wenzelm@14706: (*  Title:      HOL/Algebra/Exponent.thy
paulson@13870:     ID:         $Id$
paulson@13870:     Author:     Florian Kammueller, with new proofs by L C Paulson
paulson@13870: 
paulson@13870:     exponent p s   yields the greatest power of p that divides s.
paulson@13870: *)
paulson@13870: 
paulson@13870: header{*The Combinatorial Argument Underlying the First Sylow Theorem*}
paulson@13870: 
haftmann@16417: theory Exponent imports Main Primes begin
paulson@13870: 
paulson@13870: constdefs
paulson@13870:   exponent      :: "[nat, nat] => nat"
nipkow@16663:   "exponent p s == if prime p then (GREATEST r. p^r dvd s) else 0"
paulson@13870: 
paulson@13870: subsection{*Prime Theorems*}
paulson@13870: 
nipkow@16663: lemma prime_imp_one_less: "prime p ==> Suc 0 < p"
paulson@13870: by (unfold prime_def, force)
paulson@13870: 
paulson@13870: lemma prime_iff:
nipkow@16663:      "(prime p) = (Suc 0 < p & (\<forall>a b. p dvd a*b --> (p dvd a) | (p dvd b)))"
paulson@13870: apply (auto simp add: prime_imp_one_less)
paulson@13870: apply (blast dest!: prime_dvd_mult)
paulson@13870: apply (auto simp add: prime_def)
paulson@13870: apply (erule dvdE)
paulson@13870: apply (case_tac "k=0", simp)
paulson@13870: apply (drule_tac x = m in spec)
paulson@13870: apply (drule_tac x = k in spec)
paulson@13870: apply (simp add: dvd_mult_cancel1 dvd_mult_cancel2, auto)
paulson@13870: done
paulson@13870: 
nipkow@16663: lemma zero_less_prime_power: "prime p ==> 0 < p^a"
paulson@13870: by (force simp add: prime_iff)
paulson@13870: 
paulson@13870: 
paulson@13870: lemma zero_less_card_empty: "[| finite S; S \<noteq> {} |] ==> 0 < card(S)"
paulson@13870: by (rule ccontr, simp)
paulson@13870: 
paulson@13870: 
paulson@13870: lemma prime_dvd_cases:
nipkow@16663:      "[| p*k dvd m*n;  prime p |]  
paulson@13870:       ==> (\<exists>x. k dvd x*n & m = p*x) | (\<exists>y. k dvd m*y & n = p*y)"
paulson@13870: apply (simp add: prime_iff)
paulson@13870: apply (frule dvd_mult_left)
paulson@13870: apply (subgoal_tac "p dvd m | p dvd n")
paulson@13870:  prefer 2 apply blast
paulson@13870: apply (erule disjE)
paulson@13870: apply (rule disjI1)
paulson@13870: apply (rule_tac [2] disjI2)
paulson@13870: apply (erule_tac n = m in dvdE)
paulson@13870: apply (erule_tac [2] n = n in dvdE, auto)
paulson@13870: apply (rule_tac [2] k = p in dvd_mult_cancel)
paulson@13870: apply (rule_tac k = p in dvd_mult_cancel)
paulson@13870: apply (simp_all add: mult_ac)
paulson@13870: done
paulson@13870: 
paulson@13870: 
nipkow@16663: lemma prime_power_dvd_cases [rule_format (no_asm)]: "prime p
paulson@13870:       ==> \<forall>m n. p^c dvd m*n -->  
paulson@13870:           (\<forall>a b. a+b = Suc c --> p^a dvd m | p^b dvd n)"
paulson@13870: apply (induct_tac "c")
paulson@13870:  apply clarify
paulson@13870:  apply (case_tac "a")
paulson@13870:   apply simp
paulson@13870:  apply simp
paulson@13870: (*inductive step*)
paulson@13870: apply simp
paulson@13870: apply clarify
paulson@13870: apply (erule prime_dvd_cases [THEN disjE], assumption, auto)
paulson@13870: (*case 1: p dvd m*)
paulson@13870:  apply (case_tac "a")
paulson@13870:   apply simp
paulson@13870:  apply clarify
paulson@13870:  apply (drule spec, drule spec, erule (1) notE impE)
