# Theory WilsonRuss

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theory WilsonRuss = EulerFermat:

```(*  Title:      HOL/NumberTheory/WilsonRuss.thy
ID:         \$Id: WilsonRuss.thy,v 1.11 2003/12/03 09:49:36 paulson Exp \$
Author:     Thomas M. Rasmussen

repaired proof of prime_g_5
*)

header {* Wilson's Theorem according to Russinoff *}

theory WilsonRuss = EulerFermat:

text {*
Wilson's Theorem following quite closely Russinoff's approach
using Boyer-Moore (using finite sets instead of lists, though).
*}

subsection {* Definitions and lemmas *}

consts
inv :: "int => int => int"
wset :: "int * int => int set"

defs
inv_def: "inv p a == (a^(nat (p - 2))) mod p"

recdef wset
"measure ((λ(a, p). nat a) :: int * int => nat)"
"wset (a, p) =
(if 1 < a then
let ws = wset (a - 1, p)
in (if a ∈ ws then ws else insert a (insert (inv p a) ws)) else {})"

text {* \medskip @{term [source] inv} *}

lemma inv_is_inv_aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"
by (subst int_int_eq [symmetric], auto)

lemma inv_is_inv:
"p ∈ zprime ==> 0 < a ==> a < p ==> [a * inv p a = 1] (mod p)"
apply (unfold inv_def)
apply (subst zcong_zmod)
apply (subst zmod_zmult1_eq [symmetric])
apply (subst zcong_zmod [symmetric])
apply (subst power_Suc [symmetric])
apply (subst inv_is_inv_aux)
apply (erule_tac [2] Little_Fermat)
apply (erule_tac [2] zdvd_not_zless)
apply (unfold zprime_def, auto)
done

lemma inv_distinct:
"p ∈ zprime ==> 1 < a ==> a < p - 1 ==> a ≠ 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, 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, auto)
done

lemma inv_not_0:
"p ∈ zprime ==> 1 < a ==> a < p - 1 ==> inv p a ≠ 0"
apply safe
apply (cut_tac a = a and p = p in inv_is_inv)
apply (unfold zcong_def, auto)
apply (subgoal_tac "¬ p dvd 1")
apply (rule_tac [2] zdvd_not_zless)
apply (subgoal_tac "p dvd 1")
prefer 2
apply (subst zdvd_zminus_iff [symmetric], auto)
done

lemma inv_not_1:
"p ∈ zprime ==> 1 < a ==> a < p - 1 ==> inv p a ≠ 1"
apply safe
apply (cut_tac a = a and p = p in inv_is_inv)
prefer 4
apply simp
apply (subgoal_tac "a = 1")
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)"
apply (unfold zcong_def)
apply (simp add: Ring_and_Field.diff_diff_eq diff_diff_eq2 zdiff_zmult_distrib2)
apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
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, auto)
done

lemma inv_not_p_minus_1:
"p ∈ zprime ==> 1 < a ==> a < p - 1 ==> inv p a ≠ p - 1"
apply safe
apply (cut_tac a = a and p = p in inv_is_inv, auto)
apply (subgoal_tac "a = p - 1")
apply (rule_tac [2] zcong_zless_imp_eq, auto)
done

lemma inv_g_1:
"p ∈ zprime ==> 1 < a ==> a < p - 1 ==> 1 < inv p a"
apply (case_tac "0≤ inv p a")
apply (subgoal_tac "inv p a ≠ 1")
apply (subgoal_tac "inv p a ≠ 0")
apply (subst order_less_le)
apply (subst order_less_le)
apply (rule_tac [2] inv_not_0)
apply (rule_tac [5] inv_not_1, auto)
apply (unfold inv_def zprime_def, simp)
done

lemma inv_less_p_minus_1:
"p ∈ zprime ==> 1 < a ==> a < p - 1 ==> inv p a < p - 1"
apply (case_tac "inv p a < p")
apply (subst order_less_le)
apply (unfold inv_def zprime_def, simp)
done

lemma inv_inv_aux: "5 ≤ p ==>
nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))"
apply (subst int_int_eq [symmetric])
done

lemma zcong_zpower_zmult:
"[x^y = 1] (mod p) ==> [x^(y * z) = 1] (mod p)"
apply (induct z)
apply (subgoal_tac "zcong (x^y * x^(y * n)) (1 * 1) p")
apply (rule_tac [2] zcong_zmult, simp_all)
done

lemma inv_inv: "p ∈ zprime ==>
5 ≤ p ==> 0 < a ==> a < p ==> inv p (inv p a) = a"
apply (unfold inv_def)
apply (subst zpower_zmod)
apply (subst zpower_zpower)
apply (rule zcong_zless_imp_eq)
prefer 5
apply (subst zcong_zmod)
apply (subst mod_mod_trivial)
apply (subst zcong_zmod [symmetric])
apply (subst inv_inv_aux)
apply (subgoal_tac [2]
