(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/generic_iter_p.ma".
+set "baseuri" "cic:/matita/nat/generic_iter_p".
include "nat/primes.ma".
include "nat/ord.ma".
]
qed.
+(* a therem slightly more general than the previous one *)
+theorem or_false_eq_baseA_to_eq_iter_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.
+(\forall a. plusA baseA a = a) \to
+n \le m \to (\forall i:nat. n \le i \to i < m \to p i = false \lor g i = baseA)
+\to iter_p_gen m p A g baseA plusA = iter_p_gen n p A g baseA plusA.
+intros 9.
+elim H1
+[reflexivity
+|apply (bool_elim ? (p n1));intro
+ [elim (H4 n1)
+ [apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H5.
+ rewrite < H6.
+ reflexivity
+ |rewrite > true_to_iter_p_gen_Sn
+ [rewrite > H6.
+ rewrite > H.
+ apply H3.intros.
+ apply H4
+ [assumption
+ |apply le_S.assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ |apply le_n
+ ]
+ |rewrite > false_to_iter_p_gen_Sn
+ [apply H3.intros.
+ apply H4
+ [assumption
+ |apply le_S.assumption
+ ]
+ |assumption
+ ]
+ ]
+]
+qed.
+
theorem iter_p_gen2 :
\forall n,m:nat.
\forall p1,p2:nat \to bool.
]
qed.
+(* old version - proved without theorem iter_p_gen_knm
+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).
+rewrite < (iter_p_gen2' ? ? ? ? ? ? ? ? H H1 H2).
+apply sym_eq.
+letin h := (\lambda x.(h11 (x/m2) (x\mod m2))*m1 + (h12 (x/m2) (x\mod m2))).
+letin h1 := (\lambda x.(h21 (x/m1) (x\mod m1))*m2 + (h22 (x/m1) (x\mod m1))).
+apply (trans_eq ? ?
+ (iter_p_gen (n2*m2) (\lambda x:nat.p21 (x/m2)\land p22 (x/m2) (x\mod m2)) A
+ (\lambda x:nat.g ((h x)/m1) ((h x)\mod m1)) baseA plusA ))
+ [clear h.clear h1.
+ apply eq_iter_p_gen1
+ [intros.reflexivity
+ |intros.
+ cut (O < m2)
+ [cut (x/m2 < n2)
+ [cut (x \mod m2 < m2)
+ [elim (and_true ? ? H6).
+ elim (H3 ? ? Hcut1 Hcut2 H7 H8).
+ elim H9.clear H9.
+ elim H11.clear H11.
+ elim H9.clear H9.
+ elim H11.clear H11.
+ apply eq_f2
+ [apply sym_eq.
+ apply div_plus_times.
+ assumption
+ | apply sym_eq.
+ apply mod_plus_times.
+ assumption
+ ]
+ |apply lt_mod_m_m.
+ assumption
+ ]
+ |apply (lt_times_n_to_lt m2)
+ [assumption
+ |apply (le_to_lt_to_lt ? x)
+ [apply (eq_plus_to_le ? ? (x \mod m2)).
+ apply div_mod.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ |apply not_le_to_lt.unfold.intro.
+ generalize in match H5.
+ apply (le_n_O_elim ? H7).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ ]
+ |apply (eq_iter_p_gen_gh ? ? ? H H1 H2 ? h h1);intros
+ [cut (O < m2)
+ [cut (i/m2 < n2)
+ [cut (i \mod m2 < m2)
+ [elim (and_true ? ? H6).
+ elim (H3 ? ? Hcut1 Hcut2 H7 H8).
+ elim H9.clear H9.
+ elim H11.clear H11.
+ elim H9.clear H9.
+ elim H11.clear H11.
+ cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))/m1 =
+ h11 (i/m2) (i\mod m2))
+ [cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))\mod m1 =
+ h12 (i/m2) (i\mod m2))
+ [rewrite > Hcut3.
+ rewrite > Hcut4.
+ rewrite > H9.
+ rewrite > H15.
+ reflexivity
+ |apply mod_plus_times.
+ assumption
+ ]
+ |apply div_plus_times.
+ assumption
+ ]
+ |apply lt_mod_m_m.
+ assumption
+ ]
+ |apply (lt_times_n_to_lt m2)
+ [assumption
+ |apply (le_to_lt_to_lt ? i)
+ [apply (eq_plus_to_le ? ? (i \mod m2)).
+ apply div_mod.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ |apply not_le_to_lt.unfold.intro.
+ generalize in match H5.
+ apply (le_n_O_elim ? H7).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ |cut (O < m2)
+ [cut (i/m2 < n2)
+ [cut (i \mod m2 < m2)
+ [elim (and_true ? ? H6).
+ elim (H3 ? ? Hcut1 Hcut2 H7 H8).
+ elim H9.clear H9.
+ elim H11.clear H11.
+ elim H9.clear H9.
+ elim H11.clear H11.
+ cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))/m1 =
+ h11 (i/m2) (i\mod m2))
+ [cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))\mod m1 =
+ h12 (i/m2) (i\mod m2))
+ [rewrite > Hcut3.
+ rewrite > Hcut4.
+ rewrite > H13.
+ rewrite > H14.
+ apply sym_eq.
+ apply div_mod.
+ assumption
+ |apply mod_plus_times.
