let TMP_12 \def ((le (S y) m) \to (let TMP_8 \def (minus O y) in (let TMP_9
-\def (S y) in (let TMP_10 \def (minus O TMP_9) in (let TMP_11 \def (S TMP_10)
-in (eq nat TMP_8 TMP_11)))))) in (let TMP_13 \def (False_ind TMP_12 H2) in
-(TMP_13 H0)))))))]) in (let TMP_21 \def (refl_equal nat O) in (H0 TMP_21)))))
-in (let TMP_40 \def (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y
-n) \to (eq nat (minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(let
-TMP_29 \def (\lambda (n0: nat).((lt n0 (S n)) \to (let TMP_23 \def (S n) in
-(let TMP_24 \def (minus TMP_23 n0) in (let TMP_25 \def (S n) in (let TMP_26
-\def (S n0) in (let TMP_27 \def (minus TMP_25 TMP_26) in (let TMP_28 \def (S
-TMP_27) in (eq nat TMP_24 TMP_28))))))))) in (let TMP_37 \def (\lambda (_:
-(lt O (S n))).(let TMP_32 \def (\lambda (n0: nat).(let TMP_30 \def (S n) in
-(let TMP_31 \def (S n0) in (eq nat TMP_30 TMP_31)))) in (let TMP_33 \def (S
-n) in (let TMP_34 \def (refl_equal nat TMP_33) in (let TMP_35 \def (minus n
-O) in (let TMP_36 \def (minus_n_O n) in (eq_ind nat n TMP_32 TMP_34 TMP_35
-TMP_36))))))) in (let TMP_39 \def (\lambda (n0: nat).(\lambda (_: (((lt n0 (S
-n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))).(\lambda (H1:
-(lt (S n0) (S n))).(let TMP_38 \def (S n0) in (let H2 \def (le_S_n TMP_38 n
-H1) in (H n0 H2)))))) in (nat_ind TMP_29 TMP_37 TMP_39 y))))))) in (nat_ind
-TMP_5 TMP_22 TMP_40 x)))).
-
-theorem lt_plus_minus:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
+(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H:
+(lt y O)).(let H0 \def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat
+(S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat
+(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda
+(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1)
+in (False_ind ((le (S y) m) \to (eq nat (minus O y) (S (minus O (S y)))))
+H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H:
+((\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S
+y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to
+(eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda (_: (lt O (S
+n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S n0))) (refl_equal nat
+(S n)) (minus n O) (minus_n_O n))) (\lambda (n0: nat).(\lambda (_: (((lt n0
+(S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))).(\lambda
+(H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) n H1) in (H n0 H2)))))
+y)))) x).
+
+lemma lt_plus_minus:
\forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus
y (S x)))))))
\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(let TMP_1 \def (S
-x) in (le_plus_minus TMP_1 y H)))).
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S
+x) y H))).
-theorem lt_plus_minus_r:
+lemma lt_plus_minus_r:
\forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y
(S x)) x)))))
\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(let TMP_1 \def (S
-x) in (let TMP_2 \def (minus y TMP_1) in (let TMP_3 \def (plus x TMP_2) in
-(let TMP_5 \def (\lambda (n: nat).(let TMP_4 \def (S n) in (eq nat y TMP_4)))
-in (let TMP_6 \def (lt_plus_minus x y H) in (let TMP_7 \def (S x) in (let
-TMP_8 \def (minus y TMP_7) in (let TMP_9 \def (plus TMP_8 x) in (let TMP_10
-\def (S x) in (let TMP_11 \def (minus y TMP_10) in (let TMP_12 \def (plus_sym
-TMP_11 x) in (eq_ind_r nat TMP_3 TMP_5 TMP_6 TMP_9 TMP_12)))))))))))))).
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat
+(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x
+y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
-theorem minus_x_SO:
+lemma minus_x_SO:
\forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
\def
- \lambda (x: nat).(\lambda (H: (lt O x)).(let TMP_1 \def (minus x O) in (let
-TMP_2 \def (\lambda (n: nat).(eq nat x n)) in (let TMP_3 \def (\lambda (n:
-nat).(eq nat x n)) in (let TMP_4 \def (refl_equal nat x) in (let TMP_5 \def
-(minus x O) in (let TMP_6 \def (minus_n_O x) in (let TMP_7 \def (eq_ind nat x
-TMP_3 TMP_4 TMP_5 TMP_6) in (let TMP_8 \def (S O) in (let TMP_9 \def (minus x
-TMP_8) in (let TMP_10 \def (S TMP_9) in (let TMP_11 \def (minus_x_Sy x O H)
-in (eq_ind nat TMP_1 TMP_2 TMP_7 TMP_10 TMP_11))))))))))))).
