--- a/src/HOL/NumberTheory/Int2.thy Thu Dec 08 12:50:03 2005 +0100
+++ b/src/HOL/NumberTheory/Int2.thy Thu Dec 08 12:50:04 2005 +0100
@@ -5,14 +5,14 @@
header {*Integers: Divisibility and Congruences*}
-theory Int2 imports Finite2 WilsonRuss begin;
+theory Int2 imports Finite2 WilsonRuss begin
text{*Note. This theory is being revised. See the web page
\url{http://www.andrew.cmu.edu/~avigad/isabelle}.*}
constdefs
MultInv :: "int => int => int"
- "MultInv p x == x ^ nat (p - 2)";
+ "MultInv p x == x ^ nat (p - 2)"
(*****************************************************************)
(* *)
@@ -20,69 +20,68 @@
(* *)
(*****************************************************************)
-lemma zpower_zdvd_prop1 [rule_format]: "((0 < n) & (p dvd y)) -->
- p dvd ((y::int) ^ n)";
- by (induct_tac n, auto simp add: zdvd_zmult zdvd_zmult2 [of p y])
+lemma zpower_zdvd_prop1:
+ "0 < n \<Longrightarrow> p dvd y \<Longrightarrow> p dvd ((y::int) ^ n)"
+ by (induct n) (auto simp add: zdvd_zmult zdvd_zmult2 [of p y])
-lemma zdvd_bounds: "n dvd m ==> (m \<le> (0::int) | n \<le> m)";
-proof -;
- assume "n dvd m";
- then have "~(0 < m & m < n)";
- apply (insert zdvd_not_zless [of m n])
- by (rule contrapos_pn, auto)
- then have "(~0 < m | ~m < n)" by auto
+lemma zdvd_bounds: "n dvd m ==> m \<le> (0::int) | n \<le> m"
+proof -
+ assume "n dvd m"
+ then have "~(0 < m & m < n)"
+ using zdvd_not_zless [of m n] by auto
then show ?thesis by auto
-qed;
-
-lemma aux4: " -(m * n) = (-m) * (n::int)";
- by auto
+qed
lemma zprime_zdvd_zmult_better: "[| zprime p; p dvd (m * n) |] ==>
- (p dvd m) | (p dvd n)";
- apply (case_tac "0 \<le> m")
+ (p dvd m) | (p dvd n)"
+ apply (cases "0 \<le> m")
apply (simp add: zprime_zdvd_zmult)
- by (insert zprime_zdvd_zmult [of "-m" p n], auto)
+ apply (insert zprime_zdvd_zmult [of "-m" p n])
+ apply auto
+ done
-lemma zpower_zdvd_prop2 [rule_format]: "zprime p --> p dvd ((y::int) ^ n)
- --> 0 < n --> p dvd y";
- apply (induct_tac n, auto)
- apply (frule zprime_zdvd_zmult_better, auto)
-done
-
-lemma stupid: "(0 :: int) \<le> y ==> x \<le> x + y";
- by arith
+lemma zpower_zdvd_prop2:
+ "zprime p \<Longrightarrow> p dvd ((y::int) ^ n) \<Longrightarrow> 0 < n \<Longrightarrow> p dvd y"
+ apply (induct n)
+ apply simp
+ apply (frule zprime_zdvd_zmult_better)
+ apply simp
+ apply force
+ done
-lemma div_prop1: "[| 0 < z; (x::int) < y * z |] ==> x div z < y";
-proof -;
- assume "0 < z";
- then have "(x div z) * z \<le> (x div z) * z + x mod z";
- apply (rule_tac x = "x div z * z" in stupid)
- by (simp add: pos_mod_sign)
- also have "... = x";
- by (auto simp add: zmod_zdiv_equality [THEN sym] zmult_ac)
- also assume "x < y * z";
- finally show ?thesis;
+lemma div_prop1: "[| 0 < z; (x::int) < y * z |] ==> x div z < y"
+proof -
+ assume "0 < z"
+ then have "(x div z) * z \<le> (x div z) * z + x mod z"
+ by arith
+ also have "... = x"
+ by (auto simp add: zmod_zdiv_equality [symmetric] zmult_ac)
+ also assume "x < y * z"
+ finally show ?thesis
by (auto simp add: prems mult_less_cancel_right, insert prems, arith)
-qed;
+qed
-lemma div_prop2: "[| 0 < z; (x::int) < (y * z) + z |] ==> x div z \<le> y";
-proof -;
- assume "0 < z" and "x < (y * z) + z";
+lemma div_prop2: "[| 0 < z; (x::int) < (y * z) + z |] ==> x div z \<le> y"
+proof -
+ assume "0 < z" and "x < (y * z) + z"
