src/HOL/NumberTheory/Quadratic_Reciprocity.thy
author wenzelm
Thu Dec 08 12:50:04 2005 +0100 (2005-12-08)
changeset 18369 694ea14ab4f2
parent 16733 236dfafbeb63
child 19670 2e4a143c73c5
permissions -rw-r--r--
tuned sources and proofs
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(*  Title:      HOL/Quadratic_Reciprocity/Quadratic_Reciprocity.thy
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    ID:         $Id$
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    Authors:    Jeremy Avigad, David Gray, and Adam Kramer
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*)
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header {* The law of Quadratic reciprocity *}
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theory Quadratic_Reciprocity
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imports Gauss
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begin
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(***************************************************************)
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(*                                                             *)
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(*  Lemmas leading up to the proof of theorem 3.3 in           *)
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(*  Niven and Zuckerman's presentation                         *)
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(*                                                             *)
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(***************************************************************)
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lemma (in GAUSS) QRLemma1: "a * setsum id A =
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  p * setsum (%x. ((x * a) div p)) A + setsum id D + setsum id E"
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proof -
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  from finite_A have "a * setsum id A = setsum (%x. a * x) A"
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    by (auto simp add: setsum_const_mult id_def)
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  also have "setsum (%x. a * x) = setsum (%x. x * a)"
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    by (auto simp add: zmult_commute)
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  also have "setsum (%x. x * a) A = setsum id B"
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    by (simp add: B_def setsum_reindex_id[OF inj_on_xa_A])
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  also have "... = setsum (%x. p * (x div p) + StandardRes p x) B"
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    by (auto simp add: StandardRes_def zmod_zdiv_equality)
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  also have "... = setsum (%x. p * (x div p)) B + setsum (StandardRes p) B"
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    by (rule setsum_addf)
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  also have "setsum (StandardRes p) B = setsum id C"
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    by (auto simp add: C_def setsum_reindex_id[OF SR_B_inj])
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  also from C_eq have "... = setsum id (D \<union> E)"
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    by auto
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  also from finite_D finite_E have "... = setsum id D + setsum id E"
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    by (rule setsum_Un_disjoint) (auto simp add: D_def E_def)
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  also have "setsum (%x. p * (x div p)) B =
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      setsum ((%x. p * (x div p)) o (%x. (x * a))) A"
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    by (auto simp add: B_def setsum_reindex inj_on_xa_A)
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  also have "... = setsum (%x. p * ((x * a) div p)) A"
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    by (auto simp add: o_def)
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  also from finite_A have "setsum (%x. p * ((x * a) div p)) A =
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    p * setsum (%x. ((x * a) div p)) A"
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    by (auto simp add: setsum_const_mult)
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  finally show ?thesis by arith
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qed
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lemma (in GAUSS) QRLemma2: "setsum id A = p * int (card E) - setsum id E +
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  setsum id D"
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proof -
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  from F_Un_D_eq_A have "setsum id A = setsum id (D \<union> F)"
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    by (simp add: Un_commute)
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  also from F_D_disj finite_D finite_F
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  have "... = setsum id D + setsum id F"
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    by (auto simp add: Int_commute intro: setsum_Un_disjoint)
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  also from F_def have "F = (%x. (p - x)) ` E"
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    by auto
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  also from finite_E inj_on_pminusx_E have "setsum id ((%x. (p - x)) ` E) =
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      setsum (%x. (p - x)) E"
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    by (auto simp add: setsum_reindex)
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  also from finite_E have "setsum (op - p) E = setsum (%x. p) E - setsum id E"
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    by (auto simp add: setsum_subtractf id_def)
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  also from finite_E have "setsum (%x. p) E = p * int(card E)"
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    by (intro setsum_const)
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  finally show ?thesis
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    by arith
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qed
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lemma (in GAUSS) QRLemma3: "(a - 1) * setsum id A =
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    p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E"
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proof -
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  have "(a - 1) * setsum id A = a * setsum id A - setsum id A"
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    by (auto simp add: zdiff_zmult_distrib)
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  also note QRLemma1
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  also from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
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     setsum id E - setsum id A =
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      p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
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      setsum id E - (p * int (card E) - setsum id E + setsum id D)"
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    by auto
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  also have "... = p * (\<Sum>x \<in> A. x * a div p) -
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      p * int (card E) + 2 * setsum id E"
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    by arith
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  finally show ?thesis
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    by (auto simp only: zdiff_zmult_distrib2)
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qed
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lemma (in GAUSS) QRLemma4: "a \<in> zOdd ==>
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    (setsum (%x. ((x * a) div p)) A \<in> zEven) = (int(card E): zEven)"
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proof -
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  assume a_odd: "a \<in> zOdd"
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  from QRLemma3 have a: "p * (setsum (%x. ((x * a) div p)) A - int(card E)) =
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      (a - 1) * setsum id A - 2 * setsum id E"
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    by arith
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  from a_odd have "a - 1 \<in> zEven"
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    by (rule odd_minus_one_even)
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  hence "(a - 1) * setsum id A \<in> zEven"
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    by (rule even_times_either)
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  moreover have "2 * setsum id E \<in> zEven"
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    by (auto simp add: zEven_def)
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  ultimately have "(a - 1) * setsum id A - 2 * setsum id E \<in> zEven"
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    by (rule even_minus_even)
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  with a have "p * (setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
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    by simp
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  hence "p \<in> zEven | (setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
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    by (rule EvenOdd.even_product)
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  with p_odd have "(setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
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    by (auto simp add: odd_iff_not_even)
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  thus ?thesis
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    by (auto simp only: even_diff [symmetric])
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qed
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lemma (in GAUSS) QRLemma5: "a \<in> zOdd ==>
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   (-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"
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proof -
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  assume "a \<in> zOdd"
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  from QRLemma4 have
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    "(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A \<in> zEven)"..
