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theory Exponent(* Title: HOL/Algebra/Exponent.thy ID: $Id: Exponent.thy,v 1.14 2007/10/23 21:27:23 nipkow Exp $ Author: Florian Kammueller, with new proofs by L C Paulson exponent p s yields the greatest power of p that divides s. *) theory Exponent imports Main Primes Binomial begin section {*The Combinatorial Argument Underlying the First Sylow Theorem*} definition exponent :: "nat => nat => nat" where "exponent p s == if prime p then (GREATEST r. p^r dvd s) else 0" subsection{*Prime Theorems*} lemma prime_imp_one_less: "prime p ==> Suc 0 < p" by (unfold prime_def, force) lemma prime_iff: "(prime p) = (Suc 0 < p & (∀a b. p dvd a*b --> (p dvd a) | (p dvd b)))" apply (auto simp add: prime_imp_one_less) apply (blast dest!: prime_dvd_mult) apply (auto simp add: prime_def) apply (erule dvdE) apply (case_tac "k=0", simp) apply (drule_tac x = m in spec) apply (drule_tac x = k in spec) apply (simp add: dvd_mult_cancel1 dvd_mult_cancel2) done lemma zero_less_prime_power: "prime p ==> 0 < p^a" by (force simp add: prime_iff) lemma zero_less_card_empty: "[| finite S; S ≠ {} |] ==> 0 < card(S)" by (rule ccontr, simp) lemma prime_dvd_cases: "[| p*k dvd m*n; prime p |] ==> (∃x. k dvd x*n & m = p*x) | (∃y. k dvd m*y & n = p*y)" apply (simp add: prime_iff) apply (frule dvd_mult_left) apply (subgoal_tac "p dvd m | p dvd n") prefer 2 apply blast apply (erule disjE) apply (rule disjI1) apply (rule_tac [2] disjI2) apply (erule_tac n = m in dvdE) apply (erule_tac [2] n = n in dvdE, auto) done lemma prime_power_dvd_cases [rule_format (no_asm)]: "prime p ==> ∀m n. p^c dvd m*n --> (∀a b. a+b = Suc c --> p^a dvd m | p^b dvd n)" apply (induct_tac "c") apply clarify apply (case_tac "a") apply simp apply simp (*inductive step*) apply simp apply clarify apply (erule prime_dvd_cases [THEN disjE], assumption, auto) (*case 1: p dvd m*) apply (case_tac "a") apply simp apply clarify apply (drule spec, drule spec, erule (1) notE impE) apply (drule_tac x = nat in spec) apply (drule_tac x = b in spec) apply simp (*case 2: p dvd n*) apply (case_tac "b") apply simp apply clarify apply (drule spec, drule spec, erule (1) notE impE) apply (drule_tac x = a in spec) apply (drule_tac x = nat in spec, simp) done (*needed in this form in Sylow.ML*) lemma div_combine: "[| prime p; ~ (p ^ (Suc r) dvd n); p^(a+r) dvd n*k |] ==> p ^ a dvd k" by (drule_tac a = "Suc r" and b = a in prime_power_dvd_cases, assumption, auto) (*Lemma for power_dvd_bound*) lemma Suc_le_power: "Suc 0 < p ==> Suc n <= p^n" apply (induct_tac "n") apply (simp (no_asm_simp)) apply simp apply (subgoal_tac "2 * n + 2 <= p * p^n", simp) apply (subgoal_tac "2 * p^n <= p * p^n") apply arith apply (drule_tac k = 2 in mult_le_mono2, simp) done (*An upper bound for the n such that p^n dvd a: needed for GREATEST to exist*) lemma power_dvd_bound: "[|p^n dvd a; Suc 0 < p; a > 0|] ==> n < a" apply (drule dvd_imp_le) apply (drule_tac [2] n = n in Suc_le_power, auto) done subsection{*Exponent Theorems*} lemma exponent_ge [rule_format]: "[|p^k dvd n; prime p; 0<n|] ==> k <= exponent p n" apply (simp add: exponent_def) apply (erule Greatest_le) apply (blast dest: prime_imp_one_less power_dvd_bound) done lemma power_exponent_dvd: "s>0 ==> (p ^ exponent p s) dvd s" apply (simp add: exponent_def) apply clarify apply (rule_tac k = 0 in GreatestI) prefer 2 apply (blast dest: prime_imp_one_less power_dvd_bound, simp) done lemma power_Suc_exponent_Not_dvd: "[|(p * p ^ exponent p s) dvd s; prime p |] ==> s=0" apply (subgoal_tac "p ^ Suc (exponent p s) dvd s") prefer 2 apply simp apply (rule ccontr) apply (drule exponent_ge, auto) done lemma exponent_power_eq [simp]: "prime p ==> exponent p (p^a) = a" apply (simp (no_asm_simp) add: exponent_def) apply (rule Greatest_equality, simp) apply (simp (no_asm_simp) add: prime_imp_one_less power_dvd_imp_le) done lemma exponent_equalityI: "!r::nat. (p^r dvd a) = (p^r dvd b) ==> exponent p a = exponent p b" by (simp (no_asm_simp) add: exponent_def) lemma exponent_eq_0 [simp]: "¬ prime p ==> exponent p s = 0" by (simp (no_asm_simp) add: exponent_def) (* exponent_mult_add, easy inclusion. Could weaken p ∈ prime to Suc 0 < p *) lemma exponent_mult_add1: "[| a > 0; b > 0 |] ==> (exponent p a) + (exponent p b) <= exponent p (a * b)" apply (case_tac "prime p") apply (rule exponent_ge) apply (auto simp add: power_add) apply (blast intro: prime_imp_one_less power_exponent_dvd mult_dvd_mono) done (* exponent_mult_add, opposite inclusion *) lemma exponent_mult_add2: "[| a > 0; b > 0 |] ==> exponent p (a * b) <= (exponent p a) + (exponent p b)" apply (case_tac "prime p") apply (rule leI, clarify) apply (cut_tac p = p and s = "a*b" in power_exponent_dvd, auto) apply (subgoal_tac "p ^ (Suc (exponent p a + exponent p b)) dvd a * b") apply (rule_tac [2] le_imp_power_dvd [THEN dvd_trans]) prefer 3 apply assumption prefer 2 apply simp apply (frule_tac a = "Suc (exponent p a) " and b = "Suc (exponent p b) " in prime_power_dvd_cases) apply (assumption, force, simp) apply (blast dest: power_Suc_exponent_Not_dvd) done lemma exponent_mult_add: "[| a > 0; b > 0 |] ==> exponent p (a * b) = (exponent p a) + (exponent p b)" by (blast intro: exponent_mult_add1 exponent_mult_add2 order_antisym) lemma not_divides_exponent_0: "~ (p dvd n) ==> exponent p n = 0" apply (case_tac "exponent p n", simp) apply (case_tac "n", simp) apply (cut_tac s = n and p = p in power_exponent_dvd) apply (auto dest: dvd_mult_left) done lemma exponent_1_eq_0 [simp]: "exponent p (Suc 0) = 0" apply (case_tac "prime p") apply (auto simp add: prime_iff not_divides_exponent_0) done subsection{*Main Combinatorial Argument*} lemma le_extend_mult: "[| c > 0; a <= b |] ==> a <= b * (c::nat)" apply (rule_tac P = "%x. x <= b * c" in subst) apply (rule mult_1_right) apply (rule mult_le_mono, auto) done lemma p_fac_forw_lemma: "[| (m::nat) > 0; k > 0; k < p^a; (p^r) dvd (p^a)* m - k |] ==> r <= a" apply (rule notnotD) apply (rule notI) apply (drule contrapos_nn [OF _ leI, THEN notnotD], assumption) apply (drule less_imp_le [of a]) apply (drule le_imp_power_dvd) apply (drule_tac n = "p ^ r" in dvd_trans, assumption) apply(metis dvd_diffD1 dvd_triv_right le_extend_mult linorder_linear linorder_not_less mult_commute nat_dvd_not_less neq0_conv) done lemma p_fac_forw: "[| (m::nat) > 0; k>0; k < p^a; (p^r) dvd (p^a)* m - k |] ==> (p^r) dvd (p^a) - k" apply (frule_tac k1 = k and i = p in p_fac_forw_lemma [THEN le_imp_power_dvd], auto) apply (subgoal_tac "p^r