theorem minus_Sn_m: \forall n,m:nat. m \leq n \to (S n)-m = S (n-m).
intros 2.
-apply nat_elim2
-(\lambda n,m.m \leq n \to (S n)-m = S (n-m)).
-intros.apply le_n_O_elim n1 H.
+apply (nat_elim2
+(\lambda n,m.m \leq n \to (S n)-m = S (n-m))).
+intros.apply (le_n_O_elim n1 H).
simplify.reflexivity.
intros.simplify.reflexivity.
intros.rewrite < H.reflexivity.
theorem plus_minus:
\forall n,m,p:nat. m \leq n \to (n-m)+p = (n+p)-m.
intros 2.
-apply nat_elim2
-(\lambda n,m.\forall p:nat.m \leq n \to (n-m)+p = (n+p)-m).
-intros.apply le_n_O_elim ? H.
+apply (nat_elim2
+(\lambda n,m.\forall p:nat.m \leq n \to (n-m)+p = (n+p)-m)).
+intros.apply (le_n_O_elim ? H).
simplify.rewrite < minus_n_O.reflexivity.
intros.simplify.reflexivity.
intros.simplify.apply H.apply le_S_S_to_le.assumption.
elim n2.simplify.
apply minus_n_n.
rewrite < plus_n_Sm.
-change with S n3 = (S n3 + n1)-n1.
+change with (S n3 = (S n3 + n1)-n1).
apply H.
qed.
theorem plus_minus_m_m: \forall n,m:nat.
m \leq n \to n = (n-m)+m.
intros 2.
-apply nat_elim2 (\lambda n,m.m \leq n \to n = (n-m)+m).
-intros.apply le_n_O_elim n1 H.
+apply (nat_elim2 (\lambda n,m.m \leq n \to n = (n-m)+m)).
+intros.apply (le_n_O_elim n1 H).
reflexivity.
intros.simplify.rewrite < plus_n_O.reflexivity.
intros.simplify.rewrite < sym_plus.simplify.
theorem minus_to_plus :\forall n,m,p:nat.m \leq n \to n-m = p \to
n = m+p.
-intros.apply trans_eq ? ? ((n-m)+m).
+intros.apply (trans_eq ? ? ((n-m)+m)).
apply plus_minus_m_m.
apply H.elim H1.
apply sym_plus.
theorem plus_to_minus :\forall n,m,p:nat.
n = m+p \to n-m = p.
intros.
-apply inj_plus_r m.
+apply (inj_plus_r m).
rewrite < H.
rewrite < sym_plus.
symmetry.
theorem minus_pred_pred : \forall n,m:nat. lt O n \to lt O m \to
eq nat (minus (pred n) (pred m)) (minus n m).
intros.
-apply lt_O_n_elim n H.intro.
-apply lt_O_n_elim m H1.intro.
+apply (lt_O_n_elim n H).intro.
+apply (lt_O_n_elim m H1).intro.
simplify.reflexivity.
qed.
theorem eq_minus_n_m_O: \forall n,m:nat.
n \leq m \to n-m = O.
intros 2.
-apply nat_elim2 (\lambda n,m.n \leq m \to n-m = O).
+apply (nat_elim2 (\lambda n,m.n \leq m \to n-m = O)).
intros.simplify.reflexivity.
intros.apply False_ind.
apply not_le_Sn_O.
qed.
theorem le_SO_minus: \forall n,m:nat.S n \leq m \to S O \leq m-n.
-intros.elim H.elim minus_Sn_n n.apply le_n.
+intros.elim H.elim (minus_Sn_n n).apply le_n.
rewrite > minus_Sn_m.
apply le_S.assumption.
apply lt_to_le.assumption.
qed.
theorem minus_le_S_minus_S: \forall n,m:nat. m-n \leq S (m-(S n)).
-intros.apply nat_elim2 (\lambda n,m.m-n \leq S (m-(S n))).
+intros.apply (nat_elim2 (\lambda n,m.m-n \leq S (m-(S n)))).
intro.elim n1.simplify.apply le_n_Sn.
simplify.rewrite < minus_n_O.apply le_n.
intros.simplify.apply le_n_Sn.
theorem lt_minus_S_n_to_le_minus_n : \forall n,m,p:nat. m-(S n) < p \to m-n \leq p.
intros 3.simplify.intro.
-apply trans_le (m-n) (S (m-(S n))) p.
+apply (trans_le (m-n) (S (m-(S n))) p).
apply minus_le_S_minus_S.
assumption.
qed.
theorem le_minus_m: \forall n,m:nat. n-m \leq n.
-intros.apply nat_elim2 (\lambda m,n. n-m \leq n).
+intros.apply (nat_elim2 (\lambda m,n. n-m \leq n)).
intros.rewrite < minus_n_O.apply le_n.
intros.simplify.apply le_n.
intros.simplify.apply le_S.assumption.
qed.
theorem lt_minus_m: \forall n,m:nat. O < n \to O < m \to n-m \lt n.
