(* *)
(**************************************************************************)
-(* include "higher_order_defs/functions.ma". *)
include "hints_declaration.ma".
include "basics/functions.ma".
-include "basics/eq.ma".
+include "basics/eq.ma".
-ninductive nat : Type[0] ≝
+ninductive nat : Type ≝
| O : nat
| S : nat → nat.
*)
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 not_eq_S: ∀n,m:nat. n ≠ m → S n ≠ S m.
-/2/; nqed.
+/3/; nqed.
ndefinition not_zero: nat → Prop ≝
λn: nat. match n with
[ O ⇒ False | (S p) ⇒ True ].
ntheorem not_eq_O_S : ∀n:nat. O ≠ S n.
-#n; #eqOS; nchange with (not_zero O); nrewrite > eqOS; //.
+#n; napply nmk; #eqOS; nchange with (not_zero O); nrewrite > eqOS; //.
nqed.
-ntheorem not_eq_n_Sn : ∀n:nat. n ≠ S n.
-#n; nelim n; /2/; nqed.
+ntheorem not_eq_n_Sn: ∀n:nat. n ≠ S n.
+#n; nelim n;/2/; nqed.
ntheorem nat_case:
∀n:nat.∀P:nat → Prop.
napply nat_elim2; #n;
##[ ncases n; /2/;
##| /3/;
- ##| #m; #Hind; ncases Hind; /3/;
+ ##| #m; #Hind; ncases Hind;/3/;
+ (*
+ ##[/2/; ##|#H; nwhd;
+ napply or_introl;
+ napply eq_f2; /3/; *)
##]
nqed.
#n; nelim n; nnormalize; //; nqed.
*)
-(*
-ntheorem plus_n_SO : ∀n:nat. S n = n+S O.
-//; nqed. *)
+(* deleterio?
+ntheorem plus_n_1 : ∀n:nat. S n = n+1.
+//; nqed.
+*)
ntheorem symmetric_plus: symmetric ? plus.
#n; nelim n; nnormalize; //; nqed.
#n; nelim n; nnormalize; //; nqed.
ntheorem symmetric_times : symmetric nat times.
-#n; nelim n; nnormalize; //; nqed.
+#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 distributive_times_plus_r :
+ ∀a,b,c:nat. (b+c)*a = b*a + c*a.
+//; nqed.
ntheorem associative_times: associative nat times.
#n; nelim n; nnormalize; //; nqed.
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.
+(* nlemma eq_lt: ∀n,m. (n < m) = (S n ≤ m).
+//; nqed. *)
+
+ndefinition ge: nat → nat → Prop ≝
+λn,m:nat.m ≤ 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.
+ndefinition gt: nat → nat → Prop ≝
+λn,m:nat.m<n.
interpretation "natural 'greater than'" 'gt x y = (gt x y).
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.
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.
+#n; #m; #lenm; nelim lenm; /2/;nqed.
ntheorem le_S_S_to_le: ∀n,m:nat. S n ≤ S m → n ≤ m.
+(* XXX *) nletin hint ≝ monotonic.
+#a; #b; #H; napplyS monotonic_pred;
/2/; nqed.
-ntheorem lt_S_S_to_lt: ∀n,m. S n < S m \to n < m.
+(* this are instances of the le versions
+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.
-/2/; nqed.
+/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.
+#n; napply nmk; #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 decidable_le: ∀n,m. decidable (n≤m).
napply nat_elim2; #n; /2/;
-#m; #dec; ncases dec;/3/; nqed.
+#m; *; /3/; 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.
+(* this is le_S_S_to_le
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/;
+ ##|#m;#Hind;#HnotleSS; napply le_S_S;/3/;
##]
nqed.
#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.
+/4/; nqed.
+
+(*
+#n; #m; #Hnlt; napply le_S_S_to_le;/2/;
+(* something strange here: /2/ fails *)
+napply not_le_to_lt; napply Hnlt; nqed. *)
ntheorem le_to_not_lt: ∀n,m:nat. n ≤ m → m ≮ n.
-/2/; nqed.
+#n; #m; #H;napply lt_to_not_le; /2/; (* /3/ *) nqed.
(* lt and le trans *)
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.
+#n; nelim n; //; #abs; napply False_ind;/2/;
+nqed.
(*
theorem lt_pred: \forall n,m.
(* not eq *)
ntheorem lt_to_not_eq : ∀n,m:nat. n < m → n ≠ m.
-/2/; nqed.
+#n; #m; #H; napply not_to_not;/2/; nqed.
(*not lt
ntheorem eq_to_not_lt: ∀a,b:nat. a = b → a ≮ b.
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.
+#Heq; /3/; nqed.
+(*
+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.
+#n; ncases n; //; #a ; #abs;
+napply False_ind; /2/;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.
+#n; ncases n; //; #a; #abs;
+napply False_ind; /2/; nqed.
ntheorem le_n_Sm_elim : ∀n,m:nat.n ≤ S m →
∀P:Prop. (S n ≤ S m → P) → (n=S m → P) → P.
