dependences update

[helm.git] / helm / software / matita / contribs / LAMBDA-TYPES / Legacy-1 / coq / props.ma

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma
index e9d29ff6dcdcb5c3e425181b7308bb110e2ddc43..0b9d97b421dd0f4cfe17747b09fcabab82899295 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma
@@ -23,6 +23,9 @@ A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y)))))))
  \lambda (A: Set).(\lambda (B: Set).(\lambda (f: ((A \to B))).(\lambda (x: 
 A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x (\lambda (a: A).(eq B 
 (f x) (f a))) (refl_equal B (f x)) y H)))))).
+(* COMMENTS
+Initial nodes: 51
+END *)
 
 theorem f_equal2:
  \forall (A1: Set).(\forall (A2: Set).(\forall (B: Set).(\forall (f: ((A1 \to 
@@ -36,6 +39,9 @@ y2)))))))))))
 x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 y2)).(eq_ind 
 A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B (f x1 x2)) y2 
 H0)) y1 H))))))))).
+(* COMMENTS
+Initial nodes: 109
+END *)
 
 theorem f_equal3:
  \forall (A1: Set).(\forall (A2: Set).(\forall (A3: Set).(\forall (B: 
@@ -53,6 +59,9 @@ A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) \to (eq B (f x1 x2
 x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 x3 (\lambda (a: 
 A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 x3)) y3 H1)) y2 
 H0)) y1 H)))))))))))).
+(* COMMENTS
+Initial nodes: 183
+END *)
 
 theorem sym_eq:
  \forall (A: Set).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y 
@@ -60,6 +69,9 @@ x))))
 \def
  \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x 
 y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))).
+(* COMMENTS
+Initial nodes: 39
+END *)
 
 theorem eq_ind_r:
  \forall (A: Set).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to 
@@ -69,6 +81,9 @@ theorem eq_ind_r:
 (P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) in 
 eq return (\lambda (a: A).(\lambda (_: (eq ? ? a)).(P a))) with [refl_equal 
 \Rightarrow H])))))).
+(* COMMENTS
+Initial nodes: 38
+END *)
 
 theorem trans_eq:
  \forall (A: Set).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A x y) 
@@ -77,6 +92,9 @@ theorem trans_eq:
  \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (z: A).(\lambda 
 (H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda (a: A).(eq A x 
 a)) H z H0)))))).
+(* COMMENTS
+Initial nodes: 45
+END *)
 
 theorem sym_not_eq:
  \forall (A: Set).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to (not 
@@ -85,6 +103,9 @@ theorem sym_not_eq:
  \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq A x 
 y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a y)) 
 (refl_equal A y) x h2)))))).
+(* COMMENTS
+Initial nodes: 51
+END *)
 
 theorem nat_double_ind:
  \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to 
@@ -99,18 +120,27 @@ nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0:
 nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind 
 (\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R 
 (S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))).
+(* COMMENTS
+Initial nodes: 111
+END *)
 
 theorem eq_add_S:
  \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S 
 m))).(f_equal nat nat pred (S n) (S m) H))).
+(* COMMENTS
+Initial nodes: 33
+END *)
 
 theorem O_S:
  \forall (n: nat).(not (eq nat O (S n)))
 \def
  \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda 
 (n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))).
+(* COMMENTS
+Initial nodes: 41
+END *)
 
 theorem not_eq_S:
  \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S 
@@ -118,11 +148,17 @@ n) (S m)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda 
 (H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))).
+(* COMMENTS
+Initial nodes: 35
+END *)
 
 theorem pred_Sn:
  \forall (m: nat).(eq nat m (pred (S m)))
 \def
  \lambda (m: nat).(refl_equal nat (pred (S m))).
+(* COMMENTS
+Initial nodes: 11
+END *)
 
 theorem S_pred:
  \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n)))))
@@ -131,6 +167,9 @@ theorem S_pred:
 (\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S 
 m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 
 (S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))).
+(* COMMENTS
+Initial nodes: 79
+END *)
 
 theorem le_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) 
@@ -140,12 +179,18 @@ theorem le_trans:
 m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H 
 (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n 
 m0 IHle)))) p H0))))).
+(* COMMENTS
+Initial nodes: 57
+END *)
 
