src/HOL/Algebra/Exponent.thy
 author wenzelm Thu Mar 26 20:08:55 2009 +0100 (2009-03-26) changeset 30729 461ee3e49ad3 parent 30242 aea5d7fa7ef5 child 31717 d1f7b6245a75 permissions -rw-r--r--
interpretation/interpret: prefixes are mandatory by default;
 wenzelm@14706 ` 1` ```(* Title: HOL/Algebra/Exponent.thy ``` paulson@13870 ` 2` ``` ID: \$Id\$ ``` paulson@13870 ` 3` ``` Author: Florian Kammueller, with new proofs by L C Paulson ``` paulson@13870 ` 4` paulson@13870 ` 5` ``` exponent p s yields the greatest power of p that divides s. ``` paulson@13870 ` 6` ```*) ``` paulson@13870 ` 7` haftmann@27105 ` 8` ```theory Exponent ``` haftmann@27105 ` 9` ```imports Main Primes Binomial ``` haftmann@27105 ` 10` ```begin ``` ballarin@20318 ` 11` ballarin@27717 ` 12` ```section {*Sylow's Theorem*} ``` ballarin@27717 ` 13` ballarin@27717 ` 14` ```subsection {*The Combinatorial Argument Underlying the First Sylow Theorem*} ``` ballarin@27717 ` 15` nipkow@25134 ` 16` ```definition exponent :: "nat => nat => nat" where ``` nipkow@25134 ` 17` ```"exponent p s == if prime p then (GREATEST r. p^r dvd s) else 0" ``` paulson@13870 ` 18` ballarin@20318 ` 19` ballarin@27717 ` 20` ```text{*Prime Theorems*} ``` paulson@13870 ` 21` nipkow@16663 ` 22` ```lemma prime_imp_one_less: "prime p ==> Suc 0 < p" ``` paulson@13870 ` 23` ```by (unfold prime_def, force) ``` paulson@13870 ` 24` paulson@13870 ` 25` ```lemma prime_iff: ``` nipkow@25134 ` 26` ``` "(prime p) = (Suc 0 < p & (\a b. p dvd a*b --> (p dvd a) | (p dvd b)))" ``` paulson@13870 ` 27` ```apply (auto simp add: prime_imp_one_less) ``` paulson@13870 ` 28` ```apply (blast dest!: prime_dvd_mult) ``` paulson@13870 ` 29` ```apply (auto simp add: prime_def) ``` paulson@13870 ` 30` ```apply (erule dvdE) ``` paulson@13870 ` 31` ```apply (case_tac "k=0", simp) ``` paulson@13870 ` 32` ```apply (drule_tac x = m in spec) ``` paulson@13870 ` 33` ```apply (drule_tac x = k in spec) ``` nipkow@16733 ` 34` ```apply (simp add: dvd_mult_cancel1 dvd_mult_cancel2) ``` paulson@13870 ` 35` ```done ``` paulson@13870 ` 36` nipkow@16663 ` 37` ```lemma zero_less_prime_power: "prime p ==> 0 < p^a" ``` paulson@13870 ` 38` ```by (force simp add: prime_iff) ``` paulson@13870 ` 39` paulson@13870 ` 40` paulson@13870 ` 41` ```lemma zero_less_card_empty: "[| finite S; S \ {} |] ==> 0 < card(S)" ``` paulson@13870 ` 42` ```by (rule ccontr, simp) ``` paulson@13870 ` 43` paulson@13870 ` 44` paulson@13870 ` 45` ```lemma prime_dvd_cases: ``` nipkow@25134 ` 46` ``` "[| p*k dvd m*n; prime p |] ``` nipkow@25134 ` 47` ``` ==> (\x. k dvd x*n & m = p*x) | (\y. k dvd m*y & n = p*y)" ``` paulson@13870 ` 48` ```apply (simp add: prime_iff) ``` paulson@13870 ` 49` ```apply (frule dvd_mult_left) ``` paulson@13870 ` 50` ```apply (subgoal_tac "p dvd m | p dvd n") ``` paulson@13870 ` 51` ``` prefer 2 apply blast ``` paulson@13870 ` 52` ```apply (erule disjE) ``` paulson@13870 ` 53` ```apply (rule disjI1) ``` paulson@13870 ` 54` ```apply (rule_tac [2] disjI2) ``` haftmann@27651 ` 55` ```apply (auto elim!: dvdE) ``` paulson@13870 ` 56` ```done ``` paulson@13870 ` 57` paulson@13870 ` 