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  moreover have "0 \<le> int(card E)"
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    by auto
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  moreover have "0 \<le> setsum (%x. ((x * a) div p)) A"
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    proof (intro setsum_nonneg)
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      show "\<forall>x \<in> A. 0 \<le> x * a div p"
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      proof
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        fix x
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        assume "x \<in> A"
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        then have "0 \<le> x"
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          by (auto simp add: A_def)
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        with a_nonzero have "0 \<le> x * a"
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          by (auto simp add: zero_le_mult_iff)
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        with p_g_2 show "0 \<le> x * a div p"
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          by (auto simp add: pos_imp_zdiv_nonneg_iff)
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      qed
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    qed
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  ultimately have "(-1::int)^nat((int (card E))) =
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      (-1)^nat(((\<Sum>x \<in> A. x * a div p)))"
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    by (intro neg_one_power_parity, auto)
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  also have "nat (int(card E)) = card E"
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    by auto
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  finally show ?thesis .
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qed
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lemma MainQRLemma: "[| a \<in> zOdd; 0 < a; ~([a = 0] (mod p)); zprime p; 2 < p;
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  A = {x. 0 < x & x \<le> (p - 1) div 2} |] ==>
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  (Legendre a p) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"
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  apply (subst GAUSS.gauss_lemma)
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  apply (auto simp add: GAUSS_def)
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  apply (subst GAUSS.QRLemma5)
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  apply (auto simp add: GAUSS_def)
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  done
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(******************************************************************)
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(*                                                                *)
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(* Stuff about S, S1 and S2...                                    *)
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(*                                                                *)
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(******************************************************************)
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locale QRTEMP =
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  fixes p     :: "int"
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  fixes q     :: "int"
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  fixes P_set :: "int set"
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  fixes Q_set :: "int set"
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  fixes S     :: "(int * int) set"
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  fixes S1    :: "(int * int) set"
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  fixes S2    :: "(int * int) set"
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  fixes f1    :: "int => (int * int) set"
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  fixes f2    :: "int => (int * int) set"
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  assumes p_prime: "zprime p"
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  assumes p_g_2: "2 < p"
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  assumes q_prime: "zprime q"
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  assumes q_g_2: "2 < q"
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  assumes p_neq_q:      "p \<noteq> q"
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  defines P_set_def: "P_set == {x. 0 < x & x \<le> ((p - 1) div 2) }"
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  defines Q_set_def: "Q_set == {x. 0 < x & x \<le> ((q - 1) div 2) }"
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  defines S_def:     "S     == P_set <*> Q_set"
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  defines S1_def:    "S1    == { (x, y). (x, y):S & ((p * y) < (q * x)) }"
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  defines S2_def:    "S2    == { (x, y). (x, y):S & ((q * x) < (p * y)) }"
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  defines f1_def:    "f1 j  == { (j1, y). (j1, y):S & j1 = j &
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                                 (y \<le> (q * j) div p) }"
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  defines f2_def:    "f2 j  == { (x, j1). (x, j1):S & j1 = j &
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                                 (x \<le> (p * j) div q) }"
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lemma (in QRTEMP) p_fact: "0 < (p - 1) div 2"
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proof -
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  from prems have "2 < p" by (simp add: QRTEMP_def)
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  then have "2 \<le> p - 1" by arith
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  then have "2 div 2 \<le> (p - 1) div 2" by (rule zdiv_mono1, auto)
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  then show ?thesis by auto
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qed
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lemma (in QRTEMP) q_fact: "0 < (q - 1) div 2"
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proof -
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  from prems have "2 < q" by (simp add: QRTEMP_def)
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  then have "2 \<le> q - 1" by arith
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  then have "2 div 2 \<le> (q - 1) div 2" by (rule zdiv_mono1, auto)
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  then show ?thesis by auto
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qed
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lemma (in QRTEMP) pb_neq_qa: "[|1 \<le> b; b \<le> (q - 1) div 2 |] ==>
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    (p * b \<noteq> q * a)"
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proof
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  assume "p * b = q * a" and "1 \<le> b" and "b \<le> (q - 1) div 2"
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  then have "q dvd (p * b)" by (auto simp add: dvd_def)
