(**************************************************************************)
(* include "higher_order_defs/functions.ma". *)
+include "hints_declaration.ma".
include "basics/functions.ma".
include "basics/eq.ma".
interpretation "Natural numbers" 'N = nat.
-default "natural numbers" cic:/matita/ng/arithmetics/nat/nat.ind.
+alias num (instance 0) = "nnatural number".
-alias num (instance 0) = "natural number".
+(*
+nrecord pos : Type ≝
+ {n:>nat; is_pos: n ≠ 0}.
+
+ncoercion nat_to_pos: ∀n:nat. n ≠0 →pos ≝ mk_pos on
+*)
+
+(* default "natural numbers" cic:/matita/ng/arithmetics/nat/nat.ind.
+*)
ndefinition pred ≝
- λn. match n with
- [ O ⇒ O
- | (S p) ⇒ p].
+ λn. match n with [ O ⇒ O | (S p) ⇒ p].
ntheorem pred_Sn : ∀n. n = pred (S n).
-//. nqed.
+//; nqed.
ntheorem injective_S : injective nat nat S.
-(* nwhd; #a; #b;#H;napplyS pred_Sn. *)
-//. nqed.
+//; nqed.
(*
ntheorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m.
ndefinition not_zero: nat → Prop ≝
λn: nat. match n with
- [ O ⇒ False
- | (S p) ⇒ True ].
+ [ O ⇒ False | (S p) ⇒ True ].
ntheorem not_eq_O_S : ∀n:nat. O ≠ S n.
#n; #eqOS; nchange with (not_zero O); nrewrite > eqOS; //.
#R; #ROn; #RSO; #RSS; #n; nelim n;//;
#n0; #Rn0m; #m; ncases m;/2/; nqed.
-ntheorem decidable_eq_nat : \forall n,m:nat.decidable (n=m).
-napply nat_elim2;#n;
+ntheorem decidable_eq_nat : ∀n,m:nat.decidable (n=m).
+napply nat_elim2; #n;
##[ ncases n; /2/;
##| /3/;
- ##| #m; #Hind; ncases Hind; /3/; (* ??? /2/;#neqnm; /3/; *)
+ ##| #m; #Hind; ncases Hind; /3/;
##]
nqed.
interpretation "natural plus" 'plus x y = (plus x y).
-ntheorem plus_n_O: ∀n:nat. n = n+O.
-#n; nelim n; /2/; nqed.
+ntheorem plus_O_n: ∀n:nat. n = 0+n.
+//; nqed.
+
+(*
+ntheorem plus_Sn_m: ∀n,m:nat. S (n + m) = S n + m.
+//; nqed.
+*)
-ntheorem plus_n_Sm : ∀n,m:nat. S (n+m) = n+(S m).
+ntheorem plus_n_O: ∀n:nat. n = n+0.
#n; nelim n; nnormalize; //; nqed.
-ntheorem plus_n_SO : ∀n:nat. S n = n+(S O).
-//; nqed.
+ntheorem plus_n_Sm : ∀n,m:nat. S (n+m) = n + S m.
+#n; nelim n; nnormalize; //; nqed.
-ntheorem sym_plus: ∀n,m:nat. n+m = m+n.
+(*
+ntheorem plus_Sn_m1: ∀n,m:nat. S m + n = n + S m.
#n; nelim n; nnormalize; //; nqed.
+*)
+
+(*
+ntheorem plus_n_SO : ∀n:nat. S n = n+S O.
+//; nqed. *)
+
+ntheorem symmetric_plus: symmetric ? plus.
+#n; nelim n; nnormalize; //; nqed.
ntheorem associative_plus : associative nat plus.
#n; nelim n; nnormalize; //; nqed.
-(* ntheorem assoc_plus : \forall n,m,p:nat. (n+m)+p = n+(m+p)
-\def associative_plus. *)
+ntheorem assoc_plus1: ∀a,b,c. c + (b + a) = b + c + a.
+//; nqed.
ntheorem injective_plus_r: ∀n:nat.injective nat nat (λm.n+m).
#n; nelim n; nnormalize; /3/; nqed.
(* ntheorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
-\def injective_plus_r. *)
+\def injective_plus_r.
ntheorem injective_plus_l: ∀m:nat.injective nat nat (λn.n+m).
-/2/; nqed.
+/2/; nqed. *)
(* ntheorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
\def injective_plus_l. *)
ntheorem times_Sn_m: ∀n,m:nat. m+n*m = S n*m.
//; nqed.
+ntheorem times_O_n: ∀n:nat. O = O*n.
+//; nqed.
+
ntheorem times_n_O: ∀n:nat. O = n*O.
#n; nelim n; //; nqed.
ntheorem times_n_Sm : ∀n,m:nat. n+(n*m) = n*(S m).
-#n; nelim n; nnormalize; /2/; nqed.
+#n; nelim n; nnormalize; //; nqed.
+
+ntheorem symmetric_times : symmetric nat times.
+#n; nelim n; nnormalize; //; nqed.
+
+(* variant sym_times : \forall n,m:nat. n*m = m*n \def
+symmetric_times. *)
+
+ntheorem distributive_times_plus : distributive nat times plus.
+#n; nelim n; nnormalize; //; nqed.
+ntheorem distributive_times_plus_r:
+\forall a,b,c:nat. (b+c)*a = b*a + c*a.
+//; nqed.
+
+ntheorem associative_times: associative nat times.
