+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/nat/generic_sigma_p.ma".
-
-include "nat/primes.ma".
-include "nat/ord.ma".
-
-
-
-(*a generic definition of summatory indexed over natural numbers:
- * n:N is the advanced end in the range of the sommatory
- * p: N -> bool is a predicate. if (p i) = true, then (g i) is summed, else it isn't
- * A is the type of the elements of the sum.
- * g:nat -> A, is the function which associates the nth element of the sum to the nth natural numbers
- * baseA (of type A) is the neutral element of A for plusA operation
- * plusA: A -> A -> A is the sum over elements in A.
- *)
-let rec sigma_p_gen (n:nat) (p: nat \to bool) (A:Type) (g: nat \to A)
- (baseA: A) (plusA: A \to A \to A) \def
- match n with
- [ O \Rightarrow baseA
- | (S k) \Rightarrow
- match p k with
- [true \Rightarrow (plusA (g k) (sigma_p_gen k p A g baseA plusA))
- |false \Rightarrow sigma_p_gen k p A g baseA plusA]
- ].
-
-theorem true_to_sigma_p_Sn_gen:
-\forall n:nat. \forall p:nat \to bool. \forall A:Type. \forall g:nat \to A.
-\forall baseA:A. \forall plusA: A \to A \to A.
-p n = true \to sigma_p_gen (S n) p A g baseA plusA =
-(plusA (g n) (sigma_p_gen n p A g baseA plusA)).
-intros.
-simplify.
-rewrite > H.
-reflexivity.
-qed.
-
-
-
-theorem false_to_sigma_p_Sn_gen:
-\forall n:nat. \forall p:nat \to bool. \forall A:Type. \forall g:nat \to A.
-\forall baseA:A. \forall plusA: A \to A \to A.
-p n = false \to sigma_p_gen (S n) p A g baseA plusA = sigma_p_gen n p A g baseA plusA.
-intros.
-simplify.
-rewrite > H.
-reflexivity.
-qed.
-
-
-theorem eq_sigma_p_gen: \forall p1,p2:nat \to bool. \forall A:Type.
-\forall g1,g2: nat \to A. \forall baseA: A.
-\forall plusA: A \to A \to A. \forall n:nat.
-(\forall x. x < n \to p1 x = p2 x) \to
-(\forall x. x < n \to g1 x = g2 x) \to
-sigma_p_gen n p1 A g1 baseA plusA = sigma_p_gen n p2 A g2 baseA plusA.
-intros 8.
-elim n
-[ reflexivity
-| apply (bool_elim ? (p1 n1))
- [ intro.
- rewrite > (true_to_sigma_p_Sn_gen ? ? ? ? ? ? H3).
- rewrite > true_to_sigma_p_Sn_gen
- [ apply eq_f2
- [ apply H2.apply le_n.
- | apply H
- [ intros.apply H1.apply le_S.assumption
- | intros.apply H2.apply le_S.assumption
- ]
- ]
- | rewrite < H3.apply sym_eq.apply H1.apply le_n
- ]
- | intro.
- rewrite > (false_to_sigma_p_Sn_gen ? ? ? ? ? ? H3).
- rewrite > false_to_sigma_p_Sn_gen
- [ apply H
- [ intros.apply H1.apply le_S.assumption
- | intros.apply H2.apply le_S.assumption
- ]
- | rewrite < H3.apply sym_eq.apply H1.apply le_n
- ]
- ]
-]
-qed.
-
-theorem eq_sigma_p1_gen: \forall p1,p2:nat \to bool. \forall A:Type.
-\forall g1,g2: nat \to A. \forall baseA: A.
-\forall plusA: A \to A \to A.\forall n:nat.
-(\forall x. x < n \to p1 x = p2 x) \to
-(\forall x. x < n \to p1 x = true \to g1 x = g2 x) \to
-sigma_p_gen n p1 A g1 baseA plusA = sigma_p_gen n p2 A g2 baseA plusA.
-intros 8.
-elim n
-[ reflexivity
-| apply (bool_elim ? (p1 n1))
- [ intro.
- rewrite > (true_to_sigma_p_Sn_gen ? ? ? ? ? ? H3).
- rewrite > true_to_sigma_p_Sn_gen
- [ apply eq_f2
- [ apply H2
- [ apply le_n
- | assumption
- ]
- | apply H
- [ intros.apply H1.apply le_S.assumption
- | intros.apply H2
- [ apply le_S.assumption
- | assumption
- ]
- ]
- ]
- | rewrite < H3.
