--- a/src/HOL/NumberTheory/WilsonRuss.thy Wed Feb 26 10:48:00 2003 +0100
+++ b/src/HOL/NumberTheory/WilsonRuss.thy Wed Feb 26 13:16:07 2003 +0100
@@ -2,6 +2,9 @@
ID: $Id$
Author: Thomas M. Rasmussen
Copyright 2000 University of Cambridge
+
+Changes by Jeremy Avigad, 2003/02/21:
+ repaired proof of prime_g_5
*)
header {* Wilson's Theorem according to Russinoff *}
@@ -33,9 +36,7 @@
text {* \medskip @{term [source] inv} *}
lemma inv_is_inv_aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"
- apply (subst int_int_eq [symmetric])
- apply auto
- done
+by (subst int_int_eq [symmetric], auto)
lemma inv_is_inv:
"p \<in> zprime \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> [a * inv p a = 1] (mod p)"
@@ -47,35 +48,30 @@
apply (subst inv_is_inv_aux)
apply (erule_tac [2] Little_Fermat)
apply (erule_tac [2] zdvd_not_zless)
- apply (unfold zprime_def)
- apply auto
+ apply (unfold zprime_def, auto)
done
lemma inv_distinct:
"p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> a \<noteq> inv p a"
apply safe
apply (cut_tac a = a and p = p in zcong_square)
- apply (cut_tac [3] a = a and p = p in inv_is_inv)
- apply auto
+ apply (cut_tac [3] a = a and p = p in inv_is_inv, auto)
apply (subgoal_tac "a = 1")
apply (rule_tac [2] m = p in zcong_zless_imp_eq)
apply (subgoal_tac [7] "a = p - 1")
- apply (rule_tac [8] m = p in zcong_zless_imp_eq)
- apply auto
+ apply (rule_tac [8] m = p in zcong_zless_imp_eq, auto)
done
lemma inv_not_0:
"p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 0"
apply safe
apply (cut_tac a = a and p = p in inv_is_inv)
- apply (unfold zcong_def)
- apply auto
+ apply (unfold zcong_def, auto)
apply (subgoal_tac "\<not> p dvd 1")
apply (rule_tac [2] zdvd_not_zless)
apply (subgoal_tac "p dvd 1")
prefer 2
- apply (subst zdvd_zminus_iff [symmetric])
- apply auto
+ apply (subst zdvd_zminus_iff [symmetric], auto)
done
lemma inv_not_1:
@@ -85,8 +81,7 @@
prefer 4
apply simp
apply (subgoal_tac "a = 1")
- apply (rule_tac [2] zcong_zless_imp_eq)
- apply auto
+ apply (rule_tac [2] zcong_zless_imp_eq, auto)
done
lemma inv_not_p_minus_1_aux: "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
@@ -97,19 +92,16 @@
apply (subst zdvd_zminus_iff)
apply (subst zdvd_reduce)
apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
- apply (subst zdvd_reduce)
- apply auto
+ apply (subst zdvd_reduce, auto)
done
lemma inv_not_p_minus_1:
"p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> p - 1"
apply safe
- apply (cut_tac a = a and p = p in inv_is_inv)
- apply auto
+ apply (cut_tac a = a and p = p in inv_is_inv, auto)
apply (simp add: inv_not_p_minus_1_aux)
apply (subgoal_tac "a = p - 1")
- apply (rule_tac [2] zcong_zless_imp_eq)
- apply auto
+ apply (rule_tac [2] zcong_zless_imp_eq, auto)
done
lemma inv_g_1:
@@ -121,20 +113,16 @@
apply (subst zle_add1_eq_le [symmetric])
apply (subst order_less_le)
apply (rule_tac [2] inv_not_0)
- apply (rule_tac [5] inv_not_1)
- apply auto
- apply (unfold inv_def zprime_def)
- apply (simp)
+ apply (rule_tac [5] inv_not_1, auto)
