theorem distributive_times_minus: distributive nat times minus.
simplify.
intros.
-apply (leb_elim z y).intro.
-cut x*(y-z)+x*z = (x*y-x*z)+x*z.
-apply inj_plus_l (x*z).
-assumption.
-apply trans_eq nat ? (x*y).
-rewrite < distr_times_plus.
-rewrite < plus_minus_m_m ? ? H.reflexivity.
-rewrite < plus_minus_m_m ? ? ?.reflexivity.
-apply le_times_r.
-assumption.
-intro.
-rewrite > eq_minus_n_m_O.
-rewrite > eq_minus_n_m_O (x*y).
-rewrite < sym_times.simplify.reflexivity.
-apply lt_to_le.
-apply not_le_to_lt.assumption.
-apply le_times_r.apply lt_to_le.
-apply not_le_to_lt.assumption.
+apply (leb_elim z y).
+ intro.cut x*(y-z)+x*z = (x*y-x*z)+x*z.
+ apply inj_plus_l (x*z).assumption.
+ apply trans_eq nat ? (x*y).
+ rewrite < distr_times_plus.rewrite < plus_minus_m_m ? ? H.reflexivity.
+ rewrite < plus_minus_m_m ? ? ?.
+ reflexivity.
+ apply le_times_r.assumption.
+ intro.rewrite > eq_minus_n_m_O.
+ rewrite > eq_minus_n_m_O (x*y).
+ rewrite < sym_times.simplify.reflexivity.
+ apply le_times_r.apply lt_to_le.apply not_le_to_lt.assumption.
+ apply lt_to_le.apply not_le_to_lt.assumption.
qed.
theorem distr_times_minus: \forall n,m,p:nat. n*(m-p) = n*m-n*p
theorem eq_minus_minus_minus_plus: \forall n,m,p:nat. (n-m)-p = n-(m+p).
intros.
cut m+p \le n \or \not m+p \le n.
-elim Hcut.
-apply sym_eq.
-apply plus_to_minus.assumption.
-rewrite > assoc_plus.
-rewrite > sym_plus p.
-rewrite < plus_minus_m_m.
-rewrite > sym_plus.
-rewrite < plus_minus_m_m.reflexivity.
-rewrite > eq_minus_n_m_O n (m+p).
-rewrite > eq_minus_n_m_O (n-m) p.reflexivity.
-apply decidable_le (m+p) n.
-apply le_plus_to_minus_r.
-rewrite > sym_plus.assumption.
-apply trans_le ? (m+p).
-rewrite < sym_plus.
-apply le_plus_n.assumption.
-apply lt_to_le.apply not_le_to_lt.assumption.
-apply le_plus_to_minus.
-apply lt_to_le.apply not_le_to_lt.
-rewrite < sym_plus.assumption.
+ elim Hcut.
+ symmetry.apply plus_to_minus.assumption.
+ rewrite > assoc_plus.rewrite > sym_plus p.rewrite < plus_minus_m_m.
+ rewrite > sym_plus.rewrite < plus_minus_m_m.
+ reflexivity.
+ apply trans_le ? (m+p).
+ rewrite < sym_plus.apply le_plus_n.
+ assumption.
+ apply le_plus_to_minus_r.rewrite > sym_plus.assumption.
+ rewrite > eq_minus_n_m_O n (m+p).
+ rewrite > eq_minus_n_m_O (n-m) p.
+ reflexivity.
+ apply le_plus_to_minus.apply lt_to_le. rewrite < sym_plus.
+ apply not_le_to_lt. assumption.
+ apply lt_to_le.apply not_le_to_lt.assumption.
+ apply decidable_le (m+p) n.
qed.
theorem eq_plus_minus_minus_minus: \forall n,m,p:nat. p \le m \to m \le n \to