- apply sym_eq.apply H1.apply le_n
- ]
- | intro.
- rewrite > (false_to_sigma_p_Sn_gen ? ? ? ? ? ? H3).
- rewrite > false_to_sigma_p_Sn_gen
- [ apply H
- [ intros.apply H1.apply le_S.assumption
- | intros.apply H2
- [ apply le_S.assumption
- | assumption
- ]
- ]
- | rewrite < H3.apply sym_eq.
- apply H1.apply le_n
- ]
- ]
-]
-qed.
-
-theorem sigma_p_false_gen: \forall A:Type. \forall g: nat \to A. \forall baseA:A.
-\forall plusA: A \to A \to A. \forall n.
-sigma_p_gen n (\lambda x.false) A g baseA plusA = baseA.
-intros.
-elim n
-[ reflexivity
-| simplify.
- assumption
-]
-qed.
-
-theorem sigma_p_plusA_gen: \forall A:Type. \forall n,k:nat.\forall p:nat \to bool.
-\forall g: nat \to A. \forall baseA:A. \forall plusA: A \to A \to A.
-(symmetric A plusA) \to (\forall a:A. (plusA a baseA) = a) \to (associative A plusA)
-\to
-sigma_p_gen (k + n) p A g baseA plusA
-= (plusA (sigma_p_gen k (\lambda x.p (x+n)) A (\lambda x.g (x+n)) baseA plusA)
- (sigma_p_gen n p A g baseA plusA)).
-intros.
-
-elim k
-[ rewrite < (plus_n_O n).
- simplify.
- rewrite > H in \vdash (? ? ? %).
- rewrite > (H1 ?).
- reflexivity
-| apply (bool_elim ? (p (n1+n)))
- [ intro.
- rewrite > (true_to_sigma_p_Sn_gen ? ? ? ? ? ? H4).
- rewrite > (true_to_sigma_p_Sn_gen n1 (\lambda x.p (x+n)) ? ? ? ? H4).
- rewrite > (H2 (g (n1 + n)) ? ?).
- rewrite < H3.
- reflexivity
- | intro.
- rewrite > (false_to_sigma_p_Sn_gen ? ? ? ? ? ? H4).
- rewrite > (false_to_sigma_p_Sn_gen n1 (\lambda x.p (x+n)) ? ? ? ? H4).
- assumption
- ]
-]
-qed.
-
-theorem false_to_eq_sigma_p_gen: \forall A:Type. \forall n,m:nat.\forall p:nat \to bool.
-\forall g: nat \to A. \forall baseA:A. \forall plusA: A \to A \to A.
-n \le m \to (\forall i:nat. n \le i \to i < m \to p i = false)
-\to sigma_p_gen m p A g baseA plusA = sigma_p_gen n p A g baseA plusA.
-intros 8.
-elim H
-[ reflexivity
-| simplify.
- rewrite > H3
- [ simplify.
- apply H2.
- intros.
- apply H3
- [ apply H4
- | apply le_S.
- assumption
- ]
- | assumption
- |apply le_n
- ]
-]
-qed.
-
-theorem sigma_p2_gen :
-\forall n,m:nat.
-\forall p1,p2:nat \to bool.
-\forall A:Type.
-\forall g: nat \to nat \to A.
-\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
-sigma_p_gen (n*m)
- (\lambda x.andb (p1 (div x m)) (p2 (mod x m)))
- A
- (\lambda x.g (div x m) (mod x m))
- baseA
- plusA =
-sigma_p_gen n p1 A
- (\lambda x.sigma_p_gen m p2 A (g x) baseA plusA)
- baseA plusA.
-intros.
-elim n
-[ simplify.
- reflexivity
-| apply (bool_elim ? (p1 n1))
- [ intro.
- rewrite > (true_to_sigma_p_Sn_gen ? ? ? ? ? ? H4).
- simplify in \vdash (? ? (? % ? ? ? ? ?) ?).
- rewrite > sigma_p_plusA_gen
- [ rewrite < H3.
- apply eq_f2
- [ apply eq_sigma_p_gen
- [ intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H5).
- rewrite > (mod_plus_times ? ? ? H5).
- rewrite > H4.
- simplify.
- reflexivity
- | intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H5).
- rewrite > (mod_plus_times ? ? ? H5).
- rewrite > H4.
- simplify.reflexivity.
- ]
- | reflexivity
- ]
- | assumption
- | assumption
- | assumption
- ]
- | intro.
