--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+include "nat/primes.ma".
+(* include "nat/ord.ma". *)
+include "nat/generic_iter_p.ma".
+(* include "nat/count.ma". necessary just to use bool_to_nat and bool_to_nat_andb*)
+include "nat/iteration2.ma".
+
+(* pi_p on nautral numbers is a specialization of iter_p_gen *)
+definition pi_p: nat \to (nat \to bool) \to (nat \to nat) \to nat \def
+\lambda n, p, g. (iter_p_gen n p nat g (S O) times).
+
+theorem true_to_pi_p_Sn:
+\forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
+p n = true \to pi_p (S n) p g = (g n)*(pi_p n p g).
+intros.
+unfold pi_p.
+apply true_to_iter_p_gen_Sn.
+assumption.
+qed.
+
+theorem false_to_pi_p_Sn:
+\forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
+p n = false \to pi_p (S n) p g = pi_p n p g.
+intros.
+unfold pi_p.
+apply false_to_iter_p_gen_Sn.
+assumption.
+qed.
+
+theorem eq_pi_p: \forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to nat.\forall n.
+(\forall x. x < n \to p1 x = p2 x) \to
+(\forall x. x < n \to g1 x = g2 x) \to
+pi_p n p1 g1 = pi_p n p2 g2.
+intros.
+unfold pi_p.
+apply eq_iter_p_gen;
+assumption.
+qed.
+
+theorem eq_pi_p1: \forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to nat.\forall n.
+(\forall x. x < n \to p1 x = p2 x) \to
+(\forall x. x < n \to p1 x = true \to g1 x = g2 x) \to
+pi_p n p1 g1 = pi_p n p2 g2.
+intros.
+unfold pi_p.
+apply eq_iter_p_gen1;
+assumption.
+qed.
+
+theorem pi_p_false:
+\forall g: nat \to nat.\forall n.pi_p n (\lambda x.false) g = S O.
+intros.
+unfold pi_p.
+apply iter_p_gen_false.
+qed.
+
+theorem pi_p_times: \forall n,k:nat.\forall p:nat \to bool.
+\forall g: nat \to nat.
+pi_p (k+n) p g
+= pi_p k (\lambda x.p (x+n)) (\lambda x.g (x+n)) * pi_p n p g.
+intros.
+unfold pi_p.
+apply (iter_p_gen_plusA nat n k p g (S O) times)
+[ apply sym_times.
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+| apply associative_times
+]
+qed.
+
+theorem false_to_eq_pi_p: \forall n,m:nat.n \le m \to
+\forall p:nat \to bool.
+\forall g: nat \to nat. (\forall i:nat. n \le i \to i < m \to
+p i = false) \to pi_p m p g = pi_p n p g.
+intros.
+unfold pi_p.
+apply (false_to_eq_iter_p_gen);
+assumption.
+qed.
+
+theorem or_false_eq_SO_to_eq_pi_p:
+\forall n,m:nat.\forall p:nat \to bool.
+\forall g: nat \to nat.
+n \le m \to (\forall i:nat. n \le i \to i < m \to p i = false \lor g i = S O)
+\to pi_p m p g = pi_p n p g.
+intros.
+unfold pi_p.
+apply or_false_eq_baseA_to_eq_iter_p_gen
+ [intros.simplify.rewrite < plus_n_O.reflexivity
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem pi_p2 :
+\forall n,m:nat.
+\forall p1,p2:nat \to bool.
+\forall g: nat \to nat \to nat.
+pi_p (n*m)
+ (\lambda x.andb (p1 (div x m)) (p2 (mod x m)))
+ (\lambda x.g (div x m) (mod x m)) =
+pi_p n p1
+ (\lambda x.pi_p m p2 (g x)).
+intros.
+unfold pi_p.
+apply (iter_p_gen2 n m p1 p2 nat g (S O) times)
+[ apply sym_times
+| apply associative_times
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+]
+qed.
+
+theorem pi_p2' :
+\forall n,m:nat.