"zcong (a * a^(nat (p - 1) * nat (p - 3))) (a * 1) p")
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, simp_all)
done

text {* \medskip @{term wset} *}

declare wset.simps [simp del]

lemma wset_induct:
"(!!a p. P {} a p) ==>
(!!a p. 1 < (a::int) ==> P (wset (a - 1, p)) (a - 1) p
==> P (wset (a, p)) a p)
==> P (wset (u, v)) u v"
proof -
case rule_context
show ?thesis
apply (rule wset.induct, safe)
apply (case_tac [2] "1 < a")
apply (rule_tac [2] rule_context, simp_all)
done
qed

lemma wset_mem_imp_or [rule_format]:
"1 < a ==> b ∉ wset (a - 1, p)
==> b ∈ wset (a, p) --> b = a ∨ b = inv p a"
apply (subst wset.simps)
apply (unfold Let_def, simp)
done

lemma wset_mem_mem [simp]: "1 < a ==> a ∈ wset (a, p)"
apply (subst wset.simps)
apply (unfold Let_def, simp)
done

lemma wset_subset: "1 < a ==> b ∈ wset (a - 1, p) ==> b ∈ wset (a, p)"
apply (subst wset.simps)
apply (unfold Let_def, auto)
done

lemma wset_g_1 [rule_format]:
"p ∈ zprime --> a < p - 1 --> b ∈ wset (a, p) --> 1 < b"
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 ∨ b = inv p a")
apply (rule_tac [4] wset_mem_imp_or)
prefer 2
apply simp
apply (rule inv_g_1, auto)
done

lemma wset_less [rule_format]:
"p ∈ zprime --> a < p - 1 --> b ∈ wset (a, p) --> b < p - 1"
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 ∨ b = inv p a")
apply (rule_tac [4] wset_mem_imp_or)
prefer 2
apply simp
apply (rule inv_less_p_minus_1, auto)
done

lemma wset_mem [rule_format]:
"p ∈ zprime -->
a < p - 1 --> 1 < b --> b ≤ a --> b ∈ wset (a, p)"
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)
apply (simp (no_asm_simp))
apply auto
done

lemma wset_mem_inv_mem [rule_format]:
"p ∈ zprime --> 5 ≤ p --> a < p - 1 --> b ∈ wset (a, p)
--> inv p b ∈ wset (a, p)"
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, auto)
apply (case_tac "b = inv p a")
apply (simp (no_asm_simp))
apply (subst inv_inv)
apply (subgoal_tac [6] "b = a ∨ b = inv p a")
apply (rule_tac [7] wset_mem_imp_or, auto)
done

lemma wset_inv_mem_mem:
"p ∈ zprime ==> 5 ≤ p ==> a < p - 1 ==> 1 < b ==> b < p - 1
==> inv p b ∈ wset (a, p) ==> b ∈ 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, 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, auto)
done

lemma wset_zcong_prod_1 [rule_format]:
"p ∈ zprime -->
5 ≤ p --> a < p - 1 --> [ssetprod (wset (a, p)) = 1] (mod p)"
apply (induct a p rule: wset_induct)
prefer 2
apply (subst wset.simps)
apply (unfold Let_def, auto)
apply (subst ssetprod_insert)
apply (tactic {* stac (thm "ssetprod_insert") 3 *})
apply (subgoal_tac [5]
"zcong (a * inv p a * ssetprod (wset (a - 1, p))) (1 * 1) p")
prefer 5
apply (rule_tac [5] zcong_zmult)
apply (rule_tac [5] inv_is_inv)
apply (tactic "Clarify_tac 4")
apply (subgoal_tac [4] "a ∈ wset (a - 1, p)")
apply (rule_tac [5] wset_inv_mem_mem)
apply (rule inv_distinct, auto)
done

lemma d22set_eq_wset: "p ∈ zprime ==> d22set (p - 2) = wset (p - 2, p)"
apply safe
apply (erule wset_mem)
apply (rule_tac [2] d22set_g_1)
apply (rule_tac [3] d22set_le)
apply (rule_tac [4] d22set_mem)
apply (erule_tac [4] wset_g_1)
prefer 6
apply (subgoal_tac "p - 2 + 1 = p - 1")
apply (simp (no_asm_simp))
apply (erule wset_less, auto)
done

subsection {* Wilson *}

lemma prime_g_5: "p ∈ zprime ==> p ≠ 2 ==> p ≠ 3 ==> 5 ≤ p"
apply (unfold zprime_def dvd_def)
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, arith)
apply (rule_tac x = 2 in exI, auto)
done

theorem Wilson_Russ:
"p ∈ zprime ==> [zfact (p - 1) = -1] (mod p)"
apply (subgoal_tac "[(p - 1) * zfact (p - 2) = -1 * 1] (mod p)")
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, auto)
apply (simp only: zcong_def)
apply (simp (no_asm_simp))
apply (case_tac "p = 2")
apply (case_tac "p = 3")
apply (subgoal_tac "5 ≤ p")
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, auto)
done

end
```

