# Theory EulerFermat

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theory EulerFermat = BijectionRel + IntFact:

```(*  Title:      HOL/NumberTheory/EulerFermat.thy
ID:         \$Id: EulerFermat.thy,v 1.11 2003/08/29 13:19:32 ballarin Exp \$
Author:     Thomas M. Rasmussen
Copyright   2000  University of Cambridge

Changes by Jeremy Avigad, 2003/02/21:
repaired proof of Bnor_prime (removed use of zprime_def)
*)

header {* Fermat's Little Theorem extended to Euler's Totient function *}

theory EulerFermat = BijectionRel + IntFact:

text {*
Fermat's Little Theorem extended to Euler's Totient function. More
abstract approach than Boyer-Moore (which seems necessary to achieve
the extended version).
*}

subsection {* Definitions and lemmas *}

consts
RsetR :: "int => int set set"
BnorRset :: "int * int => int set"
norRRset :: "int => int set"
noXRRset :: "int => int => int set"
phi :: "int => nat"
is_RRset :: "int set => int => bool"
RRset2norRR :: "int set => int => int => int"

inductive "RsetR m"
intros
empty [simp]: "{} ∈ RsetR m"
insert: "A ∈ RsetR m ==> zgcd (a, m) = 1 ==>
∀a'. a' ∈ A --> ¬ zcong a a' m ==> insert a A ∈ RsetR m"

recdef BnorRset
"measure ((λ(a, m). nat a) :: int * int => nat)"
"BnorRset (a, m) =
(if 0 < a then
let na = BnorRset (a - 1, m)
in (if zgcd (a, m) = 1 then insert a na else na)
else {})"

defs
norRRset_def: "norRRset m == BnorRset (m - 1, m)"
noXRRset_def: "noXRRset m x == (λa. a * x) ` norRRset m"
phi_def: "phi m == card (norRRset m)"
is_RRset_def: "is_RRset A m == A ∈ RsetR m ∧ card A = phi m"
RRset2norRR_def:
"RRset2norRR A m a ==
(if 1 < m ∧ is_RRset A m ∧ a ∈ A then
SOME b. zcong a b m ∧ b ∈ norRRset m
else 0)"

constdefs
zcongm :: "int => int => int => bool"
"zcongm m == λa b. zcong a b m"

lemma abs_eq_1_iff [iff]: "(abs z = (1::int)) = (z = 1 ∨ z = -1)"
-- {* LCP: not sure why this lemma is needed now *}
by (auto simp add: zabs_def)

text {* \medskip @{text norRRset} *}

declare BnorRset.simps [simp del]

lemma BnorRset_induct:
"(!!a m. P {} a m) ==>
(!!a m. 0 < (a::int) ==> P (BnorRset (a - 1, m::int)) (a - 1) m
==> P (BnorRset(a,m)) a m)
==> P (BnorRset(u,v)) u v"
proof -
case rule_context
show ?thesis
apply (rule BnorRset.induct, safe)
apply (case_tac [2] "0 < a")
apply (rule_tac [2] rule_context, simp_all)
apply (simp_all add: BnorRset.simps rule_context)
done
qed

lemma Bnor_mem_zle [rule_format]: "b ∈ BnorRset (a, m) --> b ≤ a"
apply (induct a m rule: BnorRset_induct)
prefer 2
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
done

lemma Bnor_mem_zle_swap: "a < b ==> b ∉ BnorRset (a, m)"
by (auto dest: Bnor_mem_zle)

lemma Bnor_mem_zg [rule_format]: "b ∈ BnorRset (a, m) --> 0 < b"
apply (induct a m rule: BnorRset_induct)
prefer 2
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
done

lemma Bnor_mem_if [rule_format]:
"zgcd (b, m) = 1 --> 0 < b --> b ≤ a --> b ∈ BnorRset (a, m)"
apply (induct a m rule: BnorRset.induct, auto)
apply (case_tac "a = b")
prefer 2
apply (simp add: order_less_le)
apply (simp (no_asm_simp))
prefer 2
apply (subst BnorRset.simps)
defer
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
done

lemma Bnor_in_RsetR [rule_format]: "a < m --> BnorRset (a, m) ∈ RsetR m"
apply (induct a m rule: BnorRset_induct, simp)
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
apply (rule RsetR.insert)