+ \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n:
+nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal
+nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
-theorem le_x_pred_y:
+lemma le_x_pred_y:
\forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
\def
- \lambda (y: nat).(let TMP_2 \def (\lambda (n: nat).(\forall (x: nat).((lt x
-n) \to (let TMP_1 \def (pred n) in (le x TMP_1))))) in (let TMP_11 \def
-(\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def (match H with [le_n
-\Rightarrow (\lambda (H0: (eq nat (S x) O)).(let TMP_7 \def (S x) in (let
-TMP_8 \def (\lambda (e: nat).(match e with [O \Rightarrow False | (S _)
-\Rightarrow True])) in (let H1 \def (eq_ind nat TMP_7 TMP_8 I O H0) in (let
-TMP_9 \def (le x O) in (False_ind TMP_9 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 (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to
+(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def
+(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let H1 \def
+(eq_ind nat (S x) (\lambda (e: nat).(match e with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H0) in (False_ind (le x O) H1))) | (le_S m H0)
+\Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m)
(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow
-True])) in (let H2 \def (eq_ind nat TMP_3 TMP_4 I O H1) in (let TMP_5 \def
-((le (S x) m) \to (le x O)) in (let TMP_6 \def (False_ind TMP_5 H2) in (TMP_6
-H0)))))))]) in (let TMP_10 \def (refl_equal nat O) in (H0 TMP_10))))) in (let
-TMP_12 \def (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt x n) \to
-(le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S n))).(le_S_n x n
-H0))))) in (nat_ind TMP_2 TMP_11 TMP_12 y)))).
-
-theorem lt_le_minus:
+True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0
+(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt
+x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S
+n))).(le_S_n x n H0))))) y).
+
+lemma lt_le_minus:
\forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(let TMP_1 \def (S
-O) in (let TMP_2 \def (S O) in (let TMP_3 \def (plus TMP_2 x) in (let TMP_4
-\def (\lambda (n: nat).(le n y)) in (let TMP_5 \def (S O) in (let TMP_6 \def
-(plus x TMP_5) in (let TMP_7 \def (S O) in (let TMP_8 \def (plus_sym x TMP_7)
-in (let TMP_9 \def (eq_ind_r nat TMP_3 TMP_4 H TMP_6 TMP_8) in (le_minus x y
-TMP_1 TMP_9)))))))))))).
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S
+O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O))
+(plus_sym x (S O)))))).
-theorem lt_le_e:
+lemma lt_le_e:
\forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P))
\to ((((le d n) \to P)) \to P))))
\def
\lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n
d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in
-(let TMP_1 \def (le d n) in (let TMP_2 \def (lt n d) in (or_ind TMP_1 TMP_2 P
-H0 H H1)))))))).
+(or_ind (le d n) (lt n d) P H0 H H1)))))).
-theorem lt_eq_e:
+lemma lt_eq_e:
\forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
\to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x
-y)).(let TMP_1 \def (lt x y) in (let TMP_2 \def (eq nat x y) in (let TMP_3
-\def (le_lt_or_eq x y H1) in (or_ind TMP_1 TMP_2 P H H0 TMP_3))))))))).
+y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
-theorem lt_eq_gt_e:
+lemma lt_eq_gt_e:
\forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x)
-\to P))).(let TMP_3 \def (\lambda (H2: (le y x)).(let TMP_2 \def (\lambda
-(H3: (eq nat y x)).(let TMP_1 \def (sym_eq nat y x H3) in (H0 TMP_1))) in
-(lt_eq_e y x P H1 TMP_2 H2))) in (lt_le_e x y P H TMP_3))))))).