then have "x < (y + 1) * z" by (auto simp add: int_distrib)
- then have "x div z < y + 1";
- by (rule_tac y = "y + 1" in div_prop1, auto simp add: prems)
+ then have "x div z < y + 1"
+ apply -
+ apply (rule_tac y = "y + 1" in div_prop1)
+ apply (auto simp add: prems)
+ done
then show ?thesis by auto
-qed;
+qed
-lemma zdiv_leq_prop: "[| 0 < y |] ==> y * (x div y) \<le> (x::int)";
-proof-;
- assume "0 < y";
+lemma zdiv_leq_prop: "[| 0 < y |] ==> y * (x div y) \<le> (x::int)"
+proof-
+ assume "0 < y"
from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto
- moreover have "0 \<le> x mod y";
+ moreover have "0 \<le> x mod y"
by (auto simp add: prems pos_mod_sign)
- ultimately show ?thesis;
+ ultimately show ?thesis
by arith
-qed;
+qed
(*****************************************************************)
(* *)
@@ -90,96 +89,102 @@
(* *)
(*****************************************************************)
-lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)";
+lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)"
by (auto simp add: zcong_def)
-lemma zcong_id: "[m = 0] (mod m)";
+lemma zcong_id: "[m = 0] (mod m)"
by (auto simp add: zcong_def zdvd_0_right)
-lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)";
+lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)"
by (auto simp add: zcong_refl zcong_zadd)
-lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)";
- by (induct_tac z, auto simp add: zcong_zmult)
+lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)"
+ by (induct z) (auto simp add: zcong_zmult)
lemma zcong_eq_trans: "[| [a = b](mod m); b = c; [c = d](mod m) |] ==>
- [a = d](mod m)";
- by (auto, rule_tac b = c in zcong_trans)
+ [a = d](mod m)"
+ apply (erule zcong_trans)
+ apply simp
+ done
-lemma aux1: "a - b = (c::int) ==> a = c + b";
+lemma aux1: "a - b = (c::int) ==> a = c + b"
by auto
lemma zcong_zmult_prop1: "[a = b](mod m) ==> ([c = a * d](mod m) =
- [c = b * d] (mod m))";
+ [c = b * d] (mod m))"
apply (auto simp add: zcong_def dvd_def)
apply (rule_tac x = "ka + k * d" in exI)
- apply (drule aux1)+;
+ apply (drule aux1)+
apply (auto simp add: int_distrib)
apply (rule_tac x = "ka - k * d" in exI)
- apply (drule aux1)+;
+ apply (drule aux1)+
apply (auto simp add: int_distrib)
-done
+ done
lemma zcong_zmult_prop2: "[a = b](mod m) ==>
- ([c = d * a](mod m) = [c = d * b] (mod m))";
+ ([c = d * a](mod m) = [c = d * b] (mod m))"
by (auto simp add: zmult_ac zcong_zmult_prop1)
lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p);
- ~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)";
+ ~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)"
apply (auto simp add: zcong_def)
apply (drule zprime_zdvd_zmult_better, auto)
-done
+ done
lemma zcong_less_eq: "[| 0 < x; 0 < y; 0 < m; [x = y] (mod m);
- x < m; y < m |] ==> x = y";
+ x < m; y < m |] ==> x = y"
apply (simp add: zcong_zmod_eq)
- apply (subgoal_tac "(x mod m) = x");
- apply (subgoal_tac "(y mod m) = y");
+ apply (subgoal_tac "(x mod m) = x")
+ apply (subgoal_tac "(y mod m) = y")
apply simp
apply (rule_tac [1-2] mod_pos_pos_trivial)
-by auto
+ apply auto
+ done
lemma zcong_neg_1_impl_ne_1: "[| 2 < p; [x = -1] (mod p) |] ==>
- ~([x = 1] (mod p))";
-proof;
+ ~([x = 1] (mod p))"
+proof
assume "2 < p" and "[x = 1] (mod p)" and "[x = -1] (mod p)"
- then have "[1 = -1] (mod p)";
+ then have "[1 = -1] (mod p)"