apply (rule_tac [3] allI)
apply (rule_tac [3] impI)
apply (rule_tac [3] zcong_not)
apply (subgoal_tac [6] "a' ≤ a - 1")
apply (rule_tac [7] Bnor_mem_zle)
apply (rule_tac [5] Bnor_mem_zg, auto)
done

lemma Bnor_fin: "finite (BnorRset (a, m))"
apply (induct a m rule: BnorRset_induct)
prefer 2
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
done

lemma norR_mem_unique_aux: "a ≤ b - 1 ==> a < (b::int)"
apply auto
done

lemma norR_mem_unique:
"1 < m ==>
zgcd (a, m) = 1 ==> ∃!b. [a = b] (mod m) ∧ b ∈ norRRset m"
apply (unfold norRRset_def)
apply (cut_tac a = a and m = m in zcong_zless_unique, auto)
apply (rule_tac [2] m = m in zcong_zless_imp_eq)
apply (auto intro: Bnor_mem_zle Bnor_mem_zg zcong_trans
order_less_imp_le norR_mem_unique_aux simp add: zcong_sym)
apply (rule_tac x = b in exI, safe)
apply (rule Bnor_mem_if)
apply (case_tac [2] "b = 0")
apply (auto intro: order_less_le [THEN iffD2])
prefer 2
apply (simp only: zcong_def)
apply (subgoal_tac "zgcd (a, m) = m")
prefer 2
apply (subst zdvd_iff_zgcd [symmetric])
apply (rule_tac [4] zgcd_zcong_zgcd)
apply (simp_all add: zdvd_zminus_iff zcong_sym)
done

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

lemma RRset_gcd [rule_format]:
"is_RRset A m ==> a ∈ A --> zgcd (a, m) = 1"
apply (unfold is_RRset_def)
apply (rule RsetR.induct, auto)
done

lemma RsetR_zmult_mono:
"A ∈ RsetR m ==>
0 < m ==> zgcd (x, m) = 1 ==> (λa. a * x) ` A ∈ RsetR m"
apply (erule RsetR.induct, simp_all)
apply (rule RsetR.insert, auto)
apply (blast intro: zgcd_zgcd_zmult)
apply (simp add: zcong_cancel)
done

lemma card_nor_eq_noX:
"0 < m ==>
zgcd (x, m) = 1 ==> card (noXRRset m x) = card (norRRset m)"
apply (unfold norRRset_def noXRRset_def)
apply (rule card_image)
apply (auto simp add: inj_on_def Bnor_fin)
apply (simp add: BnorRset.simps)
done

lemma noX_is_RRset:
"0 < m ==> zgcd (x, m) = 1 ==> is_RRset (noXRRset m x) m"
apply (unfold is_RRset_def phi_def)
apply (auto simp add: card_nor_eq_noX)
apply (unfold noXRRset_def norRRset_def)
apply (rule RsetR_zmult_mono)
apply (rule Bnor_in_RsetR, simp_all)
done

lemma aux_some:
"1 < m ==> is_RRset A m ==> a ∈ A
==> zcong a (SOME b. [a = b] (mod m) ∧ b ∈ norRRset m) m ∧
(SOME b. [a = b] (mod m) ∧ b ∈ norRRset m) ∈ norRRset m"
apply (rule norR_mem_unique [THEN ex1_implies_ex, THEN someI_ex])
apply (rule_tac [2] RRset_gcd, simp_all)
done

lemma RRset2norRR_correct:
"1 < m ==> is_RRset A m ==> a ∈ A ==>
[a = RRset2norRR A m a] (mod m) ∧ RRset2norRR A m a ∈ norRRset m"
apply (unfold RRset2norRR_def, simp)
apply (rule aux_some, simp_all)
done

lemmas RRset2norRR_correct1 =
RRset2norRR_correct [THEN conjunct1, standard]
lemmas RRset2norRR_correct2 =
RRset2norRR_correct [THEN conjunct2, standard]

lemma RsetR_fin: "A ∈ RsetR m ==> finite A"
by (erule RsetR.induct, auto)

lemma RRset_zcong_eq [rule_format]:
"1 < m ==>
is_RRset A m ==> [a = b] (mod m) ==> a ∈ A --> b ∈ A --> a = b"
apply (unfold is_RRset_def)
apply (rule RsetR.induct)
apply (auto simp add: zcong_sym)
done

lemma aux:
"P (SOME a. P a) ==> Q (SOME a. Q a) ==>
(SOME a. P a) = (SOME a. Q a) ==> ∃a. P a ∧ Q a"
apply auto
done

lemma RRset2norRR_inj:
"1 < m ==> is_RRset A m ==> inj_on (RRset2norRR A m) A"
apply (unfold RRset2norRR_def inj_on_def, auto)
apply (subgoal_tac "∃b. ([x = b] (mod m) ∧ b ∈ norRRset m) ∧
([y = b] (mod m) ∧ b ∈ norRRset m)")
