\ /
V_______________________________________________________________ *)
-(* To be ported
+include "arithmetics/primes.ma".
include "arithmetics/bigops.ma".
-definition natAop ≝ 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 natACop ≝ mk_ACop nat 0 natAop commutative_plus.
-
-definition natDop ≝ mk_Dop nat 0 natACop times (λn.(sym_eq ??? (times_n_O n)))
- distributive_times_plus.
-
-unification hint 0 ≔ ;
- S ≟ natAop
-(* ---------------------------------------- *) ⊢
- plus ≡ op ? ? S.
-
-unification hint 0 ≔ ;
- S ≟ natACop
-(* ---------------------------------------- *) ⊢
- plus ≡ op ? ? S.
-
-unification hint 0 ≔ ;
- S ≟ natDop
-(* ---------------------------------------- *) ⊢
- plus ≡ sum ? ? S.
-
-unification hint 0 ≔ ;
- S ≟ natDop
-(* ---------------------------------------- *) ⊢
- times ≡ prod ? ? S.
-
(* 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)}.
+for @{'bigop $n plus 0 (λ${ident i}. $p) (λ${ident i}. $f)}.
notation "∑_{ ident i < n } f"
with precedence 80
for @{'bigop ($b-$a) times 1 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
(λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
-
-(*
-
-definition p_ord_times \def
-\lambda p,m,x.
- match p_ord x p with
- [pair q r \Rightarrow r*m+q].
-
-theorem eq_p_ord_times: \forall p,m,x.
-p_ord_times p m x = (ord_rem x p)*m+(ord x p).
-intros.unfold p_ord_times. unfold ord_rem.
-unfold ord.
-elim (p_ord x p).
-reflexivity.
-qed.
+(* instances of associative and commutative operations *)
-theorem div_p_ord_times:
-\forall p,m,x. ord x p < m \to p_ord_times p m x / m = ord_rem x p.
-intros.rewrite > eq_p_ord_times.
-apply div_plus_times.
-assumption.
-qed.
+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.
-theorem mod_p_ord_times:
-\forall p,m,x. ord x p < m \to p_ord_times p m x \mod m = ord x p.
-intros.rewrite > eq_p_ord_times.
-apply mod_plus_times.
-assumption.
-qed.
+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.
-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.
+definition natD ≝ mk_Dop nat 0 plusAC times (λn.(sym_eq ??? (times_n_O n)))
+ distributive_times_plus.
+
+(********************************************************)
-theorem iter_p_gen_knm:
-\forall A:Type.
-\forall baseA: A.
-\forall plusA: A \to A \to A.
-(symmetric A plusA) \to
-(associative A plusA) \to
-(\forall a:A.(plusA a baseA) = a)\to
-\forall g: nat \to A.
-\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
-iter_p_gen k p1 A g baseA plusA =
-iter_p_gen n p21 A (\lambda x:nat.iter_p_gen m (p22 x) A (\lambda y. g (h2 x y)) baseA plusA) baseA plusA.
-intros.
-rewrite < (iter_p_gen2' n m p21 p22 ? ? ? ? H H1 H2).
-apply sym_eq.
-apply (eq_iter_p_gen_gh A baseA plusA H H1 H2 g ? (\lambda x.(h11 x)*m+(h12 x)))
- [intros.
- elim (H4 (i/m) (i \mod m));clear H4
- [elim H7.clear H7.
- elim H4.clear H4.
- assumption
- |apply (lt_times_to_lt_div ? ? ? H5)
- |apply lt_mod_m_m.
- apply (lt_times_to_lt_O ? ? ? H5)
- |apply (andb_true_true ? ? H6)
- |apply (andb_true_true_r ? ? H6)
- ]
- |intros.
- elim (H4 (i/m) (i \mod m));clear H4
- [elim H7.clear H7.
- elim H4.clear H4.
- rewrite > H10.
- rewrite > H9.
- apply sym_eq.
- apply div_mod.
- apply (lt_times_to_lt_O ? ? ? H5)
- |apply (lt_times_to_lt_div ? ? ? H5)
- |apply lt_mod_m_m.