+ apply (unfold inv_def zprime_def, simp)
done
lemma inv_less_p_minus_1:
"p \<in> zprime \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a < p - 1"
apply (case_tac "inv p a < p")
apply (subst order_less_le)
- apply (simp add: inv_not_p_minus_1)
- apply auto
- apply (unfold inv_def zprime_def)
- apply (simp)
+ apply (simp add: inv_not_p_minus_1, auto)
+ apply (unfold inv_def zprime_def, simp)
done
lemma inv_inv_aux: "5 \<le> p ==>
@@ -149,8 +137,7 @@
apply (induct z)
apply (auto simp add: zpower_zadd_distrib)
apply (subgoal_tac "zcong (x^y * x^(y * n)) (1 * 1) p")
- apply (rule_tac [2] zcong_zmult)
- apply simp_all
+ apply (rule_tac [2] zcong_zmult, simp_all)
done
lemma inv_inv: "p \<in> zprime \<Longrightarrow>
@@ -169,8 +156,7 @@
apply (rule_tac [3] zcong_zmult)
apply (rule_tac [4] zcong_zpower_zmult)
apply (erule_tac [4] Little_Fermat)
- apply (rule_tac [4] zdvd_not_zless)
- apply (simp_all)
+ apply (rule_tac [4] zdvd_not_zless, simp_all)
done
@@ -186,11 +172,9 @@
proof -
case rule_context
show ?thesis
- apply (rule wset.induct)
- apply safe
+ apply (rule wset.induct, safe)
apply (case_tac [2] "1 < a")
- apply (rule_tac [2] rule_context)
- apply simp_all
+ apply (rule_tac [2] rule_context, simp_all)
apply (simp_all add: wset.simps rule_context)
done
qed
@@ -199,55 +183,47 @@
"1 < a \<Longrightarrow> b \<notin> wset (a - 1, p)
==> b \<in> wset (a, p) --> b = a \<or> b = inv p a"
apply (subst wset.simps)
- apply (unfold Let_def)
- apply simp
+ apply (unfold Let_def, simp)
done
lemma wset_mem_mem [simp]: "1 < a ==> a \<in> wset (a, p)"
apply (subst wset.simps)
- apply (unfold Let_def)
- apply simp
+ apply (unfold Let_def, simp)
done
lemma wset_subset: "1 < a \<Longrightarrow> b \<in> wset (a - 1, p) ==> b \<in> wset (a, p)"
apply (subst wset.simps)
- apply (unfold Let_def)
- apply auto
+ apply (unfold Let_def, auto)
done
lemma wset_g_1 [rule_format]:
"p \<in> zprime --> a < p - 1 --> b \<in> wset (a, p) --> 1 < b"
- apply (induct a p rule: wset_induct)
- apply auto
+ apply (induct a p rule: wset_induct, auto)
apply (case_tac "b = a")
apply (case_tac [2] "b = inv p a")
apply (subgoal_tac [3] "b = a \<or> b = inv p a")
apply (rule_tac [4] wset_mem_imp_or)
prefer 2
apply simp
- apply (rule inv_g_1)
- apply auto
+ apply (rule inv_g_1, auto)
done
lemma wset_less [rule_format]:
"p \<in> zprime --> a < p - 1 --> b \<in> wset (a, p) --> b < p - 1"
- apply (induct a p rule: wset_induct)
- apply auto
+ apply (induct a p rule: wset_induct, auto)
apply (case_tac "b = a")
apply (case_tac [2] "b = inv p a")
apply (subgoal_tac [3] "b = a \<or> b = inv p a")
apply (rule_tac [4] wset_mem_imp_or)
prefer 2
apply simp
- apply (rule inv_less_p_minus_1)
- apply auto
+ apply (rule inv_less_p_minus_1, auto)
done
lemma wset_mem [rule_format]:
"p \<in> zprime -->
a < p - 1 --> 1 < b --> b \<le> a --> b \<in> wset (a, p)"
- apply (induct a p rule: wset.induct)
- apply auto
+ apply (induct a p rule: wset.induct, auto)
apply (subgoal_tac "b = a")
apply (rule_tac [2] zle_anti_sym)
apply (rule_tac [4] wset_subset)
