(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/minus.ma".
-include "nat/orders_op.ma".
-include "nat/times.ma".
+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 (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.eq nat n (minus n O).
+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.eq nat O (minus n n).
+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.eq nat (S O) (minus (S n) n).
+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.
-le m n \to eq nat (minus (S n) m) (S (minus n m)).
+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.le m n \to eq nat (minus (S n) m) (S (minus 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.
qed.
theorem plus_minus:
-\forall n,m,p:nat. le m n \to eq nat (plus (minus n m) p) (minus (plus n p) m).
+\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.le m n \to eq nat (plus (minus n m) p) (minus (plus 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.
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.
-le m n \to eq nat n (plus (minus n m) m).
+m \leq n \to n = (n-m)+m.
intros 2.
-apply nat_elim2 (\lambda n,m.le m n \to eq nat n (plus (minus 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.
apply le_S_S_to_le.assumption.
qed.
-theorem minus_to_plus :\forall n,m,p:nat.le m n \to eq nat (minus n m) p \to
-eq nat n (plus m p).
-intros.apply trans_eq ? ? (plus (minus n m) m) ?.
+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.le m n \to
-eq nat n (plus m p) \to eq nat (minus n m) p.
+theorem plus_to_minus :\forall n,m,p:nat.
+n = m+p \to n-m = p.
intros.
-apply inj_plus_r m.
-rewrite < H1.
+apply (inj_plus_r m).
+rewrite < H.
rewrite < sym_plus.
symmetry.
-apply plus_minus_m_m.assumption.
+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 minus_ge_O: \forall n,m:nat.
-le n m \to eq nat (minus n m) O.
+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.le n m \to eq nat (minus n m) O).
+apply (nat_elim2 (\lambda n,m.n \leq m \to n-m = O)).
intros.simplify.reflexivity.
intros.apply False_ind.
-(* ancora problemi con il not *)
-apply not_le_Sn_O n1 H.
+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.le (S n) m \to le (S O) (minus m n).
-intros.elim H.elim minus_Sn_n n.apply le_n.
+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 le_plus_minus: \forall n,m,p. (le (plus n m) p) \to (le n (minus p m)).
-intros 3.
-elim p.simplify.apply trans_le ? (plus n m) ?.
-elim sym_plus ? ?.
-apply plus_le.assumption.
-apply le_n_Sm_elim ? ? H1.
+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.
-*)
-check distributive.
+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 times_minus_distr: \forall n,m,p:nat.
-eq nat (times n (minus m p)) (minus (times n m) (times n p)).
+theorem distributive_times_minus: distributive nat times minus.
+unfold distributive.
intros.
-apply (leb_ind p m).intro.
-cut eq nat (plus (times n (minus m p)) (times n p)) (plus (minus (times n m) (times n p)) (times n p)).
-apply plus_injective_right ? ? (times n p).
-assumption.
-apply trans_eq nat ? (times n m).
-elim (times_plus_distr ? ? ?).
-elim (minus_plus ? ? H).apply refl_equal.
-elim (minus_plus ? ? ?).apply refl_equal.
-apply times_le_monotony_left.
-assumption.
-intro.
-elim sym_eq ? ? ? (minus_ge_O ? ? ?).
-elim sym_eq ? ? ? (minus_ge_O ? ? ?).
-elim (sym_times ? ?).simplify.apply refl_equal.
-simplify.
-apply times_le_monotony_left.
-cut (lt m p) \to (le m p).
-apply Hcut.simplify.apply not_le_lt ? ? H.
-intro.apply lt_le.apply H1.
-cut (lt m p) \to (le m p).
-apply Hcut.simplify.apply not_le_lt ? ? H.
-intro.apply lt_le.apply H1.
-qed.
-
-theorem minus_le: \forall n,m:nat. le (minus n m) n.
-intro.elim n.simplify.apply le_n.
-elim m.simplify.apply le_n.
-simplify.apply le_S.apply H.
+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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