paulson@13870:  apply (drule_tac x = nat in spec)
paulson@13870:  apply (drule_tac x = b in spec)
paulson@13870:  apply simp
paulson@13870:  apply (blast intro: dvd_refl mult_dvd_mono)
paulson@13870: (*case 2: p dvd n*)
paulson@13870: apply (case_tac "b")
paulson@13870:  apply simp
paulson@13870: apply clarify
paulson@13870: apply (drule spec, drule spec, erule (1) notE impE)
paulson@13870: apply (drule_tac x = a in spec)
paulson@13870: apply (drule_tac x = nat in spec, simp)
paulson@13870: apply (blast intro: dvd_refl mult_dvd_mono)
paulson@13870: done
paulson@13870: 
paulson@13870: (*needed in this form in Sylow.ML*)
paulson@13870: lemma div_combine:
nipkow@16663:      "[| prime p; ~ (p ^ (Suc r) dvd n);  p^(a+r) dvd n*k |]  
paulson@13870:       ==> p ^ a dvd k"
paulson@13870: by (drule_tac a = "Suc r" and b = a in prime_power_dvd_cases, assumption, auto)
paulson@13870: 
paulson@13870: (*Lemma for power_dvd_bound*)
paulson@13870: lemma Suc_le_power: "Suc 0 < p ==> Suc n <= p^n"
paulson@13870: apply (induct_tac "n")
paulson@13870: apply (simp (no_asm_simp))
paulson@13870: apply simp
paulson@13870: apply (subgoal_tac "2 * n + 2 <= p * p^n", simp)
paulson@13870: apply (subgoal_tac "2 * p^n <= p * p^n")
paulson@13870: (*?arith_tac should handle all of this!*)
paulson@13870: apply (rule order_trans)
paulson@13870: prefer 2 apply assumption
paulson@13870: apply (drule_tac k = 2 in mult_le_mono2, simp)
paulson@13870: apply (rule mult_le_mono1, simp)
paulson@13870: done
paulson@13870: 
paulson@13870: (*An upper bound for the n such that p^n dvd a: needed for GREATEST to exist*)
paulson@13870: lemma power_dvd_bound: "[|p^n dvd a;  Suc 0 < p;  0 < a|] ==> n < a"
paulson@13870: apply (drule dvd_imp_le)
paulson@13870: apply (drule_tac [2] n = n in Suc_le_power, auto)
paulson@13870: done
paulson@13870: 
paulson@13870: 
paulson@13870: subsection{*Exponent Theorems*}
paulson@13870: 
paulson@13870: lemma exponent_ge [rule_format]:
nipkow@16663:      "[|p^k dvd n;  prime p;  0<n|] ==> k <= exponent p n"
paulson@13870: apply (simp add: exponent_def)
paulson@13870: apply (erule Greatest_le)
paulson@13870: apply (blast dest: prime_imp_one_less power_dvd_bound)
paulson@13870: done
paulson@13870: 
paulson@13870: lemma power_exponent_dvd: "0<s ==> (p ^ exponent p s) dvd s"
paulson@13870: apply (simp add: exponent_def)
paulson@13870: apply clarify
paulson@13870: apply (rule_tac k = 0 in GreatestI)
paulson@13870: prefer 2 apply (blast dest: prime_imp_one_less power_dvd_bound, simp)
paulson@13870: done
paulson@13870: 
paulson@13870: lemma power_Suc_exponent_Not_dvd:
nipkow@16663:      "[|(p * p ^ exponent p s) dvd s;  prime p |] ==> s=0"
paulson@13870: apply (subgoal_tac "p ^ Suc (exponent p s) dvd s")
paulson@13870:  prefer 2 apply simp 
paulson@13870: apply (rule ccontr)
paulson@13870: apply (drule exponent_ge, auto)
paulson@13870: done
paulson@13870: 
nipkow@16663: lemma exponent_power_eq [simp]: "prime p ==> exponent p (p^a) = a"
paulson@13870: apply (simp (no_asm_simp) add: exponent_def)
paulson@13870: apply (rule Greatest_equality, simp)
paulson@13870: apply (simp (no_asm_simp) add: prime_imp_one_less power_dvd_imp_le)
paulson@13870: done
paulson@13870: 
paulson@13870: lemma exponent_equalityI:
