include "nat/gcd.ma".
include "nat/nth_prime.ma".
+
+theorem prova :
+ \forall n,m:nat.
+ \forall P:nat \to Prop.
+ \forall H:P (S (S O)).
+ \forall H:P (S (S (S O))).
+ \forall H1: \forall x.P x \to O = x.
+ O = S (S (S (S (S O)))).
+ intros.
+ auto paramodulation.
+ qed.
+
+theorem example2:
+\forall x: nat. (x+S O)*(x-S O) = x*x - S O.
+intro;
+apply (nat_case x);
+ [ auto paramodulation.|intro.auto paramodulation.]
+qed.
+
+theorem prova3:
+ \forall A:Set.
+ \forall m:A \to A \to A.
+ \forall divides: A \to A \to Prop.
+ \forall o,a,b,two:A.
+ \forall H1:\forall x.m o x = x.
+ \forall H1:\forall x.m x o = x.
+ \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
+ \forall H1:\forall x.m x o = x.
+ \forall H2:\forall x,y.m x y = m y x.
+ \forall H3:\forall x,y,z. m x y = m x z \to y = z.
+ (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
+ \forall H4:\forall x,y.(divides x y \to (\exists z.m x z = y)).
+ \forall H4:\forall x,y,z.m x z = y \to divides x y.
+ \forall H4:\forall x,y.divides two (m x y) \to divides two x ∨ divides two y.
+ \forall H5:m a a = m two (m b b).
+ \forall H6:\forall x.divides x a \to divides x b \to x = o.
+ two = o.
+ intros.
+ cut (divides two a);
+ [2:elim (H8 a a);[assumption.|assumption|rewrite > H9.auto.]
+ |elim (H6 ? ? Hcut).
+ cut (divides two b);
+ [ apply (H10 ? Hcut Hcut1).
+ | elim (H8 b b);[assumption.|assumption|
+ apply (H7 ? ? (m a1 a1));
+ apply (H5 two ? ?);rewrite < H9.
+ rewrite < H11.rewrite < H2.
+ apply eq_f.rewrite > H2.rewrite > H4.reflexivity.
+ ]
+ ]
+ ]
+ qed.
+
+theorem prova31:
+ \forall A:Set.
+ \forall m,f:A \to A \to A.
+ \forall divides: A \to A \to Prop.
+ \forall o,a,b,two:A.
+ \forall H1:\forall x.m o x = x.
+ \forall H1:\forall x.m x o = x.
+ \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
+ \forall H1:\forall x.m x o = x.
+ \forall H2:\forall x,y.m x y = m y x.
+ \forall H3:\forall x,y,z. m x y = m x z \to y = z.
+ (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
+ \forall H4:\forall x,y.(divides x y \to m x (f x y) = y).
+ \forall H4:\forall x,y,z.m x z = y \to divides x y.
+ \forall H4:\forall x,y.divides two (m x y) \to divides two x ∨ divides two y.
+ \forall H5:m a a = m two (m b b).
+ \forall H6:\forall x.divides x a \to divides x b \to x = o.
+ two = o.
+ intros.
+ cut (divides two a);
+ [2:elim (H8 a a);[assumption.|assumption|rewrite > H9.auto.]
+ |(*elim (H6 ? ? Hcut). *)
+ cut (divides two b);
+ [ apply (H10 ? Hcut Hcut1).
+ | elim (H8 b b);[assumption.|assumption|
+
+ apply (H7 ? ? (m (f two a) (f two a)));
+ apply (H5 two ? ?);
+ rewrite < H9.
+ rewrite < (H6 two a Hcut) in \vdash (? ? ? %).
+ rewrite < H2.apply eq_f.
+ rewrite < H4 in \vdash (? ? ? %).
+ rewrite > H2.reflexivity.
+ ]
+ ]
+ ]
+ qed.
+
+theorem prova32:
+ \forall A:Set.
+ \forall m,f:A \to A \to A.
+ \forall divides: A \to A \to Prop.
+ \forall o,a,b,two:A.
+ \forall H1:\forall x.m o x = x.
+ \forall H1:\forall x.m x o = x.
+ \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
+ \forall H1:\forall x.m x o = x.
+ \forall H2:\forall x,y.m x y = m y x.
+ \forall H3:\forall x,y,z. m x y = m x z \to y = z.
+ (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
+ \forall H4:\forall x,y.(divides x y \to m x (f x y) = y).
+ \forall H4:\forall x,y,z.m x z = y \to divides x y.
+ \forall H4:\forall x.divides two (m x x) \to divides two x.
+ \forall H5:m a a = m two (m b b).
+ \forall H6:\forall x.divides x a \to divides x b \to x = o.
+ two = o.
+ intros.
+ cut (divides two a);[|apply H8;rewrite > H9.auto].
