# Theory WilsonBij

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theory WilsonBij = BijectionRel + IntFact:

```(*  Title:      HOL/NumberTheory/WilsonBij.thy
ID:         \$Id: WilsonBij.thy,v 1.6 2003/12/03 09:49:35 paulson Exp \$
Author:     Thomas M. Rasmussen
*)

header {* Wilson's Theorem using a more abstract approach *}

theory WilsonBij = BijectionRel + IntFact:

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 *}

constdefs
reciR :: "int => int => int => bool"
"reciR p ==
λa b. zcong (a * b) 1 p ∧ 1 < a ∧ a < p - 1 ∧ 1 < b ∧ b < p - 1"
inv :: "int => int => int"
"inv p a ==
if p ∈ zprime ∧ 0 < a ∧ a < p then
(SOME x. 0 ≤ x ∧ x < p ∧ zcong (a * x) 1 p)
else 0"

text {* \medskip Inverse *}

lemma inv_correct:
"p ∈ zprime ==> 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, standard]
lemmas inv_less = inv_correct [THEN conjunct2, THEN conjunct1, standard]
lemmas inv_is_inv = inv_correct [THEN conjunct2, THEN conjunct2, standard]

lemma inv_not_0:
"p ∈ zprime ==> 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
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])
apply auto
done

lemma inv_not_1:
"p ∈ zprime ==> 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: 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)
apply auto
done

lemma inv_not_p_minus_1:
"p ∈ zprime ==> 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: "p ∈ zprime ==> 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:
"p ∈ zprime ==> 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: "p ∈ zprime ==> inj_on (inv p) (d22set (p - 2))"
apply (unfold inj_on_def)
apply auto
apply (rule zcong_zless_imp_eq)
apply (tactic {* stac (thm "zcong_cancel" RS sym) 5 *})
apply (rule_tac [7] zcong_trans)
apply (tactic {* stac (thm "zcong_sym") 8 *})
apply (erule_tac [7] inv_is_inv)
apply (tactic "Asm_simp_tac 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:
"p ∈ zprime ==> 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:
"p ∈ zprime ==> (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: "p ∈ zprime ==> bijP (reciR p) (d22set (p - 2))"
apply (unfold reciR_def bijP_def)
apply auto
apply (rule d22set_mem)
apply auto
done

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

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

lemma bijER_d22set: "p ∈ zprime ==> 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:
"p ∈ zprime ==> A ∈ bijER (reciR p) ==> [ssetprod 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 ssetprod_insert)
prefer 3
apply (subst ssetprod_insert)
apply (subgoal_tac "zcong ((a * b) * ssetprod A) (1 * 1) p")
apply (rule zcong_zmult)
apply auto
done

theorem Wilson_Bij: "p ∈ zprime ==> [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
```

### Definitions and lemmas

lemma inv_correct:

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

lemmas inv_ge:

`  [| p ∈ zprime; 0 < a; a < p |] ==> 0 ≤ WilsonBij.inv p a`

lemmas inv_less:

`  [| p ∈ zprime; 0 < a; a < p |] ==> WilsonBij.inv p a < p`

lemmas inv_is_inv:

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

lemma inv_not_0:

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

lemma inv_not_1:

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

lemma 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 |] ==> WilsonBij.inv p a ≠ p - 1`

lemma inv_g_1:

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

lemma inv_less_p_minus_1:

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

lemma aux1:

`  1 < x ==> 0 ≤ x`

lemma aux2:

`  1 < x ==> 0 < x`

lemma aux3:

`  x ≤ p - 2 ==> x < p`

lemma aux4:

`  x ≤ p - 2 ==> x < p - 1`

lemma inv_inj:

`  p ∈ zprime ==> inj_on (WilsonBij.inv p) (d22set (p - 2))`

lemma inv_d22set_d22set:

`  p ∈ zprime ==> WilsonBij.inv p ` d22set (p - 2) = d22set (p - 2)`

lemma d22set_d22set_bij:

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

lemma reciP_bijP:

`  p ∈ zprime ==> bijP (reciR p) (d22set (p - 2))`

lemma reciP_uniq:

`  p ∈ zprime ==> uniqP (reciR p)`

lemma reciP_sym:

`  p ∈ zprime ==> symP (reciR p)`

lemma bijER_d22set:

`  p ∈ zprime ==> d22set (p - 2) ∈ bijER (reciR p)`

### Wilson

lemma bijER_zcong_prod_1:

`  [| p ∈ zprime; A ∈ bijER (reciR p) |] ==> [ssetprod A = 1] (mod p)`

theorem Wilson_Bij:

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