paulson@13870:      "!r::nat. (p^r dvd a) = (p^r dvd b) ==> exponent p a = exponent p b"
paulson@13870: by (simp (no_asm_simp) add: exponent_def)
paulson@13870: 
nipkow@16663: lemma exponent_eq_0 [simp]: "\<not> prime p ==> exponent p s = 0"
paulson@13870: by (simp (no_asm_simp) add: exponent_def)
paulson@13870: 
paulson@13870: 
paulson@13870: (* exponent_mult_add, easy inclusion.  Could weaken p \<in> prime to Suc 0 < p *)
paulson@13870: lemma exponent_mult_add1:
paulson@13870:      "[| 0 < a; 0 < b |]   
paulson@13870:       ==> (exponent p a) + (exponent p b) <= exponent p (a * b)"
nipkow@16663: apply (case_tac "prime p")
paulson@13870: apply (rule exponent_ge)
paulson@13870: apply (auto simp add: power_add)
paulson@13870: apply (blast intro: prime_imp_one_less power_exponent_dvd mult_dvd_mono)
paulson@13870: done
paulson@13870: 
paulson@13870: (* exponent_mult_add, opposite inclusion *)
paulson@13870: lemma exponent_mult_add2: "[| 0 < a; 0 < b |]  
paulson@13870:       ==> exponent p (a * b) <= (exponent p a) + (exponent p b)"
nipkow@16663: apply (case_tac "prime p")
paulson@13870: apply (rule leI, clarify)
paulson@13870: apply (cut_tac p = p and s = "a*b" in power_exponent_dvd, auto)
paulson@13870: apply (subgoal_tac "p ^ (Suc (exponent p a + exponent p b)) dvd a * b")
paulson@13870: apply (rule_tac [2] le_imp_power_dvd [THEN dvd_trans])
paulson@13870:   prefer 3 apply assumption
paulson@13870:  prefer 2 apply simp 
paulson@13870: apply (frule_tac a = "Suc (exponent p a) " and b = "Suc (exponent p b) " in prime_power_dvd_cases)
paulson@13870:  apply (assumption, force, simp)
paulson@13870: apply (blast dest: power_Suc_exponent_Not_dvd)
paulson@13870: done
paulson@13870: 
paulson@13870: lemma exponent_mult_add:
paulson@13870:      "[| 0 < a; 0 < b |]  
paulson@13870:       ==> exponent p (a * b) = (exponent p a) + (exponent p b)"
paulson@13870: by (blast intro: exponent_mult_add1 exponent_mult_add2 order_antisym)
paulson@13870: 
paulson@13870: 
paulson@13870: lemma not_divides_exponent_0: "~ (p dvd n) ==> exponent p n = 0"
paulson@13870: apply (case_tac "exponent p n", simp)
paulson@13870: apply (case_tac "n", simp)
paulson@13870: apply (cut_tac s = n and p = p in power_exponent_dvd)
paulson@13870: apply (auto dest: dvd_mult_left)
paulson@13870: done
paulson@13870: 
paulson@13870: lemma exponent_1_eq_0 [simp]: "exponent p (Suc 0) = 0"
nipkow@16663: apply (case_tac "prime p")
paulson@13870: apply (auto simp add: prime_iff not_divides_exponent_0)
paulson@13870: done
paulson@13870: 
paulson@13870: 
paulson@13870: subsection{*Lemmas for the Main Combinatorial Argument*}
paulson@13870: 
paulson@14889: lemma le_extend_mult: "[| 0 < c; a <= b |] ==> a <= b * (c::nat)"
paulson@14889: apply (rule_tac P = "%x. x <= b * c" in subst)
paulson@14889: apply (rule mult_1_right)
paulson@14889: apply (rule mult_le_mono, auto)
paulson@14889: done
paulson@14889: 
paulson@13870: lemma p_fac_forw_lemma:
paulson@13870:      "[| 0 < (m::nat); 0<k; k < p^a; (p^r) dvd (p^a)* m - k |] ==> r <= a"
paulson@13870: apply (rule notnotD)
paulson@13870: apply (rule notI)
paulson@13870: apply (drule contrapos_nn [OF _ leI, THEN notnotD], assumption)
paulson@13870: apply (drule_tac m = a in less_imp_le)