 theorem le_trans_S:
  \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S 
 n) m (le_S n n (le_n n)) H))).
+(* COMMENTS
+Initial nodes: 33
+END *)
 
 theorem le_n_S:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m))))
@@ -153,12 +198,18 @@ theorem le_n_S:
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
 (n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le 
 n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))).
+(* COMMENTS
+Initial nodes: 65
+END *)
 
 theorem le_O_n:
  \forall (n: nat).(le O n)
 \def
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda 
 (n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n).
+(* COMMENTS
+Initial nodes: 33
+END *)
 
 theorem le_S_n:
  \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m)))
@@ -167,6 +218,9 @@ theorem le_S_n:
 n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0: 
 nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S 
 n m0 H0)))) (S m) H))).
+(* COMMENTS
+Initial nodes: 69
+END *)
 
 theorem le_Sn_O:
  \forall (n: nat).(not (le (S n) O))
@@ -174,6 +228,9 @@ theorem le_Sn_O:
  \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: 
 nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: 
 (IsSucc m)).I))) O H)).
+(* COMMENTS
+Initial nodes: 43
+END *)
 
 theorem le_Sn_n:
  \forall (n: nat).(not (le (S n) n))
@@ -181,6 +238,9 @@ theorem le_Sn_n:
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O 
 O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le 
 (S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n).
+(* COMMENTS
+Initial nodes: 57
+END *)
 
 theorem le_antisym:
  \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n 
@@ -192,11 +252,17 @@ nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n)
 \to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S 
 m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) 
 in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))).
+(* COMMENTS
+Initial nodes: 119
+END *)
 
 theorem le_n_O_eq:
  \forall (n: nat).((le n O) \to (eq nat O n))
 \def
  \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)).
+(* COMMENTS
+Initial nodes: 19
+END *)
 
 theorem le_elim_rel:
  \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to 
@@ -213,33 +279,51 @@ m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn
 n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: 
 (P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 
 H1)))))) m Le))))) n)))).
+(* COMMENTS
+Initial nodes: 181
+END *)
 
 theorem lt_n_n:
  \forall (n: nat).(not (lt n n))
 \def
  le_Sn_n.
+(* COMMENTS
+Initial nodes: 1
+END *)
 
 theorem lt_n_S:
  \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m 
 H))).
+(* COMMENTS
+Initial nodes: 19
+END *)
 
 theorem lt_n_Sn:
  \forall (n: nat).(lt n (S n))
 \def
  \lambda (n: nat).(le_n (S n)).
+(* COMMENTS
+Initial nodes: 7
+END *)
 
 theorem lt_S_n:
  \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S 
 n) m H))).
+(* COMMENTS
+Initial nodes: 23
+END *)
 
 theorem lt_n_O:
  \forall (n: nat).(not (lt n O))
 \def
  le_Sn_O.
+(* COMMENTS
+Initial nodes: 1
+END *)
 
 theorem lt_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) 
@@ -249,16 +333,25 @@ theorem lt_trans:
 m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S 
 (S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt 
 n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+(* COMMENTS
+Initial nodes: 71
+END *)
 
 theorem lt_O_Sn:
  \forall (n: nat).(lt O (S n))
 \def
  \lambda (n: nat).(le_n_S O n (le_O_n n)).
+(* COMMENTS
+Initial nodes: 11
+END *)
 
 theorem lt_le_S:
  \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p)))
 \def
  \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)).
+(* COMMENTS
+Initial nodes: 11
+END *)
 
 theorem le_not_lt:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n))))
@@ -267,11 +360,17 @@ theorem le_not_lt:
 (n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n 
 m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle 
 (le_trans_S (S m0) n H1)))))) m H))).
+(* COMMENTS
+Initial nodes: 67
+END *)
 
 theorem le_lt_n_Sm:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))).
+(* COMMENTS
+Initial nodes: 17
+END *)
 
 theorem le_lt_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) 
@@ -281,6 +380,9 @@ theorem le_lt_trans:
 m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) 
 (le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: 
 (lt n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+(* COMMENTS
+Initial nodes: 69
+END *)
 
 theorem lt_le_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) 
@@ -290,18 +392,27 @@ theorem lt_le_trans:
 m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H 
 (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S 
 (S n) m0 IHle)))) p H0))))).
+(* COMMENTS
+Initial nodes: 59
+END *)
 