apply (rule_tac [2] aux)
apply (rule_tac [3] aux_some)
apply (rule_tac [2] aux_some)
apply (rule RRset_zcong_eq, auto)
apply (rule_tac b = b in zcong_trans)
apply (simp_all add: zcong_sym)
done

lemma RRset2norRR_eq_norR:
"1 < m ==> is_RRset A m ==> RRset2norRR A m ` A = norRRset m"
apply (rule card_seteq)
prefer 3
apply (subst card_image)
apply (rule_tac [2] RRset2norRR_inj, auto)
apply (rule_tac [4] RRset2norRR_correct2, auto)
apply (unfold is_RRset_def phi_def norRRset_def)
apply (auto simp add: RsetR_fin Bnor_fin)
done

lemma Bnor_prod_power_aux: "a ∉ A ==> inj f ==> f a ∉ f ` A"
by (unfold inj_on_def, auto)

lemma Bnor_prod_power [rule_format]:
"x ≠ 0 ==> a < m --> ssetprod ((λa. a * x) ` BnorRset (a, m)) =
ssetprod (BnorRset(a, m)) * x^card (BnorRset (a, m))"
apply (induct a m rule: BnorRset_induct)
prefer 2
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
apply (simp add: Bnor_fin Bnor_mem_zle_swap)
apply (subst ssetprod_insert)
apply (rule_tac [2] Bnor_prod_power_aux)
apply (unfold inj_on_def)
apply (simp_all add: zmult_ac Bnor_fin finite_imageI
Bnor_mem_zle_swap)
done

subsection {* Fermat *}

lemma bijzcong_zcong_prod:
"(A, B) ∈ bijR (zcongm m) ==> [ssetprod A = ssetprod B] (mod m)"
apply (unfold zcongm_def)
apply (erule bijR.induct)
apply (subgoal_tac [2] "a ∉ A ∧ b ∉ B ∧ finite A ∧ finite B")
apply (auto intro: fin_bijRl fin_bijRr zcong_zmult)
done

lemma Bnor_prod_zgcd [rule_format]:
"a < m --> zgcd (ssetprod (BnorRset (a, m)), m) = 1"
apply (induct a m rule: BnorRset_induct)
prefer 2
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
apply (simp add: Bnor_fin Bnor_mem_zle_swap)
apply (blast intro: zgcd_zgcd_zmult)
done

theorem Euler_Fermat:
"0 < m ==> zgcd (x, m) = 1 ==> [x^(phi m) = 1] (mod m)"
apply (unfold norRRset_def phi_def)
apply (case_tac "x = 0")
apply (case_tac [2] "m = 1")
apply (rule_tac [3] iffD1)
apply (rule_tac [3] k = "ssetprod (BnorRset (m - 1, m))"
in zcong_cancel2)
prefer 5
apply (subst Bnor_prod_power [symmetric])
apply (rule_tac [7] Bnor_prod_zgcd, simp_all)
apply (rule bijzcong_zcong_prod)
apply (fold norRRset_def noXRRset_def)
apply (subst RRset2norRR_eq_norR [symmetric])
apply (rule_tac [3] inj_func_bijR, auto)
apply (unfold zcongm_def)
apply (rule_tac [2] RRset2norRR_correct1)
apply (rule_tac [5] RRset2norRR_inj)
apply (auto intro: order_less_le [THEN iffD2]
apply (unfold noXRRset_def norRRset_def)
apply (rule finite_imageI)
apply (rule Bnor_fin)
done

lemma Bnor_prime [rule_format (no_asm)]:
"p ∈ zprime ==>
a < p --> (∀b. 0 < b ∧ b ≤ a --> zgcd (b, p) = 1)
--> card (BnorRset (a, p)) = nat a"
apply (auto simp add: zless_zprime_imp_zrelprime)
apply (induct a p rule: BnorRset.induct)
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
apply (subgoal_tac "finite (BnorRset (a - 1,m))")
apply (subgoal_tac "a ~: BnorRset (a - 1,m)")
apply (frule Bnor_mem_zle, arith)
apply (frule Bnor_fin)
done

lemma phi_prime: "p ∈ zprime ==> phi p = nat (p - 1)"
apply (unfold phi_def norRRset_def)
apply (rule Bnor_prime, auto)
apply (erule zless_zprime_imp_zrelprime, simp_all)
done

theorem Little_Fermat:
"p ∈ zprime ==> ¬ p dvd x ==> [x^(nat (p - 1)) = 1] (mod p)"
apply (subst phi_prime [symmetric])
apply (rule_tac [2] Euler_Fermat)
apply (erule_tac [3] zprime_imp_zrelprime)
apply (unfold zprime_def, auto)
done

end
```