+#n; nelim n; nnormalize; //; nqed.
+
+nlemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
+//; nqed.
+
+(* ci servono questi risultati?
ntheorem times_O_to_O: ∀n,m:nat.n*m=O → n=O ∨ m=O.
-napply nat_elim2; (* /2/ slow! *)
- ##[#n; #H; @1; //;
- ##|#n; #H; @2; //; (* ?? *)
- ##|#n; #m; #H; #H1; napply False_ind;napply not_eq_O_S;
- (* why two steps? *) /2/;
- ##]
-nqed.
+napply nat_elim2; /2/;
+#n; #m; #H; nnormalize; #H1; napply False_ind;napply not_eq_O_S;
+//; nqed.
ntheorem times_n_SO : ∀n:nat. n = n * S O.
+#n; //; nqed.
+
+ntheorem times_SSO_n : ∀n:nat. n + n = (S(S O)) * n.
+nnormalize; //; nqed.
+
+nlemma times_SSO: \forall n.(S(S O))*(S n) = S(S((S(S O))*n)).
//; nqed.
+ntheorem or_eq_eq_S: \forall n.\exists m.
+n = (S(S O))*m \lor n = S ((S(S O))*m).
+#n; nelim n;
+ ##[@; /2/;
+ ##|#a; #H; nelim H; #b;#or;nelim or;#aeq;
+ ##[@ b; @ 2; //;
+ ##|@ (S b); @ 1; /2/;
+ ##]
+nqed.
+*)
+
+(******************** ordering relations ************************)
+
+ninductive le (n:nat) : nat → Prop ≝
+ | le_n : le n n
+ | le_S : ∀ m:nat. le n m → le n (S m).
+
+interpretation "natural 'less or equal to'" 'leq x y = (le x y).
+
+interpretation "natural 'neither less nor equal to'" 'nleq x y = (Not (le x y)).
+
+ndefinition lt: nat → nat → Prop ≝
+λn,m:nat. S n ≤ m.
+
+interpretation "natural 'less than'" 'lt x y = (lt x y).
+
+interpretation "natural 'not less than'" 'nless x y = (Not (lt x y)).
+
+ndefinition ge: nat \to nat \to Prop \def
+\lambda n,m:nat.m \leq n.
+
+interpretation "natural 'greater or equal to'" 'geq x y = (ge x y).
+
+ndefinition gt: nat \to nat \to Prop \def
+\lambda n,m:nat.m<n.
+
+interpretation "natural 'greater than'" 'gt x y = (gt x y).
+
+interpretation "natural 'not greater than'" 'ngtr x y = (Not (gt x y)).
+
+ntheorem transitive_le : transitive nat le.
+#a; #b; #c; #leab; #lebc;nelim lebc;/2/;
+nqed.
+
+(*
+ntheorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
+\def transitive_le. *)
+
+ntheorem transitive_lt: transitive nat lt.
+#a; #b; #c; #ltab; #ltbc;nelim ltbc;/2/;nqed.
+
(*
+theorem trans_lt: \forall n,m,p:nat. lt n m \to lt m p \to lt n p
+\def transitive_lt. *)
+
+ntheorem le_S_S: ∀n,m:nat. n ≤ m → S n ≤ S m.
+#n; #m; #lenm; nelim lenm; /2/; nqed.
+
+ntheorem le_O_n : ∀n:nat. O ≤ n.
+#n; nelim n; /2/; nqed.
+
+ntheorem le_n_Sn : ∀n:nat. n ≤ S n.
+/2/; nqed.
+
+ntheorem le_pred_n : ∀n:nat. pred n ≤ n.
+#n; nelim n; //; nqed.
+
+ntheorem monotonic_pred: monotonic ? le pred.
+#n; #m; #lenm; nelim lenm; //; /2/; nqed.
+
+ntheorem le_S_S_to_le: ∀n,m:nat. S n ≤ S m → n ≤ m.
+/2/; nqed.
+
+ntheorem lt_S_S_to_lt: ∀n,m. S n < S m \to n < m.
+/2/; nqed.
+
+ntheorem lt_to_lt_S_S: ∀n,m. n < m → S n < S m.
+/2/; nqed.
+
+ntheorem lt_to_not_zero : ∀n,m:nat. n < m → not_zero m.
+#n; #m; #Hlt; nelim Hlt;//; nqed.
+
+(* lt vs. le *)
+ntheorem not_le_Sn_O: ∀ n:nat. S n ≰ O.
+#n; #Hlen0; napply (lt_to_not_zero ?? Hlen0); nqed.
+
+ntheorem not_le_to_not_le_S_S: ∀ n,m:nat. n ≰ m → S n ≰ S m.
+/3/; nqed.
+
+ntheorem not_le_S_S_to_not_le: ∀ n,m:nat. S n ≰ S m → n ≰ m.
+/3/; nqed.
+
+ntheorem decidable_le: ∀n,m. decidable (n≤m).
+napply nat_elim2; #n; /2/;
+#m; #dec; ncases dec;/3/; nqed.
+
+ntheorem decidable_lt: ∀n,m. decidable (n < m).
+#n; #m; napply decidable_le ; nqed.
-ntheorem times_SSO_n : ∀n:nat. n + n = 2 * n.
+ntheorem not_le_Sn_n: ∀n:nat. S n ≰ n.
+#n; nelim n; /2/; nqed.
