include "arithmetics/primes.ma".
include "arithmetics/bigops.ma".
+(* Sigma e Pi *)
+
+notation "∑_{ ident i < n | p } f"
+ with precedence 80
+for @{'bigop $n plus 0 (λ${ident i}. $p) (λ${ident i}. $f)}.
+
+notation "∑_{ ident i < n } f"
+ with precedence 80
+for @{'bigop $n plus 0 (λ${ident i}.true) (λ${ident i}. $f)}.
+
+notation "∑_{ ident j ∈ [a,b[ } f"
+ with precedence 80
+for @{'bigop ($b-$a) plus 0 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
+ (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+
+notation "∑_{ ident j ∈ [a,b[ | p } f"
+ with precedence 80
+for @{'bigop ($b-$a) plus 0 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
+ (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+
+notation "∏_{ ident i < n | p} f"
+ with precedence 80
+for @{'bigop $n times 1 (λ${ident i}.$p) (λ${ident i}. $f)}.
+
+notation "∏_{ ident i < n } f"
+ with precedence 80
+for @{'bigop $n times 1 (λ${ident i}.true) (λ${ident i}. $f)}.
+
+notation "∏_{ ident j ∈ [a,b[ } f"
+ with precedence 80
+for @{'bigop ($b-$a) times 1 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
+ (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+
+notation "∏_{ ident j ∈ [a,b[ | p } f"
+ with precedence 80
+for @{'bigop ($b-$a) times 1 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
+ (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+
+(* instances of associative and commutative operations *)
+
+definition plusA ≝ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n))
+ (λa,b,c.sym_eq ??? (associative_plus a b c)).
+
+definition plusAC ≝ mk_ACop nat 0 plusA commutative_plus.
+
+definition timesA ≝ mk_Aop nat 1 times
+ (λa.sym_eq ??? (plus_n_O a)) (λn.sym_eq ??? (times_n_1 n))
+ (λa,b,c.sym_eq ??? (associative_times a b c)).
+
+definition timesAC ≝ mk_ACop nat 1 timesA commutative_times.
+
+definition natD ≝ mk_Dop nat 0 plusAC times (λn.(sym_eq ??? (times_n_O n)))
+ distributive_times_plus.
+
+(********************************************************)
+
theorem sigma_const: ∀n:nat. ∑_{i<n} 1 = n.
#n elim n // #n1 >bigop_Strue //
qed.
|#m1 #Hind >times_pi >Hind %
]
qed.
-
-(*
-theorem true_to_pi_p_Sn: ∀n,p,g.
- 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.
-
-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 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 ∧ 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) →
-(*
-Pi z < k | p1 z. g z =
-Pi x < n | p21 x. Pi y < m | p22 x y.g (h2 x y).
-*)
-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