theorem max_O_f : \forall f: nat \to bool. max O f = O.
intro. simplify.
-elim f O.
+elim (f O).
simplify.reflexivity.
simplify.reflexivity.
qed.
n\le m \to max n f \le max m f.
intros.elim H.
apply le_n.
-apply trans_le ? (max n1 f).apply H2.
-cut (f (S n1) = true \land max (S n1) f = (S n1)) \lor
-(f (S n1) = false \land max (S n1) f = max n1 f).
+apply (trans_le ? (max n1 f)).apply H2.
+cut ((f (S n1) = true \land max (S n1) f = (S n1)) \lor
+(f (S n1) = false \land max (S n1) f = max n1 f)).
elim Hcut.elim H3.
rewrite > H5.
apply le_S.apply le_max_n.
theorem f_m_to_le_max: \forall f: nat \to bool. \forall n,m:nat.
m\le n \to f m = true \to m \le max n f.
-intros 3.elim n.apply le_n_O_elim m H.
+intros 3.elim n.apply (le_n_O_elim m H).
apply le_O_n.
-apply le_n_Sm_elim m n1 H1.
-intro.apply trans_le ? (max n1 f).
+apply (le_n_Sm_elim m n1 H1).
+intro.apply (trans_le ? (max n1 f)).
apply H.apply le_S_S_to_le.assumption.assumption.
apply le_to_le_max.apply le_n_Sn.
intro.simplify.rewrite < H3.
(\exists i:nat. le i n \land f i = true) \to f (max n f) = true.
intros 2.
elim n.elim H.elim H1.generalize in match H3.
-apply le_n_O_elim a H2.intro.simplify.rewrite > H4.
+apply (le_n_O_elim a H2).intro.simplify.rewrite > H4.
simplify.assumption.
simplify.
-apply bool_ind (\lambda b:bool.
-(f (S n1) = b) \to (f ([\lambda b:bool.nat] match b in bool with
+apply (bool_ind (\lambda b:bool.
+(f (S n1) = b) \to (f (match b in bool with
[ true \Rightarrow (S n1)
-| false \Rightarrow (max n1 f)])) = true).
+| false \Rightarrow (max n1 f)])) = true)).
simplify.intro.assumption.
simplify.intro.apply H.
elim H1.elim H3.generalize in match H5.
-apply le_n_Sm_elim a n1 H4.
+apply (le_n_Sm_elim a n1 H4).
intros.
-apply ex_intro nat ? a.
+apply (ex_intro nat ? a).
split.apply le_S_S_to_le.assumption.assumption.
intros.apply False_ind.apply not_eq_true_false.
rewrite < H2.rewrite < H7.rewrite > H6. reflexivity.
theorem lt_max_to_false : \forall f:nat \to bool.
\forall n,m:nat. (max n f) < m \to m \leq n \to f m = false.
intros 2.
-elim n.absurd le m O.assumption.
-cut O < m.apply lt_O_n_elim m Hcut.exact not_le_Sn_O.
-rewrite < max_O_f f.assumption.
+elim n.absurd (le m O).assumption.
+cut (O < m).apply (lt_O_n_elim m Hcut).exact not_le_Sn_O.
+rewrite < (max_O_f f).assumption.
generalize in match H1.
-elim max_S_max f n1.
+elim (max_S_max f n1).
elim H3.
-absurd m \le S n1.assumption.
+absurd (m \le S n1).assumption.
apply lt_to_not_le.rewrite < H6.assumption.
elim H3.
-apply le_n_Sm_elim m n1 H2.
+apply (le_n_Sm_elim m n1 H2).
intro.
apply H.rewrite < H6.assumption.
apply le_S_S_to_le.assumption.
theorem min_aux_O_f: \forall f:nat \to bool. \forall i :nat.
min_aux O i f = i.
intros.simplify.rewrite < minus_n_O.
-elim f i.
-simplify.reflexivity.