- apply (lt_times_to_lt_O ? ? ? H5)
- |apply (andb_true_true ? ? H6)
- |apply (andb_true_true_r ? ? H6)
- ]
- |intros.
- elim (H4 (i/m) (i \mod m));clear H4
- [elim H7.clear H7.
- elim H4.clear H4.
- assumption
- |apply (lt_times_to_lt_div ? ? ? H5)
- |apply lt_mod_m_m.
- apply (lt_times_to_lt_O ? ? ? H5)
- |apply (andb_true_true ? ? H6)
- |apply (andb_true_true_r ? ? H6)
- ]
- |intros.
- elim (H3 j H5 H6).
- elim H7.clear H7.
- elim H9.clear H9.
- elim H7.clear H7.
- rewrite > div_plus_times
- [rewrite > mod_plus_times
- [rewrite > H9.
- rewrite > H12.
- reflexivity.
- |assumption
- ]
- |assumption
- ]
- |intros.
- elim (H3 j H5 H6).
- elim H7.clear H7.
- elim H9.clear H9.
- elim H7.clear H7.
- rewrite > div_plus_times
- [rewrite > mod_plus_times
- [assumption
- |assumption
- ]
- |assumption
- ]
- |intros.
- elim (H3 j H5 H6).
- elim H7.clear H7.
- elim H9.clear H9.
- elim H7.clear H7.
- apply (lt_to_le_to_lt ? ((h11 j)*m+m))
- [apply monotonic_lt_plus_r.
- assumption
- |rewrite > sym_plus.
- change with ((S (h11 j)*m) \le n*m).
- apply monotonic_le_times_l.
- assumption
- ]
- ]
+theorem sigma_const: ∀n:nat. ∑_{i<n} 1 = n.
+#n elim n // #n1 >bigop_Strue //
qed.
-theorem iter_p_gen_divides:
-\forall A:Type.
-\forall baseA: A.
-\forall plusA: A \to A \to A.
-\forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to
-\forall g: nat \to A.
-(symmetric A plusA) \to (associative A plusA) \to (\forall a:A.(plusA a baseA) = a)
-
-\to
-
-iter_p_gen (S (n*(exp p m))) (\lambda x.divides_b x (n*(exp p m))) A g baseA plusA =
-iter_p_gen (S n) (\lambda x.divides_b x n) A
- (\lambda x.iter_p_gen (S m) (\lambda y.true) A (\lambda y.g (x*(exp p y))) baseA plusA) baseA plusA.
-intros.
-cut (O < p)
- [rewrite < (iter_p_gen2 ? ? ? ? ? ? ? ? H3 H4 H5).
- apply (trans_eq ? ?
- (iter_p_gen (S n*S m) (\lambda x:nat.divides_b (x/S m) n) A
- (\lambda x:nat.g (x/S m*(p)\sup(x\mod S m))) baseA plusA) )
- [apply sym_eq.
- apply (eq_iter_p_gen_gh ? ? ? ? ? ? g ? (p_ord_times p (S m)))
- [ assumption
- | assumption
- | assumption
- |intros.
- lapply (divides_b_true_to_lt_O ? ? H H7).
- apply divides_to_divides_b_true
- [rewrite > (times_n_O O).
- apply lt_times
- [assumption
- |apply lt_O_exp.assumption
- ]
- |apply divides_times
- [apply divides_b_true_to_divides.assumption
- |apply (witness ? ? (p \sup (m-i \mod (S m)))).
- rewrite < exp_plus_times.
- apply eq_f.
- rewrite > sym_plus.
- apply plus_minus_m_m.
- autobatch by le_S_S_to_le, lt_mod_m_m, lt_O_S;
- ]
- ]
- |intros.
- lapply (divides_b_true_to_lt_O ? ? H H7).
- unfold p_ord_times.
- rewrite > (p_ord_exp1 p ? (i \mod (S m)) (i/S m))
- [change with ((i/S m)*S m+i \mod S m=i).
- apply sym_eq.
- apply div_mod.
