(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/Z/sigma_p".
-
include "Z/times.ma".
include "nat/primes.ma".
include "nat/ord.ma".
apply (iter_p_gen_plusA Z n k p g OZ Zplus)
[ apply symmetricZPlus.
| intros.
- apply cic:/matita/Z/plus/Zplus_z_OZ.con
+ apply Zplus_z_OZ.
| apply associative_Zplus
]
qed.
(\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 n p1 (\lambda x.g(h x)) = sigma_p n1 (\lambda x.p2 x) g.
+sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 p2 g.
intros.
unfold sigma_p.
apply (eq_iter_p_gen_gh Z OZ Zplus ? ? ? g h h1 n n1 p1 p2)
|intros.apply sym_Ztimes
]
qed.
+
+
+theorem sigma_p_knm:
+\forall g: nat \to Z.
+\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
+sigma_p k p1 g=
+sigma_p n p21 (\lambda x:nat.sigma_p m (p22 x) (\lambda y. g (h2 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+apply iter_p_gen_knm
+ [ apply symmetricZPlus
+ |apply associative_Zplus
+ | intro.
+ apply (Zplus_z_OZ a)
+ | exact h11
+ | exact h12
+ | assumption
+ | assumption
+ ]
+qed.
+
+
+theorem sigma_p2_eq:
+\forall g: nat \to nat \to Z.
+\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
+sigma_p n1 p11 (\lambda x:nat .sigma_p m1 (p12 x) (\lambda y. g x y)) =
+sigma_p n2 p21 (\lambda x:nat .sigma_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? (? ? ? ? (\lambda x:?.%) ? ?) ?).
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+
+apply(iter_p_gen_2_eq Z OZ Zplus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12 p22)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intro.
+ apply (Zplus_z_OZ a)
+| assumption
+| assumption
+]
+qed.
+
+
+
+
+(*
+
+
+
+
+
+rewrite < sigma_p2'.
+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 ? ?
+(sigma_p n2 p21 (\lambda x:nat. sigma_p m2 (p22 x)
+ (\lambda y:nat.(g (((h11 x y)*m1+(h12 x y))/m1) (((h11 x y)*m1+(h12 x y))\mod m1)) ) ) ))
+[
+ apply (sigma_p_knm (\lambda e. (g (e/m1) (e \mod m1))) ha ha12 ha22);intros
+ [ elim (and_true ? ? H3).
+ cut(O \lt m1)
+ [ cut(x/m1 < n1)
+ [ cut((x \mod m1) < m1)
+ [ elim (H1 ? ? Hcut1 Hcut2 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ split
+ [ split
+ [ split
+ [ split
+ [ assumption
+ | assumption
+ ]
+ | rewrite > H11.
+ rewrite > H10.
+ 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 H2.
+ apply (le_n_O_elim ? H6).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n.
+ ]
+ | elim (H ? ? H2 H3 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ 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 H9.
+ apply (le_n_O_elim ? H8).
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ | apply not_le_to_lt.unfold.intro.
+ generalize in match H7.
+ apply (le_n_O_elim ? H8).
+ 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_sigma_p1)
+ [ intros. reflexivity
+ | intros.
+ apply (eq_sigma_p1)
+ [ intros. reflexivity
+ | intros.
+ rewrite > (div_plus_times)
+ [ rewrite > (mod_plus_times)
+ [ reflexivity
+ | elim (H x x1 H2 H4 H3 H5).
+ assumption
+ ]
+ | elim (H x x1 H2 H4 H3 H5).
+ assumption
+ ]
+ ]
+ ]
+]
+qed.
+
+rewrite < sigma_p2' in \vdash (? ? ? %).
+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 ? ?
+ (sigma_p (n2*m2) (\lambda x:nat.p21 (x/m2)\land p22 (x/m2) (x\mod m2))
+ (\lambda x:nat.g ((h x)/m1) ((h x)\mod m1))))
+ [clear h.clear h1.
+ apply eq_sigma_p1
+ [intros.reflexivity
+ |intros.
+ cut (O < m2)
+ [cut (x/m2 < n2)
+ [cut (x \mod m2 < m2)
+ [elim (and_true ? ? H3).
+ elim (H ? ? Hcut1 Hcut2 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ 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 H2.
+ apply (le_n_O_elim ? H4).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ ]
+ |apply (eq_sigma_p_gh ? h h1);intros
+ [cut (O < m2)
+ [cut (i/m2 < n2)
+ [cut (i \mod m2 < m2)
+ [elim (and_true ? ? H3).
+ elim (H ? ? Hcut1 Hcut2 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ 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 > H6.
+ rewrite > H12.
+ 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 H2.
+ apply (le_n_O_elim ? H4).
+ 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 ? ? H3).
+ elim (H ? ? Hcut1 Hcut2 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ 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 > H10.
+ rewrite > H11.
+ 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 H2.
+ apply (le_n_O_elim ? H4).
+ 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 ? ? H3).
+ elim (H ? ? Hcut1 Hcut2 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ 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 H2.
+ apply (le_n_O_elim ? H4).
+ 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 ? ? H3).
+ elim (H1 ? ? Hcut1 Hcut2 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ 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 > H6.
+ rewrite > H12.
+ 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 H2.
+ apply (le_n_O_elim ? H4).
+ 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 ? ? H3).
+ elim (H1 ? ? Hcut1 Hcut2 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ 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 > H10.
+ rewrite > H11.
+ 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 H2.
+ apply (le_n_O_elim ? H4).
+ 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 ? ? H3).
+ elim (H1 ? ? Hcut1 Hcut2 H4 H5).
+ elim H6.clear H6.
+ elim H8.clear H8.
+ elim H6.clear H6.
+ elim H8.clear H8.
+ 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 H2.
+ apply (le_n_O_elim ? H4).
+ rewrite < times_n_O.
+ apply le_to_not_lt.
+ apply le_O_n
+ ]
+ ]
+ ]
+qed.
+*)
+
+
projects / helm.git / blobdiff