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  with q_prime p_g_2 have "q dvd p | q dvd b"
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    by (auto simp add: zprime_zdvd_zmult)
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  moreover have "~ (q dvd p)"
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  proof
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    assume "q dvd p"
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    with p_prime have "q = 1 | q = p"
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      apply (auto simp add: zprime_def QRTEMP_def)
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      apply (drule_tac x = q and R = False in allE)
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      apply (simp add: QRTEMP_def)
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      apply (subgoal_tac "0 \<le> q", simp add: QRTEMP_def)
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      apply (insert prems)
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      apply (auto simp add: QRTEMP_def)
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      done
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    with q_g_2 p_neq_q show False by auto
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  qed
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  ultimately have "q dvd b" by auto
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  then have "q \<le> b"
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  proof -
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    assume "q dvd b"
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    moreover from prems have "0 < b" by auto
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    ultimately show ?thesis using zdvd_bounds [of q b] by auto
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  qed
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  with prems have "q \<le> (q - 1) div 2" by auto
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  then have "2 * q \<le> 2 * ((q - 1) div 2)" by arith
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  then have "2 * q \<le> q - 1"
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  proof -
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    assume "2 * q \<le> 2 * ((q - 1) div 2)"
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    with prems have "q \<in> zOdd" by (auto simp add: QRTEMP_def zprime_zOdd_eq_grt_2)
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    with odd_minus_one_even have "(q - 1):zEven" by auto
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    with even_div_2_prop2 have "(q - 1) = 2 * ((q - 1) div 2)" by auto
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    with prems show ?thesis by auto
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  qed
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  then have p1: "q \<le> -1" by arith
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  with q_g_2 show False by auto
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qed
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lemma (in QRTEMP) P_set_finite: "finite (P_set)"
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  using p_fact by (auto simp add: P_set_def bdd_int_set_l_le_finite)
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lemma (in QRTEMP) Q_set_finite: "finite (Q_set)"
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  using q_fact by (auto simp add: Q_set_def bdd_int_set_l_le_finite)
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lemma (in QRTEMP) S_finite: "finite S"
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  by (auto simp add: S_def  P_set_finite Q_set_finite finite_cartesian_product)
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lemma (in QRTEMP) S1_finite: "finite S1"
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proof -
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  have "finite S" by (auto simp add: S_finite)
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  moreover have "S1 \<subseteq> S" by (auto simp add: S1_def S_def)
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  ultimately show ?thesis by (auto simp add: finite_subset)
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qed
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lemma (in QRTEMP) S2_finite: "finite S2"
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proof -
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  have "finite S" by (auto simp add: S_finite)
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  moreover have "S2 \<subseteq> S" by (auto simp add: S2_def S_def)
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  ultimately show ?thesis by (auto simp add: finite_subset)
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qed
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lemma (in QRTEMP) P_set_card: "(p - 1) div 2 = int (card (P_set))"
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  using p_fact by (auto simp add: P_set_def card_bdd_int_set_l_le)
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lemma (in QRTEMP) Q_set_card: "(q - 1) div 2 = int (card (Q_set))"
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  using q_fact by (auto simp add: Q_set_def card_bdd_int_set_l_le)
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lemma (in QRTEMP) S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
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  using P_set_card Q_set_card P_set_finite Q_set_finite
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  by (auto simp add: S_def zmult_int setsum_constant)
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lemma (in QRTEMP) S1_Int_S2_prop: "S1 \<inter> S2 = {}"
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  by (auto simp add: S1_def S2_def)
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lemma (in QRTEMP) S1_Union_S2_prop: "S = S1 \<union> S2"
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  apply (auto simp add: S_def P_set_def Q_set_def S1_def S2_def)
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proof -
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  fix a and b
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  assume "~ q * a < p * b" and b1: "0 < b" and b2: "b \<le> (q - 1) div 2"
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  with zless_linear have "(p * b < q * a) | (p * b = q * a)" by auto
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  moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> q * a)" by auto
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  ultimately show "p * b < q * a" by auto
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qed
paulson@13871
   287
wenzelm@18369
   288
lemma (in QRTEMP) card_sum_S1_S2: "((p - 1) div 2) * ((q - 1) div 2) =
nipkow@15392
   289
    int(card(S1)) + int(card(S2))"
wenzelm@18369
   290
proof -
nipkow@15392
   291
  have "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
paulson@13871
   292
    by (auto simp add: S_card)
nipkow@15392
   293
  also have "... = int( card(S1) + card(S2))"
paulson@13871
   294
    apply (insert S1_finite S2_finite S1_Int_S2_prop S1_Union_S2_prop)
paulson@13871
   295
    apply (drule card_Un_disjoint, auto)
wenzelm@18369
   296
    done
paulson@13871
   297
  also have "... = int(card(S1)) + int(card(S2))" by auto
nipkow@15392
   298
  finally show ?thesis .