+\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda
+(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
-theorem lt_gen_xS:
+lemma lt_gen_xS:
\forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2
nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
\def
- \lambda (x: nat).(let TMP_6 \def (\lambda (n: nat).(\forall (n0: nat).((lt n
-(S n0)) \to (let TMP_1 \def (eq nat n O) in (let TMP_3 \def (\lambda (m:
-nat).(let TMP_2 \def (S m) in (eq nat n TMP_2))) in (let TMP_4 \def (\lambda
-(m: nat).(lt m n0)) in (let TMP_5 \def (ex2 nat TMP_3 TMP_4) in (or TMP_1
-TMP_5)))))))) in (let TMP_13 \def (\lambda (n: nat).(\lambda (_: (lt O (S
-n))).(let TMP_7 \def (eq nat O O) in (let TMP_9 \def (\lambda (m: nat).(let
-TMP_8 \def (S m) in (eq nat O TMP_8))) in (let TMP_10 \def (\lambda (m:
-nat).(lt m n)) in (let TMP_11 \def (ex2 nat TMP_9 TMP_10) in (let TMP_12 \def
-(refl_equal nat O) in (or_introl TMP_7 TMP_11 TMP_12)))))))) in (let TMP_30
-\def (\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((lt n (S n0)) \to
-(or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) (\lambda (m:
-nat).(lt m n0)))))))).(\lambda (n0: nat).(\lambda (H0: (lt (S n) (S
-n0))).(let TMP_14 \def (S n) in (let TMP_15 \def (eq nat TMP_14 O) in (let
-TMP_18 \def (\lambda (m: nat).(let TMP_16 \def (S n) in (let TMP_17 \def (S
-m) in (eq nat TMP_16 TMP_17)))) in (let TMP_19 \def (\lambda (m: nat).(lt m
-n0)) in (let TMP_20 \def (ex2 nat TMP_18 TMP_19) in (let TMP_23 \def (\lambda
-(m: nat).(let TMP_21 \def (S n) in (let TMP_22 \def (S m) in (eq nat TMP_21
-TMP_22)))) in (let TMP_24 \def (\lambda (m: nat).(lt m n0)) in (let TMP_25
-\def (S n) in (let TMP_26 \def (refl_equal nat TMP_25) in (let TMP_27 \def (S
-n) in (let TMP_28 \def (le_S_n TMP_27 n0 H0) in (let TMP_29 \def (ex_intro2
-nat TMP_23 TMP_24 n TMP_26 TMP_28) in (or_intror TMP_15 TMP_20
-TMP_29))))))))))))))))) in (nat_ind TMP_6 TMP_13 TMP_30 x)))).
-
-theorem le_lt_false:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S
+n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m)))
+(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S
+n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m)))
+(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda
+(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda
+(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0:
+nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat
+(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0)))
+(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt
+m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
+
+lemma le_lt_false:
\forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P:
Prop).P))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt
-y x)).(\lambda (P: Prop).(let TMP_1 \def (le_not_lt x y H H0) in (False_ind P
-TMP_1)))))).
+y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
-theorem lt_neq:
+lemma lt_neq:
\forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq
-nat x y)).(let TMP_1 \def (\lambda (n: nat).(lt n y)) in (let H1 \def (eq_ind
-nat x TMP_1 H y H0) in (lt_n_n y H1)))))).
+nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in
+(lt_n_n y H1))))).
-theorem arith0:
+lemma arith0:
\forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n)
\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
\def
\lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le
-(plus d2 h2) n)).(\lambda (h1: nat).(let TMP_1 \def (plus d2 h1) in (let
-TMP_2 \def (plus h2 TMP_1) in (let TMP_3 \def (minus TMP_2 h2) in (let TMP_6
-\def (\lambda (n0: nat).(let TMP_4 \def (plus n h1) in (let TMP_5 \def (minus
-TMP_4 h2) in (le n0 TMP_5)))) in (let TMP_7 \def (plus d2 h1) in (let TMP_8
-\def (plus h2 TMP_7) in (let TMP_9 \def (plus d2 h1) in (let TMP_10 \def
-(le_plus_l h2 TMP_9) in (let TMP_11 \def (plus n h1) in (let TMP_12 \def
-(plus h2 d2) in (let TMP_13 \def (plus TMP_12 h1) in (let TMP_15 \def
-(\lambda (n0: nat).(let TMP_14 \def (plus n h1) in (le n0 TMP_14))) in (let
-TMP_16 \def (plus d2 h2) in (let TMP_19 \def (\lambda (n0: nat).(let TMP_17
