# Theory Primes

theory Primes
imports ZF
```(*  Title:      ZF/ex/Primes.thy
Author:     Christophe Tabacznyj and Lawrence C Paulson
*)

section‹The Divides Relation and Euclid's algorithm for the GCD›

theory Primes imports ZF begin

definition
divides :: "[i,i]=>o"              (infixl "dvd" 50)  where
"m dvd n == m ∈ nat & n ∈ nat & (∃k ∈ nat. n = m#*k)"

definition
is_gcd  :: "[i,i,i]=>o"     ―‹definition of great common divisor›  where
"is_gcd(p,m,n) == ((p dvd m) & (p dvd n))   &
(∀d∈nat. (d dvd m) & (d dvd n) ⟶ d dvd p)"

definition
gcd     :: "[i,i]=>i"       ―‹Euclid's algorithm for the gcd›  where
"gcd(m,n) == transrec(natify(n),
%n f. λm ∈ nat.
if n=0 then m else f`(m mod n)`n) ` natify(m)"

definition
coprime :: "[i,i]=>o"       ―‹the coprime relation›  where
"coprime(m,n) == gcd(m,n) = 1"

definition
prime   :: i                ―‹the set of prime numbers›  where
"prime == {p ∈ nat. 1<p & (∀m ∈ nat. m dvd p ⟶ m=1 | m=p)}"

subsection‹The Divides Relation›

lemma dvdD: "m dvd n ==> m ∈ nat & n ∈ nat & (∃k ∈ nat. n = m#*k)"
by (unfold divides_def, assumption)

lemma dvdE:
"[|m dvd n;  !!k. [|m ∈ nat; n ∈ nat; k ∈ nat; n = m#*k|] ==> P|] ==> P"
by (blast dest!: dvdD)

lemmas dvd_imp_nat1 = dvdD [THEN conjunct1]
lemmas dvd_imp_nat2 = dvdD [THEN conjunct2, THEN conjunct1]

lemma dvd_0_right [simp]: "m ∈ nat ==> m dvd 0"
apply (fast intro: nat_0I mult_0_right [symmetric])
done

lemma dvd_0_left: "0 dvd m ==> m = 0"

lemma dvd_refl [simp]: "m ∈ nat ==> m dvd m"
apply (fast intro: nat_1I mult_1_right [symmetric])
done

lemma dvd_trans: "[| m dvd n; n dvd p |] ==> m dvd p"
by (auto simp add: divides_def intro: mult_assoc mult_type)

lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m=n"
apply (force dest: mult_eq_self_implies_10
done

lemma dvd_mult_left: "[|(i#*j) dvd k; i ∈ nat|] ==> i dvd k"
by (auto simp add: divides_def mult_assoc)

lemma dvd_mult_right: "[|(i#*j) dvd k; j ∈ nat|] ==> j dvd k"
apply (rule_tac x = "i#*ka" in bexI)
apply (rule mult_type)
done

subsection‹Euclid's Algorithm for the GCD›

lemma gcd_0 [simp]: "gcd(m,0) = natify(m)"
apply (subst transrec, simp)
done

lemma gcd_natify1 [simp]: "gcd(natify(m),n) = gcd(m,n)"

lemma gcd_natify2 [simp]: "gcd(m, natify(n)) = gcd(m,n)"

lemma gcd_non_0_raw:
"[| 0<n;  n ∈ nat |] ==> gcd(m,n) = gcd(n, m mod n)"
apply (rule_tac P = "%z. left (z) = right" for left right in transrec [THEN ssubst])
apply (simp add: ltD [THEN mem_imp_not_eq, THEN not_sym]
mod_less_divisor [THEN ltD])
done

