# Theory WilsonBij

theory WilsonBij
imports BijectionRel IntFact
```(*  Title:      HOL/Old_Number_Theory/WilsonBij.thy
Author:     Thomas M. Rasmussen
*)

section ‹Wilson's Theorem using a more abstract approach›

theory WilsonBij
imports BijectionRel IntFact
begin

text ‹
Wilson's Theorem using a more ``abstract'' approach based on
bijections between sets.  Does not use Fermat's Little Theorem
(unlike Russinoff).
›

subsection ‹Definitions and lemmas›

definition reciR :: "int => int => int => bool"
where "reciR p = (λa b. zcong (a * b) 1 p ∧ 1 < a ∧ a < p - 1 ∧ 1 < b ∧ b < p - 1)"

definition inv :: "int => int => int" where
"inv p a =
(if zprime p ∧ 0 < a ∧ a < p then
(SOME x. 0 ≤ x ∧ x < p ∧ zcong (a * x) 1 p)
else 0)"

text ‹\medskip Inverse›

lemma inv_correct:
"zprime p ==> 0 < a ==> a < p
==> 0 ≤ inv p a ∧ inv p a < p ∧ [a * inv p a = 1] (mod p)"
apply (unfold inv_def)
apply (simp (no_asm_simp))
apply (rule zcong_lineq_unique [THEN ex1_implies_ex, THEN someI_ex])
apply (erule_tac [2] zless_zprime_imp_zrelprime)
apply (unfold zprime_def)
apply auto
done

lemmas inv_ge = inv_correct [THEN conjunct1]
lemmas inv_less = inv_correct [THEN conjunct2, THEN conjunct1]
lemmas inv_is_inv = inv_correct [THEN conjunct2, THEN conjunct2]

lemma inv_not_0:
"zprime p ==> 1 < a ==> a < p - 1 ==> inv p a ≠ 0"
-- ‹same as @{text WilsonRuss}›
apply safe
apply (cut_tac a = a and p = p in inv_is_inv)
apply (unfold zcong_def)
apply auto
done

lemma inv_not_1:
"zprime p ==> 1 < a ==> a < p - 1 ==> inv p a ≠ 1"
-- ‹same as @{text WilsonRuss}›
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)
apply auto
done

lemma aux: "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
-- ‹same as @{text WilsonRuss}›
apply (unfold zcong_def)
apply (simp add: diff_diff_eq diff_diff_eq2 right_diff_distrib)
apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
apply (subst dvd_minus_iff)
apply (subst zdvd_reduce)
apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
apply (subst zdvd_reduce)
apply auto
done

lemma inv_not_p_minus_1:
"zprime p ==> 1 < a ==> a < p - 1 ==> inv p a ≠ p - 1"
-- ‹same as @{text WilsonRuss}›
apply safe
apply (cut_tac a = a and p = p in inv_is_inv)
apply auto
apply (subgoal_tac "a = p - 1")
apply (rule_tac [2] zcong_zless_imp_eq)
apply auto
done

text ‹
Below is slightly different as we don't expand @{term [source] inv}
but use ``@{text correct}'' theorems.