- rewrite > (false_to_sigma_p_Sn_gen ? ? ? ? ? ? H4).
- simplify in \vdash (? ? (? % ? ? ? ? ?) ?).
- rewrite > sigma_p_plusA_gen
- [ rewrite > H3.
- apply (trans_eq ? ? (plusA baseA
- (sigma_p_gen n1 p1 A (\lambda x:nat.sigma_p_gen m p2 A (g x) baseA plusA) baseA plusA )))
- [ apply eq_f2
- [ rewrite > (eq_sigma_p_gen ? (\lambda x.false) A ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
- [ apply sigma_p_false_gen
- | intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H5).
- rewrite > (mod_plus_times ? ? ? H5).
- rewrite > H4.
- simplify.reflexivity
- | intros.reflexivity.
- ]
- | reflexivity
- ]
- | rewrite < H.
- rewrite > H2.
- reflexivity.
- ]
- | assumption
- | assumption
- | assumption
- ]
- ]
-]
-qed.
-
-
-theorem sigma_p2_gen':
-\forall n,m:nat.
-\forall p1: nat \to bool.
-\forall p2: nat \to nat \to bool.
-\forall A:Type.
-\forall g: nat \to nat \to A.
-\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
-sigma_p_gen (n*m)
- (\lambda x.andb (p1 (div x m)) (p2 (div x m)(mod x m)))
- A
- (\lambda x.g (div x m) (mod x m))
- baseA
- plusA =
-sigma_p_gen n p1 A
- (\lambda x.sigma_p_gen m (p2 x) A (g x) baseA plusA)
- baseA plusA.
-intros.
-elim n
-[ simplify.
- reflexivity
-| apply (bool_elim ? (p1 n1))
- [ intro.
- rewrite > (true_to_sigma_p_Sn_gen ? ? ? ? ? ? H4).
- simplify in \vdash (? ? (? % ? ? ? ? ?) ?).
- rewrite > sigma_p_plusA_gen
- [ rewrite < H3.
- apply eq_f2
- [ apply eq_sigma_p_gen
- [ intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H5).
- rewrite > (mod_plus_times ? ? ? H5).
- rewrite > H4.
- simplify.
- reflexivity
- | intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H5).
- rewrite > (mod_plus_times ? ? ? H5).
- rewrite > H4.
- simplify.reflexivity.
- ]
- | reflexivity
- ]
- | assumption
- | assumption
- | assumption
- ]
- | intro.
- rewrite > (false_to_sigma_p_Sn_gen ? ? ? ? ? ? H4).
- simplify in \vdash (? ? (? % ? ? ? ? ?) ?).
- rewrite > sigma_p_plusA_gen
- [ rewrite > H3.
- apply (trans_eq ? ? (plusA baseA
- (sigma_p_gen n1 p1 A (\lambda x:nat.sigma_p_gen m (p2 x) A (g x) baseA plusA) baseA plusA )))
- [ apply eq_f2
- [ rewrite > (eq_sigma_p_gen ? (\lambda x.false) A ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
- [ apply sigma_p_false_gen
- | intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H5).
- rewrite > (mod_plus_times ? ? ? H5).
- rewrite > H4.
- simplify.reflexivity
- | intros.reflexivity.
- ]
- | reflexivity
- ]
- | rewrite < H.
- rewrite > H2.
- reflexivity.
- ]
- | assumption
- | assumption
- | assumption
- ]
- ]
-]
-qed.
-
-lemma sigma_p_gi_gen:
-\forall A:Type.
-\forall g: nat \to A.
-\forall baseA:A.
-\forall plusA: A \to A \to A.
-\forall n,i:nat.
-\forall p:nat \to bool.
-(symmetric A plusA) \to (associative A plusA) \to (\forall a:A.(plusA a baseA) = a)
- \to
-
-i < n \to p i = true \to
-(sigma_p_gen n p A g baseA plusA) =
-(plusA (g i) (sigma_p_gen n (\lambda x:nat. andb (p x) (notb (eqb x i))) A g baseA plusA)).
-intros 5.
-elim n
-[ apply False_ind.
- apply (not_le_Sn_O i).
- assumption
-| apply (bool_elim ? (p n1));intro
- [ elim (le_to_or_lt_eq i n1)
- [ rewrite > true_to_sigma_p_Sn_gen
- [ rewrite > true_to_sigma_p_Sn_gen
- [ rewrite < (H2 (g i) ? ?).