lemma gcd_non_0: "0 < natify(n) ==> gcd(m,n) = gcd(n, m mod n)"
apply (cut_tac m = m and n = "natify (n) " in gcd_non_0_raw)
apply auto
done

lemma gcd_1 [simp]: "gcd(m,1) = 1"

lemma dvd_add: "[| k dvd a; k dvd b |] ==> k dvd (a #+ b)"
done

lemma dvd_mult: "k dvd n ==> k dvd (m #* n)"
apply (fast intro: mult_left_commute mult_type)
done

lemma dvd_mult2: "k dvd m ==> k dvd (m #* n)"
apply (subst mult_commute)
apply (blast intro: dvd_mult)
done

(* k dvd (m*k) *)
lemmas dvdI1 [simp] = dvd_refl [THEN dvd_mult]
lemmas dvdI2 [simp] = dvd_refl [THEN dvd_mult2]

lemma dvd_mod_imp_dvd_raw:
"[| a ∈ nat; b ∈ nat; k dvd b; k dvd (a mod b) |] ==> k dvd a"
apply (case_tac "b=0")
apply (blast intro: mod_div_equality [THEN subst]
elim: dvdE
intro!: dvd_add dvd_mult mult_type mod_type div_type)
done

lemma dvd_mod_imp_dvd: "[| k dvd (a mod b); k dvd b; a ∈ nat |] ==> k dvd a"
apply (cut_tac b = "natify (b)" in dvd_mod_imp_dvd_raw)
apply auto
done

(*Imitating TFL*)
lemma gcd_induct_lemma [rule_format (no_asm)]: "[| n ∈ nat;
∀m ∈ nat. P(m,0);
∀m ∈ nat. ∀n ∈ nat. 0<n ⟶ P(n, m mod n) ⟶ P(m,n) |]
==> ∀m ∈ nat. P (m,n)"
apply (erule_tac i = n in complete_induct)
apply (case_tac "x=0")
apply (simp (no_asm_simp))
apply clarify
apply (drule_tac x1 = m and x = x in bspec [THEN bspec])
apply (blast intro: mod_less_divisor [THEN ltD])
done

lemma gcd_induct: "!!P. [| m ∈ nat; n ∈ nat;
!!m. m ∈ nat ==> P(m,0);
!!m n. [|m ∈ nat; n ∈ nat; 0<n; P(n, m mod n)|] ==> P(m,n) |]
==> P (m,n)"
by (blast intro: gcd_induct_lemma)

subsection‹Basic Properties of @{term gcd}›

text‹type of gcd›
lemma gcd_type [simp,TC]: "gcd(m, n) ∈ nat"
apply (subgoal_tac "gcd (natify (m), natify (n)) ∈ nat")
apply simp
apply (rule_tac m = "natify (m)" and n = "natify (n)" in gcd_induct)
apply auto
done

text‹Property 1: gcd(a,b) divides a and b›

lemma gcd_dvd_both:
"[| m ∈ nat; n ∈ nat |] ==> gcd (m, n) dvd m & gcd (m, n) dvd n"
apply (rule_tac m = m and n = n in gcd_induct)
apply (blast intro: dvd_mod_imp_dvd_raw nat_into_Ord [THEN Ord_0_lt])
done

lemma gcd_dvd1 [simp]: "m ∈ nat ==> gcd(m,n) dvd m"
apply (cut_tac m = "natify (m)" and n = "natify (n)" in gcd_dvd_both)
apply auto
done

lemma gcd_dvd2 [simp]: "n ∈ nat ==> gcd(m,n) dvd n"
apply (cut_tac m = "natify (m)" and n = "natify (n)" in gcd_dvd_both)
apply auto
done

text‹if f divides a and b then f divides gcd(a,b)›

lemma dvd_mod: "[| f dvd a; f dvd b |] ==> f dvd (a mod b)"
apply (case_tac "b=0")
apply (blast intro: mod_mult_distrib2 [symmetric])
done

text‹Property 2: for all a,b,f naturals,