apply (auto simp add: zcong_sym)
apply (drule zcong_trans, auto)
- done
- then have "[1 + 1 = -1 + 1] (mod p)";
+ done
+ then have "[1 + 1 = -1 + 1] (mod p)"
by (simp only: zcong_shift)
- then have "[2 = 0] (mod p)";
+ then have "[2 = 0] (mod p)"
by auto
- then have "p dvd 2";
+ then have "p dvd 2"
by (auto simp add: dvd_def zcong_def)
- with prems show False;
+ with prems show False
by (auto simp add: zdvd_not_zless)
-qed;
+qed
-lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)";
+lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)"
by (auto simp add: zcong_def)
lemma zcong_zprime_prod_zero: "[| zprime p; 0 < a |] ==>
- [a * b = 0] (mod p) ==> [a = 0] (mod p) | [b = 0] (mod p)";
+ [a * b = 0] (mod p) ==> [a = 0] (mod p) | [b = 0] (mod p)"
by (auto simp add: zcong_zero_equiv_div zprime_zdvd_zmult)
lemma zcong_zprime_prod_zero_contra: "[| zprime p; 0 < a |] ==>
- ~[a = 0](mod p) & ~[b = 0](mod p) ==> ~[a * b = 0] (mod p)";
+ ~[a = 0](mod p) & ~[b = 0](mod p) ==> ~[a * b = 0] (mod p)"
apply auto
apply (frule_tac a = a and b = b and p = p in zcong_zprime_prod_zero)
-by auto
+ apply auto
+ done
-lemma zcong_not_zero: "[| 0 < x; x < m |] ==> ~[x = 0] (mod m)";
+lemma zcong_not_zero: "[| 0 < x; x < m |] ==> ~[x = 0] (mod m)"
by (auto simp add: zcong_zero_equiv_div zdvd_not_zless)
-lemma zcong_zero: "[| 0 \<le> x; x < m; [x = 0](mod m) |] ==> x = 0";
+lemma zcong_zero: "[| 0 \<le> x; x < m; [x = 0](mod m) |] ==> x = 0"
apply (drule order_le_imp_less_or_eq, auto)
-by (frule_tac m = m in zcong_not_zero, auto)
+ apply (frule_tac m = m in zcong_not_zero)
+ apply auto
+ done
lemma all_relprime_prod_relprime: "[| finite A; \<forall>x \<in> A. (zgcd(x,y) = 1) |]
- ==> zgcd (setprod id A,y) = 1";
- by (induct set: Finites, auto simp add: zgcd_zgcd_zmult)
+ ==> zgcd (setprod id A,y) = 1"
+ by (induct set: Finites) (auto simp add: zgcd_zgcd_zmult)
(*****************************************************************)
(* *)
@@ -188,69 +193,69 @@
(*****************************************************************)
lemma MultInv_prop1: "[| 2 < p; [x = y] (mod p) |] ==>
- [(MultInv p x) = (MultInv p y)] (mod p)";
+ [(MultInv p x) = (MultInv p y)] (mod p)"
by (auto simp add: MultInv_def zcong_zpower)
lemma MultInv_prop2: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
- [(x * (MultInv p x)) = 1] (mod p)";
-proof (simp add: MultInv_def zcong_eq_zdvd_prop);
- assume "2 < p" and "zprime p" and "~ p dvd x";
- have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)";
+ [(x * (MultInv p x)) = 1] (mod p)"
+proof (simp add: MultInv_def zcong_eq_zdvd_prop)
+ assume "2 < p" and "zprime p" and "~ p dvd x"
+ have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)"
by auto
- also from prems have "nat (p - 2) + 1 = nat (p - 2 + 1)";
+ also from prems have "nat (p - 2) + 1 = nat (p - 2 + 1)"
by (simp only: nat_add_distrib, auto)
also have "p - 2 + 1 = p - 1" by arith
- finally have "[x * x ^ nat (p - 2) = x ^ nat (p - 1)] (mod p)";
+ finally have "[x * x ^ nat (p - 2) = x ^ nat (p - 1)] (mod p)"
by (rule ssubst, auto)
- also from prems have "[x ^ nat (p - 1) = 1] (mod p)";
+ also from prems have "[x ^ nat (p - 1) = 1] (mod p)"
by (auto simp add: Little_Fermat)
- finally (zcong_trans) show "[x * x ^ nat (p - 2) = 1] (mod p)";.;
-qed;
+ finally (zcong_trans) show "[x * x ^ nat (p - 2) = 1] (mod p)" .