+ apply H10;
+ [ assumption.
+ | apply (H8 b);
+ apply (H7 ? ? (m (f two a) (f two a)));
+ apply (H5 two ? ?);
+ auto paramodulation.
+ (*
+ rewrite < H9.
+ rewrite < (H6 two a Hcut) in \vdash (? ? ? %).
+ rewrite < H2.apply eq_f.
+ rewrite < H4 in \vdash (? ? ? %).
+ rewrite > H2.reflexivity.
+ *)
+ ]
+qed.
+
(* the following factorization algorithm looks for the largest prime
factor. *)
definition max_prime_factor \def \lambda n:nat.
[ apply (trans_lt ? (S O));
[ unfold lt; apply le_n;
| apply lt_SO_smallest_factor; assumption; ]
- | apply divides_smallest_factor_n;
+ | letin x \def le.auto.
+ (*
+ apply divides_smallest_factor_n;
apply (trans_lt ? (S O));
[ unfold lt; apply le_n;
- | assumption; ] ] ]
- | apply prime_to_nth_prime;
+ | assumption; ] *) ] ]
+ | letin x \def prime. auto.
+ (*
+ apply prime_to_nth_prime;
apply prime_smallest_factor_n;
- assumption; ] ]
+ assumption; *) ] ]
qed.
theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to
cut (prime (nth_prime (max_prime_factor n))).
apply lt_O_nth_prime_n.apply prime_nth_prime.
cut (nth_prime (max_prime_factor n) \divides n).
-apply (transitive_divides ? n).
-apply divides_max_prime_factor_n.
-assumption.assumption.
-apply divides_b_true_to_divides.
-apply lt_O_nth_prime_n.
-apply divides_to_divides_b_true.
-apply lt_O_nth_prime_n.
-apply divides_max_prime_factor_n.
-assumption.
+auto.
+auto.
+(*
+ [ apply (transitive_divides ? n);
+ [ apply divides_max_prime_factor_n.
+ assumption.
+ | assumption.
+ ]
+ | apply divides_b_true_to_divides;
+ [ apply lt_O_nth_prime_n.
+ | apply divides_to_divides_b_true;
+ [ apply lt_O_nth_prime_n.
+ | apply divides_max_prime_factor_n.
+ assumption.
+ ]
+ ]
+ ]
+*)
qed.
theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
apply divides_max_prime_factor_n.
assumption.unfold Not.
intro.
-cut (r \mod (nth_prime (max_prime_factor n)) \neq O).
-apply Hcut1.apply divides_to_mod_O.
-apply lt_O_nth_prime_n.assumption.
-apply (p_ord_aux_to_not_mod_O n n ? q r).
-apply lt_SO_nth_prime_n.assumption.
-apply le_n.
-rewrite < H1.assumption.
+cut (r \mod (nth_prime (max_prime_factor n)) \neq O);
+ [unfold Not in Hcut1.auto.
+ (*
+ apply Hcut1.apply divides_to_mod_O;
+ [ apply lt_O_nth_prime_n.
+ | assumption.
+ ]
+ *)
+ |letin z \def le.
+ cut(pair nat nat q r=p_ord_aux n n (nth_prime (max_prime_factor n)));
+ [2: rewrite < H1.assumption.|letin x \def le.auto width = 4]
+ (* CERCA COME MAI le_n non lo applica se lo trova come Const e non Rel *)
+ ].
+(*
+ apply (p_ord_aux_to_not_mod_O n n ? q r);
+ [ apply lt_SO_nth_prime_n.
+ | assumption.
+ | apply le_n.
+ | rewrite < H1.assumption.
+ ]
+ ].
+*)
+cut (n=r*(nth_prime p)\sup(q));
+ [letin www \def le.letin www1 \def divides.
+ auto.
+(*
apply (le_to_or_lt_eq (max_prime_factor r) (max_prime_factor n)).
apply divides_to_max_prime_factor.
assumption.assumption.
apply (witness r n ((nth_prime p) \sup q)).
+*)
+ |
rewrite < sym_times.
apply (p_ord_aux_to_exp n n ? q r).
apply lt_O_nth_prime_n.assumption.
+]
qed.
theorem p_ord_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
left.split.assumption.reflexivity.
intro.right.rewrite > Hcut2.
simplify.unfold lt.apply le_S_S.apply le_O_n.
-cut (r \lt (S O) \or r=(S O)).
+cut (r < (S O) ∨ r=(S O)).
elim Hcut2.absurd (O=r).
apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
unfold Not.intro.
cut (O=n1).
apply (not_le_Sn_O O).
-rewrite > Hcut3 in \vdash (? ? %).
+rewrite > Hcut3 in ⊢ (? ? %).
assumption.rewrite > Hcut.
rewrite < H6.reflexivity.
assumption.
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