+elim (f i).reflexivity.
simplify.reflexivity.
qed.
theorem min_O_f : \forall f:nat \to bool.
min O f = O.
-intro.apply min_aux_O_f f O.
+intro.apply (min_aux_O_f f O).
qed.
theorem min_aux_S : \forall f: nat \to bool. \forall i,n:nat.
f (min_aux off m f) = true.
intros 2.
elim off.elim H.elim H1.elim H2.
-cut a = m.
-rewrite > min_aux_O_f f.rewrite < Hcut.assumption.
-apply antisym_le a m .assumption.rewrite > minus_n_O m.assumption.
+cut (a = m).
+rewrite > (min_aux_O_f f).rewrite < Hcut.assumption.
+apply (antisym_le a m).assumption.rewrite > (minus_n_O m).assumption.
simplify.
-apply bool_ind (\lambda b:bool.
-(f (m-(S n)) = b) \to (f ([\lambda b:bool.nat] match b in bool with
+apply (bool_ind (\lambda b:bool.
+(f (m-(S n)) = b) \to (f (match b in bool with
[ true \Rightarrow m-(S n)
-| false \Rightarrow (min_aux n m f)])) = true).
+| false \Rightarrow (min_aux n m f)])) = true)).
simplify.intro.assumption.
simplify.intro.apply H.
elim H1.elim H3.elim H4.
elim (le_to_or_lt_eq (m-(S n)) a H6).
-apply ex_intro nat ? a.
+apply (ex_intro nat ? a).
split.split.
apply lt_minus_S_n_to_le_minus_n.assumption.
assumption.assumption.
-absurd f a = false.rewrite < H8.assumption.
+absurd (f a = false).rewrite < H8.assumption.
rewrite > H5.
apply not_eq_true_false.
reflexivity.
theorem lt_min_aux_to_false : \forall f:nat \to bool.
\forall n,off,m:nat. (n-off) \leq m \to m < (min_aux off n f) \to f m = false.
intros 3.
-elim off.absurd le n m.rewrite > minus_n_O.assumption.
-apply lt_to_not_le.rewrite < min_aux_O_f f n.assumption.
+elim off.absurd (le n m).rewrite > minus_n_O.assumption.
+apply lt_to_not_le.rewrite < (min_aux_O_f f n).assumption.
generalize in match H1.
-elim min_aux_S f n1 n.
+elim (min_aux_S f n1 n).
elim H3.
-absurd n - S n1 \le m.assumption.
+absurd (n - S n1 \le m).assumption.
apply lt_to_not_le.rewrite < H6.assumption.
elim H3.
-elim le_to_or_lt_eq (n -(S n1)) m.
+elim (le_to_or_lt_eq (n -(S n1)) m).
apply H.apply lt_minus_S_n_to_le_minus_n.assumption.
rewrite < H6.assumption.
rewrite < H7.assumption.
\forall n,off:nat. (n-off) \leq (min_aux off n f).
intros 3.
elim off.rewrite < minus_n_O.
-rewrite > min_aux_O_f f n.apply le_n.
-elim min_aux_S f n1 n.
+rewrite > (min_aux_O_f f n).apply le_n.
+elim (min_aux_S f n1 n).
elim H1.rewrite > H3.apply le_n.
elim H1.rewrite > H3.
-apply trans_le (n-(S n1)) (n-n1).
+apply (trans_le (n-(S n1)) (n-n1)).
apply monotonic_le_minus_r.
apply le_n_Sn.
assumption.
qed.
+theorem le_min_aux_r : \forall f:nat \to bool.
+\forall n,off:nat. (min_aux off n f) \le n.
+intros.
+elim off.simplify.rewrite < minus_n_O.
+elim (f n).simplify.apply le_n.
+simplify.apply le_n.
+simplify.elim (f (n -(S n1))).
+simplify.apply le_plus_to_minus.
+rewrite < sym_plus.apply le_plus_n.
+simplify.assumption.
+qed.