dvd p^a*m") prefer 2 apply (blast intro: dvd_mult2) apply (drule dvd_diffD1) apply assumption prefer 2 apply (blast intro: dvd_diff) apply (drule gr0_implies_Suc, auto) done lemma r_le_a_forw: "[| (k::nat) > 0; k < p^a; p>0; (p^r) dvd (p^a) - k |] ==> r <= a" by (rule_tac m = "Suc 0" in p_fac_forw_lemma, auto) lemma p_fac_backw: "[| m>0; k>0; (p::nat)≠0; k < p^a; (p^r) dvd p^a - k |] ==> (p^r) dvd (p^a)*m - k" apply (frule_tac k1 = k and i = p in r_le_a_forw [THEN le_imp_power_dvd], auto) apply (subgoal_tac "p^r dvd p^a*m") prefer 2 apply (blast intro: dvd_mult2) apply (drule dvd_diffD1) apply assumption prefer 2 apply (blast intro: dvd_diff) apply (drule less_imp_Suc_add, auto) done lemma exponent_p_a_m_k_equation: "[| m>0; k>0; (p::nat)≠0; k < p^a |] ==> exponent p (p^a * m - k) = exponent p (p^a - k)" apply (blast intro: exponent_equalityI p_fac_forw p_fac_backw) done text{*Suc rules that we have to delete from the simpset*} lemmas bad_Sucs = binomial_Suc_Suc mult_Suc mult_Suc_right (*The bound K is needed; otherwise it's too weak to be used.*) lemma p_not_div_choose_lemma [rule_format]: "[| ∀i. Suc i < K --> exponent p (Suc i) = exponent p (Suc(j+i))|] ==> k<K --> exponent p ((j+k) choose k) = 0" apply (case_tac "prime p") prefer 2 apply simp apply (induct_tac "k") apply (simp (no_asm)) (*induction step*) apply (subgoal_tac "(Suc (j+n) choose Suc n) > 0") prefer 2 apply (simp add: zero_less_binomial_iff, clarify) apply (subgoal_tac "exponent p ((Suc (j+n) choose Suc n) * Suc n) = exponent p (Suc n)") txt{*First, use the assumed equation. We simplify the LHS to @{term "exponent p (Suc (j + n) choose Suc n) + exponent p (Suc n)"} the common terms cancel, proving the conclusion.*} apply (simp del: bad_Sucs add: exponent_mult_add) txt{*Establishing the equation requires first applying @{text Suc_times_binomial_eq} ...*} apply (simp del: bad_Sucs add: Suc_times_binomial_eq [symmetric]) txt{*...then @{text exponent_mult_add} and the quantified premise.*} apply (simp del: bad_Sucs add: zero_less_binomial_iff exponent_mult_add) done (*The lemma above, with two changes of variables*) lemma p_not_div_choose: "[| k<K; k<=n; ∀j. 0<j & j<K --> exponent p (n - k + (K - j)) = exponent p (K - j)|] ==> exponent p (n choose k) = 0" apply (cut_tac j = "n-k" and k = k and p = p in p_not_div_choose_lemma) prefer 3 apply simp prefer 2 apply assumption apply (drule_tac x = "K - Suc i" in spec) apply (simp add: Suc_diff_le) done lemma const_p_fac_right: "m>0 ==> exponent p ((p^a * m - Suc 0) choose (p^a - Suc 0)) = 0" apply (case_tac "prime p") prefer 2 apply simp apply (frule_tac a = a in zero_less_prime_power) apply (rule_tac K = "p^a" in p_not_div_choose) apply simp apply simp apply (case_tac "m") apply (case_tac [2] "p^a") apply auto (*now the hard case, simplified to exponent p (Suc (p ^ a * m + i - p ^ a)) = exponent p (Suc i) *) apply (subgoal_tac "0<p") prefer 2 apply (force dest!: prime_imp_one_less) apply (subst exponent_p_a_m_k_equation, auto) done lemma const_p_fac: "m>0 ==> exponent p (((p^a) * m) choose p^a) = exponent p m" apply (case_tac "prime p") prefer 2 apply simp apply (subgoal_tac "0 < p^a * m & p^a <= p^a * m") prefer 2 apply (force simp add: prime_iff) txt{*A similar trick to the one used in @{text p_not_div_choose_lemma}: insert an equation; use @{text exponent_mult_add} on the LHS; on the RHS, first transform the binomial coefficient, then use @{text exponent_mult_add}.