+ assumption
+ ]
+ |apply div_plus_times.
+ assumption
+ ]
+ |apply lt_mod_m_m.
+ assumption
+ ]
+ |apply (lt_times_n_to_lt m2)
+ [assumption
+ |apply (le_to_lt_to_lt ? i)
+ [apply (eq_plus_to_le ? ? (i \mod m2)).
+ apply div_mod.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ |apply not_le_to_lt.unfold.intro.
+ generalize in match H5.
+ apply (le_n_O_elim ? H7).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ |cut (O < m2)
+ [cut (i/m2 < n2)
+ [cut (i \mod m2 < m2)
+ [elim (and_true ? ? H6).
+ elim (H3 ? ? Hcut1 Hcut2 H7 H8).
+ elim H9.clear H9.
+ elim H11.clear H11.
+ elim H9.clear H9.
+ elim H11.clear H11.
+ apply lt_times_plus_times
+ [assumption|assumption]
+ |apply lt_mod_m_m.
+ assumption
+ ]
+ |apply (lt_times_n_to_lt m2)
+ [assumption
+ |apply (le_to_lt_to_lt ? i)
+ [apply (eq_plus_to_le ? ? (i \mod m2)).
+ apply div_mod.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ |apply not_le_to_lt.unfold.intro.
+ generalize in match H5.
+ apply (le_n_O_elim ? H7).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ |cut (O < m1)
+ [cut (j/m1 < n1)
+ [cut (j \mod m1 < m1)
+ [elim (and_true ? ? H6).
+ elim (H4 ? ? Hcut1 Hcut2 H7 H8).
+ elim H9.clear H9.
+ elim H11.clear H11.
+ elim H9.clear H9.
+ elim H11.clear H11.
+ cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))/m2 =
+ h21 (j/m1) (j\mod m1))
+ [cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))\mod m2 =
+ h22 (j/m1) (j\mod m1))
+ [rewrite > Hcut3.
+ rewrite > Hcut4.
+ rewrite > H9.
+ rewrite > H15.
+ reflexivity
+ |apply mod_plus_times.
+ assumption
+ ]
+ |apply div_plus_times.
+ assumption
+ ]
+ |apply lt_mod_m_m.
+ assumption
+ ]
+ |apply (lt_times_n_to_lt m1)
+ [assumption
+ |apply (le_to_lt_to_lt ? j)
+ [apply (eq_plus_to_le ? ? (j \mod m1)).
+ apply div_mod.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ |apply not_le_to_lt.unfold.intro.
+ generalize in match H5.
+ apply (le_n_O_elim ? H7).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ |cut (O < m1)
+ [cut (j/m1 < n1)
+ [cut (j \mod m1 < m1)
+ [elim (and_true ? ? H6).
+ elim (H4 ? ? Hcut1 Hcut2 H7 H8).
+ elim H9.clear H9.
+ elim H11.clear H11.
+ elim H9.clear H9.
+ elim H11.clear H11.
+ cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))/m2 =
+ h21 (j/m1) (j\mod m1))
+ [cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))\mod m2 =
+ h22 (j/m1) (j\mod m1))
+ [rewrite > Hcut3.
+ rewrite > Hcut4.
+ rewrite > H13.
+ rewrite > H14.
+ apply sym_eq.
+ apply div_mod.
+ assumption
+ |apply mod_plus_times.
+ assumption
+ ]
+ |apply div_plus_times.
+ assumption
+ ]
+ |apply lt_mod_m_m.
+ assumption
+ ]
+ |apply (lt_times_n_to_lt m1)
+ [assumption
+ |apply (le_to_lt_to_lt ? j)
+ [apply (eq_plus_to_le ? ? (j \mod m1)).
+ apply div_mod.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ |apply not_le_to_lt.unfold.intro.
+ generalize in match H5.
+ apply (le_n_O_elim ? H7).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ |cut (O < m1)
+ [cut (j/m1 < n1)
+ [cut (j \mod m1 < m1)
+ [elim (and_true ? ? H6).
+ elim (H4 ? ? Hcut1 Hcut2 H7 H8).
+ elim H9.clear H9.
+ elim H11.clear H11.
+ elim H9.clear H9.
+ elim H11.clear H11.
+ apply (lt_times_plus_times ? ? ? m2)
+ [assumption|assumption]
+ |apply lt_mod_m_m.
+ assumption
+ ]
+ |apply (lt_times_n_to_lt m1)
+ [assumption
+ |apply (le_to_lt_to_lt ? j)
+ [apply (eq_plus_to_le ? ? (j \mod m1)).
+ apply div_mod.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ |apply not_le_to_lt.unfold.intro.
+ generalize in match H5.
+ apply (le_n_O_elim ? H7).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ ]
+ ]
+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
+ ]
+ | 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
+ ]
+ | rewrite > Hcut1.
+ rewrite > Hcut.
+ assumption
+ ]
+ | rewrite > Hcut1.
+ rewrite > Hcut.
+ assumption
+ ]
+ | cut(O \lt m1)
+ [ cut(O \lt n1)
+ [ apply (lt_to_le_to_lt ? ((h11 i j)*m1 + m1) )
+ [ 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
+ ]
+ ]
+ ]
+]
+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
+ ]
+ ]
+qed.
+
+
projects / helm.git / blobdiff