- apply lt_O_S
- |assumption
- |unfold Not.intro.
- apply H2.
- apply (trans_divides ? (i/ S m))
- [assumption|
- apply divides_b_true_to_divides;assumption]
- |apply sym_times.
- ]
- |intros.
- apply le_S_S.
- apply le_times
- [apply le_S_S_to_le.
- change with ((i/S m) < S n).
- apply (lt_times_to_lt_l m).
- apply (le_to_lt_to_lt ? i);[2:assumption]
- autobatch by eq_plus_to_le, div_mod, lt_O_S.
- |apply le_exp
- [assumption
- |apply le_S_S_to_le.
- apply lt_mod_m_m.
- apply lt_O_S
- ]
- ]
- |intros.
- cut (ord j p < S m)
- [rewrite > div_p_ord_times
- [apply divides_to_divides_b_true
- [apply lt_O_ord_rem
- [elim H1.assumption
- |apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |cut (n = ord_rem (n*(exp p m)) p)
- [rewrite > Hcut2.
- apply divides_to_divides_ord_rem
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord_rem.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |assumption
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |intros.
- cut (ord j p < S m)
- [rewrite > div_p_ord_times
- [rewrite > mod_p_ord_times
- [rewrite > sym_times.
- apply sym_eq.
- apply exp_ord
- [elim H1.assumption
- |apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut2.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |assumption
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |intros.
- rewrite > eq_p_ord_times.
- rewrite > sym_plus.
- apply (lt_to_le_to_lt ? (S m +ord_rem j p*S m))
- [apply lt_plus_l.
- apply le_S_S.
- cut (m = ord (n*(p \sup m)) p)
- [rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > sym_times.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |reflexivity
- ]
- ]
- |change with (S (ord_rem j p)*S m \le S n*S m).
- apply le_times_l.
- apply le_S_S.
- cut (n = ord_rem (n*(p \sup m)) p)
- [rewrite > Hcut1.
- apply divides_to_le
- [apply lt_O_ord_rem
- [elim H1.assumption
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |apply divides_to_divides_ord_rem
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- ]
- |unfold ord_rem.
- rewrite > sym_times.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |reflexivity
- ]
- ]
- ]
- ]
- |apply eq_iter_p_gen
-
- [intros.
- elim (divides_b (x/S m) n);reflexivity
- |intros.reflexivity
+(* monotonicity; these roperty should be expressed at a more
+genral level *)
+
+theorem le_pi:
+∀n.∀p:nat → bool.∀g1,g2:nat → nat.
+ (∀i.i<n → p i = true → g1 i ≤ g2 i ) →
+ ∏_{i < n | p i} (g1 i) ≤ ∏_{i < n | p i} (g2 i).
+#n #p #g1 #g2 elim n
+ [#_ @le_n
+ |#n1 #Hind #Hle cases (true_or_false (p n1)) #Hcase
+ [ >bigop_Strue // >bigop_Strue // @le_times
+ [@Hle // |@Hind #i #lti #Hpi @Hle [@lt_to_le @le_S_S @lti|@Hpi]]
+ |>bigop_Sfalse // >bigop_Sfalse // @Hind
+ #i #lti #Hpi @Hle [@lt_to_le @le_S_S @lti|@Hpi]
]
]
-|elim H1.apply lt_to_le.assumption
-]
qed.
+theorem exp_sigma: ∀n,a,p.
+ ∏_{i < n | p i} a = exp a (∑_{i < n | p i} 1).
+#n #a #p elim n // #n1 cases (true_or_false (p n1)) #Hcase
+ [>bigop_Strue // >bigop_Strue // |>bigop_Sfalse // >bigop_Sfalse //]
+qed.
-
-theorem iter_p_gen_2_eq:
-\forall A:Type.
-\forall baseA: A.
-\forall plusA: A \to A \to A.
-(symmetric A plusA) \to
-(associative A plusA) \to
-(\forall a:A.(plusA a baseA) = a)\to
-\forall g: nat \to nat \to A.
