+theorem eq_sigma_p_pred:
+\forall n,p,g. p O = true \to
+sigma_p (S n) (\lambda i.p (pred i)) (\lambda i.g(pred i)) =
+plus (sigma_p n p g) (g O).
+intros.
+unfold sigma_p.
+apply eq_iter_p_gen_pred
+ [assumption
+ |apply symmetricIntPlus
+ |apply associative_plus
+ |intros.apply sym_eq.apply plus_n_O
+ ]
+qed.
+
+(* monotonicity *)
+theorem le_sigma_p:
+\forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to p i = true \to g1 i \le g2 i ) \to
+sigma_p n p g1 \le sigma_p n p g2.
+intros.
+elim n in H ⊢ %
+ [apply le_n.
+ |apply (bool_elim ? (p n1));intros
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply le_plus
+ [apply H1[apply le_n|assumption]
+ |apply H.
+ intros.
+ apply H1[apply le_S.assumption|assumption]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply H.
+ intros.
+ apply H1[apply le_S.assumption|assumption]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+(* a slightly more general result *)
+theorem le_sigma_p1:
+\forall n:nat. \forall p1,p2:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to
+bool_to_nat (p1 i)*(g1 i) \le bool_to_nat (p2 i)*g2 i) \to
+sigma_p n p1 g1 \le sigma_p n p2 g2.
+intros.
+elim n in H ⊢ %
+ [apply le_n.
+ |apply (bool_elim ? (p1 n1));intros
+ [apply (bool_elim ? (p2 n1));intros
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply le_plus
+ [lapply (H1 n1) as H5
+ [rewrite > H2 in H5.
+ rewrite > H3 in H5.
+ simplify in H5.
+ rewrite < plus_n_O in H5.
+ rewrite < plus_n_O in H5.
+ assumption
+ |apply le_S_S.apply le_n
+ ]
+ |apply H.intros.
+ apply H1.apply le_S.assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > true_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [change in ⊢ (? ? %) with (O + sigma_p n1 p2 g2).
+ apply le_plus
+ [lapply (H1 n1) as H5
+ [rewrite > H2 in H5.
+ rewrite > H3 in H5.
+ simplify in H5.
+ rewrite < plus_n_O in H5.
+ assumption
+ |apply le_S_S.apply le_n
+ ]
+ |apply H.intros.
+ apply H1.apply le_S.assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ |apply (bool_elim ? (p2 n1));intros
+ [rewrite > false_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [change in ⊢ (? % ?) with (O + sigma_p n1 p1 g1).
+ apply le_plus
+ [lapply (H1 n1) as H5
+ [rewrite > H2 in H5.
+ rewrite > H3 in H5.
+ simplify in H5.
+ rewrite < plus_n_O in H5.
+ assumption
+ |apply le_S_S.apply le_n
+ ]
+ |apply H.intros.
+ apply H1.apply le_S.assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply H.intros.
+ apply H1.apply le_S.assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ ]
+qed.