+
+ntheorem lt_S_to_le: ∀n,m:nat. n < S m → n ≤ m.
+/2/; nqed.
+
+ntheorem not_le_to_lt: ∀n,m. n ≰ m → m < n.
+napply nat_elim2; #n;
+ ##[#abs; napply False_ind;/2/;
+ ##|/2/;
+ ##|#m;#Hind;#HnotleSS; napply lt_to_lt_S_S;/3/;
+ ##]
+nqed.
+
+ntheorem lt_to_not_le: ∀n,m. n < m → m ≰ n.
+#n; #m; #Hltnm; nelim Hltnm;/3/; nqed.
+
+ntheorem not_lt_to_le: ∀n,m:nat. n ≮ m → m ≤ n.
+#n; #m; #Hnlt; napply lt_S_to_le;
+(* something strange here: /2/ fails:
+ we need an extra depths for unfolding not *)
+napply not_le_to_lt; napply Hnlt; nqed.
+
+ntheorem le_to_not_lt: ∀n,m:nat. n ≤ m → m ≮ n.
+/2/; nqed.
+
+(* lt and le trans *)
+
+ntheorem lt_to_le_to_lt: ∀n,m,p:nat. n < m → m ≤ p → n < p.
+#n; #m; #p; #H; #H1; nelim H1; /2/; nqed.
+
+ntheorem le_to_lt_to_lt: ∀n,m,p:nat. n ≤ m → m < p → n < p.
+#n; #m; #p; #H; nelim H; /3/; nqed.
+
+ntheorem lt_S_to_lt: ∀n,m. S n < m → n < m.
+/2/; nqed.
+
+ntheorem ltn_to_ltO: ∀n,m:nat. n < m → O < m.
+/2/; nqed.
+
+(*
+theorem lt_SO_n_to_lt_O_pred_n: \forall n:nat.
+(S O) \lt n \to O \lt (pred n).
intros.
-simplify.
-rewrite < plus_n_O.
-reflexivity.
+apply (ltn_to_ltO (pred (S O)) (pred n) ?).
+ apply (lt_pred (S O) n);
+ [ apply (lt_O_S O)
+ | assumption
+ ]
+qed. *)
+
+ntheorem lt_O_n_elim: ∀n:nat. O < n →
+ ∀P:nat → Prop.(∀m:nat.P (S m)) → P n.
+#n; nelim n; //; #abs; napply False_ind; /2/; nqed.
+
+(*
+theorem lt_pred: \forall n,m.
+ O < n \to n < m \to pred n < pred m.
+apply nat_elim2
+ [intros.apply False_ind.apply (not_le_Sn_O ? H)
+ |intros.apply False_ind.apply (not_le_Sn_O ? H1)
+ |intros.simplify.unfold.apply le_S_S_to_le.assumption
+ ]
qed.
-alias num (instance 0) = "natural number".
-lemma times_SSO: \forall n.2*(S n) = S(S(2*n)).
-intro.
-simplify.rewrite < plus_n_Sm.reflexivity.
+theorem S_pred: \forall n:nat.lt O n \to eq nat n (S (pred n)).
+intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
+apply eq_f.apply pred_Sn.
qed.
-theorem or_eq_eq_S: \forall n.\exists m.
-n = (S(S O))*m \lor n = S ((S(S O))*m).
-intro.elim n
- [apply (ex_intro ? ? O).
- left.reflexivity
- |elim H. elim H1
- [apply (ex_intro ? ? a).
- right.apply eq_f.assumption
- |apply (ex_intro ? ? (S a)).
- left. rewrite > H2.
- apply sym_eq.
- apply times_SSO
- ]
+theorem le_pred_to_le:
+ ∀n,m. O < m → pred n ≤ pred m → n ≤ m.
+intros 2;
+elim n;
+[ apply le_O_n
+| simplify in H2;
+ rewrite > (S_pred m);
+ [ apply le_S_S;
+ assumption
+ | assumption
]
+].
+qed.
+
+*)
+
+(* le to lt or eq *)
+ntheorem le_to_or_lt_eq: ∀n,m:nat. n ≤ m → n < m ∨ n = m.
+#n; #m; #lenm; nelim lenm; /3/; nqed.
+
+(* not eq *)
+ntheorem lt_to_not_eq : ∀n,m:nat. n < m → n ≠ m.
+/2/; nqed.
+
+(*not lt
+ntheorem eq_to_not_lt: ∀a,b:nat. a = b → a ≮ b.
+intros.
+unfold Not.
+intros.
+rewrite > H in H1.
+apply (lt_to_not_eq b b)
+[ assumption
+| reflexivity
+]
+qed.
+
+theorem lt_n_m_to_not_lt_m_Sn: ∀n,m. n < m → m ≮ S n.
+intros;
+unfold Not;
+intro;
+unfold lt in H;
+unfold lt in H1;
+generalize in match (le_S_S ? ? H);
+intro;
+generalize in match (transitive_le ? ? ? H2 H1);
+intro;
+apply (not_le_Sn_n ? H3).
+qed. *)
+
+ntheorem not_eq_to_le_to_lt: ∀n,m. n≠m → n≤m → n<m.
+#n; #m; #Hneq; #Hle; ncases (le_to_or_lt_eq ?? Hle); //;
+#Heq; nelim (Hneq Heq); nqed.
+
+(* le elimination *)
+ntheorem le_n_O_to_eq : ∀n:nat. n ≤ O → O=n.