*} apply (subgoal_tac "exponent p ((( (p^a) * m) choose p^a) * p^a) = a + exponent p m") apply (simp del: bad_Sucs add: zero_less_binomial_iff exponent_mult_add prime_iff) txt{*one subgoal left!*} apply (subst times_binomial_minus1_eq, simp, simp) apply (subst exponent_mult_add, simp) apply (simp (no_asm_simp) add: zero_less_binomial_iff) apply arith apply (simp del: bad_Sucs add: exponent_mult_add const_p_fac_right) done end
lemma prime_imp_one_less:
prime p ==> Suc 0 < p
lemma prime_iff:
prime p = (Suc 0 < p ∧ (∀a b. p dvd a * b --> p dvd a ∨ p dvd b))
lemma zero_less_prime_power:
prime p ==> 0 < p ^ a
lemma zero_less_card_empty:
[| finite S; S ≠ {} |] ==> 0 < card S
lemma prime_dvd_cases:
[| p * k dvd m * n; prime p |]
==> (∃x. k dvd x * n ∧ m = p * x) ∨ (∃y. k dvd m * y ∧ n = p * y)
lemma prime_power_dvd_cases:
[| prime p; p ^ c dvd m * n; a + b = Suc c |] ==> p ^ a dvd m ∨ p ^ b dvd n
lemma div_combine:
[| prime p; ¬ p ^ Suc r dvd n; p ^ (a + r) dvd n * k |] ==> p ^ a dvd k
lemma Suc_le_power:
Suc 0 < p ==> Suc n ≤ p ^ n
lemma power_dvd_bound:
[| p ^ n dvd a; Suc 0 < p; 0 < a |] ==> n < a
lemma exponent_ge:
[| p ^ k dvd n; prime p; 0 < n |] ==> k ≤ exponent p n
lemma power_exponent_dvd:
0 < s ==> p ^ exponent p s dvd s
lemma power_Suc_exponent_Not_dvd:
[| p * p ^ exponent p s dvd s; prime p |] ==> s = 0
lemma exponent_power_eq:
prime p ==> exponent p (p ^ a) = a
lemma exponent_equalityI:
∀r. (p ^ r dvd a) = (p ^ r dvd b) ==> exponent p a = exponent p b
lemma exponent_eq_0:
¬ prime p ==> exponent p s = 0
lemma exponent_mult_add1:
[| 0 < a; 0 < b |] ==> exponent p a + exponent p b ≤ exponent p (a * b)
lemma exponent_mult_add2:
[| 0 < a; 0 < b |] ==> exponent p (a * b) ≤ exponent p a + exponent p b
lemma exponent_mult_add:
[| 0 < a; 0 < b |] ==> exponent p (a * b) = exponent p a + exponent p b
lemma not_divides_exponent_0:
¬ p dvd n ==> exponent p n = 0
lemma exponent_1_eq_0:
exponent p (Suc 0) = 0
lemma le_extend_mult:
[| 0 < c; a ≤ b |] ==> a ≤ b * c
lemma p_fac_forw_lemma:
[| 0 < m; 0 < k; k < p ^ a; p ^ r dvd p ^ a * m - k |] ==> r ≤ a
lemma p_fac_forw:
[| 0 < m; 0 < k; k < p ^ a; p ^ r dvd p ^ a * m - k |] ==> p ^ r dvd p ^ a - k
lemma r_le_a_forw:
[| 0 < k; k < p ^ a; 0 < p; p ^ r dvd p ^ a - k |] ==> r ≤ a
lemma p_fac_backw:
[| 0 < m; 0 < k; p ≠ 0; k < p ^ a; p ^ r dvd p ^ a - k |]
==> p ^ r dvd p ^ a * m - k
lemma exponent_p_a_m_k_equation:
[| 0 < m; 0 < k; p ≠ 0; k < p ^ a |]
==> exponent p (p ^ a * m - k) = exponent p (p ^ a - k)
lemma bad_Sucs:
Suc n choose Suc k = n choose k + (n choose Suc k)
Suc m * n = n + m * n
m * Suc n = m + m * n
lemma p_not_div_choose_lemma:
[| !!i. Suc i < K ==> exponent p (Suc i) = exponent p (Suc (j + i)); k < K |]
==> exponent p (j + k choose k) = 0
lemma p_not_div_choose:
[| k < K; k ≤ n;
∀j. 0 < j ∧ j < K --> exponent p (n - k + (K - j)) = exponent p (K - j) |]
==> exponent p (n choose k) = 0
lemma const_p_fac_right:
0 < m ==> exponent p (p ^ a * m - Suc 0 choose (p ^ a - Suc 0)) = 0
lemma const_p_fac:
0 < m ==> exponent p (p ^ a * m choose p ^ a) = exponent p m