-\def (plus n0 h1) in (let TMP_18 \def (plus n h1) in (le TMP_17 TMP_18)))) in
-(let TMP_20 \def (plus d2 h2) in (let TMP_21 \def (plus TMP_20 h1) in (let
-TMP_22 \def (plus n h1) in (let TMP_23 \def (plus d2 h2) in (let TMP_24 \def
-(plus TMP_23 h1) in (let TMP_25 \def (plus n h1) in (let TMP_26 \def (plus d2
-h2) in (let TMP_27 \def (le_n h1) in (let TMP_28 \def (le_plus_plus TMP_26 n
-h1 h1 H TMP_27) in (let TMP_29 \def (le_n_S TMP_24 TMP_25 TMP_28) in (let
-TMP_30 \def (le_S_n TMP_21 TMP_22 TMP_29) in (let TMP_31 \def (plus h2 d2) in
-(let TMP_32 \def (plus_sym h2 d2) in (let TMP_33 \def (eq_ind_r nat TMP_16
-TMP_19 TMP_30 TMP_31 TMP_32) in (let TMP_34 \def (plus d2 h1) in (let TMP_35
-\def (plus h2 TMP_34) in (let TMP_36 \def (plus_assoc_l h2 d2 h1) in (let
-TMP_37 \def (eq_ind_r nat TMP_13 TMP_15 TMP_33 TMP_35 TMP_36) in (let TMP_38
-\def (le_minus_minus h2 TMP_8 TMP_10 TMP_11 TMP_37) in (let TMP_39 \def (plus
-d2 h1) in (let TMP_40 \def (plus d2 h1) in (let TMP_41 \def (minus_plus h2
-TMP_40) in (eq_ind nat TMP_3 TMP_6 TMP_38 TMP_39
-TMP_41))))))))))))))))))))))))))))))))))))))))).
-
-theorem O_minus:
+(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1))
+h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2
+(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat
+(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat
+(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus
+(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1)
+(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2
+d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1)
+(minus_plus h2 (plus d2 h1))))))).
+
+lemma O_minus:
\forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
\def
- \lambda (x: nat).(let TMP_2 \def (\lambda (n: nat).(\forall (y: nat).((le n
-y) \to (let TMP_1 \def (minus n y) in (eq nat TMP_1 O))))) in (let TMP_3 \def
-(\lambda (y: nat).(\lambda (_: (le O y)).(refl_equal nat O))) in (let TMP_20
-\def (\lambda (x0: nat).(\lambda (H: ((\forall (y: nat).((le x0 y) \to (eq
-nat (minus x0 y) O))))).(\lambda (y: nat).(let TMP_5 \def (\lambda (n:
-nat).((le (S x0) n) \to (let TMP_4 \def (match n with [O \Rightarrow (S x0) |
-(S l) \Rightarrow (minus x0 l)]) in (eq nat TMP_4 O)))) in (let TMP_17 \def
-(\lambda (H0: (le (S x0) O)).(let TMP_7 \def (\lambda (n: nat).(let TMP_6
-\def (S n) in (eq nat O TMP_6))) in (let TMP_8 \def (\lambda (n: nat).(le x0
-n)) in (let TMP_9 \def (S x0) in (let TMP_10 \def (eq nat TMP_9 O) in (let
-TMP_15 \def (\lambda (x1: nat).(\lambda (H1: (eq nat O (S x1))).(\lambda (_:
-(le x0 x1)).(let TMP_11 \def (\lambda (ee: nat).(match ee with [O \Rightarrow
-True | (S _) \Rightarrow False])) in (let TMP_12 \def (S x1) in (let H3 \def
-(eq_ind nat O TMP_11 I TMP_12 H1) in (let TMP_13 \def (S x0) in (let TMP_14
-\def (eq nat TMP_13 O) in (False_ind TMP_14 H3))))))))) in (let TMP_16 \def
-(le_gen_S x0 O H0) in (ex2_ind nat TMP_7 TMP_8 TMP_10 TMP_15 TMP_16))))))))
-in (let TMP_19 \def (\lambda (n: nat).(\lambda (_: (((le (S x0) n) \to (eq
-nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0 l)])
-O)))).(\lambda (H1: (le (S x0) (S n))).(let TMP_18 \def (le_S_n x0 n H1) in
-(H n TMP_18))))) in (nat_ind TMP_5 TMP_17 TMP_19 y))))))) in (nat_ind TMP_2
-TMP_3 TMP_20 x)))).
-
-theorem minus_minus:
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
+(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O
+y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y:
+nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind
+(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
+x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0)
+O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0
+n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S
+x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1)
+H1) in (False_ind (eq nat (S x0) O) H3))))) (le_gen_S x0 O H0))) (\lambda (n:
+nat).(\lambda (_: (((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
+x0) | (S l) \Rightarrow (minus x0 l)]) O)))).(\lambda (H1: (le (S x0) (S
+n))).(H n (le_S_n x0 n H1))))) y)))) x).