+#n; ncases n; //; #a ; #abs; nelim (not_le_Sn_O ? abs); nqed.
+
+ntheorem le_n_O_elim: ∀n:nat. n ≤ O → ∀P: nat →Prop. P O → P n.
+#n; ncases n; //; #a; #abs; nelim (not_le_Sn_O ? abs); nqed.
+
+ntheorem le_n_Sm_elim : ∀n,m:nat.n ≤ S m →
+∀P:Prop. (S n ≤ S m → P) → (n=S m → P) → P.
+#n; #m; #Hle; #P; nelim Hle; /3/; nqed.
+
+(* le and eq *)
+
+ntheorem le_to_le_to_eq: ∀n,m. n ≤ m → m ≤ n → n = m.
+napply nat_elim2; /4/; nqed.
+
+ntheorem lt_O_S : ∀n:nat. O < S n.
+/2/; nqed.
+
+(*
+(* other abstract properties *)
+theorem antisymmetric_le : antisymmetric nat le.
+unfold antisymmetric.intros 2.
+apply (nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m))).
+intros.apply le_n_O_to_eq.assumption.
+intros.apply False_ind.apply (not_le_Sn_O ? H).
+intros.apply eq_f.apply H.
+apply le_S_S_to_le.assumption.
+apply le_S_S_to_le.assumption.
qed.
-theorem symmetric_times : symmetric nat times.
-unfold symmetric.
-intros.elim x
- [simplify.apply times_n_O.
- | simplify.rewrite > H.apply times_n_Sm.]
+theorem antisym_le: \forall n,m:nat. n \leq m \to m \leq n \to n=m
+\def antisymmetric_le.
+
+theorem le_n_m_to_lt_m_Sn_to_eq_n_m: ∀n,m. n ≤ m → m < S n → n=m.
+intros;
+unfold lt in H1;
+generalize in match (le_S_S_to_le ? ? H1);
+intro;
+apply antisym_le;
+assumption.
qed.
+*)
-variant sym_times : \forall n,m:nat. n*m = m*n \def
-symmetric_times.
+(* well founded induction principles *)
-theorem distributive_times_plus : distributive nat times plus.
-unfold distributive.
-intros.elim x.
-simplify.reflexivity.
-simplify.
-demodulate all.
+ntheorem nat_elim1 : ∀n:nat.∀P:nat → Prop.
+(∀m.(∀p. p < m → P p) → P m) → P n.
+#n; #P; #H;
+ncut (∀q:nat. q ≤ n → P q);/2/;
+nelim n;
+ ##[#q; #HleO; (* applica male *)
+ napply (le_n_O_elim ? HleO);
+ napply H; #p; #ltpO;
+ napply False_ind; /2/;
+ ##|#p; #Hind; #q; #HleS;
+ napply H; #a; #lta; napply Hind;
+ napply le_S_S_to_le;/2/;
+ ##]
+nqed.
+
+(* some properties of functions *)
(*
-rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
-apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
-rewrite > assoc_plus.reflexivity. *)
+definition increasing \def \lambda f:nat \to nat.
+\forall n:nat. f n < f (S n).
+
+theorem increasing_to_monotonic: \forall f:nat \to nat.
+increasing f \to monotonic nat lt f.
+unfold monotonic.unfold lt.unfold increasing.unfold lt.intros.elim H1.apply H.
+apply (trans_le ? (f n1)).
+assumption.apply (trans_le ? (S (f n1))).
+apply le_n_Sn.
+apply H.
+qed.
+
+theorem le_n_fn: \forall f:nat \to nat. (increasing f)
+\to \forall n:nat. n \le (f n).
+intros.elim n.
+apply le_O_n.
+apply (trans_le ? (S (f n1))).
+apply le_S_S.apply H1.
+simplify in H. unfold increasing in H.unfold lt in H.apply H.
+qed.
+
+theorem increasing_to_le: \forall f:nat \to nat. (increasing f)
+\to \forall m:nat. \exists i. m \le (f i).
+intros.elim m.
+apply (ex_intro ? ? O).apply le_O_n.
+elim H1.
+apply (ex_intro ? ? (S a)).
+apply (trans_le ? (S (f a))).
+apply le_S_S.assumption.
+simplify in H.unfold increasing in H.unfold lt in H.
+apply H.
+qed.
+
+theorem increasing_to_le2: \forall f:nat \to nat. (increasing f)
+\to \forall m:nat. (f O) \le m \to
+\exists i. (f i) \le m \land m <(f (S i)).
+intros.elim H1.
+apply (ex_intro ? ? O).
+split.apply le_n.apply H.
+elim H3.elim H4.
+cut ((S n1) < (f (S a)) \lor (S n1) = (f (S a))).
+elim Hcut.
+apply (ex_intro ? ? a).
+split.apply le_S. assumption.assumption.
+apply (ex_intro ? ? (S a)).
+split.rewrite < H7.apply le_n.
+rewrite > H7.
+apply H.
+apply le_to_or_lt_eq.apply H6.
qed.
+*)
+
+(*********************** monotonicity ***************************)
+ntheorem monotonic_le_plus_r:
+∀n:nat.monotonic nat le (λm.n + m).
+#n; #a; #b; nelim n; nnormalize; //;
+#m; #H; #leab;napply le_S_S; /2/; nqed.
