# Theory EulerFermat

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

section ‹Fermat's Little Theorem extended to Euler's Totient function›

theory EulerFermat
imports BijectionRel IntFact
begin

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›

inductive_set RsetR :: "int => int set set" for m :: int
where
empty [simp]: "{} ∈ RsetR m"
| insert: "A ∈ RsetR m ==> zgcd a m = 1 ==>
∀a'. a' ∈ A --> ¬ zcong a a' m ==> insert a A ∈ RsetR m"

fun BnorRset :: "int ⇒ int => int set" where
"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 {})"

definition norRRset :: "int => int set"
where "norRRset m = BnorRset (m - 1) m"

definition noXRRset :: "int => int => int set"
where "noXRRset m x = (λa. a * x) ` norRRset m"

definition phi :: "int => nat"
where "phi m = card (norRRset m)"

definition is_RRset :: "int set => int => bool"
where "is_RRset A m = (A ∈ RsetR m ∧ card A = phi m)"

definition RRset2norRR :: "int set => int => int => int"
where
"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)"

definition zcongm :: "int => int => int => bool"
where "zcongm m = (λa b. zcong a b m)"

lemma abs_eq_1_iff [iff]: "(¦z¦ = (1::int)) = (z = 1 ∨ z = -1)"
-- ‹LCP: not sure why this lemma is needed now›

text ‹\medskip @{text norRRset}›

declare BnorRset.simps [simp del]

lemma BnorRset_induct:
assumes "!!a m. P {} a m"
and "!!a m :: int. 0 < a ==> P (BnorRset (a - 1) m) (a - 1) m
==> P (BnorRset a m) a m"
shows "P (BnorRset u v) u v"
apply (rule BnorRset.induct)
apply (case_tac "0 < a")
apply (rule_tac assms)
apply simp_all
done

lemma Bnor_mem_zle [rule_format]: "b ∈ BnorRset a m ⟶ b ≤ a"
apply (induct a m rule: BnorRset_induct)
apply simp
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 (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
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)
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)
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)
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 (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]
lemmas RRset2norRR_correct2 = RRset2norRR_correct [THEN conjunct2]

lemma RsetR_fin: "A ∈ RsetR m ==> finite A"
by (induct set: RsetR) 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)
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)
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 RRset2norRR_inj, auto)
apply (rule_tac [3] RRset2norRR_correct2, auto)
apply (unfold is_RRset_def phi_def norRRset_def)
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 --> ∏((λa. a * x) ` BnorRset a m) =
∏(BnorRset a m) * x^card (BnorRset a m)"
apply (induct a m rule: BnorRset_induct)
prefer 2
apply (simplesubst BnorRset.simps)  --‹multiple redexes›
apply (unfold Let_def, auto)
apply (subst setprod.insert)
apply (rule_tac [2] Bnor_prod_power_aux)
apply (unfold inj_on_def)
apply (simp_all add: ac_simps Bnor_fin Bnor_mem_zle_swap)
done

subsection ‹Fermat›

lemma bijzcong_zcong_prod:
"(A, B) ∈ bijR (zcongm m) ==> [∏A = ∏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 (∏(BnorRset a m)) m = 1"
apply (induct a m rule: BnorRset_induct)
prefer 2
apply (subst BnorRset.simps)
apply (unfold Let_def, auto)
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 = "∏(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, fold 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:
"⟦ zprime p; a < p ⟧ ⟹ card (BnorRset a p) = nat a"
apply (induct a p rule: BnorRset.induct)
apply (subst BnorRset.simps)
apply (unfold Let_def, auto simp add:zless_zprime_imp_zrelprime)
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: "zprime p ==> phi p = nat (p - 1)"
apply (unfold phi_def norRRset_def)
apply (rule Bnor_prime, auto)
done

theorem Little_Fermat:
"zprime p ==> ¬ 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
```