58` nipkow@16663 ` 59` ```lemma prime_power_dvd_cases [rule_format (no_asm)]: "prime p ``` nipkow@25134 ` 60` ``` ==> \m n. p^c dvd m*n --> ``` nipkow@25134 ` 61` ``` (\a b. a+b = Suc c --> p^a dvd m | p^b dvd n)" ``` haftmann@27105 ` 62` ```apply (induct c) ``` paulson@13870 ` 63` ``` apply clarify ``` paulson@13870 ` 64` ``` apply (case_tac "a") ``` paulson@13870 ` 65` ``` apply simp ``` paulson@13870 ` 66` ``` apply simp ``` paulson@13870 ` 67` ```(*inductive step*) ``` paulson@13870 ` 68` ```apply simp ``` paulson@13870 ` 69` ```apply clarify ``` paulson@13870 ` 70` ```apply (erule prime_dvd_cases [THEN disjE], assumption, auto) ``` paulson@13870 ` 71` ```(*case 1: p dvd m*) ``` paulson@13870 ` 72` ``` apply (case_tac "a") ``` paulson@13870 ` 73` ``` apply simp ``` paulson@13870 ` 74` ``` apply clarify ``` paulson@13870 ` 75` ``` apply (drule spec, drule spec, erule (1) notE impE) ``` paulson@13870 ` 76` ``` apply (drule_tac x = nat in spec) ``` paulson@13870 ` 77` ``` apply (drule_tac x = b in spec) ``` paulson@13870 ` 78` ``` apply simp ``` paulson@13870 ` 79` ```(*case 2: p dvd n*) ``` paulson@13870 ` 80` ```apply (case_tac "b") ``` paulson@13870 ` 81` ``` apply simp ``` paulson@13870 ` 82` ```apply clarify ``` paulson@13870 ` 83` ```apply (drule spec, drule spec, erule (1) notE impE) ``` paulson@13870 ` 84` ```apply (drule_tac x = a in spec) ``` paulson@13870 ` 85` ```apply (drule_tac x = nat in spec, simp) ``` paulson@13870 ` 86` ```done ``` paulson@13870 ` 87` paulson@13870 ` 88` ```(*needed in this form in Sylow.ML*) ``` paulson@13870 ` 89` ```lemma div_combine: ``` nipkow@25134 ` 90` ``` "[| prime p; ~ (p ^ (Suc r) dvd n); p^(a+r) dvd n*k |] ``` nipkow@25134 ` 91` ``` ==> p ^ a dvd k" ``` paulson@13870 ` 92` ```by (drule_tac a = "Suc r" and b = a in prime_power_dvd_cases, assumption, auto) ``` paulson@13870 ` 93` paulson@13870 ` 94` ```(*Lemma for power_dvd_bound*) ``` paulson@13870 ` 95` ```lemma Suc_le_power: "Suc 0 < p ==> Suc n <= p^n" ``` haftmann@27105 ` 96` ```apply (induct n) ``` paulson@13870 ` 97` ```apply (simp (no_asm_simp)) ``` paulson@13870 ` 98` ```apply simp ``` paulson@13870 ` 99` ```apply (subgoal_tac "2 * n + 2 <= p * p^n", simp) ``` paulson@13870 ` 100` ```apply (subgoal_tac "2 * p^n <= p * p^n") ``` nipkow@25134 ` 101` ```apply arith ``` paulson@13870 ` 102` ```apply (drule_tac k = 2 in mult_le_mono2, simp) ``` paulson@13870 ` 103` ```done ``` paulson@13870 ` 104` paulson@13870 ` 105` ```(*An upper bound for the n such that p^n dvd a: needed for GREATEST to exist*) ``` nipkow@25162 ` 106` ```lemma power_dvd_bound: "[|p^n dvd a; Suc 0 < p; a > 0|] ==> n < a" ``` paulson@13870 ` 107` ```apply (drule dvd_imp_le) ``` paulson@13870 ` 108` ```apply (drule_tac [2] n = n in Suc_le_power, auto) ``` paulson@13870 ` 109` ```done ``` paulson@13870 ` 110` paulson@13870 ` 111` ballarin@27717 ` 112` ```text{*Exponent Theorems*} ``` paulson@13870 ` 113` paulson@13870 ` 114` ```lemma exponent_ge [rule_format]: ``` nipkow@25134 ` 115` ``` "[|p^k dvd n; prime p; 0 k <= exponent p n" ``` paulson@13870 ` 116` ```apply (simp add: exponent_def) ``` paulson@13870 ` 117` ```apply (erule Greatest_le) ``` paulson@13870 ` 118` ```apply (blast dest: prime_imp_one_less power_dvd_bound) ``` paulson@13870 ` 119` ```done ``` paulson@13870 ` 120` nipkow@25162 ` 121` ```lemma power_exponent_dvd: "s>0 ==> (p ^ exponent p s) dvd s" ``` paulson@13870 ` 122` ```apply (simp add: exponent_def) ``` paulson@13870 ` 123` ```apply clarify ``` paulson@13870 ` 124` ```apply (rule_tac k = 0 in GreatestI) ``` paulson@13870 ` 125` ```prefer 2 apply (blast dest: prime_imp_one_less power_dvd_bound, simp) ``` paulson@13870 ` 126` ```done ``` paulson@13870 ` 127` paulson@13870 ` 128` ```lemma power_Suc_exponent_Not_dvd: ``` nipkow@25134 ` 129` ``` "[|(p * p ^ exponent p s) dvd s; prime p |] ==> s=0" ``` paulson@13870 ` 130` ```apply (subgoal_tac "p ^ Suc (exponent p s) dvd s") ``` paulson@13870 ` 131` ``` prefer 2 apply simp ``` paulson@13870 ` 132` ```apply (rule ccontr) ``` paulson@13870 ` 133` ```apply (drule exponent_ge, auto) ``` paulson@13870 ` 134` ```done ``` paulson@13870 ` 135` nipkow@16663 ` 136` ```lemma exponent_power_eq [simp]: "prime p ==> exponent p (p^a) = a" ``` paulson@13870 ` 137` ```apply (simp (no_asm_simp) add: exponent_def) ``` paulson@13870 ` 138` ```apply (rule Greatest_equality, simp) ``` paulson@13870 ` 139` ```apply (simp (no_asm_simp) add: prime_imp_one_less power_dvd_imp_le) ``` paulson@13870 ` 140` ```done ``` paulson@13870 ` 141` paulson@13870 ` 142` ```lemma exponent_equalityI: ``` nipkow@25134 ` 143` ``` "!r::nat. (p^r dvd a) = (p^r dvd b) ==> exponent p a = exponent p b" ``` paulson@13870 ` 144` ```by (simp (no_asm_simp) add: exponent_def) ``` paulson@13870 ` 145` nipkow@16663 ` 146` ```lemma exponent_eq_0 [simp]: "\ prime p ==> exponent p s = 0" ``` paulson@13870 ` 147` ```by (simp (no_asm_simp) add: exponent_def) ``` paulson@13870 ` 148` paulson@13870 ` 149` paulson@13870 ` 150` ```(* exponent_mult_add, easy inclusion. Could weaken p \ prime to Suc 0 < p *) ``` nipkow@25162 ` 151` ```lemma exponent_mult_add1: "[| a > 0; b > 0 |] ``` nipkow@25134 ` 152` ``` ==> (exponent p a) + (exponent p b) <= exponent p (a * b)" ``` nipkow@16663 ` 153` ```apply (case_tac "prime p") ``` paulson@13870 ` 154` ```apply (rule exponent_ge) ``` paulson@13870 ` 155` ```apply (auto simp add: power_add) ``` paulson@13870 ` 156` ```apply (blast intro: prime_imp_one_less power_exponent_dvd mult_dvd_mono) ``` paulson@13870 ` 157` ```done ``` paulson@13870 ` 158` paulson@13870 ` 159` ```(* exponent_mult_add, opposite inclusion *) ``` nipkow@25162 ` 160` ```lemma exponent_mult_add2: "[| a > 0; b > 0 |] ``` nipkow@25134 ` 161` ``` ==> exponent p (a * b) <= (exponent p a) + (exponent p b)" ``` nipkow@16663 ` 162` ```apply (case_tac "prime p") ``` paulson@13870 ` 163` ```apply (rule leI, clarify) ``` paulson@13870 ` 164` ```apply (cut_tac p = p and s = "a*b" in power_exponent_dvd, auto) ``` paulson@13870 ` 165` ```apply (subgoal_tac "p ^ (Suc (exponent p a + exponent p b)) dvd a * b") ``` paulson@13870 ` 166` ```apply (rule_tac [2] le_imp_power_dvd [THEN dvd_trans]) ``` paulson@13870 ` 167` ``` prefer 3 apply assumption ``` paulson@13870 ` 