nipkow@15392
   299
qed
paulson@13871
   300
wenzelm@18369
   301
lemma (in QRTEMP) aux1a: "[| 0 < a; a \<le> (p - 1) div 2;
paulson@13871
   302
                             0 < b; b \<le> (q - 1) div 2 |] ==>
nipkow@15392
   303
                          (p * b < q * a) = (b \<le> q * a div p)"
nipkow@15392
   304
proof -
nipkow@15392
   305
  assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2"
nipkow@15392
   306
  have "p * b < q * a ==> b \<le> q * a div p"
nipkow@15392
   307
  proof -
nipkow@15392
   308
    assume "p * b < q * a"
paulson@13871
   309
    then have "p * b \<le> q * a" by auto
nipkow@15392
   310
    then have "(p * b) div p \<le> (q * a) div p"
wenzelm@18369
   311
      by (rule zdiv_mono1) (insert p_g_2, auto)
nipkow@15392
   312
    then show "b \<le> (q * a) div p"
paulson@13871
   313
      apply (subgoal_tac "p \<noteq> 0")
paulson@13871
   314
      apply (frule zdiv_zmult_self2, force)
wenzelm@18369
   315
      apply (insert p_g_2, auto)
wenzelm@18369
   316
      done
nipkow@15392
   317
  qed
nipkow@15392
   318
  moreover have "b \<le> q * a div p ==> p * b < q * a"
nipkow@15392
   319
  proof -
nipkow@15392
   320
    assume "b \<le> q * a div p"
nipkow@15392
   321
    then have "p * b \<le> p * ((q * a) div p)"
wenzelm@18369
   322
      using p_g_2 by (auto simp add: mult_le_cancel_left)
nipkow@15392
   323
    also have "... \<le> q * a"
wenzelm@18369
   324
      by (rule zdiv_leq_prop) (insert p_g_2, auto)
nipkow@15392
   325
    finally have "p * b \<le> q * a" .
nipkow@15392
   326
    then have "p * b < q * a | p * b = q * a"
paulson@13871
   327
      by (simp only: order_le_imp_less_or_eq)
nipkow@15392
   328
    moreover have "p * b \<noteq> q * a"
wenzelm@18369
   329
      by (rule  pb_neq_qa) (insert prems, auto)
paulson@13871
   330
    ultimately show ?thesis by auto
nipkow@15392
   331
  qed
nipkow@15392
   332
  ultimately show ?thesis ..
nipkow@15392
   333
qed
paulson@13871
   334
wenzelm@18369
   335
lemma (in QRTEMP) aux1b: "[| 0 < a; a \<le> (p - 1) div 2;
paulson@13871
   336
                             0 < b; b \<le> (q - 1) div 2 |] ==>
nipkow@15392
   337
                          (q * a < p * b) = (a \<le> p * b div q)"
nipkow@15392
   338
proof -
nipkow@15392
   339
  assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2"
nipkow@15392
   340
  have "q * a < p * b ==> a \<le> p * b div q"
nipkow@15392
   341
  proof -
nipkow@15392
   342
    assume "q * a < p * b"
paulson@13871
   343
    then have "q * a \<le> p * b" by auto
nipkow@15392
   344
    then have "(q * a) div q \<le> (p * b) div q"
wenzelm@18369
   345
      by (rule zdiv_mono1) (insert q_g_2, auto)
nipkow@15392
   346
    then show "a \<le> (p * b) div q"
paulson@13871
   347
      apply (subgoal_tac "q \<noteq> 0")
paulson@13871
   348
      apply (frule zdiv_zmult_self2, force)
wenzelm@18369
   349
      apply (insert q_g_2, auto)
wenzelm@18369
   350
      done
nipkow@15392
   351
  qed
nipkow@15392
   352
  moreover have "a \<le> p * b div q ==> q * a < p * b"
nipkow@15392
   353
  proof -
nipkow@15392
   354
    assume "a \<le> p * b div q"
nipkow@15392
   355
    then have "q * a \<le> q * ((p * b) div q)"
wenzelm@18369
   356
      using q_g_2 by (auto simp add: mult_le_cancel_left)
nipkow@15392
   357
    also have "... \<le> p * b"
wenzelm@18369
   358
      by (rule zdiv_leq_prop) (insert q_g_2, auto)
nipkow@15392
   359
    finally have "q * a \<le> p * b" .
nipkow@15392
   360
    then have "q * a < p * b | q * a = p * b"
paulson@13871
   361
      by (simp only: order_le_imp_less_or_eq)
nipkow@15392
   362
    moreover have "p * b \<noteq> q * a"
wenzelm@18369
   363
      by (rule  pb_neq_qa) (insert prems, auto)
paulson@13871
   364
    ultimately show ?thesis by auto
nipkow@15392
   365
  qed
nipkow@15392
   366
  ultimately show ?thesis ..