+
+lemma minus_minus:
\forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y)
\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
\def
- \lambda (z: nat).(let TMP_1 \def (\lambda (n: nat).(\forall (x:
-nat).(\forall (y: nat).((le n x) \to ((le n y) \to ((eq nat (minus x n)
-(minus y n)) \to (eq nat x y))))))) in (let TMP_9 \def (\lambda (x:
-nat).(\lambda (y: nat).(\lambda (_: (le O x)).(\lambda (_: (le O y)).(\lambda
-(H1: (eq nat (minus x O) (minus y O))).(let TMP_2 \def (minus x O) in (let
-TMP_4 \def (\lambda (n: nat).(let TMP_3 \def (minus y O) in (eq nat n
-TMP_3))) in (let TMP_5 \def (minus_n_O x) in (let H2 \def (eq_ind_r nat TMP_2
-TMP_4 H1 x TMP_5) in (let TMP_6 \def (minus y O) in (let TMP_7 \def (\lambda
-(n: nat).(eq nat x n)) in (let TMP_8 \def (minus_n_O y) in (let H3 \def
-(eq_ind_r nat TMP_6 TMP_7 H2 y TMP_8) in H3))))))))))))) in (let TMP_40 \def
-(\lambda (z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0
-x) \to ((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x
-y)))))))).(\lambda (x: nat).(let TMP_10 \def (\lambda (n: nat).(\forall (y:
-nat).((le (S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y
-(S z0))) \to (eq nat n y)))))) in (let TMP_20 \def (\lambda (y: nat).(\lambda
-(H: (le (S z0) O)).(\lambda (_: (le (S z0) y)).(\lambda (_: (eq nat (minus O
-(S z0)) (minus y (S z0)))).(let TMP_12 \def (\lambda (n: nat).(let TMP_11
-\def (S n) in (eq nat O TMP_11))) in (let TMP_13 \def (\lambda (n: nat).(le
-z0 n)) in (let TMP_14 \def (eq nat O y) in (let TMP_18 \def (\lambda (x0:
-nat).(\lambda (H2: (eq nat O (S x0))).(\lambda (_: (le z0 x0)).(let TMP_15
-\def (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _)
-\Rightarrow False])) in (let TMP_16 \def (S x0) in (let H4 \def (eq_ind nat O
-TMP_15 I TMP_16 H2) in (let TMP_17 \def (eq nat O y) in (False_ind TMP_17
-H4)))))))) in (let TMP_19 \def (le_gen_S z0 O H) in (ex2_ind nat TMP_12
-TMP_13 TMP_14 TMP_18 TMP_19)))))))))) in (let TMP_39 \def (\lambda (x0:
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y:
+nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq
+nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O
+x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y
+O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n
+(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O)
+(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda
+(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to
+((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x
+y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le
+(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0)))
+\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda
+(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S
+z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le
+z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S
+x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x0)
+H2) in (False_ind (eq nat O y) H4))))) (le_gen_S z0 O H)))))) (\lambda (x0:
nat).(\lambda (_: ((\forall (y: nat).((le (S z0) x0) \to ((le (S z0) y) \to
((eq nat (minus x0 (S z0)) (minus y (S z0))) \to (eq nat x0 y))))))).(\lambda
-(y: nat).(let TMP_22 \def (\lambda (n: nat).((le (S z0) (S x0)) \to ((le (S
-z0) n) \to ((eq nat (minus (S x0) (S z0)) (minus n (S z0))) \to (let TMP_21
-\def (S x0) in (eq nat TMP_21 n)))))) in (let TMP_34 \def (\lambda (H: (le (S
-z0) (S x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0)
-(S z0)) (minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (let TMP_24 \def
-(\lambda (n: nat).(let TMP_23 \def (S n) in (eq nat O TMP_23))) in (let
-TMP_25 \def (\lambda (n: nat).(le z0 n)) in (let TMP_26 \def (S x0) in (let
-TMP_27 \def (eq nat TMP_26 O) in (let TMP_32 \def (\lambda (x1: nat).(\lambda
-(H2: (eq nat O (S x1))).(\lambda (_: (le z0 x1)).(let TMP_28 \def (\lambda
-(ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) in
-(let TMP_29 \def (S x1) in (let H4 \def (eq_ind nat O TMP_28 I TMP_29 H2) in
-(let TMP_30 \def (S x0) in (let TMP_31 \def (eq nat TMP_30 O) in (False_ind
-TMP_31 H4))))))))) in (let TMP_33 \def (le_gen_S z0 O H0) in (ex2_ind nat
-TMP_24 TMP_25 TMP_27 TMP_32 TMP_33))))))))))) in (let TMP_38 \def (\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)))).(let TMP_35 \def (le_S_n z0
-x0 H) in (let TMP_36 \def (le_S_n z0 y0 H0) in (let TMP_37 \def (IH x0 y0
-TMP_35 TMP_36 H1) in (f_equal nat nat S x0 y0 TMP_37))))))))) in (nat_ind
-TMP_22 TMP_34 TMP_38 y))))))) in (nat_ind TMP_10 TMP_20 TMP_39 x))))))) in
-(nat_ind TMP_1 TMP_9 TMP_40 z)))).