-\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
-iter_p_gen n1 p11 A
- (\lambda x:nat .iter_p_gen m1 (p12 x) A (\lambda y. g x y) baseA plusA)
- baseA plusA =
-iter_p_gen n2 p21 A
- (\lambda x:nat .iter_p_gen m2 (p22 x) A (\lambda y. g (h11 x y) (h12 x y)) baseA plusA )
- baseA plusA.
-
-intros.
-rewrite < (iter_p_gen2' ? ? ? ? ? ? ? ? H H1 H2).
-letin ha:= (\lambda x,y.(((h11 x y)*m1) + (h12 x y))).
-letin ha12:= (\lambda x.(h21 (x/m1) (x \mod m1))).
-letin ha22:= (\lambda x.(h22 (x/m1) (x \mod m1))).
-
-apply (trans_eq ? ?
-(iter_p_gen n2 p21 A (\lambda x:nat. iter_p_gen m2 (p22 x) A
- (\lambda y:nat.(g (((h11 x y)*m1+(h12 x y))/m1) (((h11 x y)*m1+(h12 x y))\mod m1))) baseA plusA ) baseA plusA))
-[
- apply (iter_p_gen_knm A baseA plusA H H1 H2 (\lambda e. (g (e/m1) (e \mod m1))) ha ha12 ha22);intros
- [ elim (and_true ? ? H6).
- cut(O \lt m1)
- [ cut(x/m1 < n1)
- [ cut((x \mod m1) < m1)
- [ elim (H4 ? ? Hcut1 Hcut2 H7 H8).
- elim H9.clear H9.
- elim H11.clear H11.
- elim H9.clear H9.
- elim H11.clear H11.
- split
- [ split
- [ split
- [ split
- [ assumption
- | assumption
- ]
- | unfold ha.
- unfold ha12.
- unfold ha22.
- rewrite > H14.
- rewrite > H13.
- apply sym_eq.
- apply div_mod.
- assumption
- ]
- | assumption
- ]
- | assumption
- ]
- | apply lt_mod_m_m.
- assumption
- ]
- | apply (lt_times_n_to_lt m1)
- [ assumption
- | apply (le_to_lt_to_lt ? x)
- [ apply (eq_plus_to_le ? ? (x \mod m1)).
- apply div_mod.
- assumption
- | assumption
- ]
- ]
- ]
- | apply not_le_to_lt.unfold.intro.
- generalize in match H5.
- apply (le_n_O_elim ? H9).
- rewrite < times_n_O.
- apply le_to_not_lt.
- apply le_O_n.
- ]
- | elim (H3 ? ? H5 H6 H7 H8).
- elim H9.clear H9.
- elim H11.clear H11.
- elim H9.clear H9.
- elim H11.clear H11.
- cut(((h11 i j)*m1 + (h12 i j))/m1 = (h11 i j))
- [ cut(((h11 i j)*m1 + (h12 i j)) \mod m1 = (h12 i j))
- [ split
- [ split
- [ split
- [ apply true_to_true_to_andb_true
- [ rewrite > Hcut.
- assumption
- | rewrite > Hcut1.
- rewrite > Hcut.
- assumption
- ]
- | unfold ha.
- unfold ha12.
- rewrite > Hcut1.
- rewrite > Hcut.
- assumption
- ]
- | unfold ha.
- unfold ha22.
- rewrite > Hcut1.
- rewrite > Hcut.
- assumption
- ]
- | cut(O \lt m1)
- [ cut(O \lt n1)
- [ apply (lt_to_le_to_lt ? ((h11 i j)*m1 + m1) )
- [ unfold ha.
- apply (lt_plus_r).
- assumption
- | rewrite > sym_plus.
- rewrite > (sym_times (h11 i j) m1).
- rewrite > times_n_Sm.
- rewrite > sym_times.
- apply (le_times_l).
- assumption
- ]
- | apply not_le_to_lt.unfold.intro.
- generalize in match H12.
- apply (le_n_O_elim ? H11).
- apply le_to_not_lt.
- apply le_O_n
- ]
- | apply not_le_to_lt.unfold.intro.
