(\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
+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 H2[apply le_n|assumption]
- |apply H1.
+ [apply H1[apply le_n|assumption]
+ |apply H.
intros.
- apply H2[apply le_S.assumption|assumption]
+ apply H1[apply le_S.assumption|assumption]
]
|assumption
]
]
|rewrite > false_to_sigma_p_Sn
[rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
- [apply H1.
+ [apply H.
intros.
- apply H2[apply le_S.assumption|assumption]
+ apply H1[apply le_S.assumption|assumption]
|assumption
]
|assumption
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.
-generalize in match H.
-elim n
+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 (H2 n1) as H5
- [rewrite > H3 in H5.
- rewrite > H4 in H5.
+ [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 H1.intros.
- apply H2.apply le_S.assumption
+ |apply H.intros.
+ apply H1.apply le_S.assumption
]
|assumption
]
[rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
[change in ⊢ (? ? %) with (O + sigma_p n1 p2 g2).
apply le_plus
- [lapply (H2 n1) as H5
- [rewrite > H3 in H5.
- rewrite > H4 in H5.
+ [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 H1.intros.
- apply H2.apply le_S.assumption
+ |apply H.intros.
+ apply H1.apply le_S.assumption
]
|assumption
]
[rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
[change in ⊢ (? % ?) with (O + sigma_p n1 p1 g1).
apply le_plus
- [lapply (H2 n1) as H5
- [rewrite > H3 in H5.
- rewrite > H4 in H5.
+ [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 H1.intros.
- apply H2.apply le_S.assumption
+ |apply H.intros.
+ apply H1.apply le_S.assumption
]
|assumption
]
]
|rewrite > false_to_sigma_p_Sn
[rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
- [apply H1.intros.
- apply H2.apply le_S.assumption
+ [apply H.intros.
+ apply H1.apply le_S.assumption
|assumption
]
|assumption
]
]
]
-qed.
+qed.
theorem lt_sigma_p:
\forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.