+qed
lemma MultInv_prop2a: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
- [(MultInv p x) * x = 1] (mod p)";
+ [(MultInv p x) * x = 1] (mod p)"
by (auto simp add: MultInv_prop2 zmult_ac)
-lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))";
+lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))"
by (simp add: nat_diff_distrib)
-lemma aux_2: "2 < p ==> 0 < nat (p - 2)";
+lemma aux_2: "2 < p ==> 0 < nat (p - 2)"
by auto
lemma MultInv_prop3: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
- ~([MultInv p x = 0](mod p))";
+ ~([MultInv p x = 0](mod p))"
apply (auto simp add: MultInv_def zcong_eq_zdvd_prop aux_1)
apply (drule aux_2)
apply (drule zpower_zdvd_prop2, auto)
-done
+ done
lemma aux__1: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==>
[(MultInv p (MultInv p x)) = (x * (MultInv p x) *
- (MultInv p (MultInv p x)))] (mod p)";
+ (MultInv p (MultInv p x)))] (mod p)"
apply (drule MultInv_prop2, auto)
- apply (drule_tac k = "MultInv p (MultInv p x)" in zcong_scalar, auto);
+ apply (drule_tac k = "MultInv p (MultInv p x)" in zcong_scalar, auto)
apply (auto simp add: zcong_sym)
-done
+ done
lemma aux__2: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==>
- [(x * (MultInv p x) * (MultInv p (MultInv p x))) = x] (mod p)";
+ [(x * (MultInv p x) * (MultInv p (MultInv p x))) = x] (mod p)"
apply (frule MultInv_prop3, auto)
apply (insert MultInv_prop2 [of p "MultInv p x"], auto)
apply (drule MultInv_prop2, auto)
apply (drule_tac k = x in zcong_scalar2, auto)
apply (auto simp add: zmult_ac)
-done
+ done
lemma MultInv_prop4: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
- [(MultInv p (MultInv p x)) = x] (mod p)";
+ [(MultInv p (MultInv p x)) = x] (mod p)"
apply (frule aux__1, auto)
apply (drule aux__2, auto)
apply (drule zcong_trans, auto)
-done
+ done
lemma MultInv_prop5: "[| 2 < p; zprime p; ~([x = 0](mod p));
~([y = 0](mod p)); [(MultInv p x) = (MultInv p y)] (mod p) |] ==>
- [x = y] (mod p)";
+ [x = y] (mod p)"
apply (drule_tac a = "MultInv p x" and b = "MultInv p y" and
m = p and k = x in zcong_scalar)
apply (insert MultInv_prop2 [of p x], simp)
@@ -261,38 +266,38 @@
apply (insert MultInv_prop2a [of p y], auto simp add: zmult_ac)
apply (insert zcong_zmult_prop2 [of "y * MultInv p y" 1 p y x])
apply (auto simp add: zcong_sym)
-done
+ done
lemma MultInv_zcong_prop1: "[| 2 < p; [j = k] (mod p) |] ==>
- [a * MultInv p j = a * MultInv p k] (mod p)";
+ [a * MultInv p j = a * MultInv p k] (mod p)"
by (drule MultInv_prop1, auto simp add: zcong_scalar2)
lemma aux___1: "[j = a * MultInv p k] (mod p) ==>
- [j * k = a * MultInv p k * k] (mod p)";
+ [j * k = a * MultInv p k * k] (mod p)"
by (auto simp add: zcong_scalar)
lemma aux___2: "[|2 < p; zprime p; ~([k = 0](mod p));
- [j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)";
+ [j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)"
apply (insert MultInv_prop2a [of p k] zcong_zmult_prop2
[of "MultInv p k * k" 1 p "j * k" a])
apply (auto simp add: zmult_ac)
-done
+ done
lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k =
- (MultInv p j) * a] (mod p)";
+ (MultInv p j) * a] (mod p)"
by (auto simp add: zmult_assoc zcong_scalar2)
lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p));
[(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |]
- ==> [k = a * (MultInv p j)] (mod p)";
+ ==> [k = a * (MultInv p j)] (mod p)"
apply (insert MultInv_prop2a [of p j] zcong_zmult_prop1
[of "MultInv p j * j" 1 p "MultInv p j * a" k])
apply (auto simp add: zmult_ac zcong_sym)
-done
+ done
lemma MultInv_zcong_prop2: "[| 2 < p; zprime p; ~([k = 0](mod p));
~([j = 0](mod p)); [j = a * MultInv p k] (mod p) |] ==>
- [k = a * MultInv p j] (mod p)";
+ [k = a * MultInv p j] (mod p)"
apply (drule aux___1)
apply (frule aux___2, auto)
by (drule aux___3, drule aux___4, auto)
@@ -300,11 +305,11 @@
lemma MultInv_zcong_prop3: "[| 2 < p; zprime p; ~([a = 0](mod p));
~([k = 0](mod p)); ~([j = 0](mod p));
[a * MultInv p j = a * MultInv p k] (mod p) |] ==>
- [j = k] (mod p)";
+ [j = k] (mod p)"
apply (auto simp add: zcong_eq_zdvd_prop [of a p])
apply (frule zprime_imp_zrelprime, auto)
apply (insert zcong_cancel2 [of p a "MultInv p j" "MultInv p k"], auto)
apply (drule MultInv_prop5, auto)
-done
+ done
end