--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+
+set "baseuri" "cic:/matita/nat/minus".
+
+include "nat/le_arith.ma".
+include "nat/compare.ma".
+
+let rec minus n m \def
+ match n with
+ [ O \Rightarrow O
+ | (S p) \Rightarrow
+ match m with
+ [O \Rightarrow (S p)
+ | (S q) \Rightarrow minus p q ]].
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural minus" 'minus x y = (cic:/matita/nat/minus/minus.con x y).
+
+theorem minus_n_O: \forall n:nat.n=n-O.
+intros.elim n.simplify.reflexivity.
+simplify.reflexivity.
+qed.
+
+theorem minus_n_n: \forall n:nat.O=n-n.
+intros.elim n.simplify.
+reflexivity.
+simplify.apply H.
+qed.
+
+theorem minus_Sn_n: \forall n:nat. S O = (S n)-n.
+intro.elim n.
+simplify.reflexivity.
+elim H.reflexivity.
+qed.
+
+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).
+simplify.reflexivity.
+intros.simplify.reflexivity.
+intros.rewrite < H.reflexivity.
+apply le_S_S_to_le. assumption.
+qed.
+
+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).
+simplify.rewrite < minus_n_O.reflexivity.
+intros.simplify.reflexivity.
+intros.simplify.apply H.apply le_S_S_to_le.assumption.
+qed.
+
+theorem minus_plus_m_m: \forall n,m:nat.n = (n+m)-m.
+intros 2.
+generalize in match n.
+elim m.
+rewrite < minus_n_O.apply plus_n_O.
+elim n2.simplify.
+apply minus_n_n.
+rewrite < plus_n_Sm.
+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).
+reflexivity.
+intros.simplify.rewrite < plus_n_O.reflexivity.
+intros.simplify.rewrite < sym_plus.simplify.
+apply eq_f.rewrite < sym_plus.apply H.
+apply le_S_S_to_le.assumption.
+qed.
+
+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)).
+apply plus_minus_m_m.
+apply H.elim H1.
+apply sym_plus.
+qed.
+
+theorem plus_to_minus :\forall n,m,p:nat.
+n = m+p \to n-m = p.
+intros.
+apply (inj_plus_r m).
+rewrite < H.
+rewrite < sym_plus.
+symmetry.
+apply plus_minus_m_m.rewrite > H.
+rewrite > sym_plus.
+apply le_plus_n.
+qed.
+
+theorem minus_S_S : \forall n,m:nat.
+eq nat (minus (S n) (S m)) (minus n m).
+intros.
+reflexivity.
+qed.
+
+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.
+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)).
+intros.simplify.reflexivity.
+intros.apply False_ind.
+apply not_le_Sn_O.
+goal 13.apply H.
+intros.
+simplify.apply H.apply le_S_S_to_le. apply H1.
+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.
+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)))).
+intro.elim n1.simplify.apply le_n_Sn.
+simplify.rewrite < minus_n_O.apply le_n.
+intros.simplify.apply le_n_Sn.
+intros.simplify.apply H.
+qed.
+
+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 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.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.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)).
+intros.apply le_O_n.
+simplify.intros. assumption.
+simplify.intros.apply le_S_S.apply H.assumption.
+qed.
+
+(* 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.
+intros.rewrite < minus_n_O.
+apply le_minus_m.
+intros.elim a.simplify.apply le_n.
+simplify.apply H.apply le_S_S_to_le.assumption.
+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.apply le_O_n.
+simplify.intros.rewrite < plus_n_O.assumption.
+intros.
+rewrite < plus_n_Sm.
+apply le_S_S.apply H.
+exact H1.
+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.simplify.apply le_O_n.
+intros 2.rewrite < plus_n_O.intro.simplify.assumption.
+intros.simplify.apply H.
+apply le_S_S_to_le.rewrite > plus_n_Sm.assumption.
+qed.
+
+(* 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)))).
+intro.rewrite < plus_n_O.rewrite < minus_n_O.intro.assumption.
+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))).
+rewrite < sym_plus.
+apply le_plus_n.assumption.
+intros.simplify.
+apply H.apply le_S_S_to_le.
+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.
+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.
+ 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
+\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).
+ elim Hcut.
+ symmetry.apply plus_to_minus.
+ 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
+p+(n-m) = n-(m-p).
+intros.
+apply sym_eq.
+apply plus_to_minus.
+rewrite < assoc_plus.
+rewrite < plus_minus_m_m.
+rewrite < sym_plus.
+rewrite < plus_minus_m_m.reflexivity.
+assumption.assumption.
+qed.
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