 theorem lt_le_weak:
  \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m 
 H))).
+(* COMMENTS
+Initial nodes: 17
+END *)
 
 theorem lt_n_Sm_le:
  \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m 
 H))).
+(* COMMENTS
+Initial nodes: 19
+END *)
 
 theorem le_lt_or_eq:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m))))
@@ -311,6 +422,9 @@ theorem le_lt_or_eq:
 (refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_: 
 (or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) 
 (le_n_S n m0 H0))))) m H))).
+(* COMMENTS
+Initial nodes: 109
+END *)
 
 theorem le_or_lt:
  \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n)))
@@ -324,6 +438,9 @@ n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S
 n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S 
 n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S 
 m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)).
+(* COMMENTS
+Initial nodes: 209
+END *)
 
 theorem plus_n_O:
  \forall (n: nat).(eq nat n (plus n O))
@@ -331,6 +448,9 @@ theorem plus_n_O:
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) 
 (refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 
 O))).(f_equal nat nat S n0 (plus n0 O) H))) n).
+(* COMMENTS
+Initial nodes: 57
+END *)
 
 theorem plus_n_Sm:
  \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m))))
@@ -339,6 +459,9 @@ theorem plus_n_Sm:
 (plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0: 
 nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat 
 S (S (plus n0 n)) (plus n0 (S n)) H))) m)).
+(* COMMENTS
+Initial nodes: 85
+END *)
 
 theorem plus_sym:
  \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n)))
@@ -348,6 +471,9 @@ n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus
 y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S 
 (plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) 
 (plus_n_Sm m y)))) n)).
+(* COMMENTS
+Initial nodes: 111
+END *)
 
 theorem plus_Snm_nSm:
  \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m))))
@@ -356,6 +482,9 @@ theorem plus_Snm_nSm:
 nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda 
 (n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n 
 (S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))).
+(* COMMENTS
+Initial nodes: 99
+END *)
 
 theorem plus_assoc_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m 
@@ -366,6 +495,9 @@ nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat
 (plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p)) 
 (plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 
 m) p) H))) n))).
+(* COMMENTS
+Initial nodes: 101
+END *)
 
 theorem plus_assoc_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n 
@@ -373,6 +505,9 @@ m) p) (plus n (plus m p)))))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n 
 (plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))).
+(* COMMENTS
+Initial nodes: 37
+END *)
 
 theorem simpl_plus_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) 
@@ -386,6 +521,9 @@ nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m)
 nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn 
 (plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) 
 (eq_add_S (plus n0 m) (plus n0 p) H))))))))) n).
+(* COMMENTS
+Initial nodes: 161
+END *)
 
 theorem minus_n_O:
  \forall (n: nat).(eq nat n (minus n O))
@@ -393,6 +531,9 @@ theorem minus_n_O:
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) 
 (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 
 O))).(refl_equal nat (S n0)))) n).
+(* COMMENTS
+Initial nodes: 47
+END *)
 
 theorem minus_n_n:
  \forall (n: nat).(eq nat O (minus n n))
@@ -400,6 +541,9 @@ theorem minus_n_n:
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) 
 (refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 
 n0))).IHn)) n).
+(* COMMENTS
+Initial nodes: 41
+END *)
 
 theorem minus_Sn_m:
  \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) 
@@ -411,6 +555,9 @@ n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p
 (minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: 
 (le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow 
 (S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))).
+(* COMMENTS
+Initial nodes: 111
+END *)
 
 theorem plus_minus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) 
@@ -426,12 +573,18 @@ p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda
 (n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq 
 nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H 
 (eq_add_S m0 (plus n0 p) H0)))))) m n))).
+(* COMMENTS
+Initial nodes: 199
+END *)
 
 theorem minus_plus:
  \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m))
 \def
  \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) 
 (plus_minus (plus n m) n m (refl_equal nat (plus n m))))).
+(* COMMENTS
+Initial nodes: 41
+END *)
 
 theorem le_pred_n:
  \forall (n: nat).(le (pred n) n)
@@ -439,6 +592,9 @@ theorem le_pred_n:
  \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) 
 (\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 
 (le_n n0)))) n).
+(* COMMENTS
+Initial nodes: 43
+END *)
 