168` ``` prefer 2 apply simp ``` paulson@13870 ` 169` ```apply (frule_tac a = "Suc (exponent p a) " and b = "Suc (exponent p b) " in prime_power_dvd_cases) ``` paulson@13870 ` 170` ``` apply (assumption, force, simp) ``` paulson@13870 ` 171` ```apply (blast dest: power_Suc_exponent_Not_dvd) ``` paulson@13870 ` 172` ```done ``` paulson@13870 ` 173` nipkow@25162 ` 174` ```lemma exponent_mult_add: "[| a > 0; b > 0 |] ``` nipkow@25134 ` 175` ``` ==> exponent p (a * b) = (exponent p a) + (exponent p b)" ``` paulson@13870 ` 176` ```by (blast intro: exponent_mult_add1 exponent_mult_add2 order_antisym) ``` paulson@13870 ` 177` paulson@13870 ` 178` paulson@13870 ` 179` ```lemma not_divides_exponent_0: "~ (p dvd n) ==> exponent p n = 0" ``` paulson@13870 ` 180` ```apply (case_tac "exponent p n", simp) ``` paulson@13870 ` 181` ```apply (case_tac "n", simp) ``` paulson@13870 ` 182` ```apply (cut_tac s = n and p = p in power_exponent_dvd) ``` paulson@13870 ` 183` ```apply (auto dest: dvd_mult_left) ``` paulson@13870 ` 184` ```done ``` paulson@13870 ` 185` paulson@13870 ` 186` ```lemma exponent_1_eq_0 [simp]: "exponent p (Suc 0) = 0" ``` nipkow@16663 ` 187` ```apply (case_tac "prime p") ``` paulson@13870 ` 188` ```apply (auto simp add: prime_iff not_divides_exponent_0) ``` paulson@13870 ` 189` ```done ``` paulson@13870 ` 190` paulson@13870 ` 191` ballarin@27717 ` 192` ```text{*Main Combinatorial Argument*} ``` paulson@13870 ` 193` nipkow@25162 ` 194` ```lemma le_extend_mult: "[| c > 0; a <= b |] ==> a <= b * (c::nat)" ``` paulson@14889 ` 195` ```apply (rule_tac P = "%x. x <= b * c" in subst) ``` paulson@14889 ` 196` ```apply (rule mult_1_right) ``` paulson@14889 ` 197` ```apply (rule mult_le_mono, auto) ``` paulson@14889 ` 198` ```done ``` paulson@14889 ` 199` paulson@13870 ` 200` ```lemma p_fac_forw_lemma: ``` nipkow@25162 ` 201` ``` "[| (m::nat) > 0; k > 0; k < p^a; (p^r) dvd (p^a)* m - k |] ==> r <= a" ``` paulson@13870 ` 202` ```apply (rule notnotD) ``` paulson@13870 ` 203` ```apply (rule notI) ``` paulson@13870 ` 204` ```apply (drule contrapos_nn [OF _ leI, THEN notnotD], assumption) ``` paulson@24742 ` 205` ```apply (drule less_imp_le [of a]) ``` paulson@13870 ` 206` ```apply (drule le_imp_power_dvd) ``` haftmann@27651 ` 207` ```apply (drule_tac b = "p ^ r" in dvd_trans, assumption) ``` nipkow@25134 ` 208` ```apply(metis dvd_diffD1 dvd_triv_right le_extend_mult linorder_linear linorder_not_less mult_commute nat_dvd_not_less neq0_conv) ``` paulson@13870 ` 209` ```done ``` paulson@13870 ` 210` nipkow@25162 ` 211` ```lemma p_fac_forw: "[| (m::nat) > 0; k>0; k < p^a; (p^r) dvd (p^a)* m - k |] ``` nipkow@25134 ` 212` ``` ==> (p^r) dvd (p^a) - k" ``` haftmann@30011 ` 213` ```apply (frule p_fac_forw_lemma [THEN le_imp_power_dvd, of _ k p], auto) ``` paulson@13870 ` 214` ```apply (subgoal_tac "p^r dvd p^a*m") ``` paulson@13870 ` 215` ``` prefer 2 apply (blast intro: dvd_mult2) ``` paulson@13870 ` 216` ```apply (drule dvd_diffD1) ``` paulson@13870 ` 217` ``` apply assumption ``` nipkow@30042 ` 218` ``` prefer 2 apply (blast intro: nat_dvd_diff) ``` nipkow@25162 ` 219` ```apply (drule gr0_implies_Suc, auto) ``` paulson@13870 ` 220` ```done ``` paulson@13870 ` 221` paulson@13870 ` 222` nipkow@25134 ` 223` ```lemma r_le_a_forw: ``` nipkow@25162 ` 224` ``` "[| (k::nat) > 0; k < p^a; p>0; (p^r) dvd (p^a) - k |] ==> r <= a" ``` paulson@13870 ` 225` ```by (rule_tac m = "Suc 0" in p_fac_forw_lemma, auto) ``` paulson@13870 ` 226` nipkow@25162 ` 227` ```lemma p_fac_backw: "[| m>0; k>0; (p::nat)\0; k < p^a; (p^r) dvd p^a - k |] ``` nipkow@25134 ` 228` ``` ==> (p^r) dvd (p^a)*m - k" ``` haftmann@30011 ` 229` ```apply (frule_tac k1 = k and p1 = p in r_le_a_forw [THEN le_imp_power_dvd], auto) ``` paulson@13870 ` 230` ```apply (subgoal_tac "p^r dvd p^a*m") ``` paulson@13870 ` 231` ``` prefer 2 apply (blast intro: dvd_mult2) ``` paulson@13870 ` 232` ```apply (drule dvd_diffD1) ``` paulson@13870 ` 233` ``` apply assumption ``` nipkow@30042 ` 234` ``` prefer 2 apply (blast intro: nat_dvd_diff) ``` paulson@13870 ` 235` ```apply (drule less_imp_Suc_add, auto) ``` paulson@13870 ` 236` ```done ``` paulson@13870 ` 237` nipkow@25162 ` 238` ```lemma exponent_p_a_m_k_equation: "[| m>0; k>0; (p::nat)\0; k < p^a |] ``` nipkow@25134 ` 239` ``` ==> exponent p (p^a * m - k) = exponent p (p^a - k)" ``` paulson@13870 ` 240` ```apply (blast intro: exponent_equalityI p_fac_forw p_fac_backw) ``` paulson@13870 ` 241` ```done ``` paulson@13870 ` 242` paulson@13870 ` 243` ```text{*Suc rules that we have to delete from the simpset*} ``` paulson@13870 ` 244` ```lemmas bad_Sucs = binomial_Suc_Suc mult_Suc mult_Suc_right ``` paulson@13870 ` 245` paulson@13870 ` 246` ```(*The bound K is needed; otherwise it's too weak to be used.*) ``` paulson@13870 ` 247` ```lemma p_not_div_choose_lemma [rule_format]: ``` nipkow@25134 ` 248` ``` "[| \i. Suc i < K --> exponent p (Suc i) = exponent p (Suc(j+i))|] ``` nipkow@25134 ` 249` ``` ==> k exponent p ((j+k) choose k) = 0" ``` haftmann@27105 ` 250` ```apply (cases "prime p") ``` paulson@13870 ` 251` ``` prefer 2 apply simp ``` haftmann@27105 ` 252` ```apply (induct k) ``` paulson@13870 ` 253` ```apply (simp (no_asm)) ``` paulson@13870 ` 254` ```(*induction step*) ``` haftmann@27105 ` 255` ```apply (subgoal_tac "(Suc (j+k) choose Suc k) > 0") ``` paulson@13870 ` 256` ``` prefer 2 apply (simp add: zero_less_binomial_iff, clarify) ``` haftmann@27105 ` 257` ```apply (subgoal_tac "exponent p ((Suc (j+k) choose Suc k) * Suc k) = ``` haftmann@27105 ` 258` ``` exponent p (Suc k)") ``` paulson@13870 ` 259` ``` txt{*First, use the assumed equation. We simplify the LHS to ``` haftmann@27105 ` 260` ``` @{term "exponent p (Suc (j + k) choose Suc k) + exponent p (Suc k)"} ``` paulson@13870 ` 261` ``` the common terms cancel, proving the conclusion.*} ``` paulson@13870 ` 262` ``` apply (simp del: bad_Sucs add: exponent_mult_add) ``` paulson@13870 ` 263` ```txt{*Establishing the equation requires first applying ``` paulson@13870 ` 264` ``` @{text Suc_times_binomial_eq} ...*} ``` paulson@13870 ` 265` ```apply (simp del: bad_Sucs add: Suc_times_binomial_eq [symmetric]) ``` paulson@13870 ` 266` ```txt{*...then @{text exponent_mult_add} and the quantified premise.