if f divides a and f divides b then f divides gcd(a,b)›

lemma gcd_greatest_raw [rule_format]:
"[| m ∈ nat; n ∈ nat; f ∈ nat |]
==> (f dvd m) ⟶ (f dvd n) ⟶ f dvd gcd(m,n)"
apply (rule_tac m = m and n = n in gcd_induct)
done

lemma gcd_greatest: "[| f dvd m;  f dvd n;  f ∈ nat |] ==> f dvd gcd(m,n)"
apply (rule gcd_greatest_raw)
done

lemma gcd_greatest_iff [simp]: "[| k ∈ nat; m ∈ nat; n ∈ nat |]
==> (k dvd gcd (m, n)) ⟷ (k dvd m & k dvd n)"
by (blast intro!: gcd_greatest gcd_dvd1 gcd_dvd2 intro: dvd_trans)

subsection‹The Greatest Common Divisor›

text‹The GCD exists and function gcd computes it.›

lemma is_gcd: "[| m ∈ nat; n ∈ nat |] ==> is_gcd(gcd(m,n), m, n)"

text‹The GCD is unique›

lemma is_gcd_unique: "[|is_gcd(m,a,b); is_gcd(n,a,b); m∈nat; n∈nat|] ==> m=n"
apply (blast intro: dvd_anti_sym)
done

lemma is_gcd_commute: "is_gcd(k,m,n) ⟷ is_gcd(k,n,m)"

lemma gcd_commute_raw: "[| m ∈ nat; n ∈ nat |] ==> gcd(m,n) = gcd(n,m)"
apply (rule is_gcd_unique)
apply (rule is_gcd)
apply (rule_tac [3] is_gcd_commute [THEN iffD1])
apply (rule_tac [3] is_gcd, auto)
done

lemma gcd_commute: "gcd(m,n) = gcd(n,m)"
apply (cut_tac m = "natify (m)" and n = "natify (n)" in gcd_commute_raw)
apply auto
done

lemma gcd_assoc_raw: "[| k ∈ nat; m ∈ nat; n ∈ nat |]
==> gcd (gcd (k, m), n) = gcd (k, gcd (m, n))"
apply (rule is_gcd_unique)
apply (rule is_gcd)
apply (blast intro: gcd_dvd1 gcd_dvd2 gcd_type intro: dvd_trans)
done

lemma gcd_assoc: "gcd (gcd (k, m), n) = gcd (k, gcd (m, n))"
apply (cut_tac k = "natify (k)" and m = "natify (m)" and n = "natify (n) "
in gcd_assoc_raw)
apply auto
done

lemma gcd_0_left [simp]: "gcd (0, m) = natify(m)"
by (simp add: gcd_commute [of 0])

lemma gcd_1_left [simp]: "gcd (1, m) = 1"
by (simp add: gcd_commute [of 1])

lemma gcd_add1 [simp]: "gcd (m #+ n, n) = gcd (m, n)"
apply (subgoal_tac "gcd (m #+ natify (n), natify (n)) = gcd (m, natify (n))")
apply simp
apply (case_tac "natify (n) = 0")
apply (auto simp add: Ord_0_lt_iff gcd_non_0)
done

lemma gcd_add2 [simp]: "gcd (m, m #+ n) = gcd (m, n)"
apply (rule gcd_commute [THEN trans])
apply (rule gcd_commute)
done

lemma gcd_add2' [simp]: "gcd (m, n #+ m) = gcd (m, n)"

lemma gcd_add_mult_raw: "k ∈ nat ==> gcd (m, k #* m #+ n) = gcd (m, n)"
apply (erule nat_induct)
done

lemma gcd_add_mult: "gcd (m, k #* m #+ n) = gcd (m, n)"
apply (cut_tac k = "natify (k)" in gcd_add_mult_raw)
apply auto
done

subsection‹Multiplication Laws›

lemma gcd_mult_distrib2_raw:
"[| k ∈ nat; m ∈ nat; n ∈ nat |]