nipkow@15392
   367
qed
paulson@13871
   368
wenzelm@18369
   369
lemma aux2: "[| zprime p; zprime q; 2 < p; 2 < q |] ==>
nipkow@15392
   370
             (q * ((p - 1) div 2)) div p \<le> (q - 1) div 2"
nipkow@15392
   371
proof-
nipkow@16663
   372
  assume "zprime p" and "zprime q" and "2 < p" and "2 < q"
paulson@13871
   373
  (* Set up what's even and odd *)
nipkow@15392
   374
  then have "p \<in> zOdd & q \<in> zOdd"
paulson@13871
   375
    by (auto simp add:  zprime_zOdd_eq_grt_2)
nipkow@15392
   376
  then have even1: "(p - 1):zEven & (q - 1):zEven"
paulson@13871
   377
    by (auto simp add: odd_minus_one_even)
nipkow@15392
   378
  then have even2: "(2 * p):zEven & ((q - 1) * p):zEven"
paulson@13871
   379
    by (auto simp add: zEven_def)
nipkow@15392
   380
  then have even3: "(((q - 1) * p) + (2 * p)):zEven"
paulson@14434
   381
    by (auto simp: EvenOdd.even_plus_even)
paulson@13871
   382
  (* using these prove it *)
nipkow@15392
   383
  from prems have "q * (p - 1) < ((q - 1) * p) + (2 * p)"
paulson@13871
   384
    by (auto simp add: int_distrib)
nipkow@15392
   385
  then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2"
nipkow@15392
   386
    apply (rule_tac x = "((p - 1) * q)" in even_div_2_l)
paulson@13871
   387
    by (auto simp add: even3, auto simp add: zmult_ac)
nipkow@15392
   388
  also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)"
paulson@13871
   389
    by (auto simp add: even1 even_prod_div_2)
nipkow@15392
   390
  also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p"
paulson@13871
   391
    by (auto simp add: even1 even2 even_prod_div_2 even_sum_div_2)
wenzelm@18369
   392
  finally show ?thesis
wenzelm@18369
   393
    apply (rule_tac x = " q * ((p - 1) div 2)" and
nipkow@15392
   394
                    y = "(q - 1) div 2" in div_prop2)
wenzelm@18369
   395
    using prems by auto
nipkow@15392
   396
qed
paulson@13871
   397
nipkow@15392
   398
lemma (in QRTEMP) aux3a: "\<forall>j \<in> P_set. int (card (f1 j)) = (q * j) div p"
nipkow@15392
   399
proof
nipkow@15392
   400
  fix j
nipkow@15392
   401
  assume j_fact: "j \<in> P_set"
nipkow@15392
   402
  have "int (card (f1 j)) = int (card {y. y \<in> Q_set & y \<le> (q * j) div p})"
nipkow@15392
   403
  proof -
nipkow@15392
   404
    have "finite (f1 j)"
nipkow@15392
   405
    proof -
paulson@13871
   406
      have "(f1 j) \<subseteq> S" by (auto simp add: f1_def)
paulson@13871
   407
      with S_finite show ?thesis by (auto simp add: finite_subset)
nipkow@15392
   408
    qed
nipkow@15392
   409
    moreover have "inj_on (%(x,y). y) (f1 j)"
paulson@13871
   410
      by (auto simp add: f1_def inj_on_def)
nipkow@15392
   411
    ultimately have "card ((%(x,y). y) ` (f1 j)) = card  (f1 j)"
paulson@13871
   412
      by (auto simp add: f1_def card_image)
nipkow@15392
   413
    moreover have "((%(x,y). y) ` (f1 j)) = {y. y \<in> Q_set & y \<le> (q * j) div p}"
wenzelm@18369
   414
      using prems by (auto simp add: f1_def S_def Q_set_def P_set_def image_def)
paulson@13871
   415
    ultimately show ?thesis by (auto simp add: f1_def)
nipkow@15392
   416
  qed
nipkow@15392
   417
  also have "... = int (card {y. 0 < y & y \<le> (q * j) div p})"
nipkow@15392
   418
  proof -
wenzelm@18369
   419
    have "{y. y \<in> Q_set & y \<le> (q * j) div p} =
nipkow@15392
   420
        {y. 0 < y & y \<le> (q * j) div p}"
paulson@13871
   421
      apply (auto simp add: Q_set_def)
wenzelm@18369
   422
    proof -
wenzelm@18369
   423
      fix x
wenzelm@18369
   424
      assume "0 < x" and "x \<le> q * j div p"
wenzelm@18369
   425
      with j_fact P_set_def  have "j \<le> (p - 1) div 2" by auto
wenzelm@18369
   426
      with q_g_2 have "q * j \<le> q * ((p - 1) div 2)"
wenzelm@18369
   427
        by (auto simp add: mult_le_cancel_left)
wenzelm@18369
   428
      with p_g_2 have "q * j div p \<le> q * ((p - 1) div 2) div p"
wenzelm@18369
   429
        by (auto simp add: zdiv_mono1)
wenzelm@18369
   430
      also from prems have "... \<le> (q - 1) div 2"
wenzelm@18369
   431
        apply simp
wenzelm@18369
   432
        apply (insert aux2)
wenzelm@18369
   433
        apply (simp add: QRTEMP_def)