paulson@13870: apply (drule le_imp_power_dvd)
paulson@13870: apply (drule_tac n = "p ^ r" in dvd_trans, assumption)
paulson@13870: apply (frule_tac m = k in less_imp_le)
paulson@13870: apply (drule_tac c = m in le_extend_mult, assumption)
paulson@13870: apply (drule_tac k = "p ^ a" and m = " (p ^ a) * m" in dvd_diffD1)
paulson@13870: prefer 2 apply assumption
paulson@13870: apply (rule dvd_refl [THEN dvd_mult2])
paulson@13870: apply (drule_tac n = k in dvd_imp_le, auto)
paulson@13870: done
paulson@13870: 
paulson@13870: lemma p_fac_forw: "[| 0 < (m::nat); 0<k; k < p^a; (p^r) dvd (p^a)* m - k |]  
paulson@13870:       ==> (p^r) dvd (p^a) - k"
paulson@13870: apply (frule_tac k1 = k and i = p in p_fac_forw_lemma [THEN le_imp_power_dvd], auto)
paulson@13870: apply (subgoal_tac "p^r dvd p^a*m")
paulson@13870:  prefer 2 apply (blast intro: dvd_mult2)
paulson@13870: apply (drule dvd_diffD1)
paulson@13870:   apply assumption
paulson@13870:  prefer 2 apply (blast intro: dvd_diff)
paulson@13870: apply (drule less_imp_Suc_add, auto)
paulson@13870: done
paulson@13870: 
paulson@13870: 
paulson@13870: lemma r_le_a_forw: "[| 0 < (k::nat); k < p^a; 0 < p; (p^r) dvd (p^a) - k |] ==> r <= a"
paulson@13870: by (rule_tac m = "Suc 0" in p_fac_forw_lemma, auto)
paulson@13870: 
paulson@13870: lemma p_fac_backw: "[| 0<m; 0<k; 0 < (p::nat);  k < p^a;  (p^r) dvd p^a - k |]  
paulson@13870:       ==> (p^r) dvd (p^a)*m - k"
paulson@13870: apply (frule_tac k1 = k and i = p in r_le_a_forw [THEN le_imp_power_dvd], auto)
paulson@13870: apply (subgoal_tac "p^r dvd p^a*m")
paulson@13870:  prefer 2 apply (blast intro: dvd_mult2)
paulson@13870: apply (drule dvd_diffD1)
paulson@13870:   apply assumption
paulson@13870:  prefer 2 apply (blast intro: dvd_diff)
paulson@13870: apply (drule less_imp_Suc_add, auto)
paulson@13870: done
paulson@13870: 
paulson@13870: lemma exponent_p_a_m_k_equation: "[| 0<m; 0<k; 0 < (p::nat);  k < p^a |]  
paulson@13870:       ==> exponent p (p^a * m - k) = exponent p (p^a - k)"
paulson@13870: apply (blast intro: exponent_equalityI p_fac_forw p_fac_backw)
paulson@13870: done
paulson@13870: 
paulson@13870: text{*Suc rules that we have to delete from the simpset*}
paulson@13870: lemmas bad_Sucs = binomial_Suc_Suc mult_Suc mult_Suc_right
paulson@13870: 
paulson@13870: (*The bound K is needed; otherwise it's too weak to be used.*)
paulson@13870: lemma p_not_div_choose_lemma [rule_format]:
paulson@13870:      "[| \<forall>i. Suc i < K --> exponent p (Suc i) = exponent p (Suc(j+i))|]  
paulson@13870:       ==> k<K --> exponent p ((j+k) choose k) = 0"
nipkow@16663: apply (case_tac "prime p")
paulson@13870:  prefer 2 apply simp 
paulson@13870: apply (induct_tac "k")
paulson@13870: apply (simp (no_asm))
paulson@13870: (*induction step*)
paulson@13870: apply (subgoal_tac "0 < (Suc (j+n) choose Suc n) ")
paulson@13870:  prefer 2 apply (simp add: zero_less_binomial_iff, clarify)
paulson@13870: apply (subgoal_tac "exponent p ((Suc (j+n) choose Suc n) * Suc n) = 
paulson@13870:                     exponent p (Suc n)")
paulson@13870:  txt{*First, use the assumed equation.  We simplify the LHS to
paulson@13870:   @{term "exponent p (Suc (j + n) choose Suc n) + exponent p (Suc n)"}
