]
qed.
+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.
+generalize in match H.
+elim n
+ [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 H2[apply le_n|assumption]
+ |apply H1.
+ intros.
+ apply H2[apply le_S.assumption|assumption]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply H1.
+ intros.
+ apply H2[apply le_S.assumption|assumption]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+theorem lt_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
+(\exists i. i < n \and (p i = true) \and (g1 i < g2 i)) \to
+sigma_p n p g1 < sigma_p n p g2.
+intros 4.
+elim n
+ [elim H1.clear H1.
+ elim H2.clear H2.
+ elim H1.clear H1.
+ apply False_ind.
+ apply (lt_to_not_le ? ? H2).
+ apply le_O_n
+ |apply (bool_elim ? (p n1));intros
+ [apply (bool_elim ? (leb (S (g1 n1)) (g2 n1)));intros
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [change with
+ (S (g1 n1)+sigma_p n1 p g1 \le g2 n1+sigma_p n1 p g2).
+ apply le_plus
+ [apply leb_true_to_le.assumption
+ |apply le_sigma_p.intros.
+ apply H1
+ [apply lt_to_le.apply le_S_S.assumption
+ |assumption
+ ]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [unfold lt.
+ rewrite > plus_n_Sm.
+ apply le_plus
+ [apply H1
+ [apply le_n
+ |assumption
+ ]
+ |apply H
+ [intros.apply H1
+ [apply lt_to_le.apply le_S_S.assumption
+ |assumption
+ ]
+ |elim H2.clear H2.
+ elim H5.clear H5.
+ elim H2.clear H2.
+ apply (ex_intro ? ? a).
+ split
+ [split
+ [elim (le_to_or_lt_eq a n1)
+ [assumption
+ |absurd (g1 a < g2 a)
+ [assumption
+ |apply leb_false_to_not_le.
+ rewrite > H2.
+ assumption
+ ]
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply H
+ [intros.apply H1
+ [apply lt_to_le.apply le_S_S.assumption
+ |assumption
+ ]
+ |elim H2.clear H2.
+ elim H4.clear H4.
+ elim H2.clear H2.
+ apply (ex_intro ? ? a).
+ split
+ [split
+ [elim (le_to_or_lt_eq a n1)
+ [assumption
+ |apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H6.
+ rewrite < H3.
+ rewrite < H2.
+ reflexivity
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
theorem sigma_p_divides:
\forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to
\forall g: nat \to nat.
|assumption
]
qed.
-
\ No newline at end of file
+
+
+theorem sigma_p2_eq:
+\forall g: nat \to nat \to nat.
+\forall h11,h12,h21,h22: nat \to nat \to nat.
+\forall n1,m1,n2,m2.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to
+p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true
+\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j
+\land h11 i j < n1 \land h12 i j < m1) \to
+(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to
+p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true
+\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j
+\land (h21 i j) < n2 \land (h22 i j) < m2) \to
+sigma_p n1 p11 (\lambda x:nat .sigma_p m1 (p12 x) (\lambda y. g x y)) =
+sigma_p n2 p21 (\lambda x:nat .sigma_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? (? ? ? ? (\lambda x:?.%) ? ?) ?).
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+
+apply(iter_p_gen_2_eq nat O plus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12 p22)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intro.
+ rewrite < (plus_n_O).
+ reflexivity
+| assumption
+| assumption
+]
+qed.
\ No newline at end of file
projects / helm.git / blobdiff