wenzelm@18369
   434
        done
wenzelm@18369
   435
      finally show "x \<le> (q - 1) div 2" using prems by auto
wenzelm@18369
   436
    qed
paulson@13871
   437
    then show ?thesis by auto
nipkow@15392
   438
  qed
nipkow@15392
   439
  also have "... = (q * j) div p"
nipkow@15392
   440
  proof -
paulson@13871
   441
    from j_fact P_set_def have "0 \<le> j" by auto
paulson@14387
   442
    with q_g_2 have "q * 0 \<le> q * j" by (auto simp only: mult_left_mono)
paulson@13871
   443
    then have "0 \<le> q * j" by auto
nipkow@15392
   444
    then have "0 div p \<le> (q * j) div p"
paulson@13871
   445
      apply (rule_tac a = 0 in zdiv_mono1)
wenzelm@18369
   446
      apply (insert p_g_2, auto)
wenzelm@18369
   447
      done
paulson@13871
   448
    also have "0 div p = 0" by auto
paulson@13871
   449
    finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
nipkow@15392
   450
  qed
nipkow@15392
   451
  finally show "int (card (f1 j)) = q * j div p" .
nipkow@15392
   452
qed
paulson@13871
   453
nipkow@15392
   454
lemma (in QRTEMP) aux3b: "\<forall>j \<in> Q_set. int (card (f2 j)) = (p * j) div q"
nipkow@15392
   455
proof
nipkow@15392
   456
  fix j
nipkow@15392
   457
  assume j_fact: "j \<in> Q_set"
nipkow@15392
   458
  have "int (card (f2 j)) = int (card {y. y \<in> P_set & y \<le> (p * j) div q})"
nipkow@15392
   459
  proof -
nipkow@15392
   460
    have "finite (f2 j)"
nipkow@15392
   461
    proof -
paulson@13871
   462
      have "(f2 j) \<subseteq> S" by (auto simp add: f2_def)
paulson@13871
   463
      with S_finite show ?thesis by (auto simp add: finite_subset)
nipkow@15392
   464
    qed
nipkow@15392
   465
    moreover have "inj_on (%(x,y). x) (f2 j)"
paulson@13871
   466
      by (auto simp add: f2_def inj_on_def)
nipkow@15392
   467
    ultimately have "card ((%(x,y). x) ` (f2 j)) = card  (f2 j)"
paulson@13871
   468
      by (auto simp add: f2_def card_image)
nipkow@15392
   469
    moreover have "((%(x,y). x) ` (f2 j)) = {y. y \<in> P_set & y \<le> (p * j) div q}"
wenzelm@18369
   470
      using prems by (auto simp add: f2_def S_def Q_set_def P_set_def image_def)
paulson@13871
   471
    ultimately show ?thesis by (auto simp add: f2_def)
nipkow@15392
   472
  qed
nipkow@15392
   473
  also have "... = int (card {y. 0 < y & y \<le> (p * j) div q})"
nipkow@15392
   474
  proof -
wenzelm@18369
   475
    have "{y. y \<in> P_set & y \<le> (p * j) div q} =
nipkow@15392
   476
        {y. 0 < y & y \<le> (p * j) div q}"
paulson@13871
   477
      apply (auto simp add: P_set_def)
wenzelm@18369
   478
    proof -
wenzelm@18369
   479
      fix x
wenzelm@18369
   480
      assume "0 < x" and "x \<le> p * j div q"
wenzelm@18369
   481
      with j_fact Q_set_def  have "j \<le> (q - 1) div 2" by auto
wenzelm@18369
   482
      with p_g_2 have "p * j \<le> p * ((q - 1) div 2)"
wenzelm@18369
   483
        by (auto simp add: mult_le_cancel_left)
wenzelm@18369
   484
      with q_g_2 have "p * j div q \<le> p * ((q - 1) div 2) div q"
wenzelm@18369
   485
        by (auto simp add: zdiv_mono1)
wenzelm@18369
   486
      also from prems have "... \<le> (p - 1) div 2"
wenzelm@18369
   487
        by (auto simp add: aux2 QRTEMP_def)
wenzelm@18369
   488
      finally show "x \<le> (p - 1) div 2" using prems by auto
nipkow@15392
   489
      qed
paulson@13871
   490
    then show ?thesis by auto
nipkow@15392
   491
  qed
nipkow@15392
   492
  also have "... = (p * j) div q"
nipkow@15392
   493
  proof -
paulson@13871
   494
    from j_fact Q_set_def have "0 \<le> j" by auto
paulson@14387
   495
    with p_g_2 have "p * 0 \<le> p * j" by (auto simp only: mult_left_mono)
paulson@13871
   496
    then have "0 \<le> p * j" by auto
nipkow@15392
   497
    then have "0 div q \<le> (p * j) div q"
paulson@13871
   498
      apply (rule_tac a = 0 in zdiv_mono1)
wenzelm@18369
   499
      apply (insert q_g_2, auto)
wenzelm@18369
   500
      done
paulson@13871
   501
    also have "0 div q = 0" by auto
paulson@13871
   502
    finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
nipkow@15392
   503
  qed
nipkow@15392
   504
  finally show "int (card (f2 j)) = p * j div q" .