›

lemma inv_g_1: "zprime p ==> 1 < a ==> a < p - 1 ==> 1 < 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)
apply auto
apply (rule inv_ge)
apply auto
done

lemma inv_less_p_minus_1:
"zprime p ==> 1 < a ==> a < p - 1 ==> inv p a < p - 1"
-- ‹ditto›
apply (subst order_less_le)
done

text ‹\medskip Bijection›

lemma aux1: "1 < x ==> 0 ≤ (x::int)"
apply auto
done

lemma aux2: "1 < x ==> 0 < (x::int)"
apply auto
done

lemma aux3: "x ≤ p - 2 ==> x < (p::int)"
apply auto
done

lemma aux4: "x ≤ p - 2 ==> x < (p::int) - 1"
apply auto
done

lemma inv_inj: "zprime p ==> inj_on (inv p) (d22set (p - 2))"
apply (unfold inj_on_def)
apply auto
apply (rule zcong_zless_imp_eq)
apply (tactic ‹stac @{context} (@{thm zcong_cancel} RS sym) 5›)
apply (rule_tac [7] zcong_trans)
apply (tactic ‹stac @{context} @{thm zcong_sym} 8›)
apply (erule_tac [7] inv_is_inv)
apply (tactic "asm_simp_tac @{context} 9")
apply (erule_tac [9] inv_is_inv)
apply (rule_tac [6] zless_zprime_imp_zrelprime)
apply (rule_tac [8] inv_less)
apply (rule_tac [7] inv_g_1 [THEN aux2])
apply (unfold zprime_def)
apply (auto intro: d22set_g_1 d22set_le
aux1 aux2 aux3 aux4)
done

lemma inv_d22set_d22set:
"zprime p ==> inv p ` d22set (p - 2) = d22set (p - 2)"
apply (rule endo_inj_surj)
apply (rule d22set_fin)
apply (erule_tac [2] inv_inj)
apply auto
apply (rule d22set_mem)
apply (erule inv_g_1)
apply (subgoal_tac [3] "inv p xa < p - 1")
apply (erule_tac [4] inv_less_p_minus_1)
apply (auto intro: d22set_g_1 d22set_le aux4)
done

lemma d22set_d22set_bij:
"zprime p ==> (d22set (p - 2), d22set (p - 2)) ∈ bijR (reciR p)"
apply (unfold reciR_def)
apply (rule_tac s = "(d22set (p - 2), inv p ` d22set (p - 2))" in subst)
apply (rule inj_func_bijR)
apply (rule_tac [3] d22set_fin)
apply (erule_tac [2] inv_inj)
apply auto
apply (erule inv_is_inv)
apply (erule_tac [5] inv_g_1)
apply (erule_tac [7] inv_less_p_minus_1)
apply (auto intro: d22set_g_1 d22set_le aux2 aux3 aux4)
done

lemma reciP_bijP: "zprime p ==> bijP (reciR p) (d22set (p - 2))"
apply (unfold reciR_def bijP_def)
apply auto
apply (rule d22set_mem)
apply auto
done

lemma reciP_uniq: "zprime p ==> uniqP (reciR p)"
apply (unfold reciR_def uniqP_def)
apply auto
apply (rule zcong_zless_imp_eq)
apply (tactic ‹stac @{context} (@{thm zcong_cancel2} RS sym) 5›)
apply (rule_tac [7] zcong_trans)
apply (tactic ‹stac @{context} @{thm zcong_sym} 8›)
apply (rule_tac [6] zless_zprime_imp_zrelprime)
apply auto
apply (rule zcong_zless_imp_eq)
apply (tactic ‹stac @{context} (@{thm zcong_cancel} RS sym) 5›)
apply (rule_tac [7] zcong_trans)
apply (tactic ‹stac @{context} @{thm zcong_sym} 8›)
apply (rule_tac [6] zless_zprime_imp_zrelprime)
apply auto
done

lemma reciP_sym: "zprime p ==> symP (reciR p)"
apply (unfold reciR_def symP_def)
apply auto
done

lemma bijER_d22set: "zprime p ==> d22set (p - 2) ∈ bijER (reciR p)"
apply (rule bijR_bijER)
apply (erule d22set_d22set_bij)
apply (erule reciP_bijP)
apply (erule reciP_uniq)
apply (erule reciP_sym)
done

subsection ‹Wilson›

lemma bijER_zcong_prod_1:
"zprime p ==> A ∈ bijER (reciR p) ==> [∏A = 1] (mod p)"
apply (unfold reciR_def)
apply (erule bijER.induct)
apply (subgoal_tac [2] "a = 1 ∨ a = p - 1")
apply (rule_tac [3] zcong_square_zless)
apply auto
apply (subst setprod.insert)
prefer 3
apply (subst setprod.insert)
apply (subgoal_tac "zcong ((a * b) * ∏A) (1 * 1) p")
apply (rule zcong_zmult)
apply auto
done

theorem Wilson_Bij: "zprime p ==> [zfact (p - 1) = -1] (mod p)"
apply (subgoal_tac "zcong ((p - 1) * zfact (p - 2)) (-1 * 1) p")
apply (rule_tac [2] zcong_zmult)
apply (subst zfact.simps)
apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst)
apply auto
apply (subst d22set_prod_zfact [symmetric])
apply (rule bijER_zcong_prod_1)
apply (rule_tac [2] bijER_d22set)
apply auto
done

end
```