+
+(*
+ntheorem le_plus_r: ∀p,n,m:nat. n ≤ m → p + n ≤ p + m
+≝ monotonic_le_plus_r. *)
+
+ntheorem monotonic_le_plus_l:
+∀m:nat.monotonic nat le (λn.n + m).
+/2/; nqed.
+
+(*
+ntheorem le_plus_l: \forall p,n,m:nat. n \le m \to n + p \le m + p
+\def monotonic_le_plus_l. *)
+
+ntheorem le_plus: ∀n1,n2,m1,m2:nat. n1 ≤ n2 \to m1 ≤ m2
+→ n1 + m1 ≤ n2 + m2.
+#n1; #n2; #m1; #m2; #len; #lem; napply transitive_le;
+/2/; nqed.
+
+ntheorem le_plus_n :∀n,m:nat. m ≤ n + m.
+/2/; nqed.
+
+ntheorem le_plus_n_r :∀n,m:nat. m ≤ m + n.
+/2/; nqed.
+
+ntheorem eq_plus_to_le: ∀n,m,p:nat.n=m+p → m ≤ n.
+//; nqed.
+
+ntheorem le_plus_to_le: ∀a,n,m. a + n ≤ a + m → n ≤ m.
+#a; nelim a; /3/; nqed.
+
+ntheorem le_plus_to_le_r: ∀a,n,m. n + a ≤ m +a → n ≤ m.
+/2/; nqed.
-variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
-\def distributive_times_plus.
+(* plus & lt *)
+ntheorem monotonic_lt_plus_r:
+∀n:nat.monotonic nat lt (λm.n+m).
+/2/; nqed.
+
+(*
+variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
+monotonic_lt_plus_r. *)
-theorem associative_times: associative nat times.
-unfold associative.
+ntheorem monotonic_lt_plus_l:
+∀n:nat.monotonic nat lt (λm.m+n).
+/2/;nqed.
+
+(*
+variant lt_plus_l: \forall n,p,q:nat. p < q \to p + n < q + n \def
+monotonic_lt_plus_l. *)
+
+ntheorem lt_plus: ∀n,m,p,q:nat. n < m → p < q → n + p < m + q.
+#n; #m; #p; #q; #ltnm; #ltpq;
+napply (transitive_lt ? (n+q));/2/; nqed.
+
+ntheorem lt_plus_to_lt_l :∀n,p,q:nat. p+n < q+n → p<q.
+/2/; nqed.
+
+ntheorem lt_plus_to_lt_r :∀n,p,q:nat. n+p < n+q → p<q.
+/2/; nqed.
+
+ntheorem le_to_lt_to_plus_lt: ∀a,b,c,d:nat.
+a ≤ c → b < d → a + b < c+d.
+(* bello /2/ un po' lento *)
+#a; #b; #c; #d; #leac; #lebd;
+nnormalize; napplyS le_plus; //; nqed.
+
+(* times *)
+ntheorem monotonic_le_times_r:
+∀n:nat.monotonic nat le (λm. n * m).
+#n; #x; #y; #lexy; nelim n; nnormalize;//;(* lento /2/;*)
+#a; #lea; napply le_plus; //;
+nqed.
+
+(*
+ntheorem le_times_r: \forall p,n,m:nat. n \le m \to p*n \le p*m
+\def monotonic_le_times_r. *)
+
+ntheorem monotonic_le_times_l:
+∀m:nat.monotonic nat le (λn.n*m).
+/2/; nqed.
+
+(*
+theorem le_times_l: \forall p,n,m:nat. n \le m \to n*p \le m*p
+\def monotonic_le_times_l. *)
+
+ntheorem le_times: ∀n1,n2,m1,m2:nat.
+n1 ≤ n2 → m1 ≤ m2 → n1*m1 ≤ n2*m2.
+#n1; #n2; #m1; #m2; #len; #lem;
+napply transitive_le; (* /2/ slow *)
+ ##[ ##| napply monotonic_le_times_l;//;
+ ##| napply monotonic_le_times_r;//;
+ ##]
+nqed.
+
+ntheorem lt_times_n: ∀n,m:nat. O < n → m ≤ n*m.
+(* bello *)
+/2/; nqed.
+
+ntheorem le_times_to_le:
+∀a,n,m. O < a → a * n ≤ a * m → n ≤ m.
+#a; napply nat_elim2; nnormalize;
+ ##[//;
+ ##|#n; #H1; #H2; napply False_ind;
+ ngeneralize in match H2;
+ napply lt_to_not_le;
+ napply (transitive_le ? (S n));/2/;
+ ##|#n; #m; #H; #lta; #le;
+ napply le_S_S; napply H; /2/;
+ ##]
+nqed.
+
+ntheorem le_S_times_2: ∀n,m.O < m → n ≤ m → n < 2*m.
+#n; #m; #posm; #lenm; (* interessante *)
+nnormalize; napplyS (le_plus n); //; nqed.
+
+(* times & lt *)
+(*
+ntheorem lt_O_times_S_S: ∀n,m:nat.O < (S n)*(S m).
+intros.simplify.unfold lt.apply le_S_S.apply le_O_n.
+qed. *)
+
+(*
+ntheorem lt_times_eq_O: \forall a,b:nat.
+O < a → a * b = O → b = O.
+intros.
+apply (nat_case1 b)
+[ intros.
+ reflexivity
+| intros.
+ rewrite > H2 in H1.
+ rewrite > (S_pred a) in H1
+ [ apply False_ind.