nipkow@15392
   505
qed
paulson@13871
   506
nipkow@15392
   507
lemma (in QRTEMP) S1_card: "int (card(S1)) = setsum (%j. (q * j) div p) P_set"
nipkow@15392
   508
proof -
nipkow@15392
   509
  have "\<forall>x \<in> P_set. finite (f1 x)"
nipkow@15392
   510
  proof
nipkow@15392
   511
    fix x
paulson@13871
   512
    have "f1 x \<subseteq> S" by (auto simp add: f1_def)
paulson@13871
   513
    with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
nipkow@15392
   514
  qed
nipkow@15392
   515
  moreover have "(\<forall>x \<in> P_set. \<forall>y \<in> P_set. x \<noteq> y --> (f1 x) \<inter> (f1 y) = {})"
paulson@13871
   516
    by (auto simp add: f1_def)
nipkow@15392
   517
  moreover note P_set_finite
wenzelm@18369
   518
  ultimately have "int(card (UNION P_set f1)) =
nipkow@15392
   519
      setsum (%x. int(card (f1 x))) P_set"
nipkow@15402
   520
    by(simp add:card_UN_disjoint int_setsum o_def)
nipkow@15392
   521
  moreover have "S1 = UNION P_set f1"
paulson@13871
   522
    by (auto simp add: f1_def S_def S1_def S2_def P_set_def Q_set_def aux1a)
wenzelm@18369
   523
  ultimately have "int(card (S1)) = setsum (%j. int(card (f1 j))) P_set"
paulson@13871
   524
    by auto
nipkow@15392
   525
  also have "... = setsum (%j. q * j div p) P_set"
nipkow@15392
   526
    using aux3a by(fastsimp intro: setsum_cong)
nipkow@15392
   527
  finally show ?thesis .
nipkow@15392
   528
qed
paulson@13871
   529
nipkow@15392
   530
lemma (in QRTEMP) S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set"
nipkow@15392
   531
proof -
nipkow@15392
   532
  have "\<forall>x \<in> Q_set. finite (f2 x)"
nipkow@15392
   533
  proof
nipkow@15392
   534
    fix x
paulson@13871
   535
    have "f2 x \<subseteq> S" by (auto simp add: f2_def)
paulson@13871
   536
    with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
nipkow@15392
   537
  qed
wenzelm@18369
   538
  moreover have "(\<forall>x \<in> Q_set. \<forall>y \<in> Q_set. x \<noteq> y -->
nipkow@15392
   539
      (f2 x) \<inter> (f2 y) = {})"
paulson@13871
   540
    by (auto simp add: f2_def)
nipkow@15392
   541
  moreover note Q_set_finite
wenzelm@18369
   542
  ultimately have "int(card (UNION Q_set f2)) =
nipkow@15392
   543
      setsum (%x. int(card (f2 x))) Q_set"
nipkow@15402
   544
    by(simp add:card_UN_disjoint int_setsum o_def)
nipkow@15392
   545
  moreover have "S2 = UNION Q_set f2"
paulson@13871
   546
    by (auto simp add: f2_def S_def S1_def S2_def P_set_def Q_set_def aux1b)
wenzelm@18369
   547
  ultimately have "int(card (S2)) = setsum (%j. int(card (f2 j))) Q_set"
paulson@13871
   548
    by auto
nipkow@15392
   549
  also have "... = setsum (%j. p * j div q) Q_set"
nipkow@15392
   550
    using aux3b by(fastsimp intro: setsum_cong)
nipkow@15392
   551
  finally show ?thesis .