+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).
-theorem plus_plus:
+lemma plus_plus:
\forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1:
nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z
x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
\def
- \lambda (z: nat).(let TMP_3 \def (\lambda (n: nat).(\forall (x1:
-nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to
-((le x2 n) \to ((eq nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to
-(let TMP_1 \def (plus x1 y2) in (let TMP_2 \def (plus x2 y1) in (eq nat TMP_1
-TMP_2))))))))))) in (let TMP_17 \def (\lambda (x1: nat).(\lambda (x2:
-nat).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda
-(H0: (le x2 O)).(\lambda (H1: (eq nat y1 y2)).(let TMP_6 \def (\lambda (n:
-nat).(let TMP_4 \def (plus x1 n) in (let TMP_5 \def (plus x2 y1) in (eq nat
-TMP_4 TMP_5)))) in (let H_y \def (le_n_O_eq x2 H0) in (let TMP_9 \def
-(\lambda (n: nat).(let TMP_7 \def (plus x1 y1) in (let TMP_8 \def (plus n y1)
-in (eq nat TMP_7 TMP_8)))) in (let H_y0 \def (le_n_O_eq x1 H) in (let TMP_12
-\def (\lambda (n: nat).(let TMP_10 \def (plus n y1) in (let TMP_11 \def (plus
-O y1) in (eq nat TMP_10 TMP_11)))) in (let TMP_13 \def (plus O y1) in (let
-TMP_14 \def (refl_equal nat TMP_13) in (let TMP_15 \def (eq_ind nat O TMP_12
-TMP_14 x1 H_y0) in (let TMP_16 \def (eq_ind nat O TMP_9 TMP_15 x2 H_y) in
-(eq_ind nat y1 TMP_6 TMP_16 y2 H1))))))))))))))))) in (let TMP_91 \def
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2:
+nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq
+nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2)
+(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1:
+nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2
+O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat
+(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O
+(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def
+(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus
+O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1))))))))
(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2:
nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to
((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus
-x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(let TMP_20 \def (\lambda
-(n: nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S
-z0)) \to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus
-(S z0) x2) y2)) \to (let TMP_18 \def (plus n y2) in (let TMP_19 \def (plus x2
-y1) in (eq nat TMP_18 TMP_19)))))))))) in (let TMP_56 \def (\lambda (x2:
-nat).(let TMP_23 \def (\lambda (n: nat).(\forall (y1: nat).(\forall (y2:
-nat).((le O (S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O)
-y1) (plus (minus (S z0) n) y2)) \to (let TMP_21 \def (plus O y2) in (let
-TMP_22 \def (plus n y1) in (eq nat TMP_21 TMP_22))))))))) in (let TMP_32 \def
-(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda
-(_: (le O (S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0
-y2)))).(let H_y \def (IH O O) in (let TMP_24 \def (minus z0 O) in (let TMP_25
-\def (\lambda (n: nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to
-((le O z0) \to ((eq nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) in
-(let TMP_26 \def (minus_n_O z0) in (let H2 \def (eq_ind_r nat TMP_24 TMP_25
-H_y z0 TMP_26) in (let TMP_27 \def (le_O_n z0) in (let TMP_28 \def (le_O_n
-z0) in (let TMP_29 \def (plus z0 y1) in (let TMP_30 \def (plus z0 y2) in (let
-TMP_31 \def (eq_add_S TMP_29 TMP_30 H1) in (H2 y1 y2 TMP_27 TMP_28
-TMP_31)))))))))))))))) in (let TMP_55 \def (\lambda (x3: nat).(\lambda (_:
-((\forall (y1: nat).(\forall (y2: nat).((le O (S z0)) \to ((le x3 (S z0)) \to
-((eq nat (S (plus z0 y1)) (plus (match x3 with [O \Rightarrow (S z0) | (S l)
-\Rightarrow (minus z0 l)]) y2)) \to (eq nat y2 (plus x3 y1))))))))).(\lambda
-(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S
-x3) (S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3)
-y2))).(let TMP_33 \def (S y1) in (let H_y \def (IH O x3 TMP_33) in (let
-TMP_34 \def (minus z0 O) in (let TMP_37 \def (\lambda (n: nat).(\forall (y3:
-nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S y1)) (plus (minus z0
-x3) y3)) \to (let TMP_35 \def (S y1) in (let TMP_36 \def (plus x3 TMP_35) in
-(eq nat y3 TMP_36)))))))) in (let TMP_38 \def (minus_n_O z0) in (let H2 \def