+ apply (eq_to_not_lt O ((S (pred a))*(S m)))
+ [ apply sym_eq.
+ assumption
+ | apply lt_O_times_S_S
+ ]
+ | assumption
+ ]
+]
+qed.
+
+theorem O_lt_times_to_O_lt: \forall a,c:nat.
+O \lt (a * c) \to O \lt a.
+intros.
+apply (nat_case1 a)
+[ intros.
+ rewrite > H1 in H.
+ simplify in H.
+ assumption
+| intros.
+ apply lt_O_S
+]
+qed.
+
+lemma lt_times_to_lt_O: \forall i,n,m:nat. i < n*m \to O < m.
intros.
-elim x. simplify.apply refl_eq.
+elim (le_to_or_lt_eq O ? (le_O_n m))
+ [assumption
+ |apply False_ind.
+ rewrite < H1 in H.
+ rewrite < times_n_O in H.
+ apply (not_le_Sn_O ? H)
+ ]
+qed. *)
+
+(*
+ntheorem monotonic_lt_times_r:
+∀n:nat.monotonic nat lt (λm.(S n)*m).
+/2/;
simplify.
-demodulate all.
-(*
-rewrite < sym_times.
->>>>>>> .r9770
-rewrite > distr_times_plus.
-rewrite < sym_times.
-rewrite < (sym_times (times n y) z).
-rewrite < H.apply refl_eq.
-*)
+intros.elim n.
+simplify.rewrite < plus_n_O.rewrite < plus_n_O.assumption.
+apply lt_plus.assumption.assumption.
+qed. *)
+
+ntheorem monotonic_lt_times_l:
+ ∀c:nat. O < c → monotonic nat lt (λt.(t*c)).
+#c; #posc; #n; #m; #ltnm;
+nelim ltnm; nnormalize;
+ ##[napplyS monotonic_lt_plus_l;//;
+ ##|#a; #_; #lt1; napply (transitive_le ??? lt1);//;
+ ##]
+nqed.
+
+ntheorem monotonic_lt_times_r:
+ ∀c:nat. O < c → monotonic nat lt (λt.(c*t)).
+(* /2/ lentissimo *)
+#c; #posc; #n; #m; #ltnm;
+(* why?? napplyS (monotonic_lt_times_l c posc n m ltnm); *)
+nrewrite > (symmetric_times c n);
+nrewrite > (symmetric_times c m);
+napply monotonic_lt_times_l;//;
+nqed.
+
+ntheorem lt_to_le_to_lt_times:
+∀n,m,p,q:nat. n < m → p ≤ q → O < q → n*p < m*q.
+#n; #m; #p; #q; #ltnm; #lepq; #posq;
+napply (le_to_lt_to_lt ? (n*q));
+ ##[napply monotonic_le_times_r;//;
+ ##|napply monotonic_lt_times_l;//;
+ ##]
+nqed.
+
+ntheorem lt_times:∀n,m,p,q:nat. n<m → p<q → n*p < m*q.
+#n; #m; #p; #q; #ltnm; #ltpq;
+napply lt_to_le_to_lt_times;/2/;
+nqed.
+
+ntheorem lt_times_n_to_lt_l:
+∀n,p,q:nat. O < n → p*n < q*n → p < q.
+#n; #p; #q; #posn; #Hlt;
+nelim (decidable_lt p q);//;
+#nltpq;napply False_ind;
+napply (lt_to_not_le ? ? Hlt);
+napply monotonic_le_times_l.
+napply not_lt_to_le; //;
+nqed.
+
+ntheorem lt_times_n_to_lt_r:
+∀n,p,q:nat. O < n → n*p < n*q → p < q.
+#n; #p; #q; #posn; #Hlt;
+napply (lt_times_n_to_lt_l ??? posn);//;
+nqed.
+
+(*
+theorem nat_compare_times_l : \forall n,p,q:nat.
+nat_compare p q = nat_compare ((S n) * p) ((S n) * q).
+intros.apply nat_compare_elim.intro.
+apply nat_compare_elim.
+intro.reflexivity.
+intro.absurd (p=q).
+apply (inj_times_r n).assumption.
+apply lt_to_not_eq. assumption.
+intro.absurd (q<p).
+apply (lt_times_to_lt_r n).assumption.
+apply le_to_not_lt.apply lt_to_le.assumption.
+intro.rewrite < H.rewrite > nat_compare_n_n.reflexivity.
+intro.apply nat_compare_elim.intro.
+absurd (p<q).
+apply (lt_times_to_lt_r n).assumption.
+apply le_to_not_lt.apply lt_to_le.assumption.
+intro.absurd (q=p).
+symmetry.
+apply (inj_times_r n).assumption.
+apply lt_to_not_eq.assumption.
+intro.reflexivity.
+qed. *)
+
+(* times and plus
+theorem lt_times_plus_times: \forall a,b,n,m:nat.
+a < n \to b < m \to a*m + b < n*m.
+intros 3.
+apply (nat_case n)
+ [intros.apply False_ind.apply (not_le_Sn_O ? H)
+ |intros.simplify.
+ rewrite < sym_plus.
+ unfold.
+ change with (S b+a*m1 \leq m1+m*m1).
+ apply le_plus
+ [assumption
+ |apply le_times
+ [apply le_S_S_to_le.assumption
+ |apply le_n
+ ]
+ ]
+ ]
+qed. *)
+
+(************************** minus ******************************)
+
+nlet rec minus n m ≝
+ match n with
+ [ O ⇒ O
+ | S p ⇒
+ match m with
+ [ O ⇒ S p
+ | S q ⇒ minus p q ]].
