(* Title: HOL/GroupTheory/Exponent
ID: $Id$
Author: Florian Kammueller, with new proofs by L C Paulson
exponent p s yields the greatest power of p that divides s.
*)
header{*The Combinatorial Argument Underlying the First Sylow Theorem*}
theory Exponent = Main + Primes:
constdefs
exponent :: "[nat, nat] => nat"
"exponent p s == if p \<in> prime then (GREATEST r. p^r dvd s) else 0"
subsection{*Prime Theorems*}
lemma prime_imp_one_less: "p \<in> prime ==> Suc 0 < p"
by (unfold prime_def, force)
lemma prime_iff:
"(p \<in> prime) = (Suc 0 < p & (\<forall>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, auto)
done
lemma zero_less_prime_power: "p \<in> prime ==> 0 < p^a"
by (force simp add: prime_iff)
lemma le_extend_mult: "[| 0 < c; 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 insert_partition:
"[| x \<notin> F; \<forall>c1\<in>insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 --> c1 \<inter> c2 = {}|]
==> x \<inter> \<Union> F = {}"
by auto
(* main cardinality theorem *)
lemma card_partition [rule_format]:
"finite C ==>
finite (\<Union> C) -->
(\<forall>c\<in>C. card c = k) -->
(\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
k * card(C) = card (\<Union> C)"
apply (erule finite_induct, simp)
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition
finite_subset [of _ "\<Union> (insert x F)"])
done
lemma zero_less_card_empty: "[| finite S; S \<noteq> {} |] ==> 0 < card(S)"
by (rule ccontr, simp)
lemma prime_dvd_cases:
"[| p*k dvd m*n; p \<in> prime |]
==> (\<exists>x. k dvd x*n & m = p*x) | (\<exists>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)
apply (rule_tac [2] k = p in dvd_mult_cancel)
apply (rule_tac k = p in dvd_mult_cancel)
apply (simp_all add: mult_ac)
done
lemma prime_power_dvd_cases [rule_format (no_asm)]: "p \<in> prime
==> \<forall>m n. p^c dvd m*n -->
(\<forall>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
apply (blast intro: dvd_refl mult_dvd_mono)
(*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)
apply (blast intro: dvd_refl mult_dvd_mono)
done
(*needed in this form in Sylow.ML*)
lemma div_combine:
"[| p \<in> prime; ~ (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")
(*?arith_tac should handle all of this!*)
apply (rule order_trans)
prefer 2 apply assumption
apply (drule_tac k = 2 in mult_le_mono2, simp)
apply (rule mult_le_mono1, 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; 0 < a|] ==> 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; p \<in> prime; 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: "0<s ==> (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; p \<in> prime |] ==> 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]: "p \<in> prime ==> 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]: "p \<notin> prime ==> exponent p s = 0"
by (simp (no_asm_simp) add: exponent_def)
(* exponent_mult_add, easy inclusion. Could weaken p \<in> prime to Suc 0 < p *)
lemma exponent_mult_add1:
"[| 0 < a; 0 < b |]
==> (exponent p a) + (exponent p b) <= exponent p (a * b)"
apply (case_tac "p \<in> prime")
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: "[| 0 < a; 0 < b |]
==> exponent p (a * b) <= (exponent p a) + (exponent p b)"
apply (case_tac "p \<in> prime")
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:
"[| 0 < a; 0 < b |]
==> 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 "p \<in> prime")
apply (auto simp add: prime_iff not_divides_exponent_0)
done
subsection{*Lemmas for the Main Combinatorial Argument*}
lemma p_fac_forw_lemma:
"[| 0 < (m::nat); 0<k; 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_tac m = a in less_imp_le)
apply (drule le_imp_power_dvd)
apply (drule_tac n = "p ^ r" in dvd_trans, assumption)
apply (frule_tac m = k in less_imp_le)
apply (drule_tac c = m in le_extend_mult, assumption)
apply (drule_tac k = "p ^ a" and m = " (p ^ a) * m" in dvd_diffD1)
prefer 2 apply assumption
apply (rule dvd_refl [THEN dvd_mult2])
apply (drule_tac n = k in dvd_imp_le, auto)
done
lemma p_fac_forw: "[| 0 < (m::nat); 0<k; 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 less_imp_Suc_add, auto)
done
lemma r_le_a_forw: "[| 0 < (k::nat); k < p^a; 0 < p; (p^r) dvd (p^a) - k |] ==> r <= a"
by (rule_tac m = "Suc 0" in p_fac_forw_lemma, auto)
lemma p_fac_backw: "[| 0<m; 0<k; 0 < (p::nat); 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: "[| 0<m; 0<k; 0 < (p::nat); 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]:
"[| \<forall>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 "p \<in> prime")
prefer 2 apply simp
apply (induct_tac "k")
apply (simp (no_asm))
(*induction step*)
apply (subgoal_tac "0 < (Suc (j+n) choose Suc n) ")
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;
\<forall>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:
"0 < m ==> exponent p ((p^a * m - Suc 0) choose (p^a - Suc 0)) = 0"
apply (case_tac "p \<in> prime")
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:
"0 < m ==> exponent p (((p^a) * m) choose p^a) = exponent p m"
apply (case_tac "p \<in> prime")
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