- generalize in match H10.
- apply (le_n_O_elim ? H11).
- apply le_to_not_lt.
- apply le_O_n
- ]
- ]
- | rewrite > (mod_plus_times m1 (h11 i j) (h12 i j)).
- reflexivity.
- assumption
- ]
- | rewrite > (div_plus_times m1 (h11 i j) (h12 i j)).
- reflexivity.
- assumption
- ]
- ]
-| apply (eq_iter_p_gen1)
- [ intros. reflexivity
- | intros.
- apply (eq_iter_p_gen1)
- [ intros. reflexivity
- | intros.
- rewrite > (div_plus_times)
- [ rewrite > (mod_plus_times)
- [ reflexivity
- | elim (H3 x x1 H5 H7 H6 H8).
- assumption
- ]
- | elim (H3 x x1 H5 H7 H6 H8).
- assumption
- ]
- ]
+theorem times_pi: ∀n,p,f,g.
+ ∏_{i<n|p i}(f i*g i) = ∏_{i<n|p i}(f i) * ∏_{i<n|p i}(g i).
+#n #p #f #g elim n // #n1 cases (true_or_false (p n1)) #Hcase #Hind
+ [>bigop_Strue // >bigop_Strue // >bigop_Strue //
+ |>bigop_Sfalse // >bigop_Sfalse // >bigop_Sfalse //
]
-]
qed.
-theorem iter_p_gen_iter_p_gen:
-\forall A:Type.
-\forall baseA: A.
-\forall plusA: A \to A \to A.
-(symmetric A plusA) \to
-(associative A plusA) \to
-(\forall a:A.(plusA a baseA) = a)\to
-\forall g: nat \to nat \to A.
-\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
-iter_p_gen n p11 A
- (\lambda x:nat.iter_p_gen m (p12 x) A (\lambda y. g x y) baseA plusA)
- baseA plusA =
-iter_p_gen m p21 A
- (\lambda y:nat.iter_p_gen n (p22 y) A (\lambda x. g x y) baseA plusA )
- baseA plusA.
-intros.
-apply (iter_p_gen_2_eq A baseA plusA H H1 H2 (\lambda x,y. g x y) (\lambda x,y.y) (\lambda x,y.x) (\lambda x,y.y) (\lambda x,y.x)
- n m m n p11 p21 p12 p22)
- [intros.split
- [split
- [split
- [split
- [split
- [apply (andb_true_true ? (p12 j i)).
- rewrite > H3
- [rewrite > H6.rewrite > H7.reflexivity
- |assumption
- |assumption
- ]
- |apply (andb_true_true_r (p11 j)).
- rewrite > H3
- [rewrite > H6.rewrite > H7.reflexivity
- |assumption
- |assumption
- ]
- ]
- |reflexivity
- ]
- |reflexivity
- ]
- |assumption
- ]
- |assumption
- ]
- |intros.split
- [split
- [split
- [split
- [split
- [apply (andb_true_true ? (p22 j i)).
- rewrite < H3
- [rewrite > H6.rewrite > H7.reflexivity
- |assumption
- |assumption
- ]
- |apply (andb_true_true_r (p21 j)).
- rewrite < H3
- [rewrite > H6.rewrite > H7.reflexivity
- |assumption
- |assumption
- ]
- ]
- |reflexivity
- ]
- |reflexivity
- ]
- |assumption
- ]
- |assumption
- ]
+theorem pi_1: ∀n,p.
+ ∏_{i < n | p i} 1 = 1.
+#n #p elim n // #n1 #Hind cases (true_or_false (p n1)) #Hc
+ [>bigop_Strue >Hind // |>bigop_Sfalse // ]
+qed.
+
+theorem exp_pi: ∀n,m,p,f.
+ ∏_{i < n | p i}(exp (f i) m) = exp (∏_{i < n | p i}(f i)) m.
+#n #m #p #f elim m
+ [@pi_1
+ |#m1 #Hind >times_pi >Hind %
]
-qed. *)*)
\ No newline at end of file
+qed.
projects / helm.git / blobdiff