+
+interpretation "natural minus" 'minus x y = (minus x y).
+
+ntheorem minus_S_S: ∀n,m:nat.S n - S m = n -m.
+//; nqed.
+
+ntheorem minus_O_n: ∀n:nat.O=O-n.
+#n; ncases n; //; nqed.
+
+ntheorem minus_n_O: ∀n:nat.n=n-O.
+#n; ncases n; //; nqed.
+
+ntheorem minus_n_n: ∀n:nat.O=n-n.
+#n; nelim n; //; nqed.
+
+ntheorem minus_Sn_n: ∀n:nat. S O = (S n)-n.
+#n; nelim n; //; nqed.
+
+ntheorem minus_Sn_m: ∀m,n:nat. m ≤ n → S n -m = S (n-m).
+(* qualcosa da capire qui
+#n; #m; #lenm; nelim lenm; napplyS refl_eq. *)
+napply nat_elim2;
+ ##[//
+ ##|#n; #abs; napply False_ind;/2/;
+ ##|#n; #m; #Hind; #c; napplyS Hind; /2/;
+ ##]
+nqed.
+
+ntheorem eq_minus_S_pred: ∀n,m. n - (S m) = pred(n -m).
+napply nat_elim2; //; nqed.
+
+ntheorem plus_minus:
+∀m,n,p:nat. m ≤ n → (n-m)+p = (n+p)-m.
+napply nat_elim2;
+ ##[//
+ ##|#n; #p; #abs; napply False_ind;/2/;
+ ##|nnormalize;/3/;
+ ##]
+nqed.
+
+ntheorem minus_plus_m_m: ∀n,m:nat.n = (n+m)-m.
+#n; #m; napplyS (plus_minus m m n); //; nqed.
+
+ntheorem plus_minus_m_m: ∀n,m:nat.
+m \leq n \to n = (n-m)+m.
+#n; #m; #lemn; napplyS symmetric_eq;
+napplyS (plus_minus m n m); //; nqed.
+
+ntheorem le_plus_minus_m_m: ∀n,m:nat. n ≤ (n-m)+m.
+#n; nelim n;
+ ##[//
+ ##|#a; #Hind; #m; ncases m;//;
+ nnormalize; #n;napplyS le_S_S;//
+ ##]
+nqed.
+
+ntheorem minus_to_plus :∀n,m,p:nat.
+ m ≤ n → n-m = p → n = m+p.
+#n; #m; #p; #lemn; #eqp; napplyS plus_minus_m_m; //;
+nqed.
+
+ntheorem plus_to_minus :∀n,m,p:nat.n = m+p → n-m = p.
+(* /4/ done in 43.5 *)
+#n; #m; #p; #eqp;
+napply symmetric_eq;
+napplyS (minus_plus_m_m p m);
+nqed.
+
+ntheorem minus_pred_pred : ∀n,m:nat. O < n → O < m →
+pred n - pred m = n - m.
+#n; #m; #posn; #posm;
+napply (lt_O_n_elim n posn);
+napply (lt_O_n_elim m posm);//.
+nqed.
+
+(*
+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;
+[2: apply H | skip].
+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.intro.
+(* autobatch *)
+(* end auto($Revision: 9739 $) proof: TIME=1.33 SIZE=100 DEPTH=100 *)
+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.
-variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
-associative_times.
+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.
-lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
-intros.
-demodulate. reflexivity.
-(* autobatch paramodulation. *)
+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.
+*)
+
+(* monotonicity and galois *)
+
+ntheorem monotonic_le_minus_l:
+∀p,q,n:nat. q ≤ p → q-n ≤ p-n.
+napply nat_elim2; #p; #q;
+ ##[#lePO; napply (le_n_O_elim ? lePO);//;
+ ##|//;
+ ##|#Hind; #n; ncases n;
+ ##[//;
+ ##|#a; #leSS; napply Hind; /2/;
+ ##]
+ ##]
+nqed.
+
+ntheorem le_minus_to_plus: ∀n,m,p. n-m ≤ p → n≤ p+m.
+#n; #m; #p; #lep;
+napply transitive_le;
+ ##[##|napply le_plus_minus_m_m
+ ##|napply monotonic_le_plus_l;//;
+ ##]
+nqed.
+
+ntheorem le_plus_to_minus: ∀n,m,p. n ≤ p+m → n-m ≤ p.
+#n; #m; #p; #lep;
+(* bello *)
+napplyS monotonic_le_minus_l;//;
+nqed.
+
+ntheorem monotonic_le_minus_r:
+∀p,q,n:nat. q ≤ p → n-p ≤ n-q.
+#p; #q; #n; #lepq;
+napply le_plus_to_minus;
+napply (transitive_le ??? (le_plus_minus_m_m ? q));/2/;
+nqed.
+
+(*********************** boolean arithmetics ********************)
+include "basics/bool.ma".
+
+nlet rec eqb n m ≝
+match n with
+ [ O ⇒ match m with [ O ⇒ true | S q ⇒ false]
+ | S p ⇒ match m with [ O ⇒ false | S q ⇒ eqb p q]
+ ].
+
+(*
+ntheorem eqb_to_Prop: ∀n,m:nat.