(* le and eq *)
ntheorem le_to_le_to_eq: ∀n,m. n ≤ m → m ≤ n → n = m.
-napply nat_elim2; /4/; nqed.
+napply nat_elim2;
+##[#n; #H1; #H2; /4/;
+ nqed.
ntheorem lt_O_S : ∀n:nat. O < S n.
/2/; nqed.
##[#q; #HleO; (* applica male *)
napply (le_n_O_elim ? HleO);
napply H; #p; #ltpO;
- napply False_ind; /2/;
+ napply False_ind; /2/; (* 3 *)
##|#p; #Hind; #q; #HleS;
napply H; #a; #lta; napply Hind;
napply le_S_S_to_le;/2/;
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
+ntheorem le_plus: ∀n1,n2,m1,m2:nat. n1 ≤ n2 → 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.
/2/; nqed.
+nlemma le_plus_a: ∀a,n,m. n ≤ m → n ≤ a + m.
+/2/; nqed.
+
+nlemma le_plus_b: ∀b,n,m. n + b ≤ m → n ≤ m.
+/2/; nqed.
+
ntheorem le_plus_n_r :∀n,m:nat. m ≤ m + n.
/2/; nqed.
//; 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.
+/2/; nqed.
(*
variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
ntheorem monotonic_lt_plus_l:
∀n:nat.monotonic nat lt (λm.m+n).
-/2/;nqed.
+/2/; nqed.
(*
variant lt_plus_l: \forall n,p,q:nat. p < q \to p + n < q + n \def
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.
+(*
+ntheorem le_to_lt_to_lt_plus: ∀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:
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
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;//;
+napply (transitive_le ? (n1*m2)); (* /2/ slow *)
+ ##[ napply monotonic_le_times_r;//;
+ ##| napplyS monotonic_le_times_r;//;
##]
nqed.
+(* interesssante *)
ntheorem lt_times_n: ∀n,m:nat. O < n → m ≤ n*m.
-(* bello *)
-/2/; nqed.
+#n; #m; #H; /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; #H1; #H2;
+ napply (transitive_le ? (a*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.
+ntheorem le_S_times_2: ∀n,m.O < m → n ≤ m → S n ≤ 2*m.
+#n; #m; #posm; #lenm; (* interessante *)
+napplyS (le_plus n m); //; nqed.
(* times & lt *)
(*
∀c:nat. O < c → monotonic nat lt (λt.(t*c)).
#c; #posc; #n; #m; #ltnm;
nelim ltnm; nnormalize;
- ##[napplyS monotonic_lt_plus_l;//;
+ ##[/2/;
##|#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.
+/2/; nqed.
ntheorem lt_to_le_to_lt_times:
∀n,m,p,q:nat. n < m → p ≤ q → O < q → n*p < m*q.
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;
+∀n,p,q:nat. p*n < q*n → p < q.
+#n; #p; #q; #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; //;
+#nltpq; napply False_ind;
+napply (absurd ? ? (lt_to_not_le ? ? Hlt));
+napplyS monotonic_le_times_r;/2/;
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.
+∀n,p,q:nat. n*p < n*q → p < q.
+/2/; nqed.
(*
theorem nat_compare_times_l : \forall n,p,q:nat.
#n; nelim n; //; nqed.
ntheorem minus_Sn_n: ∀n:nat. S O = (S n)-n.
-#n; nelim n; //; nqed.
+#n; nelim n; nnormalize; //; 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; #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.
+napply nat_elim2; nnormalize; //; 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/;
+ ##|#n; #p; #abs; napply False_ind; /2/;
##|nnormalize;/3/;
##]
nqed.
#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;
+ m ≤ n → n = (n-m)+m.
+#n; #m; #lemn; napplyS sym_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;//
+ nnormalize; #n;/2/;
##]
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;
+napply sym_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 n posn);
napply (lt_O_n_elim m posm);//.
nqed.
#n; #m; #p; #lep;
(* bello *)
napplyS monotonic_le_minus_l;//;
+(* /2/; *)
nqed.
ntheorem monotonic_le_minus_r:
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_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.
-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; //;
- ##]
+#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.
-napply nat_elim2;
- ##[#n; ncases n; nnormalize; /2/;
- ##|/2/;
- ##|nnormalize;/2/;
- ##]
+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.
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/;
+napply eqb_elim;//;
+#Heq; napply False_ind; /2/;
nqed.
nlet rec leb n m ≝
##|/3/;
##|#n; #m; #Hind; #P; #Pt; #Pf; napply Hind;
##[#lenm; napply Pt; napply le_S_S;//;
- ##|#nlenm; napply Pf; #leSS; /3/;
+ ##|#nlenm; napply Pf; /2/;
##]
##]
nqed.
#H; #H1; napply False_ind; /2/;
nqed.
-ntheorem lt_to_leb_false: ∀n,m. m < n → leb n m = false.
+ntheorem not_le_to_leb_false: ∀n,m. n ≰ m → leb n m = false.
#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.
+/3/; nqed.
+
(* serve anche ltb?
ndefinition ltb ≝λn,m. leb (S n) m.
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