- rewrite > (H1 (g i) (g n1)).
- rewrite > (H2 (g n1) ? ?).
- apply eq_f2
- [ reflexivity
- | apply H
- [ assumption
- | assumption
- | assumption
- | assumption
- | assumption
- ]
- ]
- | rewrite > H6.simplify.
- change with (notb (eqb n1 i) = notb false).
- apply eq_f.
- apply not_eq_to_eqb_false.
- unfold Not.intro.
- apply (lt_to_not_eq ? ? H7).
- apply sym_eq.assumption
- ]
- | assumption
- ]
- | rewrite > true_to_sigma_p_Sn_gen
- [ rewrite > H7.
- apply eq_f2
- [ reflexivity
- | rewrite > false_to_sigma_p_Sn_gen
- [ apply eq_sigma_p_gen
- [ intros.
- elim (p x)
- [ simplify.
- change with (notb false = notb (eqb x n1)).
- apply eq_f.
- apply sym_eq.
- apply not_eq_to_eqb_false.
- apply (lt_to_not_eq ? ? H8)
- | reflexivity
- ]
- | intros.
- reflexivity
- ]
- | rewrite > H6.
- rewrite > (eq_to_eqb_true ? ? (refl_eq ? n1)).
- reflexivity
- ]
- ]
- | assumption
- ]
- | apply le_S_S_to_le.
- assumption
- ]
- | rewrite > false_to_sigma_p_Sn_gen
- [ elim (le_to_or_lt_eq i n1)
- [ rewrite > false_to_sigma_p_Sn_gen
- [ apply H
- [ assumption
- | assumption
- | assumption
- | assumption
- | assumption
- ]
- | rewrite > H6.reflexivity
- ]
- | apply False_ind.
- apply not_eq_true_false.
- rewrite < H5.
- rewrite > H7.
- assumption
- | apply le_S_S_to_le.
- assumption
- ]
- | assumption
- ]
- ]
-]
-qed.
-
-
-theorem eq_sigma_p_gh_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 A.
-\forall h,h1: nat \to nat.
-\forall n,n1:nat.
-\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
-
-sigma_p_gen n p1 A (\lambda x.g(h x)) baseA plusA =
-sigma_p_gen n1 (\lambda x.p2 x) A g baseA plusA.
-intros 10.
-elim n
-[ generalize in match H8.
- elim n1
- [ reflexivity
- | apply (bool_elim ? (p2 n2));intro
- [ apply False_ind.
- apply (not_le_Sn_O (h1 n2)).
- apply H10
- [ apply le_n
- | assumption
- ]
- | rewrite > false_to_sigma_p_Sn_gen
- [ apply H9.
- intros.
- apply H10
- [ apply le_S.
- apply H12
- | assumption
- ]
- | assumption
- ]
- ]
- ]
-| apply (bool_elim ? (p1 n1));intro
- [ rewrite > true_to_sigma_p_Sn_gen
- [ rewrite > (sigma_p_gi_gen A g baseA plusA n2 (h n1))
- [ apply eq_f2
- [ reflexivity
- | apply H3
- [ intros.
- rewrite > H4
- [ simplify.
- change with ((\not eqb (h i) (h n1))= \not false).
- apply eq_f.
- apply not_eq_to_eqb_false.
- unfold Not.
- intro.
- apply (lt_to_not_eq ? ? H11).
- rewrite < H5
- [ rewrite < (H5 n1)
- [ apply eq_f.
- assumption
- | apply le_n
- | assumption
- ]
- | apply le_S.
- assumption
- | assumption
- ]
- | apply le_S.assumption
- | assumption
- ]
- | intros.
- apply H5
- [ apply le_S.
- assumption
- | assumption
- ]
- | intros.
- apply H6
- [ apply le_S.assumption
- | assumption
- ]
- | intros.
- apply H7
- [ assumption
- | generalize in match H12.
- elim (p2 j)
- [ reflexivity
- | assumption
- ]
- ]
- | intros.
- apply H8
- [ assumption
- | generalize in match H12.
- elim (p2 j)
- [ reflexivity
- | assumption
- ]
- ]
- | intros.
- elim (le_to_or_lt_eq (h1 j) n1)
- [ assumption
- | generalize in match H12.
- elim (p2 j)
- [ simplify in H13.
- absurd (j = (h n1))
- [ rewrite < H13.
- rewrite > H8
- [ reflexivity
- | assumption
- | autobatch
- ]
- | apply eqb_false_to_not_eq.