### Definitions and lemmas

lemma inv_is_inv_aux:

`  1 < m ==> Suc (nat (m - 2)) = nat (m - 1)`

lemma inv_is_inv:

`  [| p ∈ zprime; 0 < a; a < p |] ==> [a * WilsonRuss.inv p a = 1] (mod p)`

lemma inv_distinct:

`  [| p ∈ zprime; 1 < a; a < p - 1 |] ==> a ≠ WilsonRuss.inv p a`

lemma inv_not_0:

`  [| p ∈ zprime; 1 < a; a < p - 1 |] ==> WilsonRuss.inv p a ≠ 0`

lemma inv_not_1:

`  [| p ∈ zprime; 1 < a; a < p - 1 |] ==> WilsonRuss.inv p a ≠ 1`

lemma inv_not_p_minus_1_aux:

`  [a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)`

lemma inv_not_p_minus_1:

`  [| p ∈ zprime; 1 < a; a < p - 1 |] ==> WilsonRuss.inv p a ≠ p - 1`

lemma inv_g_1:

`  [| p ∈ zprime; 1 < a; a < p - 1 |] ==> 1 < WilsonRuss.inv p a`

lemma inv_less_p_minus_1:

`  [| p ∈ zprime; 1 < a; a < p - 1 |] ==> WilsonRuss.inv p a < p - 1`

lemma inv_inv_aux:

`  5 ≤ p ==> nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))`

lemma zcong_zpower_zmult:

`  [x ^ y = 1] (mod p) ==> [x ^ (y * z) = 1] (mod p)`

lemma inv_inv:

```  [| p ∈ zprime; 5 ≤ p; 0 < a; a < p |]
==> WilsonRuss.inv p (WilsonRuss.inv p a) = a```

lemma wset_induct:

```  [| !!a p. P {} a p;
!!a p. [| 1 < a; P (wset (a - 1, p)) (a - 1) p |] ==> P (wset (a, p)) a p |]
==> P (wset (u, v)) u v```

lemma wset_mem_imp_or:

```  [| 1 < a; b ∉ wset (a - 1, p); b ∈ wset (a, p) |]
==> b = a ∨ b = WilsonRuss.inv p a```

lemma wset_mem_mem:

`  1 < a ==> a ∈ wset (a, p)`

lemma wset_subset:

`  [| 1 < a; b ∈ wset (a - 1, p) |] ==> b ∈ wset (a, p)`

lemma wset_g_1:

`  [| p ∈ zprime; a < p - 1; b ∈ wset (a, p) |] ==> 1 < b`

lemma wset_less:

`  [| p ∈ zprime; a < p - 1; b ∈ wset (a, p) |] ==> b < p - 1`

lemma wset_mem:

`  [| p ∈ zprime; a < p - 1; 1 < b; b ≤ a |] ==> b ∈ wset (a, p)`

lemma wset_mem_inv_mem:

```  [| p ∈ zprime; 5 ≤ p; a < p - 1; b ∈ wset (a, p) |]
==> WilsonRuss.inv p b ∈ wset (a, p)```

lemma wset_inv_mem_mem:

```  [| p ∈ zprime; 5 ≤ p; a < p - 1; 1 < b; b < p - 1;
WilsonRuss.inv p b ∈ wset (a, p) |]
==> b ∈ wset (a, p)```

lemma wset_fin:

`  finite (wset (a, p))`

lemma wset_zcong_prod_1:

`  [| p ∈ zprime; 5 ≤ p; a < p - 1 |] ==> [ssetprod (wset (a, p)) = 1] (mod p)`

lemma d22set_eq_wset:

`  p ∈ zprime ==> d22set (p - 2) = wset (p - 2, p)`

### Wilson

lemma prime_g_5:

`  [| p ∈ zprime; p ≠ 2; p ≠ 3 |] ==> 5 ≤ p`

theorem Wilson_Russ:

`  p ∈ zprime ==> [zfact (p - 1) = -1] (mod p)`