### Definitions and lemmas

lemma abs_eq_1_iff:

`  (¦z¦ = 1) = (z = 1 ∨ z = -1)`

lemma BnorRset_induct:

```  [| !!a m. P {} a m;
!!a m. [| 0 < a; P (BnorRset (a - 1, m)) (a - 1) m |]
==> P (BnorRset (a, m)) a m |]
==> P (BnorRset (u, v)) u v```

lemma Bnor_mem_zle:

`  b ∈ BnorRset (a, m) ==> b ≤ a`

lemma Bnor_mem_zle_swap:

`  a < b ==> b ∉ BnorRset (a, m)`

lemma Bnor_mem_zg:

`  b ∈ BnorRset (a, m) ==> 0 < b`

lemma Bnor_mem_if:

`  [| zgcd (b, m) = 1; 0 < b; b ≤ a |] ==> b ∈ BnorRset (a, m)`

lemma Bnor_in_RsetR:

`  a < m ==> BnorRset (a, m) ∈ RsetR m`

lemma Bnor_fin:

`  finite (BnorRset (a, m))`

lemma norR_mem_unique_aux:

`  a ≤ b - 1 ==> a < b`

lemma norR_mem_unique:

`  [| 1 < m; zgcd (a, m) = 1 |] ==> ∃!b. [a = b] (mod m) ∧ b ∈ norRRset m`

lemma RRset_gcd:

`  [| is_RRset A m; a ∈ A |] ==> zgcd (a, m) = 1`

lemma RsetR_zmult_mono:

`  [| A ∈ RsetR m; 0 < m; zgcd (x, m) = 1 |] ==> (%a. a * x) ` A ∈ RsetR m`

lemma card_nor_eq_noX:

`  [| 0 < m; zgcd (x, m) = 1 |] ==> card (noXRRset m x) = card (norRRset m)`

lemma noX_is_RRset:

`  [| 0 < m; zgcd (x, m) = 1 |] ==> is_RRset (noXRRset m x) m`

lemma aux_some:

```  [| 1 < m; is_RRset A m; a ∈ A |]
==> [a = SOME b. [a = b] (mod m) ∧ b ∈ norRRset m] (mod m) ∧
(SOME b. [a = b] (mod m) ∧ b ∈ norRRset m) ∈ norRRset m```

lemma RRset2norRR_correct:

```  [| 1 < m; is_RRset A m; a ∈ A |]
==> [a = RRset2norRR A m a] (mod m) ∧ RRset2norRR A m a ∈ norRRset m```

lemmas RRset2norRR_correct1:

`  [| 1 < m; is_RRset A m; a ∈ A |] ==> [a = RRset2norRR A m a] (mod m)`

lemmas RRset2norRR_correct2:

`  [| 1 < m; is_RRset A m; a ∈ A |] ==> RRset2norRR A m a ∈ norRRset m`

lemma RsetR_fin:

`  A ∈ RsetR m ==> finite A`

lemma RRset_zcong_eq:

`  [| 1 < m; is_RRset A m; [a = b] (mod m); a ∈ A; b ∈ A |] ==> a = b`

lemma aux:

```  [| P (SOME a. P a); Q (SOME a. Q a); (SOME a. P a) = (SOME a. Q a) |]
==> ∃a. P a ∧ Q a```

lemma RRset2norRR_inj:

`  [| 1 < m; is_RRset A m |] ==> inj_on (RRset2norRR A m) A`

lemma RRset2norRR_eq_norR:

`  [| 1 < m; is_RRset A m |] ==> RRset2norRR A m ` A = norRRset m`

lemma Bnor_prod_power_aux:

`  [| a ∉ A; inj f |] ==> f a ∉ f ` A`

lemma Bnor_prod_power:

```  [| x ≠ 0; a < m |]
==> ssetprod ((%a. a * x) ` BnorRset (a, m)) =
ssetprod (BnorRset (a, m)) * x ^ card (BnorRset (a, m))```

### Fermat

lemma bijzcong_zcong_prod:

`  (A, B) ∈ bijR (zcongm m) ==> [ssetprod A = ssetprod B] (mod m)`

lemma Bnor_prod_zgcd:

`  a < m ==> zgcd (ssetprod (BnorRset (a, m)), m) = 1`

theorem Euler_Fermat:

`  [| 0 < m; zgcd (x, m) = 1 |] ==> [x ^ phi m = 1] (mod m)`

lemma Bnor_prime:

```  [| p ∈ zprime; a < p; ∀b. 0 < b ∧ b ≤ a --> zgcd (b, p) = 1 |]
==> card (BnorRset (a, p)) = nat a```

lemma phi_prime:

`  p ∈ zprime ==> phi p = nat (p - 1)`

theorem Little_Fermat:

`  [| p ∈ zprime; ¬ p dvd x |] ==> [x ^ nat (p - 1) = 1] (mod p)`