- generalize in match H14.
- elim (eqb j (h n1))
- [ apply sym_eq.assumption
- | reflexivity
- ]
- ]
- | simplify in H14.
- apply False_ind.
- apply not_eq_true_false.
- apply sym_eq.assumption
- ]
- | apply le_S_S_to_le.
- apply H9
- [ assumption
- | generalize in match H12.
- elim (p2 j)
- [ reflexivity
- | assumption
- ]
- ]
- ]
- ]
- ]
- | assumption
- | assumption
- | assumption
- | apply H6
- [ apply le_n
- | assumption
- ]
- | apply H4
- [ apply le_n
- | assumption
- ]
- ]
- | assumption
- ]
- | rewrite > false_to_sigma_p_Sn_gen
- [ apply H3
- [ intros.
- apply H4[apply le_S.assumption|assumption]
- | intros.
- apply H5[apply le_S.assumption|assumption]
- | intros.
- apply H6[apply le_S.assumption|assumption]
- | intros.
- apply H7[assumption|assumption]
- | intros.
- apply H8[assumption|assumption]
- | intros.
- elim (le_to_or_lt_eq (h1 j) n1)
- [ assumption
- | absurd (j = (h n1))
- [ rewrite < H13.
- rewrite > H8
- [ reflexivity
- | assumption
- | assumption
- ]
- | unfold Not.intro.
- apply not_eq_true_false.
- rewrite < H10.
- rewrite < H13.
- rewrite > H7
- [ reflexivity
- | assumption
- | assumption
- ]
- ]
- | apply le_S_S_to_le.
- apply H9
- [ assumption
- | assumption
- ]
- ]
- ]
- | assumption
- ]
- ]
-]
-qed.
-
-
-
-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.
-
-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.
-
-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.
-
-theorem sigma_p_divides_gen:
-\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
-
-sigma_p_gen (S (n*(exp p m))) (\lambda x.divides_b x (n*(exp p m))) A g baseA plusA =
-sigma_p_gen (S n) (\lambda x.divides_b x n) A
- (\lambda x.sigma_p_gen (S m) (\lambda y.true) A (\lambda y.g (x*(exp p y))) baseA plusA) baseA plusA.
-intros.
-cut (O < p)
- [rewrite < (sigma_p2_gen ? ? ? ? ? ? ? ? H3 H4 H5).
- apply (trans_eq ? ?
- (sigma_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_sigma_p_gh_gen ? ? ? ? ? ? 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
- ]
- ]
- |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)
- [autobatch|assumption]
- |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_sigma_p_gen
-
- [intros.
- elim (divides_b (x/S m) n);reflexivity
- |intros.reflexivity
- ]
- ]
-|elim H1.apply lt_to_le.assumption
-]
-qed.
-
-(*distributive property for sigma_p_gen*)
-theorem distributive_times_plus_sigma_p_generic: \forall A:Type.
-\forall plusA: A \to A \to A. \forall basePlusA: A.
-\forall timesA: A \to A \to A.
-\forall n:nat. \forall k:A. \forall p:nat \to bool. \forall g:nat \to A.
-(symmetric A plusA) \to (associative A plusA) \to (\forall a:A.(plusA a basePlusA) = a) \to
-(symmetric A timesA) \to (distributive A timesA plusA) \to
-(\forall a:A. (timesA a basePlusA) = basePlusA)
-
- \to
-
-((timesA k (sigma_p_gen n p A g basePlusA plusA)) =
- (sigma_p_gen n p A (\lambda i:nat.(timesA k (g i))) basePlusA plusA)).
-intros.
-elim n
-[ simplify.
- apply H5
-| cut( (p n1) = true \lor (p n1) = false)
- [ elim Hcut
- [ rewrite > (true_to_sigma_p_Sn_gen ? ? ? ? ? ? H7).
- rewrite > (true_to_sigma_p_Sn_gen ? ? ? ? ? ? H7) in \vdash (? ? ? %).
- rewrite > (H4).
- rewrite > (H3 k (g n1)).
- apply eq_f.
- assumption
- | rewrite > (false_to_sigma_p_Sn_gen ? ? ? ? ? ? H7).
- rewrite > (false_to_sigma_p_Sn_gen ? ? ? ? ? ? H7) in \vdash (? ? ? %).
- assumption
- ]
- | elim ((p n1))
- [ left.
- reflexivity
- | right.
- reflexivity
- ]
- ]
-]
-qed.
-
-
projects / helm.git / commitdiff