#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).
+ntheorem decidable_eq_nat : ∀n,m:nat.decidable (n=m).
napply nat_elim2; #n;
- ##[ ncases n; /2/;
+ ##[ ncases n; /3/;
##| /3/;
##| #m; #Hind; ncases Hind; /3/;
##]
#n; nelim n; nnormalize; //; nqed.
*)
-(*
+(* deleterio
ntheorem plus_n_SO : ∀n:nat. S n = n+S O.
//; nqed. *)
interpretation "natural 'not less than'" 'nless x y = (Not (lt x y)).
+(* nlemma eq_lt: ∀n,m. (n < m) = (S n ≤ m).
+//; nqed. *)
+
ndefinition ge: nat \to nat \to Prop \def
\lambda n,m:nat.m \leq n.
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.
+
+naxiom 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
ntheorem le_pred_n : ∀n:nat. pred n ≤ n.
#n; nelim n; //; nqed.
+(* XXX global problem *)
+nlemma my_trans_le : ∀x,y,z:nat.x ≤ y → y ≤ z → x ≤ z.
+napply transitive_le.
+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.
+(* XXX *) nletin hint ≝ monotonic. /2/; nqed.
-ntheorem lt_S_S_to_lt: ∀n,m. S n < S m \to n < m.
+ntheorem lt_S_S_to_lt: ∀n,m. S n < S m → n < m.
/2/; nqed.
ntheorem lt_to_lt_S_S: ∀n,m. n < m → S n < S m.
ntheorem not_le_S_S_to_not_le: ∀ n,m:nat. S n ≰ S m → n ≰ m.
/3/; nqed.
+naxiom decidable_le: ∀n,m. decidable (n≤m).
+(*
ntheorem decidable_le: ∀n,m. decidable (n≤m).
-napply nat_elim2; #n; /2/;
-#m; #dec; ncases dec;/3/; nqed.
+napply nat_elim2; #n; /3/;
+#m; #dec; ncases dec;/4/; nqed. *)
ntheorem decidable_lt: ∀n,m. decidable (n < m).
#n; #m; napply decidable_le ; nqed.
ntheorem not_le_Sn_n: ∀n:nat. S n ≰ n.
-#n; nelim n; /2/; nqed.
+#n; nelim n; /3/; nqed.
ntheorem lt_S_to_le: ∀n,m:nat. n < S m → n ≤ m.
/2/; nqed.
napply nat_elim2; #n;
##[#abs; napply False_ind;/2/;
##|/2/;
- ##|#m;#Hind;#HnotleSS; napply lt_to_lt_S_S;/3/;
+ ##|#m;#Hind;#HnotleSS; napply lt_to_lt_S_S;/4/;
##]
nqed.
(* le and eq *)
ntheorem le_to_le_to_eq: ∀n,m. n ≤ m → m ≤ n → n = m.
-napply nat_elim2; /3/; nqed.
+napply nat_elim2; /4/; nqed.
-ntheorem lt_O_S : \forall n:nat. O < S n.
+ntheorem lt_O_S : ∀n:nat. O < S n.
/2/; nqed.
(*
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;
+#n1; #n2; #m1; #m2; #len; #lem; napply (transitive_le ? (n1+m2));
/2/; nqed.
ntheorem le_plus_n :∀n,m:nat. m ≤ n + m.
//; nqed.
ntheorem le_plus_to_le: ∀a,n,m. a + n ≤ a + m → n ≤ m.
-#a; nelim a; /3/; nqed.
+#a; nelim a; nnormalize; /3/; nqed.
ntheorem le_plus_to_le_r: ∀a,n,m. n + a ≤ m +a → n ≤ m.
/2/; nqed.
+(* 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. *)
+
+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).
nqed.
ntheorem lt_times_n: ∀n,m:nat. O < n → m ≤ n*m.
-(* bello *)
+#n; #m; #H; napplyS monotonic_le_times_l;
/2/; nqed.
ntheorem le_times_to_le:
##]
nqed.
-ntheorem le_S_times_2: ∀n,m.O < m → n ≤ m → n < 2*m.
+ntheorem le_S_times_2: ∀n,m.O < m → n ≤ m → S n ≤ 2*m.
#n; #m; #posm; #lenm; (* interessante *)
-nnormalize; napplyS (le_plus n); //; nqed.
+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 (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.
+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;/3/;
+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 ******************************)
#n; #m; #lenm; nelim lenm; napplyS refl_eq. *)
napply nat_elim2;
##[//
- ##|#n; #abs; napply False_ind;/2/;
- ##|/3/;
+ ##|#n; #abs; napply False_ind; /2/.
+ ##|#n; #m; #Hind; #c; napplyS Hind; /2/;
##]
nqed.
+ntheorem not_eq_to_le_to_le_minus:
+ ∀n,m.n ≠ m → n ≤ m → n ≤ m - 1.
+#n; #m; ncases m;//; #m; nnormalize;
+#H; #H1; napply le_S_S_to_le;
+napplyS (not_eq_to_le_to_lt n (S m) H H1);
+nqed.
+
ntheorem eq_minus_S_pred: ∀n,m. n - (S m) = pred(n -m).
napply nat_elim2; //; nqed.