*} ``` paulson@13870 ` 267` ```apply (simp del: bad_Sucs add: zero_less_binomial_iff exponent_mult_add) ``` paulson@13870 ` 268` ```done ``` paulson@13870 ` 269` paulson@13870 ` 270` ```(*The lemma above, with two changes of variables*) ``` paulson@13870 ` 271` ```lemma p_not_div_choose: ``` nipkow@25134 ` 272` ``` "[| kj. 0 exponent p (n - k + (K - j)) = exponent p (K - j)|] ``` nipkow@25134 ` 274` ``` ==> exponent p (n choose k) = 0" ``` paulson@13870 ` 275` ```apply (cut_tac j = "n-k" and k = k and p = p in p_not_div_choose_lemma) ``` paulson@13870 ` 276` ``` prefer 3 apply simp ``` paulson@13870 ` 277` ``` prefer 2 apply assumption ``` paulson@13870 ` 278` ```apply (drule_tac x = "K - Suc i" in spec) ``` paulson@13870 ` 279` ```apply (simp add: Suc_diff_le) ``` paulson@13870 ` 280` ```done ``` paulson@13870 ` 281` paulson@13870 ` 282` paulson@13870 ` 283` ```lemma const_p_fac_right: ``` nipkow@25162 ` 284` ``` "m>0 ==> exponent p ((p^a * m - Suc 0) choose (p^a - Suc 0)) = 0" ``` nipkow@16663 ` 285` ```apply (case_tac "prime p") ``` paulson@13870 ` 286` ``` prefer 2 apply simp ``` paulson@13870 ` 287` ```apply (frule_tac a = a in zero_less_prime_power) ``` paulson@13870 ` 288` ```apply (rule_tac K = "p^a" in p_not_div_choose) ``` paulson@13870 ` 289` ``` apply simp ``` paulson@13870 ` 290` ``` apply simp ``` paulson@13870 ` 291` ``` apply (case_tac "m") ``` paulson@13870 ` 292` ``` apply (case_tac [2] "p^a") ``` paulson@13870 ` 293` ``` apply auto ``` paulson@13870 ` 294` ```(*now the hard case, simplified to ``` paulson@13870 ` 295` ``` exponent p (Suc (p ^ a * m + i - p ^ a)) = exponent p (Suc i) *) ``` paulson@13870 ` 296` ```apply (subgoal_tac "00 ==> exponent p (((p^a) * m) choose p^a) = exponent p m" ``` nipkow@16663 ` 303` ```apply (case_tac "prime p") ``` paulson@13870 ` 304` ``` prefer 2 apply simp ``` paulson@13870 ` 305` ```apply (subgoal_tac "0 < p^a * m & p^a <= p^a * m") ``` paulson@13870 ` 306` ``` prefer 2 apply (force simp add: prime_iff) ``` paulson@13870 ` 307` ```txt{*A similar trick to the one used in @{text p_not_div_choose_lemma}: ``` paulson@13870 ` 308` ``` insert an equation; use @{text exponent_mult_add} on the LHS; on the RHS, ``` paulson@13870 ` 309` ``` first ``` paulson@13870 ` 310` ``` transform the binomial coefficient, then use @{text exponent_mult_add}.*} ``` paulson@13870 ` 311` ```apply (subgoal_tac "exponent p ((( (p^a) * m) choose p^a) * p^a) = ``` paulson@13870 ` 312` ``` a + exponent p m") ``` paulson@13870 ` 313` ``` apply (simp del: bad_Sucs add: zero_less_binomial_iff exponent_mult_add prime_iff) ``` paulson@13870 ` 314` ```txt{*one subgoal left!*} ``` paulson@13870 ` 315` ```apply (subst times_binomial_minus1_eq, simp, simp) ``` paulson@13870 ` 316` ```apply (subst exponent_mult_add, simp) ``` paulson@13870 ` 317` ```apply (simp (no_asm_simp) add: zero_less_binomial_iff) ``` webertj@20432 ` 318` ```apply arith ``` paulson@13870 ` 319` ```apply (simp del: bad_Sucs add: exponent_mult_add const_p_fac_right) ``` paulson@13870 ` 320` ```done ``` paulson@13870 ` 321` paulson@13870 ` 322` paulson@13870 ` 323` ```end ```