Theory Primes

(*  Title:      ZF/ex/Primes.thy
    Author:     Christophe Tabacznyj and Lawrence C Paulson
    Copyright   1996  University of Cambridge
*)

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),
                        λn 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 (simp add: divides_def)
apply (fast intro: nat_0I mult_0_right [symmetric])
done

lemma dvd_0_left: "0 dvd m ⟹ m = 0"
by (simp add: divides_def)

lemma dvd_refl [simp]: "m ∈ nat ⟹ m dvd m"
apply (simp add: divides_def)
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 (simp add: divides_def)
apply (force dest: mult_eq_self_implies_10
             simp add: mult_assoc mult_eq_1_iff)
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 (simp add: divides_def, clarify)
apply (rule_tac x = "i#*ka" in bexI)
apply (simp add: mult_ac)
apply (rule mult_type)
done


subsection‹Euclid's Algorithm for the GCD›

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

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

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

lemma gcd_non_0_raw: 
    "⟦0<n;  n ∈ nat⟧ ⟹ gcd(m,n) = gcd(n, m mod n)"
apply (simp add: gcd_def)
apply (rule_tac P = "λz. 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"
by (simp (no_asm_simp) add: gcd_non_0)

lemma dvd_add: "⟦k dvd a; k dvd b⟧ ⟹ k dvd (a #+ b)"
apply (simp add: divides_def)
apply (fast intro: add_mult_distrib_left [symmetric] add_type)
done

lemma dvd_mult: "k dvd n ⟹ k dvd (m #* n)"
apply (simp add: divides_def)
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 (simp add: DIVISION_BY_ZERO_MOD)
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
apply (simp add: divides_def)
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 (simp_all add: Ord_0_lt_iff)
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
apply (simp add: gcd_non_0)
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 (simp_all add: gcd_non_0)
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 (simp add: divides_def)
apply (case_tac "b=0")
 apply (simp add: DIVISION_BY_ZERO_MOD, auto)
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)
apply (simp_all add: gcd_non_0 dvd_mod)
done

lemma gcd_greatest: "⟦f dvd m;  f dvd n;  f ∈ nat⟧ ⟹ f dvd gcd(m,n)"
apply (rule gcd_greatest_raw)
apply (auto simp add: divides_def)
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)"
by (simp add: is_gcd_def)

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 (simp add: is_gcd_def)
apply (blast intro: dvd_anti_sym)
done

lemma is_gcd_commute: "is_gcd(k,m,n) ⟷ is_gcd(k,n,m)"
by (simp add: is_gcd_def, blast)

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 (simp_all add: is_gcd_def)
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])


subsection‹Addition laws›

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 (subst add_commute, simp)
apply (rule gcd_commute)
done

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

lemma gcd_add_mult_raw: "k ∈ nat ⟹ gcd (m, k #* m #+ n) = gcd (m, n)"
apply (erule nat_induct)
apply (auto simp add: gcd_add2 add_assoc)
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)
apply (simp add: dvd_imp_nat1)
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 (simp add: prime_def, clarify)
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"
by (simp add: prime_def)

lemma prime_nonzero: "p ∈ prime ⟹ p≠0"
by (auto simp add: prime_def)


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 (simp add: gcd_assoc)
apply (simp add: gcd_commute)
apply (simp_all add: mult_commute)
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)
 apply (auto simp add: prime_def)
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 (simp add: not_lt_iff_le prime_into_nat)
apply (erule disjE)
 apply (frule mult_le_mono, assumption+)
apply (simp add: mult_ac)
apply (auto dest!: natify_eqE 
            simp add: not_lt_iff_le prime_into_nat mult_le_cancel_le1)
apply (simp add: prime_def)
apply (blast dest: lt_trans1)
done

lemma rearrange: "j #* (p#*j) = k#*k ⟹ k#*k = p#*(j#*j)"
by (simp add: mult_ac)

lemma prime_not_square:
     "⟦m ∈ nat; p ∈ prime⟧ ⟹ ∀k ∈ nat. 0<k ⟶ m#*m ≠ p#*(k#*k)"
apply (erule complete_induct, clarify)
apply (frule prime_dvd_other_side, assumption)
apply assumption
apply (erule dvdE)
apply (simp add: mult_assoc mult_cancel1 prime_nonzero prime_into_nat)
apply (blast dest: rearrange reduction ltD)
done

end