-(eq_ind_r nat TMP_34 TMP_37 H_y z0 TMP_38) in (let TMP_39 \def (S y1) in (let
-TMP_40 \def (plus z0 TMP_39) in (let TMP_43 \def (\lambda (n: nat).(\forall
-(y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus (minus z0 x3) y3))
-\to (let TMP_41 \def (S y1) in (let TMP_42 \def (plus x3 TMP_41) in (eq nat
-y3 TMP_42)))))))) in (let TMP_44 \def (plus z0 y1) in (let TMP_45 \def (S
-TMP_44) in (let TMP_46 \def (plus_n_Sm z0 y1) in (let H3 \def (eq_ind_r nat
-TMP_40 TMP_43 H2 TMP_45 TMP_46) in (let TMP_47 \def (S y1) in (let TMP_48
-\def (plus x3 TMP_47) in (let TMP_49 \def (\lambda (n: nat).(\forall (y3:
-nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus z0 y1)) (plus (minus z0
-x3) y3)) \to (eq nat y3 n)))))) in (let TMP_50 \def (plus x3 y1) in (let
-TMP_51 \def (S TMP_50) in (let TMP_52 \def (plus_n_Sm x3 y1) in (let H4 \def
-(eq_ind_r nat TMP_48 TMP_49 H3 TMP_51 TMP_52) in (let TMP_53 \def (le_O_n z0)
-in (let TMP_54 \def (le_S_n x3 z0 H0) in (H4 y2 TMP_53 TMP_54
-H1)))))))))))))))))))))))))))))) in (nat_ind TMP_23 TMP_32 TMP_55 x2))))) in
-(let TMP_90 \def (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall
-(y1: nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat
+x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n:
+nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0))
+\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S
+z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2:
+nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O
+(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus
+(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda
+(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O
+(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y
+\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
+nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq
+nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0))
+in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2)
+H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2:
+nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus
+(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2))
+\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2:
+nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda
+(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O
+x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S
+y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0
+(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n:
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus
+(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1))
+(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda
+(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus
+z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1))
+(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1))))))))))))
+x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1:
+nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat
(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2
-y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(let TMP_60 \def (\lambda (n:
+y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n:
nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S
z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2))
-\to (let TMP_57 \def (S x2) in (let TMP_58 \def (plus TMP_57 y2) in (let
-TMP_59 \def (plus n y1) in (eq nat TMP_58 TMP_59)))))))))) in (let TMP_83
-\def (\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S
-z0))).(\lambda (_: (le O (S z0))).(\lambda (H1: (eq nat (plus (minus z0 x2)
-y1) (S (plus z0 y2)))).(let TMP_61 \def (S y2) in (let H_y \def (IH x2 O y1
-TMP_61) in (let TMP_62 \def (minus z0 O) in (let TMP_65 \def (\lambda (n:
-nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) (plus n
-(S y2))) \to (let TMP_63 \def (S y2) in (let TMP_64 \def (plus x2 TMP_63) in
-(eq nat TMP_64 y1))))))) in (let TMP_66 \def (minus_n_O z0) in (let H2 \def
-(eq_ind_r nat TMP_62 TMP_65 H_y z0 TMP_66) in (let TMP_67 \def (S y2) in (let
-TMP_68 \def (plus z0 TMP_67) in (let TMP_71 \def (\lambda (n: nat).((le x2
-z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (let TMP_69
-\def (S y2) in (let TMP_70 \def (plus x2 TMP_69) in (eq nat TMP_70 y1)))))))
-in (let TMP_72 \def (plus z0 y2) in (let TMP_73 \def (S TMP_72) in (let