@@ -258,19 +234,16 @@
lemma wset_mem_inv_mem [rule_format]:
"p \<in> zprime --> 5 \<le> p --> a < p - 1 --> b \<in> wset (a, p)
--> inv p b \<in> wset (a, p)"
- apply (induct a p rule: wset_induct)
- apply auto
+ apply (induct a p rule: wset_induct, auto)
apply (case_tac "b = a")
apply (subst wset.simps)
apply (unfold Let_def)
- apply (rule_tac [3] wset_subset)
- apply auto
+ apply (rule_tac [3] wset_subset, auto)
apply (case_tac "b = inv p a")
apply (simp (no_asm_simp))
apply (subst inv_inv)
apply (subgoal_tac [6] "b = a \<or> b = inv p a")
- apply (rule_tac [7] wset_mem_imp_or)
- apply auto
+ apply (rule_tac [7] wset_mem_imp_or, auto)
done
lemma wset_inv_mem_mem:
@@ -278,16 +251,14 @@
\<Longrightarrow> inv p b \<in> wset (a, p) \<Longrightarrow> b \<in> wset (a, p)"
apply (rule_tac s = "inv p (inv p b)" and t = b in subst)
apply (rule_tac [2] wset_mem_inv_mem)
- apply (rule inv_inv)
- apply simp_all
+ apply (rule inv_inv, simp_all)
done
lemma wset_fin: "finite (wset (a, p))"
apply (induct a p rule: wset_induct)
prefer 2
apply (subst wset.simps)
- apply (unfold Let_def)
- apply auto
+ apply (unfold Let_def, auto)
done
lemma wset_zcong_prod_1 [rule_format]:
@@ -296,8 +267,7 @@
apply (induct a p rule: wset_induct)
prefer 2
apply (subst wset.simps)
- apply (unfold Let_def)
- apply auto
+ apply (unfold Let_def, auto)
apply (subst setprod_insert)
apply (tactic {* stac (thm "setprod_insert") 3 *})
apply (subgoal_tac [5]
@@ -310,8 +280,7 @@
apply (subgoal_tac [4] "a \<in> wset (a - 1, p)")
apply (rule_tac [5] wset_inv_mem_mem)
apply (simp_all add: wset_fin)
- apply (rule inv_distinct)
- apply auto
+ apply (rule inv_distinct, auto)
done
lemma d22set_eq_wset: "p \<in> zprime ==> d22set (p - 2) = wset (p - 2, p)"
@@ -325,8 +294,7 @@
apply (subst zle_add1_eq_le [symmetric])
apply (subgoal_tac "p - 2 + 1 = p - 1")
apply (simp (no_asm_simp))
- apply (erule wset_less)
- apply auto
+ apply (erule wset_less, auto)
done
@@ -334,16 +302,14 @@
lemma prime_g_5: "p \<in> zprime \<Longrightarrow> p \<noteq> 2 \<Longrightarrow> p \<noteq> 3 ==> 5 \<le> p"
apply (unfold zprime_def dvd_def)
- apply (case_tac "p = 4")
- apply auto
+ apply (case_tac "p = 4", auto)
apply (rule notE)
prefer 2
apply assumption
apply (simp (no_asm))
- apply (rule_tac x = "2" in exI)
- apply safe
- apply (rule_tac x = "2" in exI)
- apply auto
+ apply (rule_tac x = 2 in exI)
+ apply (safe, arith)
+ apply (rule_tac x = 2 in exI, auto)
done
theorem Wilson_Russ:
@@ -352,8 +318,7 @@
apply (rule_tac [2] zcong_zmult)
apply (simp only: zprime_def)
apply (subst zfact.simps)
- apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst)
- apply auto
+ apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst, auto)
apply (simp only: zcong_def)
apply (simp (no_asm_simp))
apply (case_tac "p = 2")
@@ -364,8 +329,7 @@
apply (erule_tac [2] prime_g_5)
apply (subst d22set_prod_zfact [symmetric])
apply (subst d22set_eq_wset)
- apply (rule_tac [2] wset_zcong_prod_1)
- apply auto
+ apply (rule_tac [2] wset_zcong_prod_1, auto)
done
end