 theorem le_plus_l:
  \forall (n: nat).(\forall (m: nat).(le n (plus n m)))
@@ -447,6 +603,9 @@ theorem le_plus_l:
 n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn: 
 ((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus 
 n0 m) (IHn m))))) n).
+(* COMMENTS
+Initial nodes: 55
+END *)
 
 theorem le_plus_r:
  \forall (n: nat).(\forall (m: nat).(le m (plus n m)))
@@ -454,6 +613,9 @@ theorem le_plus_r:
  \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus 
 n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m 
 (plus n0 m) H))) n)).
+(* COMMENTS
+Initial nodes: 47
+END *)
 
 theorem simpl_le_plus_l:
  \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p 
@@ -466,6 +628,9 @@ nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n)
 (plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda 
 (H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus 
 p0 m) H))))))) p).
+(* COMMENTS
+Initial nodes: 113
+END *)
 
 theorem le_plus_trans:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n 
@@ -473,6 +638,9 @@ theorem le_plus_trans:
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
 m)).(le_trans n m (plus m p) H (le_plus_l m p))))).
+(* COMMENTS
+Initial nodes: 31
+END *)
 
 theorem le_reg_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus 
@@ -483,6 +651,9 @@ nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H)
 (\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0 
 m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
 p))).
+(* COMMENTS
+Initial nodes: 85
+END *)
 
 theorem le_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
@@ -493,6 +664,9 @@ nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0:
 nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0: 
 nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S 
 (plus n p) (plus m0 q) H2)))) m H)))))).
+(* COMMENTS
+Initial nodes: 91
+END *)
 
 theorem le_plus_minus:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m 
@@ -503,6 +677,9 @@ n)))))
 (\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda 
 (_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat 
 S q (plus p (minus q p)) H0))))) n m Le))).
+(* COMMENTS
+Initial nodes: 91
+END *)
 
 theorem le_plus_minus_r:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m 
@@ -510,6 +687,9 @@ n)) m)))
 \def
  \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m 
 (plus n (minus m n)) (le_plus_minus n m H)))).
+(* COMMENTS
+Initial nodes: 33
+END *)
 
 theorem simpl_lt_plus_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p 
@@ -520,6 +700,9 @@ nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H)
 (\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n 
 m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S 
 (plus p0 n)) (plus p0 m) H))))) p))).
+(* COMMENTS
+Initial nodes: 99
+END *)
 
 theorem lt_reg_l:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
@@ -530,6 +713,9 @@ nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H)
 (\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 
 m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
 p))).
+(* COMMENTS
+Initial nodes: 85
+END *)
 
 theorem lt_reg_r:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
@@ -541,6 +727,9 @@ nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0:
 nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt 
 (plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m 
 p)) (plus n p) (plus_sym n p))))).
+(* COMMENTS
+Initial nodes: 129
+END *)
 
 theorem le_lt_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
@@ -550,6 +739,9 @@ n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
 nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n 
 (S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) 
 (plus (S n) p) (plus_Snm_nSm n p))))))).
+(* COMMENTS
+Initial nodes: 75
+END *)
 
 theorem lt_le_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
@@ -558,6 +750,9 @@ n m) \to ((le p q) \to (lt (plus n p) (plus m q)))))))
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
 nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m 
 p q H H0)))))).
+(* COMMENTS
+Initial nodes: 37
+END *)
 
 theorem lt_plus_plus:
  \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
@@ -566,6 +761,9 @@ n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
  \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
 nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q 
 H (lt_le_weak p q H0))))))).
+(* COMMENTS
+Initial nodes: 39
+END *)
 
 theorem well_founded_ltof:
  \forall (A: Set).(\forall (f: ((A \to nat))).(well_founded A (ltof A f)))
@@ -580,11 +778,17 @@ nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A
 (ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb 
 (lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a 
 (le_n (S (f a))))))).
+(* COMMENTS
+Initial nodes: 189
+END *)
 
 theorem lt_wf:
  well_founded nat lt
 \def
  well_founded_ltof nat (\lambda (m: nat).m).
+(* COMMENTS
+Initial nodes: 7
+END *)
 
 theorem lt_wf_ind:
  \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: 
@@ -595,4 +799,7 @@ nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt
 (\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y: 
 nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y 
 x) \to (P y))))).(H x H1)))) p (lt_wf p)))).
+(* COMMENTS
+Initial nodes: 77
+END *)