paulson@13870:   the common terms cancel, proving the conclusion.*}
paulson@13870:  apply (simp del: bad_Sucs add: exponent_mult_add)
paulson@13870: txt{*Establishing the equation requires first applying 
paulson@13870:    @{text Suc_times_binomial_eq} ...*}
paulson@13870: apply (simp del: bad_Sucs add: Suc_times_binomial_eq [symmetric])
paulson@13870: txt{*...then @{text exponent_mult_add} and the quantified premise.*}
paulson@13870: apply (simp del: bad_Sucs add: zero_less_binomial_iff exponent_mult_add)
paulson@13870: done
paulson@13870: 
paulson@13870: (*The lemma above, with two changes of variables*)
paulson@13870: lemma p_not_div_choose:
paulson@13870:      "[| k<K;  k<=n;   
paulson@13870:        \<forall>j. 0<j & j<K --> exponent p (n - k + (K - j)) = exponent p (K - j)|]  
paulson@13870:       ==> exponent p (n choose k) = 0"
paulson@13870: apply (cut_tac j = "n-k" and k = k and p = p in p_not_div_choose_lemma)
paulson@13870:   prefer 3 apply simp
paulson@13870:  prefer 2 apply assumption
paulson@13870: apply (drule_tac x = "K - Suc i" in spec)
paulson@13870: apply (simp add: Suc_diff_le)
paulson@13870: done
paulson@13870: 
paulson@13870: 
paulson@13870: lemma const_p_fac_right:
paulson@13870:      "0 < m ==> exponent p ((p^a * m - Suc 0) choose (p^a - Suc 0)) = 0"
nipkow@16663: apply (case_tac "prime p")
paulson@13870:  prefer 2 apply simp 
paulson@13870: apply (frule_tac a = a in zero_less_prime_power)
paulson@13870: apply (rule_tac K = "p^a" in p_not_div_choose)
paulson@13870:    apply simp
paulson@13870:   apply simp
paulson@13870:  apply (case_tac "m")
paulson@13870:   apply (case_tac [2] "p^a")
paulson@13870:    apply auto
paulson@13870: (*now the hard case, simplified to
paulson@13870:     exponent p (Suc (p ^ a * m + i - p ^ a)) = exponent p (Suc i) *)
paulson@13870: apply (subgoal_tac "0<p")
paulson@13870:  prefer 2 apply (force dest!: prime_imp_one_less)
paulson@13870: apply (subst exponent_p_a_m_k_equation, auto)
paulson@13870: done
paulson@13870: 
paulson@13870: lemma const_p_fac:
paulson@13870:      "0 < m ==> exponent p (((p^a) * m) choose p^a) = exponent p m"
nipkow@16663: apply (case_tac "prime p")
paulson@13870:  prefer 2 apply simp 
paulson@13870: apply (subgoal_tac "0 < p^a * m & p^a <= p^a * m")
paulson@13870:  prefer 2 apply (force simp add: prime_iff)
paulson@13870: txt{*A similar trick to the one used in @{text p_not_div_choose_lemma}:
paulson@13870:   insert an equation; use @{text exponent_mult_add} on the LHS; on the RHS,
paulson@13870:   first
paulson@13870:   transform the binomial coefficient, then use @{text exponent_mult_add}.*}
paulson@13870: apply (subgoal_tac "exponent p ((( (p^a) * m) choose p^a) * p^a) = 
paulson@13870:                     a + exponent p m")
paulson@13870:  apply (simp del: bad_Sucs add: zero_less_binomial_iff exponent_mult_add prime_iff)
paulson@13870: txt{*one subgoal left!*}
paulson@13870: apply (subst times_binomial_minus1_eq, simp, simp)
paulson@13870: apply (subst exponent_mult_add, simp)
paulson@13870: apply (simp (no_asm_simp) add: zero_less_binomial_iff)
paulson@13870: apply arith
paulson@13870: apply (simp del: bad_Sucs add: exponent_mult_add const_p_fac_right)
paulson@13870: done
paulson@13870: 
paulson@13870: 
paulson@13870: end