-intros.apply lt_O_n_elim n H.intro.
-apply lt_O_n_elim m H1.intro.
-simplify.apply le_S_S.apply le_minus_m.
+intros.apply (lt_O_n_elim n H).intro.
+apply (lt_O_n_elim m H1).intro.
+simplify.unfold lt.apply le_S_S.apply le_minus_m.
qed.
theorem minus_le_O_to_le: \forall n,m:nat. n-m \leq O \to n \leq m.
intros 2.
-apply nat_elim2 (\lambda n,m:nat.n-m \leq O \to n \leq m).
+apply (nat_elim2 (\lambda n,m:nat.n-m \leq O \to n \leq m)).
intros.apply le_O_n.
simplify.intros. assumption.
simplify.intros.apply le_S_S.apply H.assumption.
(* galois *)
theorem monotonic_le_minus_r:
\forall p,q,n:nat. q \leq p \to n-p \le n-q.
-simplify.intros 2.apply nat_elim2
-(\lambda p,q.\forall a.q \leq p \to a-p \leq a-q).
-intros.apply le_n_O_elim n H.apply le_n.
+simplify.intros 2.apply (nat_elim2
+(\lambda p,q.\forall a.q \leq p \to a-p \leq a-q)).
+intros.apply (le_n_O_elim n H).apply le_n.
intros.rewrite < minus_n_O.
apply le_minus_m.
intros.elim a.simplify.apply le_n.
qed.
theorem le_minus_to_plus: \forall n,m,p. (le (n-m) p) \to (le n (p+m)).
-intros 2.apply nat_elim2 (\lambda n,m.\forall p.(le (n-m) p) \to (le n (p+m))).
+intros 2.apply (nat_elim2 (\lambda n,m.\forall p.(le (n-m) p) \to (le n (p+m)))).
intros.apply le_O_n.
simplify.intros.rewrite < plus_n_O.assumption.
intros.
qed.
theorem le_plus_to_minus: \forall n,m,p. (le n (p+m)) \to (le (n-m) p).
-intros 2.apply nat_elim2 (\lambda n,m.\forall p.(le n (p+m)) \to (le (n-m) p)).
+intros 2.apply (nat_elim2 (\lambda n,m.\forall p.(le n (p+m)) \to (le (n-m) p))).
intros.simplify.apply le_O_n.
intros 2.rewrite < plus_n_O.intro.simplify.assumption.
intros.simplify.apply H.
(* the converse of le_plus_to_minus does not hold *)
theorem le_plus_to_minus_r: \forall n,m,p. (le (n+m) p) \to (le n (p-m)).
-intros 3.apply nat_elim2 (\lambda m,p.(le (n+m) p) \to (le n (p-m))).
+intros 3.apply (nat_elim2 (\lambda m,p.(le (n+m) p) \to (le n (p-m)))).
intro.rewrite < plus_n_O.rewrite < minus_n_O.intro.assumption.
-intro.intro.cut n=O.rewrite > Hcut.apply le_O_n.
+intro.intro.cut (n=O).rewrite > Hcut.apply le_O_n.
apply sym_eq. apply le_n_O_to_eq.
-apply trans_le ? (n+(S n1)).
+apply (trans_le ? (n+(S n1))).
rewrite < sym_plus.
apply le_plus_n.assumption.
intros.simplify.
rewrite > plus_n_Sm.assumption.
qed.
+(* minus and lt - to be completed *)
+theorem lt_minus_to_plus: \forall n,m,p. (lt n (p-m)) \to (lt (n+m) p).
+intros 3.apply (nat_elim2 (\lambda m,p.(lt n (p-m)) \to (lt (n+m) p))).
+intro.rewrite < plus_n_O.rewrite < minus_n_O.intro.assumption.
+simplify.intros.apply False_ind.apply (not_le_Sn_O n H).
+simplify.intros.unfold lt.
+apply le_S_S.
+rewrite < plus_n_Sm.
+apply H.apply H1.
+qed.
theorem distributive_times_minus: distributive nat times minus.
-simplify.
+unfold distributive.
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.
+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 > (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.
theorem distr_times_minus: \forall n,m,p:nat. n*(m-p) = n*m-n*p
\def distributive_times_minus.
+theorem eq_minus_plus_plus_minus: \forall n,m,p:nat. p \le m \to (n+m)-p = n+(m-p).
+intros.
+apply plus_to_minus.
+rewrite > sym_plus in \vdash (? ? ? %).
+rewrite > assoc_plus.
+rewrite < plus_minus_m_m.
+reflexivity.assumption.
+qed.
+
theorem eq_minus_minus_minus_plus: \forall n,m,p:nat. (n-m)-p = n-(m+p).
intros.
-cut m+p \le n \or m+p \nleq n.
+cut (m+p \le n \or m+p \nleq n).
elim Hcut.
symmetry.apply plus_to_minus.
- rewrite > assoc_plus.rewrite > sym_plus p.rewrite < plus_minus_m_m.
+ 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).
+ 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.
+ 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.
+ 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