==> k #* gcd (m, n) = gcd (k #* m, k #* n)"
apply (erule_tac m = m and n = n in gcd_induct, assumption)
apply simp
apply (case_tac "k = 0", simp)
apply (simp add: mod_geq gcd_non_0 mod_mult_distrib2 Ord_0_lt_iff)
done

lemma gcd_mult_distrib2: "k #* gcd (m, n) = gcd (k #* m, k #* n)"
apply (cut_tac k = "natify (k)" and m = "natify (m)" and n = "natify (n) "
in gcd_mult_distrib2_raw)
apply auto
done

lemma gcd_mult [simp]: "gcd (k, k #* n) = natify(k)"
by (cut_tac k = k and m = 1 and n = n in gcd_mult_distrib2, auto)

lemma gcd_self [simp]: "gcd (k, k) = natify(k)"
by (cut_tac k = k and n = 1 in gcd_mult, auto)

lemma relprime_dvd_mult:
"[| gcd (k,n) = 1;  k dvd (m #* n);  m ∈ nat |] ==> k dvd m"
apply (cut_tac k = m and m = k and n = n in gcd_mult_distrib2, auto)
apply (erule_tac b = m in ssubst)
done

lemma relprime_dvd_mult_iff:
"[| gcd (k,n) = 1;  m ∈ nat |] ==> k dvd (m #* n) ⟷ k dvd m"
by (blast intro: dvdI2 relprime_dvd_mult dvd_trans)

lemma prime_imp_relprime:
"[| p ∈ prime;  ~ (p dvd n);  n ∈ nat |] ==> gcd (p, n) = 1"
apply (drule_tac x = "gcd (p,n)" in bspec)
apply auto
apply (cut_tac m = p and n = n in gcd_dvd2, auto)
done

lemma prime_into_nat: "p ∈ prime ==> p ∈ nat"

lemma prime_nonzero: "p ∈ prime ⟹ p≠0"

text‹This theorem leads immediately to a proof of the uniqueness of
factorization.  If @{term p} divides a product of primes then it is
one of those primes.›

lemma prime_dvd_mult:
"[|p dvd m #* n; p ∈ prime; m ∈ nat; n ∈ nat |] ==> p dvd m ∨ p dvd n"
by (blast intro: relprime_dvd_mult prime_imp_relprime prime_into_nat)

lemma gcd_mult_cancel_raw:
"[|gcd (k,n) = 1; m ∈ nat; n ∈ nat|] ==> gcd (k #* m, n) = gcd (m, n)"
apply (rule dvd_anti_sym)
apply (rule gcd_greatest)
apply (rule relprime_dvd_mult [of _ k])
apply (blast intro: dvdI1 gcd_dvd1 dvd_trans)
done

lemma gcd_mult_cancel: "gcd (k,n) = 1 ==> gcd (k #* m, n) = gcd (m, n)"
apply (cut_tac m = "natify (m)" and n = "natify (n)" in gcd_mult_cancel_raw)
apply auto
done

subsection‹The Square Root of a Prime is Irrational: Key Lemma›

lemma prime_dvd_other_side:
"⟦n#*n = p#*(k#*k); p ∈ prime; n ∈ nat⟧ ⟹ p dvd n"
apply (subgoal_tac "p dvd n#*n")
apply (blast dest: prime_dvd_mult)
apply (rule_tac j = "k#*k" in dvd_mult_left)
done

lemma reduction:
"⟦k#*k = p#*(j#*j); p ∈ prime; 0 < k; j ∈ nat; k ∈ nat⟧
⟹ k < p#*j & 0 < j"
apply (rule ccontr)
apply (erule disjE)
apply (frule mult_le_mono, assumption+)
apply (auto dest!: natify_eqE
apply (blast dest: lt_trans1)
done

lemma rearrange: "j #* (p#*j) = k#*k ⟹ k#*k = p#*(j#*j)"