-TMP_74 \def (plus_n_Sm z0 y2) in (let H3 \def (eq_ind_r nat TMP_68 TMP_71 H2
-TMP_73 TMP_74) in (let TMP_75 \def (S y2) in (let TMP_76 \def (plus x2
-TMP_75) in (let TMP_77 \def (\lambda (n: nat).((le x2 z0) \to ((le O z0) \to
-((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) in
-(let TMP_78 \def (plus x2 y2) in (let TMP_79 \def (S TMP_78) in (let TMP_80
-\def (plus_n_Sm x2 y2) in (let H4 \def (eq_ind_r nat TMP_76 TMP_77 H3 TMP_79
-TMP_80) in (let TMP_81 \def (le_S_n x2 z0 H) in (let TMP_82 \def (le_O_n z0)
-in (H4 TMP_81 TMP_82 H1)))))))))))))))))))))))))))) in (let TMP_89 \def
-(\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: nat).((le
-(S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus z0 x2) y1) (plus
-(match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2))
-\to (eq nat (S (plus x2 y2)) (plus x4 y1))))))))).(\lambda (y1: nat).(\lambda
-(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (H0: (le (S x4) (S
-z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (plus (minus z0 x4)
-y2))).(let TMP_84 \def (plus x2 y2) in (let TMP_85 \def (plus x4 y1) in (let
-TMP_86 \def (le_S_n x2 z0 H) in (let TMP_87 \def (le_S_n x4 z0 H0) in (let
-TMP_88 \def (IH x2 x4 y1 y2 TMP_86 TMP_87 H1) in (f_equal nat nat S TMP_84
-TMP_85 TMP_88))))))))))))) in (nat_ind TMP_60 TMP_83 TMP_89 x3))))))) in
-(nat_ind TMP_20 TMP_56 TMP_90 x1))))))) in (nat_ind TMP_3 TMP_17 TMP_91 z)))).
-
-theorem le_S_minus:
+\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda
+(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S
+z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let
+H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O)
+(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2)
+y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O
+z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2
+z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus
+x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def
+(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0)
+\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1)))))
+H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0)
+H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall
+(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus
+z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow
+(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4
+y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S
+z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0
+x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
+y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3))))
+x1)))) z).
+
+lemma le_S_minus:
\forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to
(le d (S (minus n h))))))
\def
\lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus
-d h) n)).(let TMP_1 \def (plus d h) in (let TMP_2 \def (le_plus_l d h) in
-(let H0 \def (le_trans d TMP_1 n TMP_2 H) in (let TMP_3 \def (\lambda (n0:
-nat).(le d n0)) in (let TMP_4 \def (minus n h) in (let TMP_5 \def (plus TMP_4
-h) in (let TMP_6 \def (plus d h) in (let TMP_7 \def (le_plus_r d h) in (let
-TMP_8 \def (le_trans h TMP_6 n TMP_7 H) in (let TMP_9 \def (le_plus_minus_sym
-h n TMP_8) in (let H1 \def (eq_ind nat n TMP_3 H0 TMP_5 TMP_9) in (let TMP_10
-\def (minus n h) in (let TMP_11 \def (le_minus d n h H) in (le_S d TMP_10
-TMP_11))))))))))))))))).
-
-theorem lt_x_pred_y:
+d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
+\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h)
+(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S
+d (minus n h) (le_minus d n h H))))))).
+
+lemma lt_x_pred_y:
\forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
\def
- \lambda (x: nat).(\lambda (y: nat).(let TMP_2 \def (\lambda (n: nat).((lt x
-(pred n)) \to (let TMP_1 \def (S x) in (lt TMP_1 n)))) in (let TMP_5 \def
-(\lambda (H: (lt x O)).(let TMP_3 \def (S x) in (let TMP_4 \def (lt TMP_3 O)
-in (lt_x_O x H TMP_4)))) in (let TMP_6 \def (\lambda (n: nat).(\lambda (_:
-(((lt x (pred n)) \to (lt (S x) n)))).(\lambda (H0: (lt x n)).(lt_n_S x n
-H0)))) in (nat_ind TMP_2 TMP_5 TMP_6 y))))).
+ \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred
+n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O)))
+(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda
+(H0: (lt x n)).(lt_n_S x n H0)))) y)).
projects / helm.git / blobdiff