+\forall p1:nat \to bool.
+\forall p2:nat \to nat \to bool.
+\forall g: nat \to nat \to nat.
+pi_p (n*m)
+ (\lambda x.andb (p1 (div x m)) (p2 (div x m) (mod x m)))
+ (\lambda x.g (div x m) (mod x m)) =
+pi_p n p1
+ (\lambda x.pi_p m (p2 x) (g x)).
+intros.
+unfold pi_p.
+apply (iter_p_gen2' n m p1 p2 nat g (S O) times)
+[ apply sym_times
+| apply associative_times
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+]
+qed.
+
+lemma pi_p_gi: \forall g: nat \to nat.
+\forall n,i.\forall p:nat \to bool.i < n \to p i = true \to
+pi_p n p g = g i * pi_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
+intros.
+unfold pi_p.
+apply (iter_p_gen_gi)
+[ apply sym_times
+| apply associative_times
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+| assumption
+| assumption
+]
+qed.
+
+theorem eq_pi_p_gh:
+\forall g,h,h1: nat \to nat.\forall n,n1.
+\forall p1,p2:nat \to bool.
+(\forall i. i < n \to p1 i = true \to p2 (h i) = true) \to
+(\forall i. i < n \to p1 i = true \to h1 (h i) = i) \to
+(\forall i. i < n \to p1 i = true \to h i < n1) \to
+(\forall j. j < n1 \to p2 j = true \to p1 (h1 j) = true) \to
+(\forall j. j < n1 \to p2 j = true \to h (h1 j) = j) \to
+(\forall j. j < n1 \to p2 j = true \to h1 j < n) \to
+pi_p n p1 (\lambda x.g(h x)) = pi_p n1 p2 g.
+intros.
+unfold pi_p.
+apply (eq_iter_p_gen_gh nat (S O) times ? ? ? g h h1 n n1 p1 p2)
+[ apply sym_times
+| apply associative_times
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+]
+qed.
+
+(* monotonicity *)
+theorem le_pi_p:
+\forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to p i = true \to g1 i \le g2 i ) \to
+pi_p n p g1 \le pi_p n p g2.
+intros.
+generalize in match H.
+elim n
+ [apply le_n.
+ |apply (bool_elim ? (p n1));intros
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn in ⊢ (? ? %)
+ [apply le_times
+ [apply H2[apply le_n|assumption]
+ |apply H1.
+ intros.
+ apply H2[apply le_S.assumption|assumption]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn in ⊢ (? ? %)
+ [apply H1.
+ intros.
+ apply H2[apply le_S.assumption|assumption]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem exp_sigma_p: \forall n,a,p.
+pi_p n p (\lambda x.a) = (exp a (sigma_p n p (\lambda x.S O))).
+intros.
+elim n
+ [reflexivity
+ |apply (bool_elim ? (p n1))
+ [intro.
+ rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_sigma_p_Sn
+ [simplify.
+ rewrite > H.
+ reflexivity.
+ |assumption
+ ]
+ |assumption
+ ]
+ |intro.
+ rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_sigma_p_Sn
+ [simplify.assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem exp_sigma_p1: \forall n,a,p,f.
+pi_p n p (\lambda x.(exp a (f x))) = (exp a (sigma_p n p f)).
+intros.
+elim n
+ [reflexivity
+ |apply (bool_elim ? (p n1))
+ [intro.
+ rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_sigma_p_Sn
+ [simplify.
+ rewrite > H.
+ rewrite > exp_plus_times.
+ reflexivity.
+ |assumption
+ ]
+ |assumption
+ ]
+ |intro.
+ rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_sigma_p_Sn
+ [simplify.assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem times_pi_p: \forall n,p,f,g.
+pi_p n p (\lambda x.f x*g x) = pi_p n p f * pi_p n p g.
+intros.
+elim n
+ [simplify.reflexivity
+ |apply (bool_elim ? (p n1))
+ [intro.
+ rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > H.autobatch
+ |assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |intro.
+ rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn;assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem pi_p_SO: \forall n,p.
+pi_p n p (\lambda i.S O) = S O.
+intros.elim n
+ [reflexivity
+ |simplify.elim (p n1)
+ [simplify.rewrite < plus_n_O.assumption
+ |simplify.assumption
+ ]
+ ]
+qed.
+
+theorem exp_pi_p: \forall n,m,p,f.
+pi_p n p (\lambda x.exp (f x) m) = exp (pi_p n p f) m.
+intros.
+elim m
+ [simplify.apply pi_p_SO
+ |simplify.
+ rewrite > times_pi_p.
+ rewrite < H.
+ reflexivity
+ ]
+qed.
+
+theorem exp_times_pi_p: \forall n,m,k,p,f.
+pi_p n p (\lambda x.exp k (m*(f x))) =
+exp (pi_p n p (\lambda x.exp k (f x))) m.
+intros.
+apply (trans_eq ? ? (pi_p n p (\lambda x.(exp (exp k (f x)) m))))
+ [apply eq_pi_p;intros
+ [reflexivity
+ |apply sym_eq.rewrite > sym_times.
+ apply exp_exp_times
+ ]
+ |apply exp_pi_p
+ ]
+qed.
+
+
+theorem pi_p_knm:
+\forall g: nat \to nat.
+\forall h2:nat \to nat \to nat.
+\forall h11,h12:nat \to nat.
+\forall k,n,m.
+\forall p1,p21:nat \to bool.
+\forall p22:nat \to nat \to bool.
+(\forall x. x < k \to p1 x = true \to
+p21 (h11 x) = true \land p22 (h11 x) (h12 x) = true
+\land h2 (h11 x) (h12 x) = x
+\land (h11 x) < n \land (h12 x) < m) \to
+(\forall i,j. i < n \to j < m \to p21 i = true \to p22 i j = true \to
+p1 (h2 i j) = true \land
+h11 (h2 i j) = i \land h12 (h2 i j) = j
+\land h2 i j < k) \to
+pi_p k p1 g =
+pi_p n p21 (\lambda x:nat.pi_p m (p22 x) (\lambda y. g (h2 x y))).
+intros.
+unfold pi_p.unfold pi_p.
+apply (iter_p_gen_knm nat (S O) times sym_times assoc_times ? ? ? h11 h12)
+ [intros.apply sym_eq.apply times_n_SO.
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem pi_p_pi_p:
+\forall g: nat \to nat \to nat.
+\forall h11,h12,h21,h22: nat \to nat \to nat.
+\forall n1,m1,n2,m2.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to
+p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true
+\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j
+\land h11 i j < n1 \land h12 i j < m1) \to
+(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to
+p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true
+\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j
+\land (h21 i j) < n2 \land (h22 i j) < m2) \to
+pi_p n1 p11
+ (\lambda x:nat .pi_p m1 (p12 x) (\lambda y. g x y)) =
+pi_p n2 p21
+ (\lambda x:nat .pi_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))).
+intros.
+unfold pi_p.unfold pi_p.
+apply (iter_p_gen_2_eq ? ? ? sym_times assoc_times ? ? ? ? h21 h22)
+ [intros.apply sym_eq.apply times_n_SO.
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem pi_p_pi_p1:
+\forall g: nat \to nat \to nat.
+\forall n,m.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall x,y. x < n \to y < m \to
+ (p11 x \land p12 x y) = (p21 y \land p22 y x)) \to
+pi_p n p11 (\lambda x:nat.pi_p m (p12 x) (\lambda y. g x y)) =
+pi_p m p21 (\lambda y:nat.pi_p n (p22 y) (\lambda x. g x y)).
+intros.
+unfold pi_p.unfold pi_p.
+apply (iter_p_gen_iter_p_gen ? ? ? sym_times assoc_times)
+ [intros.apply sym_eq.apply times_n_SO.
+ |assumption
+ ]
+qed.
\ No newline at end of file
projects / helm.git / blobdiff