∀m,n,p:nat. m ≤ n → (n-m)+p = (n+p)-m.
napply nat_elim2;
##[//
- ##|#n; #p; #abs; napply False_ind;/2/;
+ ##|#n; #p; #abs; napply False_ind; /2/;
##|nnormalize;/3/;
##]
nqed.
ntheorem le_plus_minus_m_m: ∀n,m:nat. n ≤ (n-m)+m.
#n; nelim n;
##[//
- ##|#a; #Hind; #m; ncases m;/2/;
+ ##|#a; #Hind; #m; ncases m;//;
+ nnormalize; #n;napplyS le_S_S;//
##]
nqed.
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.
+*)
+
+ntheorem eqb_elim : ∀ n,m:nat.∀ P:bool → Prop.
+(n=m → (P true)) → (n ≠ m → (P false)) → (P (eqb n m)).
+napply nat_elim2;
+ ##[#n; ncases n; nnormalize; /3/;
+ ##|nnormalize; /3/;
+ ##|nnormalize; /4/;
+ ##]
+nqed.
+
+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.
+#n; #m; napply (eqb_elim n m);//;
+#_; #abs; napply False_ind; /2/;
+nqed.
+
+ntheorem eqb_false_to_not_eq: ∀n,m:nat. eqb n m = false → n ≠ m.
+#n; #m; napply (eqb_elim n m);/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/;
+ ##|/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.
+
+ntheorem le_to_ltb_false: ∀n,m. m \le n → ltb n m = false.
+#n; #m; #Hltb; napply lt_to_leb_false; /2/;
+nqed. *)
+
+ninductive compare : Type[0] ≝
+| LT : compare
+| EQ : compare
+| GT : compare.
+
+ndefinition compare_invert: compare → compare ≝
+ λc.match c with
+ [ LT ⇒ GT
+ | EQ ⇒ EQ
+ | GT ⇒ LT ].
+
+nlet rec nat_compare n m: compare ≝
+match n with
+[ O ⇒ match m with
+ [ O ⇒ EQ
+ | (S q) ⇒ LT ]
+| S p ⇒ match m with
+ [ O ⇒ GT
+ | S q ⇒ nat_compare p q]].
+
+ntheorem nat_compare_n_n: ∀n. nat_compare n n = EQ.
+#n;nelim n
+##[//
+##|#m;#IH;nnormalize;//]
+nqed.
+
+ntheorem nat_compare_S_S: ∀n,m:nat.nat_compare n m = nat_compare (S n) (S m).
+//;
+nqed.
+
+ntheorem nat_compare_pred_pred:
+ ∀n,m.O < n → O < m → nat_compare n m = nat_compare (pred n) (pred m).
+#n;#m;#Hn;#Hm;
+napply (lt_O_n_elim n Hn);
+napply (lt_O_n_elim m Hm);
+#p;#q;//;
+nqed.
+
+ntheorem nat_compare_to_Prop:
+ ∀n,m.match (nat_compare n m) with
+ [ LT ⇒ n < m
+ | EQ ⇒ n = m
+ | GT ⇒ m < n ].
+#n;#m;
+napply (nat_elim2 (λn,m.match (nat_compare n m) with
+ [ LT ⇒ n < m
+ | EQ ⇒ n = m
+ | GT ⇒ m < n ]) ?????) (* FIXME: don't want to put all these ?, especially when … does not work! *)
+##[##1,2:#n1;ncases n1;//;
+##|#n1;#m1;nnormalize;ncases (nat_compare n1 m1);
+ ##[##1,3:nnormalize;#IH;napply le_S_S;//;
+ ##|nnormalize;#IH;nrewrite > IH;//]
+nqed.
+
+ntheorem nat_compare_n_m_m_n:
+ ∀n,m:nat.nat_compare n m = compare_invert (nat_compare m n).
+#n;#m;
+napply (nat_elim2 (λn,m. nat_compare n m = compare_invert (nat_compare m n)))
+##[##1,2:#n1;ncases n1;//;
+##|#n1;#m1;#IH;nnormalize;napply IH]
+nqed.
+
+ntheorem nat_compare_elim :
+ ∀n,m. ∀P:compare → Prop.
+ (n < m → P LT) → (n=m → P EQ) → (m < n → P GT) → P (nat_compare n m).
+#n;#m;#P;#Hlt;#Heq;#Hgt;
+ncut (match (nat_compare n m) with
+ [ LT ⇒ n < m
+ | EQ ⇒ n=m
+ | GT ⇒ m < n] →
+ P (nat_compare n m))
+##[ncases (nat_compare n m);
+ ##[napply Hlt
+ ##|napply Heq
+ ##|napply Hgt]
+##|#Hcut;napply Hcut;//;
+nqed.
+
+ninductive cmp_cases (n,m:nat) : CProp[0] ≝
+ | cmp_le : n ≤ m → cmp_cases n m
+ | cmp_gt : m < n → cmp_cases n m.
+
+ntheorem lt_to_le : ∀n,m:nat. n < m → n ≤ m.
+#n;#m;#H;nelim H
+##[//
+##|/2/]
+nqed.
+
+nlemma cmp_nat: ∀n,m.cmp_cases n m.
+#n;#m; nlapply (nat_compare_to_Prop n m);
+ncases (nat_compare n m);#H
+##[@;napply lt_to_le;//
+##|@;//
+##|@2;//]
+nqed.
projects / helm.git / blobdiff