+match (eqb n m) with
+[ true \Rightarrow n = m
+| false \Rightarrow n \neq m].
+intros.
+apply (nat_elim2
+(\lambda n,m:nat.match (eqb n m) with
+[ true \Rightarrow n = m
+| false \Rightarrow n \neq m])).
+intro.elim n1.
+simplify.reflexivity.
+simplify.apply not_eq_O_S.
+intro.
+simplify.unfold Not.
+intro. apply (not_eq_O_S n1).apply sym_eq.assumption.
+intros.simplify.
+generalize in match H.
+elim ((eqb n1 m1)).
+simplify.apply eq_f.apply H1.
+simplify.unfold Not.intro.apply H1.apply inj_S.assumption.
+qed.
+
+theorem eqb_elim : \forall n,m:nat.\forall P:bool \to Prop.
+(n=m \to (P true)) \to (n \neq m \to (P false)) \to (P (eqb n m)).
+intros.
+cut
+(match (eqb n m) with
+[ true \Rightarrow n = m
+| false \Rightarrow n \neq m] \to (P (eqb n m))).
+apply Hcut.apply eqb_to_Prop.
+elim (eqb n m).
+apply ((H H2)).
+apply ((H1 H2)).
+qed.
+
+*)
+
+ntheorem eqb_n_n: ∀n. eqb n n = true.
+#n; nelim n; nnormalize; //.
+nqed.
+
+ntheorem eqb_true_to_eq: ∀n,m:nat. eqb n m = true → n = m.
+napply nat_elim2;
+ ##[#n; ncases n; nnormalize; //;
+ #m; #abs; napply False_ind;/2/;
+ ##|nnormalize; #m; #abs; napply False_ind;/2/;
+ ##|nnormalize;
+ #n; #m; #Hind; #eqnm; napply eq_f; napply Hind; //;
+ ##]
+nqed.
+
+ntheorem eqb_false_to_not_eq: ∀n,m:nat.
+ eqb n m = false → n ≠ m.
+napply nat_elim2;
+ ##[#n; ncases n; nnormalize; /2/;
+ ##|/2/;
+ ##|nnormalize;/2/;
+ ##]
+nqed.
+
+ntheorem eq_to_eqb_true: ∀n,m:nat.
+ n = m → eqb n m = true.
+//; nqed.
+
+ntheorem not_eq_to_eqb_false: ∀n,m:nat.
+ n ≠ m → eqb n m = false.
+#n; #m; #noteq;
+nelim (true_or_false (eqb n m)); //;
+#Heq; napply False_ind; napply noteq;/2/;
+nqed.
+
+nlet rec leb n m ≝
+match n with
+ [ O ⇒ true
+ | (S p) ⇒
+ match m with
+ [ O ⇒ false
+ | (S q) ⇒ leb p q]].
+
+ntheorem leb_elim: ∀n,m:nat. ∀P:bool → Prop.
+(n ≤ m → P true) → (n ≰ m → P false) → P (leb n m).
+napply nat_elim2; nnormalize;
+ ##[/2/
+ ##|/3/;
+ ##|#n; #m; #Hind; #P; #Pt; #Pf; napply Hind;
+ ##[#lenm; napply Pt; napply le_S_S;//;
+ ##|#nlenm; napply Pf; #leSS; /3/;
+ ##]
+ ##]
+nqed.
+
+ntheorem leb_true_to_le:∀n,m.leb n m = true → n ≤ m.
+#n; #m; napply leb_elim;
+ ##[//;
+ ##|#_; #abs; napply False_ind; /2/;
+ ##]
+nqed.
+
+ntheorem leb_false_to_not_le:∀n,m.
+ leb n m = false → n ≰ m.
+#n; #m; napply leb_elim;
+ ##[#_; #abs; napply False_ind; /2/;
+ ##|//;
+ ##]
+nqed.
+
+ntheorem le_to_leb_true: ∀n,m. n ≤ m → leb n m = true.
+#n; #m; napply leb_elim; //;
+#H; #H1; napply False_ind; /2/;
+nqed.
+
+ntheorem lt_to_leb_false: ∀n,m. m < n → leb n m = false.
+#n; #m; napply leb_elim; //;
+#H; #H1; napply False_ind; /2/;
+nqed.
+
+(* serve anche ltb?
+ndefinition ltb ≝λn,m. leb (S n) m.
+
+ntheorem ltb_elim: ∀n,m:nat. ∀P:bool → Prop.
+(n < m → P true) → (n ≮ m → P false) → P (ltb n m).
+#n; #m; #P; #Hlt; #Hnlt;
+napply leb_elim; /3/; nqed.
+
+ntheorem ltb_true_to_lt:∀n,m.ltb n m = true → n < m.
+#n; #m; #Hltb; napply leb_true_to_le; nassumption;
+nqed.
+
+ntheorem ltb_false_to_not_lt:∀n,m.
+ ltb n m = false → n ≮ m.
+#n; #m; #Hltb; napply leb_false_to_not_le; nassumption;
+nqed.
+
+ntheorem lt_to_ltb_true: ∀n,m. n < m → ltb n m = true.
+#n; #m; #Hltb; napply le_to_leb_true; nassumption;
+nqed.
-*)
\ No newline at end of file
+ntheorem le_to_ltb_false: ∀n,m. m \le n → ltb n m = false.
+#n; #m; #Hltb; napply lt_to_leb_false; /2/;
+nqed. *)
projects / helm.git / blobdiff