nipkow@15392
   552
qed
paulson@13871
   553
wenzelm@18369
   554
lemma (in QRTEMP) S1_carda: "int (card(S1)) =
nipkow@15392
   555
    setsum (%j. (j * q) div p) P_set"
paulson@13871
   556
  by (auto simp add: S1_card zmult_ac)
paulson@13871
   557
wenzelm@18369
   558
lemma (in QRTEMP) S2_carda: "int (card(S2)) =
nipkow@15392
   559
    setsum (%j. (j * p) div q) Q_set"
paulson@13871
   560
  by (auto simp add: S2_card zmult_ac)
paulson@13871
   561
wenzelm@18369
   562
lemma (in QRTEMP) pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) +
nipkow@15392
   563
    (setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)"
nipkow@15392
   564
proof -
wenzelm@18369
   565
  have "(setsum (%j. (j * p) div q) Q_set) +
nipkow@15392
   566
      (setsum (%j. (j * q) div p) P_set) = int (card S2) + int (card S1)"
paulson@13871
   567
    by (auto simp add: S1_carda S2_carda)
nipkow@15392
   568
  also have "... = int (card S1) + int (card S2)"
paulson@13871
   569
    by auto
nipkow@15392
   570
  also have "... = ((p - 1) div 2) * ((q - 1) div 2)"
paulson@13871
   571
    by (auto simp add: card_sum_S1_S2)
nipkow@15392
   572
  finally show ?thesis .
nipkow@15392
   573
qed
paulson@13871
   574
nipkow@16663
   575
lemma pq_prime_neq: "[| zprime p; zprime q; p \<noteq> q |] ==> (~[p = 0] (mod q))"
paulson@13871
   576
  apply (auto simp add: zcong_eq_zdvd_prop zprime_def)
paulson@13871
   577
  apply (drule_tac x = q in allE)
paulson@13871
   578
  apply (drule_tac x = p in allE)
wenzelm@18369
   579
  apply auto
wenzelm@18369
   580
  done
paulson@13871
   581
wenzelm@18369
   582
lemma (in QRTEMP) QR_short: "(Legendre p q) * (Legendre q p) =
nipkow@15392
   583
    (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))"
nipkow@15392
   584
proof -
nipkow@15392
   585
  from prems have "~([p = 0] (mod q))"
paulson@13871
   586
    by (auto simp add: pq_prime_neq QRTEMP_def)
wenzelm@18369
   587
  with prems have a1: "(Legendre p q) = (-1::int) ^
nipkow@15392
   588
      nat(setsum (%x. ((x * p) div q)) Q_set)"
paulson@13871
   589
    apply (rule_tac p = q in  MainQRLemma)
wenzelm@18369
   590
    apply (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
wenzelm@18369
   591
    done
nipkow@15392
   592
  from prems have "~([q = 0] (mod p))"
paulson@13871
   593
    apply (rule_tac p = q and q = p in pq_prime_neq)
nipkow@15392
   594
    apply (simp add: QRTEMP_def)+
nipkow@16733
   595
    done
wenzelm@18369
   596
  with prems have a2: "(Legendre q p) =
nipkow@15392
   597
      (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)"
paulson@13871
   598
    apply (rule_tac p = p in  MainQRLemma)
wenzelm@18369
   599
    apply (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
wenzelm@18369
   600
    done
wenzelm@18369
   601
  from a1 a2 have "(Legendre p q) * (Legendre q p) =
paulson@13871
   602
      (-1::int) ^ nat(setsum (%x. ((x * p) div q)) Q_set) *
nipkow@15392
   603
        (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)"
paulson@13871
   604
    by auto
wenzelm@18369
   605
  also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) +
nipkow@15392
   606
                   nat(setsum (%x. ((x * q) div p)) P_set))"
paulson@13871
   607
    by (auto simp add: zpower_zadd_distrib)
wenzelm@18369
   608
  also have "nat(setsum (%x. ((x * p) div q)) Q_set) +
paulson@13871
   609
      nat(setsum (%x. ((x * q) div p)) P_set) =
wenzelm@18369
   610
        nat((setsum (%x. ((x * p) div q)) Q_set) +
nipkow@15392
   611
          (setsum (%x. ((x * q) div p)) P_set))"
wenzelm@18369
   612
    apply (rule_tac z1 = "setsum (%x. ((x * p) div q)) Q_set" in
wenzelm@18369
   613
      nat_add_distrib [symmetric])
wenzelm@18369
   614
    apply (auto simp add: S1_carda [symmetric] S2_carda [symmetric])
wenzelm@18369
   615
    done
nipkow@15392
   616
  also have "... = nat(((p - 1) div 2) * ((q - 1) div 2))"
paulson@13871
   617
    by (auto simp add: pq_sum_prop)
nipkow@15392
   618
  finally show ?thesis .
nipkow@15392
   619
qed
paulson@13871
   620
paulson@13871
   621
theorem Quadratic_Reciprocity:
wenzelm@18369
   622
     "[| p \<in> zOdd; zprime p; q \<in> zOdd; zprime q;
wenzelm@18369
   623
         p \<noteq> q |]
wenzelm@18369
   624
      ==> (Legendre p q) * (Legendre q p) =
nipkow@15392
   625
          (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))"
wenzelm@18369
   626
  by (auto simp add: QRTEMP.QR_short zprime_zOdd_eq_grt_2 [symmetric]
paulson@13871
   627
                     QRTEMP_def)
paulson@13871
   628
paulson@13871
   629
end